Download Ray, S.N.; (1963)Some sequential Bayes procedures for comparing two binomial parameters when observations are taken in pairs."

•
••
UNIVERSITY OF NORTH CAROLINA
Department of Stat~stics
Chapel Hill, N. C•
~a ,."
. h;'l~SE:QUEN.I'IAL BAlES PROCEDURES FOR CCMPARI11G TWO
~
PARAMETERS WHEN OBSERVATIONS ARE TAKEN IN PAIRS
.
•. '_"'_ -0-
/
by
Sudhindra N.Ray
June 1963
This research was supported in part by Public Health
Service Research Grant GM-10397-01, from the Division
of General Medical Sciences, National Institutes of
Health.
Institute of Statistics
Mimeo Series No. 369
•
TABLE OF CONTENTS
CHAPTER
PAGE
v
ACKNOWLEDGME1'l'TS - - - - I
INTRODUCTION
1.0 Summary
- - - -
- - - - -
1
- - - -
1.1 Statement of the problem 1.2 Existing methods - - - - - -
1
2
- - - -
3
1.3 Criticism of the existing procedures - - - - 10
.-
1.4 A decision-theoretic formulation -
13
1.5 Reduction of the original problem
20
1.6 A decision-theoretic formulation for
25
the binomial problem - - - - - - - -
II
1.7 Review of some earlier relevant works
28
1.8 Chapter by chapter summary of the thesis
29
BAYES PROCEDURES WITH LINEAR LOSS STRUCTURE -
33
2.0
33
Summary
- - - - - - - - - - - - - -
2.1 Some well-known results about Bayes
sequential procedures with a general model -
33
2.2 Binomial model - - - - -
38
2.3 Beta prior distribution
44
2.4 Determination of the point of truncation
-
47
boundary in the binomial problem - - - - - -
56
in the binomial problem - - - - - -
•
2.5 Procedure for obtaining the optimum
iii
•
PAGE
CHAPTER
2.6 Some points useful for computation
of the optimum boundary - -
- - - -
59
2.7 Trinomial model - - - - - - - - -
62
2.8 Dirichlet a priori distribution -
64
2.9 Determination of the points of truncation
in the trinomial problem - - - - - - - - - -
65
2.10 Procedure for obtaining the optimum boundary
in the trinomial problem - - - - - - - -
71
2.11 Some points useful for computation of the
III
--
optimum boundary- - - - - - - - - - - - -
73
BAYES PROCEDURES WITH MODIFIED LOSS STRUCTURE-
76
3.0 Introduction and summary - - - - - - - -
76
3.1 Characterization of the Bayes procedure
With a general model
IV
- - - - - - - - - - - -
77
3.2 Modified loss structure With absolute deviation cost
86
3.3 Binomial model
88
3.4 Trinomial model - -
93
~OTIC
96
BEHAVIOUR OF OPTIMAL SOLUTION - -
4.0 Introduction and sUlDIIlary
96
4.1 Binomial problem - - - -
98
4.2 Series expansion for the optimal solution
for large x (binomial problem) - -
4.3 Trinomial problem - - - - - - - -
107
- 114
4.4 Series expansion for the optimal solution
for large
x (trinomial problem)
- - - - - - 122
iv
PAGE
CHAPTER
v
BOUNDS ON ASYMPTC'JrIC SOWTION -
- - - -
5.0 Introduction and summary -
129
129
5.1 Binomial problem with absolute deviation cost - - - -
·e
132
5.2 Trinomial problem - - - - -
141
BIBLIOGRAPHY - - - - - - - - - - - - - - - - - -
150
v
..
ACKNOvlLEDGME~'TS
I am deeply indebted to Professor William Jackson
Hall for proposing the problems in this thesis, for his
numerous suggestions and comments and above all for his
unfailing confidence in me even in those bleakest moments
when without his sympathetic encouragement and personal
warmth I could hardly proceed.
I wish to express my sincere thanks to Professor
Wassily Hoeffding, Professor Walter L. Smith and Professor Norman L. Johnson who read the manuscript and
offered many helpful suggestions.
I wish to thank the Government of India and the United
States Public Health Service for the financial support.
I am most grateful to Mrs. Doris Gardner for her quick
and accurate typing of the manuscript.
Also, my sincerest
thanks to Martha Jordan for her help in guiding me through
the labyrinth of rules, regulations and assorted pitfalls.
CHAPTER I
I:NTRODUCTION
1.0 Summary
In section 1.1 the basic problem of the thesis is stated, namely
that of comparing the parameters of two Bernoulli processes from paired
samples.
Section 1.2 contains a brief survey of eXisting methods of solving
the problem, most of which are based on the Neyman-Pearson theory.
In section
l.~
same shortcomings of the existing methods are dis-
cussed.
In section 1.4 a decision-theoretic
fo~ulation
of the basic
problem along with same realistic specifications of loss and cost is
proposed.
In section 1.5 the original problem of comparing two binomial probabilities is reduced to that of comparing two cell probabilities in a
certain trinomial problem.
Section 1.6 contains a corresponding
fo~ulation
for a single
Bernoulli process.
.
Section 1.7 contains a brief review of some results that are relevant for the analysis of the various. decision problems
fo~ulated.
Finally, a brief chapter by chapter summary of the thesis is presented in section 1.8, which is concluded with a schematic table of
contents.
1.1 statement of the problem.
Consider two independent data-generating processes.
Each process
yields a sequence of independent observations of dichotomous type, say,
successes and failures.
Suppose also that the observations are obtained
in pairs from the two processes.
Let
Pl and P2
bilities of a success for the two processes.
be the unknown proba-
NoW consider the problem
of selecting the process with the larger success probability on the
basis of a sequence of pairs of observations.
The method of sampling
being sequential, our primary problem is that of obtaining a good sampling rule that tells us when to stop sampling.
The statistical formulation as stated above is typical of many
practical problems arising in various fields of application.
In clini-
cal trials, for example, one may wish to compare two alternative treatments or drugs, say T
l
and T •
2
Observations may be obtained in pairs,
one member of each pair for each treatment.
Suppose they are catego-
rized into successes and failures according as the patients are cured
or not, or say, according as the blood pressure readings of the patients
are below a certain level or not.
Here
Pl and P2 are the unknown
probabilities of a success with the treatments
T
l
and T
2
respectively.
Similar situations arise in industrial production processes where
the effectiveness of two alternative production methods
say, are to be compared.
P
l
and P2 '
The effectiveness of a process is measured
in terms of the proportion of effective units produced" all units being
classified simply as effective or defective.
knO'WIl
Here Pl(P2)
is the un-
probability of a unit being effective when it is produced using
method Pl(P 2 ).
For the sake of a descriptive terminology" but without any loss of
generality" we shall sometimes refer to the tvo situations described
above.
1.2 Existing methods.
The formulation stated in the previous section ial
a$
yet"
~lete.
The specification of further details including the ultimate objective
to which the results of the investigation are to be put makes all the
difference in the particular approach to be used in analysing such
data.
In a:llllost all of the existing methods of analysis" this problem
of double dichotomies l
has been posed as a hypothesis testing problem
in the framework of the Neyman-Pearson theory.
A
very brief review of
the important existing methods is given below.
(a)
Fixed sample size procedures.
Perhaps the most common method is the classical one of taking
samples of fixed size and analysing the sample proportions of successes
as if they were normally distributed" using a one-sided or a two-sided
test of the difference between the two means.
The two-sided test is, in
effect, a chi-square test.
lwe have used the term 'double dichotomies' in the general sense as
used for example in Wald r397* and not in the restricted sense of
Barnard £17* ·
- -
*The
numbers in square brackets refer to the bibliography.
4
Another common non-sequential method is to apply Fisher's exact
test, as mentioned briefly below. The data is put in a
2 x 2 table
as follows.
Process I
Success
al
Failure
n .. a
Total
Process I I i
a
al + a 2
2n - a l - a 2
2
n - a2
l
n
n
Total
I
.j
2n
The exact probability of such a configuration, conditional on the marginals
being fixed and the null hypothesis Pl
= P2
being true, is given by the
nypergeametric term
•
Probabilities of "more extreme" (depending on whether a one-sided or a
two-sided test is wanted2 ) configurations are obtained similarly. The
observed result is judged to be insignificant or not at a fixed level
of significance, according as the total probability of the observed configuration along with more extreme configurations exceeds the fixed
level, or not.
The nypergeametric distribution being discrete, the size of Fisher's
exact test may not attain the traditional levels of significance. The
corresponding randomized version of the test ~27, p. l4~7 is a simi2Fieher's e3act test as described in ~18, p. 9§7 is one-sided.
5
lar test of Neyman structure and is essentially a uniformly most
1 against u > 1
powerful unbiased test of the hypothesis
u
~
the one-sided case, or of the hypothesis
u
=1
for
I 1 for
against u
the two-sided case, wbere
u
_
J~p_2__
El
- 1 - plf i - 1'2
is called the ratio of odds and u
O<u<co
'
<
,
<
> 1 according as 1'1 > P2 •
Fisher's exact test is a conditional one in the sense tr..at in the
proper "frame of referenc elf the marginal totals of the above
table are kept fixed.
2 x 2
A different sort of conditioning may be applied
to the data to obtain another exact, though not so efficient, test3
that is particularly relevant when only successes (or equivalently
failures) are observable.
Keeping the total number of successes fixed,
the number of successes with treatment T
l
wi th the parameter
is distributed binomially
O<ill<l
UJ
>
< 1/2
according as
1'1
>
< 1'2.
,
The problem of testing the dif-
ference between the two binomial parameters
1'1 and 1'2 is thus re-·
duced to that of testing the departure of the single binomial parameter
ill
from
1/2.
One-sided or two-sided exact tests of the hypothesis
ill = 1/2 maybe constructed in the usual way.
The methods described so far do not take account of the natural
pairing of the observations present in our formulation of the problem
in section 1.1.
Each pair of observations coming. from the two
3This test is reminiscent of a test for the equality of two Poisson para-
r27,
meters
p. 1417, and, along with its sequential analog, was brought
to our attention by Professor W. J. Hall.
6
processes is one of the four types
the term
~
SS, FF, SF, and FS..
We shall use
pairs for those recorded as SS or FF and untied pairs
for those recorded as
SF or FS.
The problem. of testing the equality
of 1'1 and 1'2 may be reduced as before to that of testing the departure from the value
1/2
of another binomial parameter v
defined
as
v
u
=
=1+ii
O<v<l
'
the probability that an untied pair is an SF.
that v -
~
1/2 according as
1'1
~
1'2.
,
It is easily verified
Wald
["39,
1'. 1017 pro-
posed an exact non-sequential test for the double dichotomies using the
above reduction.
The test of course depends on a given pairing of the
observations from the two processes.
It is to be noted that any test, sequential or non-sequential, for
the double dichotomies when based on the first type of reduction, depends only on the successes (or only on the failures) irrespective of
\
any pairing, while when based on the second type of reduction, depends
only on the untied pairs for a given pairing.
Inasmuch as these tests
fail to utilize all the relevant data they cannot be fully efficient.
In constructing a test for the double dichotomies, it is, however,
quite a standard practice to take account of any natural pairing of
observations for some well-understood reasons also explained by
Wald
["39, p.
10§]e
It would be interesting to know the relative
merits of the two types of reduction and to make a comparison between
7
the two associated tests as to their relative performances vis a vis
..
other tests, viz., the classical test and Fisher's exact test for the
non-sequential case.
(b)
Sequential procedures.
In the absence of any natural pairing of observations the
problem of testing PI = 1>2 sequentially may be reduced to that of
testing w
= 1/2
sequentially. Since by the very formulation of our
problem we assume the eXistence of such a pairing and since most of
the existing sequential procedures make use of this fact by basing the
tests on the untied pairs we shall no longer consider any test based
on successes alone.
It is to be noted, however, that corresponding to
each of the tests based on untied pairs to be described below w could
have a test based on successes alone.
The existing sequential procedures may be considered in two broad
groups, according as the number of hypotheses
~onsidered being
two or
three giving rise to the so-called two-decision or three-decision
approaches.
(i) Two-decision approaches.
Wald's sequential procedure
1 39,
p. 102.7 for analyzing
double dichotomies consists in selecting two parameter points v
and vI (v0 < 1/2 < vI)
(a, f' > 0,
and two error probabilities a
o
and f'
a + f' < 1) and then to set up a sequential probability
ratio test (SPRT) of the hypothesis
in the usual vre.y.
H2 : v = v 0
against HI: v
:=
vI
8
Since the v-parameter is not intuitively very meaningful Wald
describes the test and its operating characteristic (ae) and average
sample number (ASN) :f'unction in terms of the parameter u
= v/(l-v)"
the ratio of odds, considered by Girshick ~121 as an appropriate
measure of divergence between the two parameters
PI and P2.
maybe possible for the experimenter to choose U and
o
(0 < Uo < 1 < u < ~) such that a divergence measure ~
l
between. Pl and P2 are worth detecting.
i
= 0,
Then Vi
It
u
l
Uo or ~ u l
= Ui/(l
+ ui )"
1.
Wald has also discussed the cone1derations that may govern the
and u • Because of the symmetrical footing in which
o
l
the two treatments stand With respect to each other in our problem"
choice of
-e
U
attention might be restricted to procedures with u
IVi - 1/21
= (1
l.
- u0
)/2(10
+ u )0
s 5 . (say)" i = 0" 1.
= l/U0 j
i.e. With
It may be veri-
fied that the resulting procedure is then the same as that of Girshick
If" moreover" the two error probabilities a"
equal we arrive at symmetrical procedures.
~
are taken to be
For this class of procedures,
Hall ["25,7 has proposed a modifica.tion to effect a reduction in the
ASN when Pl and P2 are
(ii)
approXima~ely
equal.
Three-decision approaches.
Besides the two decisions of judging one of the two treat-
ments to be better than the other, the third one of making no such
judgment and proclaiming them to be the same for all practical purposes, is noV' introduced.
This is the approach of the common two-
sided non-sequential tests. Most of the procedures considered by
9
Armitage also have this feature.
..
The three decisions
PI > P2' PI < P2
v > 1/2, v
80 ,
0
80
1
,
80
2
corresponding to PI::: P2'
respectively correspond in terms of v
to v; 1/2,
< 1/2 respectively. Armitage £'2.7 has suggested a pro-
cedure obtained by running simultaneously two one-sided SPRT's, viz.,
Testing
H:
v
o
= 1/2
against
Testing H,..,:
v = 1/2 - 8 0
c;
= 1/2
+ 80
,
H:
v = 1/2
0
•
HI: v
against
The CC and ASN of this procedure might be obtained from the results of'
Boer fl~7 who developed the theory for such procedures for the binomial
parameter, just as Sobel and Wald
1 327
did for the normal mean problem.
All the sequential procedures described so far (except Hall' s J:2~7)
are ~ in the sense that they are not truncated (although they terminate with probability one).
Armitage
£!!:.7 has
considered a class of
restricted (truncated) procedures in Which sampling is continued until
ene of three specified boundaries is reached leading to the three
possible decisions about v
and hence about
PI and P2"
He con-
siders various types of boundaries and finds the performance of' the resuIting procedures in terms of the usual criteria of the Neyman-Pearson
theoretic formulation.
The sequential procedures described so far were concerned with the
derived problem in terms of the single binomial parameter v
are based solely on the untied observations.
Darwin
fl§}
and hence
bas con-
sidered within the three-decision approach a symmetrical truncated procedure which utilizes the tied pairs without discriminating between
10
the two types of tied pairs.
If
(Xli I X2i )
is the sequence of
pairs of observations where
for a success
\:i =
for a failure
i
= 1,
2, ••• , M.
= 1,
k
Zi
S
n
=
2,
and
Xli .. X2i , i
n
n
= I: Zi'
i=l
..., M ,
= 1,
2,
= 1,
2, ••• , M
,
then, Darwin's procedure accepts the three decisions
according as
where
Sn crosses the boundaries
Sn
= ~,
hand M are certain positive integers.
aI' a , a
2
o
Sn
= -h
and n:: M,
He considers the
problem of choosing hand M such that the procedure has a high
probability of leading to the correct decision.
We shall also pro-
pose in section 1.5 procedures based on Z IS.
n
1.3 Criticism of the existing procedures.
As mentioned and explained by Armitage
1 5,
p.
5"J.7,
the first
difficulty "may be caused by the fact that the measurements in which
one is primarily interested are the probabilities of success
P2'
PI and
whereas the sequential design is specified in terms of the
derived quantity v".
This difficulty becomes much more relevant when
one considers a decision-theoretic formulation of the problem.
it is agreed that v
PI and P2'
Once
is the primary parameter of interest rather than
all the usual decision-theoretic specifications regarding
lOSS, cost, prior distribution, etc. might well center around v, a
binomial parameter as considered in section 1. 6.
11
In all the sequential procedures mentioned in the previous
section where the original problem is reduced to one about a single
binomial parameter v, the ASN function as well as the terminologies
'truncation' or 'termination', etc., refer only to the untied portion
of the sequence of pairs of all observations.
The total amount of
expected sampling can be obtained in each case by dividing the ASN by
where
3{0
is the probability of a pair being tied.
Thus the total
amount of sampling before reaching a decision might be quite high
if
3{0
is sufficiently large.
The motivation behind introducing the
class of restricted procedures is that unlike those of Wald and
Girshick they have a relatively more attractive feature of possessing
a known upper bound, but only on the number of untied pairs and not
on the total number of pairs.
Thus these procedures of Ami tage (as
well as of Hall) are really not truncated in the sense that Darwin's
procedure is truncated.
It is qUite reasonable that the tied observations do not in·
tuitively contain much, if any, information about the difference
Pl - P2 and hence may not seem relevant for any test of Pl - P2.
'1"betY cannot be discarded by sufficiency or invariance considerations,
however,
b:t;tt they certainly involve some cost.
It might happen in all
the sequential procedures mentioned above except Darwin t s that the
available resources for continuing sampling are exhausted before a
sufficient number of untied observations are obtained to reach a
12
decision.
Also in Darwin1s approach,
the choice of a particular pro-
cedure with desirable power properties depends on the unknown probability
~o.
It is true, of course, that in practice, the experimenter has a
rough
kno~71edge
of the magnitude of
shown by Armitage
£5,
p.
5!:!.7
~.
o
and may utilize this, as
in choosing a suitable seQ.uential plan.
But tb±s is exactly what a Bayesian analysis attempts to do in a more,
formal way.
We may mention here that informal Bayesian analyses of
double dichotomies are to some extent available.
B;V assigning a prior
distribution for the two parameters, one may calculate the posterior
distribution after any amount of sampling, and discontinue the experiment when the posterior distribution seems adeQ.uately concentrated to
permit a decision.
Novick
£3]} has discussed such approaches in the
context of clinical trials.
Finally, Anscombe
£g7
has Q.uestioned the appropriateness for our
problem of the very foundations on which the existing procedures of the
previous section rest, viz., the operating-characteristic concepts of
the Neyman-Pearson theory of tests.
His basic tbesis in £~7 is that the
eXisting procedures might be appropriate in devising a routine decision
procedure that will work sufficiently well in the long run, but not so
in a problem where we have to analyze, interpret and make some recommendations on the basis of just one series of observations. Whatever
happened is relevant for our present action and not whatever could have
happened but did not.
Anscombe also criticizes the consideration of the third decision
of taking no positive action.
In some applications one has to recom-
mend only one of the two treatments or one of the two production pro-
13
cesses to be used in the future.
1.4 A decision-theoretic formulation.
Anscombe
£37
suggests a way in which the problem should be
tackled and illustrates to some extent with the corresponding normal
distribution problem.
He suggests a decision-theoretic formulation of
the problem in terms of loss and cost functions, which is adapted and
elaborated for our problem given below.
(a)
actions:
that treatment
The action 80 (80 ) corresponds to the recommendation
1 2
T (T ) is to be used with future patients.
l 2
801 (802 ) is preferable if Pl - P2; g > 0
(b)
costs per observation:
functions, i.e., the cost of n
(Q ~
Action
0).
We shall consider only linear cost
observations will be proportional to
n, the constant of proportionality, denoted by C, being termed the
cost per observation.
When interpreted as the usual cost of performing
an experiment, C may be regarded as a constant.
There arise, however,
certain situations where C might Mpend on the unknown parameter
Q.
We illustrate this possibility below.
Suppose in the industrial process example that the effective
i tams produced during the period of trial (to determine which one is
the better process) are marketable.
Suppose the monetary utility per
effective unit (ViZ, profit) is, say k
in some monetary unit.
Had
we know which one was the better process and allocated the two units
to it, the expected utility would have been
we expect only k(Pl + P2)
2k max (Pl' P2).
by allocating one unit to each process.
Each pair of observations thus costs (on the average)
in unrealized profit.
Instead
14
In the drug-testing situation, apart from the monetary costs of
.
experimentation, we incur a different type of cost peculiar to this
situation.
For each pair of patients allocated to the two treatments,
Qne patient is subjected to the inferior treatment giving rise to a
so-called ethical~, considered earlier by Anscombe £~7.
Although
it is difficult, if not impossible, to assess this cost in terms of
any monetary unit, it may not be too unreasonable to assume that the
ethical cost, thus involved, is "greater, the greater the difference
in the efficacy of the two treatments.
By
similar considerations
~s
in the previous situation, it turns out thattbe ethical cost per pair
of observations is increasing in
k IQ", where k
Igl-
Suppose, for simplicity it is
is any arbitrary positive constant.
We shall ignore
in this situation any monetary cost that might be incurred in obtaining
a pair of observations" as it cannot easily be combined with the ethical
cost •.
It is to be noted that the symbol k arising above in different
connections is generic and stands for any arbitrary positive constant.
For reasons to be explained in the next subsection we may take k = 1
Thus taking
B.
pair of observations as the new unit of observation, we
shall be concerned with the two following special types of cost per
observation, viz.,
Case (i):
Case (ii):
C{Q)
C(Q)
=1
, for all Q,
= IQI,
called constant cost
called absolute deviation cost.
15
(c)
losses due to wrong action:
It is well-known that a testing
of hypothesis approach on which some of the current procedures are
.
based,.· when formulated as a
simple loss functions.
decision-theoretic problem, involves only
We shall consider below two different types of
loss structure that may be more realistic.
I.
Linear loss structure:
L(Q, a )
l
(1.1)
L(Q, a )
2
:::
t:
The losses for the respective actions are
IQ J ,
if Q< 0;
,
if Q > 0 •
,
if Q < 0
,
if Q > 0 •
JO
=
l2A
Q
-
.
J
It is instructive to wri te them as regret functions, viz.,
i
The constant of proportionality
2A (the factor
2
= 1,
2.
is introduced as a
notational convenience to be understood subsequently) may be interpreted as same sort of index of the future use envisaged for the recammended drug or process (Anscombe f~7).
Since it is the relative magni-
tude of the loss vis a vis the cost per observation that matters in the
ultimate analysis, the constant k
in C(Q) , as mentioned earlier,
can always be (and henceforth is assumed to be) standardized to unity,
by adjusting the factor
2A.
After such adJustment, the constant
2A
may be interpreted as the number of future patients who will be the
potential users of the recommended drug in the drug-testing example,
or the number of future items to be produced by the recommended process
16
in the industrial process example.
perhaps, in practice, the experimenter will be able to assess
this constant, just as in the Neyman-Pearson-theoretic formulation
he is required to assess the least difference in the parameters considered to be worth detecting.
It will follow from the asymptotic
theory to be developed in Chapter TV that the sensitivity of the
optimum procedures to it diminishes as it increases.
Anseombe
£"~7
discusses how to make some guess about this constant in the context
of the clinical trial situation.
Anyhow, this type of loss structure
has been considered by so many authors £11, 26, 29, 33, 217 to name
only a few that it seems to be well-established in this branch of the
statistical literature.
II.
Modified loss structure:
We shall have occasion to consider
a slightly altered version of the linear loss structure that is especially relevant in the clinical trial situation, as envisaged by
Anscombe £~7.
= Ql
is the difference
2
between the unkno'WIl means of two normal popUlations 'With kno'WIl variances,
and where
Yn
For the problem where
Q
-
Q
stands for the cumulative sum of the differences of n
successive pairs of observations, Anscambe introduce. this type of
loss structure as follows:
ttpermps
2A
should be assessed, not as a constant, but as an
increasing function of
Iynl/n, since the more striking the treatment
difference indicated the more likely it is that the experiment will be
noticed and so will affect the treatment given ·to the future patients.
One way of introducing such a dependence of 2A on
IYnlln is to
17
assess
2A + 2n as a constant, as though the sum of the number of
patients in the trial and of the number of later patients directly
affected by the trial were fixed. 11
If we denote this latter constant a.s 2A (WhiCh should not be confused 'With the earlier 2A in the definition of the linear loss structure), and if an action is taken after performing tests on n pairs
of patients, then the numberof prospective patients to use the recommended drug is reduced to
2A - 2n.
The constant of proportionality
in the proposed loss function 'Will thus depend on
sampling.
n, the stage of
Explicitly then, we define the modified loss structure as
follows:
..
~
2(A-n) JQJ
e
L(Q, alin)
=
0
0
(1.;)
,~
)
L(Q, a ;
2 n)
=\
0
0
2(A-n)Q
,
,
,
,
,
,
if Q < 0,
n <A
if Q ~ 0,
n
~ Aj
if Q > 0,
n
= 0,1,2, •••
if
Q ~ 0, n
j
= 0,1,2, ... ;
if Q> 0, n > A
j
if Q > 0, n < A •
The considerations that led to the formulation of the modified
loss function suggest that the absolute deviation cost combines more
naturally 'With this type of loss structure.
Hence the modified loss
function 'Will be considered in Chapter III only in conjunction 'With
C(Q) = IQ J and not With constant cost •
•
18
Remark:
It seems worthwhile at this stage to point out an
interesting connection between the decision-theoretic formulation
of our problem, with the modified loss and absolute deviation cost,
and the classical two-armed bandit problem with a certain restricted
class of strategies as considered by Vogel £;9.
The original two-
armed bandit problem is concerned with devising a sampling plan (i.e.
prescribing a policy of selecting one of the two possible binomial
experiments with unknown probabilities of a success,
Pl and P2' at
each of a fixed total of 2A experiments) in order to maximize the
total return.
Vogel considers, and also gives some justification for
doing so, a class of strategies as follows.
In the first
2m
steps
each of the two kinds of experiments, I and II, say, is performed an
equal number, N, of times.
formed either with I
results of the first
The remaining
2A - 2m
steps are per-
or II alone selected on the basis of the
N pairs of observations.
Nowihe decision
whether to continue taking one more pair of observations or to discontinue taking pairs and performing the remaining steps with I
is made with the help of a sequential procedure,
0, say.
or II,
N is thus
a random variable bounded above by A.
If E(N)
denotes the expected number of pairs of steps before
deciding to continue with a single type of experiment, if Y£l -
17
denotes the probability of continuing with I£I]} when the procedure
o
is used, then the total expected return following this plan is
given by
19
while the best possible expected reutnr would have been
Thus the regret in following
5
is given by
which may easily be verified to be the same as the risk function (defined as the expected loss plus expected cost) associated with the pX'ocedure
0
for our problem with modified loss and absolute deviation
cost.
(d)
a priori distributions:
Although it is theoretically possi-
ble to carry out a Bayesian analysis for any a priori distribution"
it has become quite customary for the sake of mathematical convenience
to deal with only those a priori distributions that are closed under
sampling
£42.7"
which have alternatively been christened as natural
conjugate priors (NaP) by Raiffa and Schlaifer
£3'2.7.
In most circum-
stances praOl' knowledge of the experimenter" which is to be reflected
in the prior distribution" is rough and can adequately be quantified
with the help of a particular member of any reasonably large family
of a priori distributions.
The natural conjugate prior distribution of
of Pl and P2
is a product of two beta distributions.
.. This speci-
fication may be satisfactory when a priori it is reasonable to con-
20
sider Pl and P2
to be independent.
If'" however" the experimenter
has some prior reasons to believe that the curative potentialities
of' the two drugs go together (which might be due to the fact that the
ingredients of the two drugs are to some extent similar), then one
should select some joint distribution of' Pl and P2 with a positive
correlation between them.
As this will frequently be the case" a
natural conjugate prior is ruled out for the model as it stands.
We
shall instead convert the problem to another one for which it is possible to specify a natural conjugate family of prior distributions with
correlations between Pl and P2.
The a priori distribution with
which we shall be mostly concerned in our analysis in Chapter II and
III will thus be specified at the end of the next subsection in which
we describe the intepded reduction of the original problem.
1.5 Reduction of the original problem.
In so far as we are interested only in the difference P1 - P2
Q
and not in the actual values of Pl and P2 (the specification of loss
and cost in the previous section involving P1 and P2
~)
only through
the problem would have been a lot easier, if, like the corresponding
normal problem, there existed a sufficient statistic Y whose distribution involved only
Q.
We can, however, represent
Q
as
the difference of the probabilities of the two types of untied observations.
Moreover, due to the symmetrical footing on which the two drugs
stand to each other" it would seem that the two types of tied observations contain little or no discriminatory information about
Q
= Pl-P2'
21
and, therefore, may be coalesced into a single group with little loss
of efficiency.
The probability of a tied pair is thus
We are thus led to consider the statistic
the difference of the two Bernoulli random variables.
Z has the
folloW1ng~al distribution:
ifZ=q
(1.4)
=
~o
p~=g =
~l
~i ~ 0, i = 0,1,2
~o
~=~
=
+
~l
+ ~2 = 1
.
~2
The specifications about the actions, losses and costs per observation described in the previous section remain the same for this modified problem if one interprets
~
not as
Pl - P2 but as
~l
-
~2·
We have thus reduced the original problem of comparing two binomial
parameters into a problem of comparing two cell probabilities in a
certain trinomial model.
The fact that for the corresponding problem of comparing two
normal means it is sufficient to restrict attention to the difference
Y
= Xl - X2 of the two observations by appealing to the principle of
invariance and sufficiency
L:2~7,
and the fact that the binomial problem
is approximated by the normal problem at least for large samples, lend
some additional justification for the present reduction.
The main argu-
22
ment for this modification is, however, the resulting mathematical
simplicity engendered out of the existence of a convenient family of
a priori distributions that is NCP for the trinomial model.
This is
defined below as:
~
priori distribution:
a,b,c positive integers
This has been called the (bivariate) Dirichlet distribution
£41,
p.
1717.
This family possesses certain properties that make it as tractable
mathematically as the beta family.
Some of these properties are re-
corded in the subsection below for future reference.
1.5.1 Some Properties of the Dirichlet distribution.
(a)
The marginal distributions of the Dirichlet distribution are
beta distributions • Explicitly,
(b)
given by
The means, variarices and covariance of "'1 and "'2 are
2;
e
(1.5a)
E("'l)
(1.5b)
E(:I12 ) =
a
= a + b + c
b
a + b + c
,
V("'l) =
,
V(:I12 ) =
a(b+c)
2
(a+b+c) (a+b+c+l)
.
1
.
b(a+c)
(a+b+c)2(a+b+c+l)
1
ab
(a+b+c)2 (a+b+c+l)
From these it follows that
a - b
a +b + c
=
(1.6)
(c)
,
4 ab + ac + bc
(a+b+c)
2
(a+b+c+l)
The family of Dirichlet distributions form a natural conjugate
family to the trinomial distribution, so that the posterior distribution
of
"'1' "'2 after observing n
independent observations each following
a trinomial distribution, is another Dirichlet distribution with new
parameters.
Specifically, if the a priori distribution is
d ~("'l' "'2!a o' b o' co) and if out of the n observations
a,b,c(a+b+c = n) belong to the three cells With probability
,
1
1 - "'1 -:112 respectively, then the a posteriori distribution is
:11
:11
2
,
d S(:I1 l , "'21 a o + a, b o + b, Co + c).
1.5.2 Reduction of prior information.
The modification of the original two-binomial problem into the
trinomial problem gives rise to the technical problem of converting
the prior information about Pl and P2 into that about
:11
1
and
:11
2•
24
Although a
p~ise
analytical transformation of this information is in-
convenient mathematically (in fact, the transformation from Pl P2 to
~l' ~2
is not
one to one, but one to two at most), any such sophisti-
cated treatment may not be necessary since a priori information is usually of a rough nature.
A few lower order moments of
l and ~2
are sufficient to fit a particular Dirichlet distribution. Now it
1C
might be easier on the part of the experimenter to form some ideas
about the moments of Pl and P2 rather than about ~l and ~2
from previous experience~ In such situations the following relations
might be helpful; these relations give the means and variances of
~l
and
of Pl
~2
in terms of those of Pl
and P2 assuming independence
and P2 •
i = 1, 2 •
It may be interesting to know, in particular, what a uniform distribution of Pl
and P2' reflecting little or no a priori knOWledge
about Pl ' P2' means in terms of ~l' ~2 and vice versa. A rough
calculation using the above formulae shows that a uniform prior distribution of Pl ' P2 over the square fO, };7 x fO, };7 (1.e.,
assuming P and P independent and unifom.) corresponds approxi..
l
2
mately to the Dirichlet distribution d ~ (2 / 2,5).
25
If the experimenter feels that Pl and P2 are not independent,
then the following formulae might be usaftLl.
E(~i) = E(Pi)~l - E(p~_i17
V(~i) ~
where
V(Pi)£l -
-
P~V(Pi) V(p~_i17l/2
E(p~_i172 + V(P~_i)
,
2
E (Pi)
P is the correlation coeff!cient between Pl and P2.
Using the above formulae it may be verified that a joint distribution of Pl and P2 with uniform marginals and a correlation
P
= 1/2
d
s(4,4,11).
corresponds approximately to the Dirichlet distribution
1.6 A decision-theoretic formulation for the binomial problem.
In this section we write down, for the sake of completeness, a
corresponding decision-theoretic formulation for the problem involving
a single binomial parameter, p.
dent interest of its own.
The binomial problem has an indepen-
In fact, the problem with linear loss struc-
ture and constant cost has already been considered by Moriguti and
Robbins ~~2.7.
Moreover, this might serve as a statistical formulation
for the original problem when it is reduced, as shown in page
problem about a single binomial parameter, neglecting the ties.
6 to the
This
model also arises with continuous responses, such as temperature or blood
pressure reading, to the two treatments in the following way.
Instead
of discretizing the continuous responses as lying above or below a certain level as mentioned in page 2, we consider the original
~eeponses
26
as such.
Suppose specifically that
Xl and X are distributed with
2
absolutely continuous distribution functions F and F respectively.
2
l
Suppose also that the experimenter does Dot wish to make any assumption
about any specific parametric form, say normal, exponential etc. for
F1 and F2' and 'Wants to test whether Xl
is larger than X2 in
some probability sense (e.g., stochastically larger
L 27,
p. 7'J.7);
then the present formulation can be applied to the following binomial
parameter
,"
= PLXl > x2_7 = l F2
p
dF1
./
The various elements of a decision-theoretic formulation are
specified below:
model:
actions:
PLX=]}=p,
a
l
PLX=9]=l-p,
05p5 l •
preferable if p > 1/2 ,
a 2 preferable if p
costs per observation:
(i)
5 1/2
•
C(p) = 1, all p:
(ii)
C(p)
= Ip-l/2/:
constant cost
absolute deviation
cost.
losses:
I.
linear loss structure:
L(p,a l )
=
i
2A Ip-l/21 , if p ~ 1/2
i,
0
.:.
'--
(1.9)
L(p,a 2 )
=
( 0
J
'\\
2A(p-l/2),
, if P > 1/2
, if p ~1/2 ,
if P > 1/2
,
27
II.
modified loss structure:
L(p,al;n)
2(A-n) Ip-l/2!, if p ~ 1/2, n < A,
= )
0
,
0
, if P > 1/2, n = 0,1,2 •••
0
, if P ~ 1/2, n
= 0,1,2, ••• ,
,0
, if P > 1/2, n
~
l
(1.10)
L(p,a 2;n)
=5
if P ~ 1/2, n
~
A,
A,
l2(A-n)(p-l/2), if p > 1/2, n < A.
a priori distribution:
(1.11)
where a,b are positive integers.
We shall use the term binomial problem to denote the decision
problem formulated above.
In Chapter II we shall consider a slightly
more general situation, viz., testing p < P
-
P
o
0
against p > P , where
0
is an arbitrary number 0 < P < 1. The above formulation corres0
ponds to the special case Po = 1/2.
The corresponding decision problem concerning the trinomial model
will be called the trinomial problem.
Another important reason for
introducing the binomial problem in this thesis is the fact that, being
mathematically simpler to solve it constitutes a first step towards the
solution of the trinomial problem.
In fact the analyses for the two
problems are so· analogous that in the remaining chapters only the
28
binomial problem is treated in details.
The results for the trinomial
problem are just stated omitting many of the details.
1.7 Review of some earlier relevant work.
We have already mentioned Anscombe ~~7 which contributes rather
heavily towards the formulation of our basic problem as stated in
section 1.4.
Anscombe considers the corresponding normal problem and
gives an easily computable outer bound on the optimum boundary.
We
are, however, concerned with the exact optimum boundary for our prob..
lema.
We have also mentioned Moriguti and Robbins ~3!2.7 in section 1.6
in connection with the binomial problem.
only linear loss, constant cost and Po
Specifically, they consider
= 1/2.
From rather elementary
considerations they show the existence of the Bayes procedure for
their problem and describe it constructively.
They have also given
the numerical values of the optimum boundary for certain values of A.
Next, they give a heuristic treatment (which we shall adapt for our
generalizations) of the limiting behavior of the optimal solution
under two different types of limiting tendencies.
For the second
type of limiting tendency that turned out to fit the observed numerical
results better than the first, they reduced the problem of finding the
optimal solution to that of solving a free boundary value problem, thus
arriving at the same mathematical problem as Chernoff ~l!=.7 and Lindley
~2§l
arrived at for the normal problem.
But they gave i'or the first
time a formal series expansion solution of the free boundary value
problem (in the permissible range) which has later been shown to be
29
an asymptotic expansion by Breakwell and Chernoff
f~7.
Moriguti and
Robbins also considered some alternative ad-hoc sampling rules and compared them with the optimal one at least in some restricted sense.
Finally, they presented some charts showing the CC curves and ASN
curves for the optimal sampling rule for some representative values of
the relevant parameters involved.
It will be seen later that the limiting behaVior of the optimal
solution for the binomial problem reduces to that of the optimal solution for the normal problem under appropriate limiting tendencies.
For
this reason, results connected with the latter problem are relevant
for our purpose.
-e
Bather
18_7 in
Towards this we mention an important contribution of
outlining some methods of obtaining certain bounds on
the optimal solution.
These bounds, of course, have nothing to do
with the outer boundary of Anscombe, mentioned earlier in this section,
for the normal problem, and the possible inner boundary that might be
obtained from the modified Bayes rule developed by Amster
.£)}.
The works reviewed above are directly relevant for our speCific
problem.
It is not intended here to review any of the rapidly g:t'mdng
volume of works connected with the general theory of sequential analysis.
An excellent summary of the current state of affairs in this field is
given in
£217
by Johnson.
1.8 Chapter by chapter summary.
Chapter I.
The basic problem of double dichotomies is stated.
Existing methods of analysing such data within the framework of NeymanPearson theory are described briefly along with their main defects and
their inappropriateness in the present context.
Following Anscombe
£5.7,
30
the problem is then formulated as a decision-theoretic one vdthin the
framework of Wald theory involving more than one realistic loss and
cost function.
The problem thus formulated is then modified, for
reasons that are explained, to one inv.olving the difference of two
cell-probabilities in a certain trinomial, distribution, called the
trinomial problem.
The corresponding binomial problem is also formu-
lated.
Chapter II.
The general theory of statistical decision as given in
Blackwell and Girshick ~27 is applied to show that the Bayes procedures
for the various decision problems (with linear loss) are truncated with
probability one.
It is also described here how to obtain the Bayes
sampling rule and hence the optimum (Bayes) bcundary constructively,
-e
for both the binomial and trinomial problems (linear loss structure
and both constant cost and absolute deviation cost).
Chapter IIf. The usual characterization of a Bayes procedure in
the general set up considered in section 2.1
is
eA~ended
in section
3.1 to take account of any "stage-dependent" loss function.
These
results are then specialized to obtain some simple recursion relations
characteriZing the optimal procedures for the various decision problems
with modified loss and absolute deviation cost.
These procedures are
also shown to be necessarily truncated at a known stage.
Chapter IV.
The limiting behavior in large samples
of the optimum
boundaries for the decision problems studied in Chapter II is considered.
Extending Morigut:i.. and Robbins'
~3g7
approach the problem of finding
the normalized optimal boundaries with appropriate normalizations is
31
reduced to that of solving certain partial differential equations
free boundary value problems.
A series expansion for a certain por-
tion of the optimum boundary is then obtained for each problem.
Ex.
pansions which enable the computation of the average risk are also
given.
Chapter V.
Chapter IV
An indirect verification of the series expansions of
is offered with the help of certain upper and lawer bounds
on the optimum boundary obtained by applying same methods of Bather
-e
f§.7.
•
e
e
C\I
"'"'
A SCHEMATIC CLASSIFICATION OF THE RESUTIrS BY SECTIONS
optimum boundary
--~-~-.
~l
Exact
~~tie asymptotic
i Asymptotic
I
--'--r---------··---- - - - - -
Model
binomial
linear loss
imodified loss linear loss
linear loss
;
iconstantiabsolute
constant absolute
cone'Gant 'absmr1.nte' _f absolute
cost
deviation
coat
deviation! deviation I cost !deviation
1
cost
cost
cost
I
i cost I
2.~.2.6
Mortguti :
and
Robbins
trinomial
2.7-2.11
I
2.2-2.6 ,i
,
I
3.3
I
i
i
1
j
: 4.1-4.2 /4.1-4.2 I
: Moriguti
and
Robbins
j
;
I
2.7- 2•11 1
j
:
3.4
5.1
Bather
4.3-4.4 /4.3-4.4
l
!
I
r
I
5.2
I
:
5·2
CHAPTER II
BAYES PROCEDURES WITH LINEAR LOSS STRUCTURE
2.0
~ry.
In section 2.1 we collect together some known results of the
general theory of statistical decisions that are applied with increasing specializations in the subsequent sections.
In section
2.2 a binomial model with linear loss and a general Po
duced.
Sufficient conditions are obtained for a Bayes procedure
to be truncated.
-e
is intro-
In section 2.3, a beta prior distribution is
introduced and the sufficient condition of section 2.2 is shown
to be satisfied.
In section 2.4 the point of truncation of a Bayes
procedure is definedj it is also show how to determine it.
Section
2.5 gives a description of how to obtain the optimal boundary from
the point of truncation by working backwards step by step.
Section
2.6 contains some points useful for computation of the optimal
boundary as described in section 2.5.
Sections 2.7-2.11 treat the
trinomial problem and correspond in the numerical order with the
sections 2.2-2.6.
2.1
Some well-known results about Bayes sequential procedures with
a general model.
In this section, we collect together some know results
12.7
that
will be useful in obtaining the Bayes sequential procedures for the
binomial and the trinomial model with linear loss structure.
Notations:
{X \ :
i
sequence of independent and identically distributed
i=1,2, •••
random variables.
fQ(~l)
probability density function of the random variable
:
with respect to some meaS\11J:!e
Xl
~) = ~ Q):
a , a
1
2
:
L(Q, a ),:
i
i=1,2
C(Q)
space of states of nature.
two possible actions or terminal decisions.
losses.
cost per observation.
g
C(~)
IJ..
: a priori distribution (on a a.fie1d of subsets of
_ (C(Q) d
~(Q)
•
a ) d
S (Q),
../
(d)
L(I , a i
po(~) :::
f~{X) ~
);
~L(g,
i
o
min
i=l, 2
i
= 1,
2 •
L(~, a i ) .
J'fg(X) d
~
(Q) .
€>
~j :::
(Xl' x2 ' .... , xj ),
j
= 1,
2,...
•
j
fg,j :: fQ,j(~j)!:\'
ft;,j :;
fg,j(~j)::
11' fQ{xi
),
j = 1, 2, . . . .
~(Q),
j = 1, 2, ••••
i=l
SfQ,j d
®
j
= 1,
2, •••
0 ).
35
N
sample size (random variable) using a specified decision
function.
0
I
,."
•
,,!~"y>,J=,
j)
t.
~j ~
*j(2:j)' j
1 ; ... :
a sampling rule, where
= 0, 1, ...
is the probability that N = j, that
is, the probability of termination after observing xl' ••• ,x
l'
¢ = ~,¢j
-e
= 0, I,
j
:
¢j;: ¢.(x.), j = 0, 1,
a terminal decision rule, where
a 2 after observing xl"",xj
~n'
n
•
is the probability of taking action
J -J
8·~ (~,
j
conditional upon N
=j
•
¢): a sequential decision function.
= 0,1,2, ••• :
class of all sequential decision functions
truncated at N = n, i.e., with
*o
+
VI
+ ••• +
*n = 1
identically in x
-n'
co
u
~
n=l n
~l
:
class of all truncated sequential decision functions.
class of all sequential decision functions which terminate
almost certainly, i.e. for which
E
j=o
Vj
= I,
a.e.
for all ~
€
G
class of all sequential decision functions subject to usual
~27
measurability conditions.
36
0e: =.
(*s'
r(~,
¢~):
Bayes decision function in a specified class
average risk associated with 5.
0)
min r(~" 0)" n
P (S)::
n
o€
b.
= 0,,1,,2,..... Bayes risk in
b. n •
n
(this notation is consistent with earlier one,
p(s);:
b..
min
o€
b. 2
r(g" 5):
Bayes risk in !J.
2
po(~».
•
We state the results in the form of the following lemmas.
Lemma 2.1.
b.
A Bayes terminal decision rule for
for any n
n
or
r
-e
¢g,j
=
<
I
or
j
where u
Lemma 2.2.
= 0,1,2, ••• n
= 0,1,2, •••
is arbitrary
If Os
,
= ( \~s"
= L(~j' a l ) <
ff
=
if
ff
= L(Sj'
if
b.=b.
n ,
b. = /),k
,
~ u ~
¢g)
b.
is
2, satisfies
It
if
(0
_or
if
u
II
j
pO(~j)
if
! 0
(2.1)
= 0,,1
!J.k for any r
b., where
0 €
L(~j' a 2 ) ,
=
ff
a 2 ) < L(~j' a l )
n
= 0,1,2, ••• ,
k
= 0,1
,
,
or 2;
1).
is a Bayes decision function in /),1,
then
and, after observing Xl' ••• ' x j
if
'
sampling is ter.minated if and only
37
then
limpn(t;)
n _> co
=
peS) •
Notations.
n> 1
vm, wm defined recursively for m = m, n-1, ••• ,1 by
(V'n =
,
( 2 .4») w
\
m
\"_ vm
Lemma 2.4.
= min
=
,
(u , v )
m m
m = n, n-1, ••• ,lj
1)' m=n-l,n-2, ••• ,1 •
)(wm+1 (x,
-m xm+1 ) dll(x
r- m+
a
such that for
,
a
u
-(2.6)
j
If there exists a least positive integer n
every n> n
-
un
n-1 < v n- l'
for all -nxl'
then
...
Notations.
i
= 1,2
•
3$
(2.8)
t\
/\. 1
1\
Lemma 2.5.
!2 /\
"=.
A1
U /\ 2:
complement of
and
/\2
stopping set •
1\
continuation set
are convex, i.e., if
1;1 ~
1;2
belong
S ~ Sl and ;2'
i' then any convex linear combination
namely
belongs to
Note.
-e
/\., i
~
= 1,
2.
1 of £~7.
Lemma 2.3 follows from Theorem
All other
lemmas are essentially contained in Blackwell and Girshick I s book
2.2
£27.
Binomial model.
In this section we apply the results of the previous section with
the following specializations and call the decision problem simplY the
binomial problem.
(2.10)
f (1)
P
a
a
where
Po
f (0) = I-p,;
P
is preferable
l
if
is preferable if
2
=
j
2A(Po-p)
1....
(2.11)
-
0
,f'
L(p,a )
2
=
{2A(P:P
o
)
O~p~l •
p>
Po '
p < P
is an arbitrary constant,
L(p,a )
l
-
= p,
O<p
•
0
<1.
0
,
if
p<p
,
if
p>p
,
,
-
0
0
,
•
if p < p ,
-
if
p>p
0
0
•
39
It follows from the above specializations that
(
rl (I;)
r 2(~') =
(2.12)
,
1.
,
PO)
i g I EgP
PO; ,.
(
-'
"-
f"lo(g) =
Po (
I Eg P ->
~ s I E~P -<
't~
=
=
and
ji A mglp-pol -
(2.l3)
po{S)
=
l
A
11 Ip-pol
-
A
E~(P-Po) ,
for g
€
rl
'
A
Eg(Po-P) , for g
e;
r2
'
for S
€
110
A EsJp-pol
,
;
which may be written compactly as
(2.l4)
¢s,j
p
o
= A E~Jp-p
J ~
0
(g)
of lemma 2.1 and peg)
AIE~(p-p)1
~
0
of lemma 2.2 (equations (2.l) and (2.2»
reduce to
(2.15)
¢t::. ,J.
=~
j
,
0
if
Eg
p>p
j
0
,
u
,
if
"
=p 0
,
l_ 1
,
if
"
<p
,
= O,l,2, ••• ,n;
0
n =0,1,2, •••
.
(2.16)
where
J
.1
(2.17)
fg{i)
=
r
i
.~
is given by
1 i
P (1 .. p) -
(2.14) •
d g(p),
i =
0, 1
•
40
We develop below a sufficient condition for a Bayes procedure
for the binomial problem to be truncated with probability one, 1. e. ,
6g
in
coincides with
t::,0
81;
in b,2.
r361
S being absolutely continuous, Sobel
Os
sequential procedure
in
t::,
2
Under the assumption of
proves that the Bayes
is truncated with probability one
for a general set up that includes ours with constant cost.
His
method of proof did not allow extension for a cost depending on the
parameter;
hence the necessity of the following theorem.
Theorem 2.1.
Os
-e
A sufficient condition for a Bayes sequential procedure
for the binomial problem to be truncated with probability one is:
(2.18)
and
(a)
(2.19)
(b)
t C(p) > 03
S
var(p J
J:im
n ->
=
1
sn)1 C(Sn) =
0
a.e.
co
uniformly in n.
Moreover,
o~
~
takes at most n
0
observations, where n
the smallest positive integer such that for every n > n
-
is
- 1
a.e.
(2.20)
Note.
0
0
Condition (a)
is introduced in order that lemma 2.3
It is trivially sat1.elfied for constant cost for any
deviation cost, viz.,
lutely continuous
tribution (1.11) •
C(p)
= Ip-pof,
~.
holds.
For absolute
it is satisfied for any abso-
S and hence in particular when S is a beta dis-
41
Proof:
Because of lemma 2.3 and lemma 2.4, it is sufficient to show
that there exists a least positive integer n
n
>
-
n
0
o
such that for every
,
(2.21)
un-1 -< vn- 1
'
for all -nxl·
Writing explicitly the definitions of uland v
n-
a
n- l ' (2.3 ) and (2.4),
(2.21) reduces to
(2.22)
<
J.r
i.e., for all If;
-e
po(gn) + n C(gn>-7 fs,n dil (Xn ), for all
n-
1
~n-l'
for a fixed ; •
Writing
(2.23 )
for convenience, (2.22) reduces under the present set up, to
(2.24)
Now from the definitions
Also, expectation being a linear operation, it follows from (2.25)
that for any integrable g(p) ,
(2.26)
42
It follows from (2.13) and (2.26) by 1dentifying g with Sn_l'
~(Sn_l)
that
is identically zero for all those ;n-l such that
(Sn-l)(~)(Sn_l)(O)
all belong to the same r i , i = 1 or 2.
1
1
(2.24) is then satisfied for all these gland for any non-negative
Sn_l'
n-
cost C(p) •
Each of the remaining S lt s belong to one and only one of the
nfollowing three mutuallY disjoint subsets:
,
-e
.D2 ={~ I E\;
\
r.11
0
'
~(s)
•
<
€
=
€
€
2
•
r 2 ' so that it follows
A fg(l) Es(l) (p-po) + A fg(O) Eg(O) (po-p)
1
(2.28)
E~il)P }Cr
<
Po
r 1 and E;l(0)
from (2.23), (2.13) and (2.26) that
If t;
(1)
then Sl
P
1
(
1
A ) (p-po)pd r;(p) + A )
~o
= 2A Var
(
1
(po-p)(l-p) d ~(p)
0
(p j g)
remembering that
p
If S
€
= EgP for
!;
.r)l ' then
sil
o
€
)€
r)
~-o
r 1 and
.
siC)
follows from (2.23), (2.13) and (2.26) that
€
r 2 ' so it again
A.(s)
=
2A fg(O)
=
2A f(PO-P)(l-P) d g (p)
EgiO~po-p)
c
=
2A LEg (l-p)
<
2A
=
2A Var (p
2
- (I-Po) Eg(1-Pt7
2
rE
(1_p)2 - E (l-p) 7
s- g
Is) .
S
It can be show similarly that for
€
1.21.'
(2.30)
-e
<
Hence
2A Var (pis) •
(2.21) is satisfied if there eXists an no
for every n > n
-
0
such that
,
Var (P(Sn_l)
C(Sn_l)
<
-
1
2A
a.e.
for which a sufficient condition is (2.19) •
The second part of the theorem follows immediately from (2.31).
The proof of the theorem is thus complete.
Remarks: (1)
It is to be noted that
no
defined in (2.20) de-
pends on the particular a priori distribution S , while the sets
..lr-).
_~
, i = 0,1,2 (2.27) are defined independently of it •
44
(2)
It follows from the proof above that for the conclusion
of the theorem to remain valid it is sufficient that (2.l9) hold good
only for those Sn which belong to
2.3
1-2:: 1-20 U
..r21 LJ 1-22 •
Beta a priori distribution.
From now onwards we shall consider an a priori distribution given by
rea +b )
a -1
b -1
o 0
P 0 (l-p) 0 dp,
r(ao)r(b o )
(2.32)
where a"
b0 are positive integers.
o
It is well-known £42,7 that the beta family of distributions is
··e
closed under sampling.
£3~7"
In the terminology of Raiffa and Schlaifer
(2.32) forms a natural conjugate prior for the model binomial
distribution.
In other words, if a
n
and b
denote the number of
n
successes and failures respectively in a sample of size
n, the a
posteriori distribution is given byd
Sn (p) =
d ~(a
.
0
+ an
, bo+n
b),
n
= l,2, ••• ,
Thus the set of all possible a posteriori distributions that can arise
from the a priori distribution (2.32) can be represented by the set
of all lattice points in a positive quadrant with origin at
(a o,b 0 ).
It is well-known £27 that the Bayes decision-function can be
completely characterized by the sets
(2.9).
1\ l' A 2'
1\
defined in (2.$),
In this special case, these sets that are associated with
(2.32) can be represented as certain sets in the space of lattice
points mentioned above.
Without any risk of confusion we shall denote
these various (a,b) sets by the same notations as the corresponding
S sets.
The optimum boundary (for the decision problem) that will
be mentioned henceforth quite often, is formally defined (unlike the
terminology in Moriguti and Robbins ~;Q7) as the set of points (a,b)
in the stopping set
1\
such that (a-l,b) ~ (a,b-l) is in the con-
tinuation set /\ •
Due to the closedness property of the beta family mentioned
above, it is not necessary to determine the optimal boundary along
with the various sets
separately.
1\ l' 1\
2'
A for
each a priori distribution
It is possible to describe them in terms of a fixed origin
say (0,0), and to obtain them for a particular a priori distribution,
like (2.;2), we need only to shift the origin to (a , b).
.
0
to considering the a priori distribution (2.;2)
with a sample of n -2; a
+ b -2
000
~s
0
It amounts
being arrived at
observations with a -1 successes
0
and b -1 failures, starting with a uniform prior.
o
In the following analysis the terms optimum boundary, stopping sets,
continuation sets, etc. are defined with respect to this fixed frame of
reference.
d g(a,b)
Accordingly, by d Sn(p), unlike previously, we shall mean
where a + b
= n. I"t is obvious "that any sequential pro-
d s(a o' b o ) and is "truncated at no (in
"the sense that it takes at most no observations before it tells one
cedures Which is Bayes for
to stop), is also trunca"ted at
n + a
+ b
000
(in "the above sense)
when referred to this fixed frame of reference.
Since any a priori beta
distribution can be represented by d s(a o' b o ) where ao,b o are finite,
the condition (2.19) for a Bayes procedure to be "truncated remains
46
valid when g
n
is given the new interpretation above. We apply this
condition in this new light to arrive at the following corollary to
Theorem 2.1.
Corollary 2.1.
A Bayes sequential procedure for the binomial
problem with a beta prior distribution is truncated with probability
one.
Proof:
As noted before, condition (a) of Theorem 2.1 is satisfied
for the particular cases we are concerned with here.
Now
Var (pla,b)
Case (i).
=
ab
(a+b)2 (a+b+l)
Constant cost, i.e.
c(a,b) = 1 for all
(a,b)
Var p a,b)
C a,b
a + b ->
uniformly as
<Xl
•
Condition (2.19) is thus ensured in this case.
1
Case (ii).
Absolute deviation cost, i.e.,
c(a,b)
= )/lp-pofd;(a,b):
"0
As noted in remark (2) after Theorem 2.1, we need only verify the
condition (2.l9) for those S belonging to
.r:).,
-J.
= 0,1,2 which
i
may be written in this special case as follows:
(2.36)
..C-0
) =
.)
,_
(a, b)
(2.37)
1-)-1 = ~
(a, b)
L
a
a +b
a
a+b+l
"I
=
Po
5
< p0 <
,
a:b
~
J
) (a b) f
l
'
L
Also, it is sufficient to show that
as
(a+b) - >
C(a.,b)
}.
a+1
Po a+b+1
(2.19) holds only asymptotically
Now
co.
=
<
a
a + b
I
a-1 ( loop )b-1 dp
rP-Po I p
B(a,b)
(2.39)
J1
I
B(p
p (a+b)-l
loop (a+b).. l
Ip-plp 0
(l-p)
0
dp
°
a+'b; loop a+6)
000
for
(a,b)
function.
€
1-1i
, i
= 0,1,2,
where
B(a,b) is the well-known beta
Also, the right hand side of (2.39) is asymptotically
£15, p. 255,7 equal to
(2.40)
Thus
(2.41)
uniformly as
a + b
>
co ,
concluding the proof of the corollary.
2.4 Determination of the point of truncation in the binomial problem.
The set .1(-)
has been called, for obvious reasons, the neutral
-0
boundary by Weatherill £42.7.
We shall call
boundary for reasons to be explained below.
O
+ b
O
is smallest and no point (a,b)
the extended neutral
The point
..rl, ~
of ..rl ~
then formally defined to be that point of
a
1-1
.2!
truncation is
o
(ao,b ) such that
a+b> a
O
+ b
O
is
tt
48
~
continuation point.
The point of truncation, thus defined, mayor
may not belong to the optimum boundary defined earlier.
belong to the optimum boundary if it is in ~r )
.
-0
= 1/2,
case, i.e., when Po
(-)
when p 0
= 1/2).
In the symmetric
the point of truncation may alternatively
be defined as the first stopping point on
.1 -0
It does not
1-1
(which is the same as
Here by ttfirst" we mean "with smallest sum of
co-ordinates" •
The determination of the point of truncation is important in the
sense that, once it is known, it is possible to obtain the optimum
boundary, as will be described afterwards, by working backwards from
this point with the help of the recursion formula (2.16).
Before we proceed further, we note the following facts about the
1-1i
sets
' i = 0,1,2.
"{-10
is empty if Po is irrational. If Po is rational,
in which case it can be written as a pure fraction, say,
(1)
where
ra
and r
b
are mutually prime to each other, then the sequence
of points {rak' rbk}
= {Sk~
, say, belong to
that the farther the ratio
in
..c-).
-0
points
(2)
ra/rb
In the special case p
0
1-10
•
It is obVious
is from unity, the fewer the points
= 1/2, -C-)-0
consists of all the
(a,a ) on the diagonal a=b.
1.h
and
1-12
are empty when Po
= 1/2. They have been
primarily introduced for the situation p .,. 1/2. They represent
o
points near
but not exactly on it, on the two sides of it,
110
1-l
i
being on the side of
/\ i' i
(-)
,
..l -0
2.
that a failure (success) at the point leads
f\
the resulting posterior distribution to
(3)
Let
Ln+1
situations,
case
Po
(-)
--
n
=nJ
Then
is
intersects
(-) "
L
n
1-1 (\
L
n
2(
Al )
•
denote the set of lattice points
- -
is not empty and
= 1/2,
empty if
= t(a,b)fa+b
a + b = n.
on the line
1-1 ()
L
n
In fact, a point in
/\ 1( 1\2)' is so near to the neu-
1-1 (f·'12), though belonging to
1
tral boundary,
= 1,
is empty if and only if
Ln+1
C 1-10
In all other
,
in exactly one point.
1-1 =1-10 , it
odd while L n (-2
n
-
since
follows that
In the special
Ln ()
1-1
is
is the point (n/2, n/2) if n
is even.
-e
(4)
of 1-1
1
If Po
is given by (2.42), then there are exactly
interwoven with exactly
tween any two points, say,
r -1 points of
b
(rak, rbk) and
1-12
in
(rak+l, l'bk+l )
ra-l points
1-1
of
be-
1-1
0
•
The results of the previous section ensure that a Bayes seguential procedure is truncated.
nO
This means that there exists an integer
sufficiently large such that the sampling region (or the continua-
tion set) lies below the straight line
L o.
Using the above facts, the
n
region of sampling can be narrowed down further as stated in the following
lemma.
Lemma 2.6.
If the Bayes sequential procedure· for the binomial problem
with a ~eta prior distribution is known to be truncated at
every point (a, b) with a+b > nO
tinuation set
A is
nO (Le.,
is a stopping point), then the con-
contained within a rectangle (with a corner re-
moved in the first two cases below):
50
(i)
If the line a+b
1-l1 ,
(ao,bo) of
(a °"b °) ~
1-12
= bO
+ 1, and
n.
= nO
(aO, b 0)
with
If the line a+b::;: nO
interse;ts
1-2
at a point, say
= bO
a::;: a °+ 1" b
(a°+1, b O-1).
-
and the
intersects ~1 at a point, say,
'-), then !\ is bounded by
(a 0, bO) of
. i -o -
s=a
° ,b=b°
If the line a+b::;: nO
..2! 1-10 '
1-2,
does not intersect
arises when the line a+b::;: nO+l intersects
(ao,bo)
at a point, say
' then /\ is bounded by
diagonal joining
(iii)
1-2
(ao.bo)
~~th (0
,
a - 1 , bO+l) •
If the line a + b
(ii)
intersects
~I\ is bounded by a::;: a O , b
+h
. ..
v e diagona 1 Jo~n~ng
(iv)
= nO
1-1
which case
at a point, say
~ 1\ is bounded by
,
The proof of this long lemma is simple and follovm readily from
lemma 2.5 and the above facts (1), (2) and
(3).
The above lemma is useful in locating the point of truncation.
it is known that all the points on a certain line
points, then so are all those on the line
in
1-1,
if any.
L
n
are stopping
L _ except perhaps the one
n l
1-1 is empty, we move on
n
In other words, if
If
L _l
n
to the next line L _ ; if, on the other hand, L _l
1-1 is not
n
n 2
empty, in 1V'hich case it consists of just one point belonging to any of
n
the
,
{-).t
.l
_~ s
i
= 0,1,2,
according to (3) above, then it needs to be
verified vlhether it is a stopping point
01'
a continuation point.
it is a stopping point, we move on to the next line, and proceed
If
51
analogously till we reach the first continuation point on 1-2 • The
next theorem gives an easily applicable criterion by which it is
possible to verify whether the point S(a, b ) in
(-)
n __
L_
n l
is
a stopping point or a continuation point.
Theorem 2.2 •
(a+l,b),
..!!. s(a,b) is a point on i-2 such that gil),~,
~
giO) ,..!..&.o, (a,b+l), are stopping points, tllim. s(a,b)
is a stopping point or a continuation point according as
"'(I; ) < e(g)
(2.43)
-
or ",(g) > e(g) ,
where
(2.44)
Proof.
"'(s)
's)
(
2A
=t
2A
~Es(1-P)2 - (l-po) E (1-p27 , if
2A
~EI; p2 - Po EgP_7
Var (p
1
s
if g
s€
, if g
€
1-20
1-21 '
1-22 •
Since the B&yes procedure is known to be truncated, the hypo-
thesis of lemma 2.2 is satisfied so that (2.16) holds.
Subtracting
po(~) from both sides of (2.16), we get
where
€
",(~)
is defined as in (2.23) •
. Since sill
and giO)are stopping points, by hypothesis,
52
Hence, (2.45) reduces to
(2.46)
p(g) • po(s) = min fO, e(g) - A(S t7
Now S is a stopping point or a continuation point according as
or
i.e., by (2.46), according as
A(g) ~ e(s)
The expressions for
A(S)
A(S)
or
> e(s) •
given in (2.44) were already obtained in
(2.28), (2.29) and (2.30). The proof of the theorem is complete.
We now derive bounds on A(s)
= A(a,b)
for
s = g(a,b).
Lemma 2.7 •
o -< A(a,b) < 2A
We are now in a
p (l-p )/(a+b) •
0
pos~tion
point of truncation.
0
to determine the exact location of the
For simplicity we consider the symmetric case
first.
Special case:
p
o
As noted earlier,
consists of points
= 1/2.
1-11
(a,a).
and
1-12
are empty in this case;
,,(-1 0
It follows from the procedure indicated
above and Theorem 2.2 that the point of truncation is given by (ao,a o )
where a o
is the smallest positive integer such that
No'" for constant cost,
e = 1 , while for abs'olUte deviation cost
53
1
C( a"a )
5I
=
p -
~I
dS (a" a)
°
1
= '2
(2a)!
a! a!
where b (a"a) denotes the central term of the symmetrical binomial
i
1
distribution with parameters (2a l' "§) •
is
Thus it follows from (2.51)" (2.52) that the truncation point
(ao" aO) where a O is the smallest positive integer such that
-12 ( -A
2 - 1)
(2.53)
,
for the case of constant cost; and such that
for the case of absolute deviation cost. Standard binomial tables
O
may be used to obtain a
from (2.54). For large values of A" we
may use instead the asymptotic relation
(a o, aO) is the point of truncation, then the
O
(a"a) ~ a < a
are continuation points.
Corollary 2.2.
points
~bis
If
follows readily fram trJe deftnitlon of tbe point
c..ation," the. mcnotoIaUally
~.spect·
to
d.ecr~s:j.ng $l:tope~ty'
a~GJ,nd lJlheorein2.2~ ,e,specially
bering t]p.q5t .p:E.g0 f or any point
(a"b) •
of'
o~
'trun-
~(I;l.ia}/e(a"a.)
wirth
the equat10n (2?45) remiarn-
General case.
O<p
°
<1
It follows fram (2.43) and (2.47) that any point (a,b) with
2A P
>
a + b
(l-p )
°
C(a,b)
is a stopping point.
smallest a+b
Thus, if
°
(a',
bf
be the point of
)
satisfying (2.56) and if
(ao,bo)
1-1
with
be the point of
truncation, then
Consider the points of
starting from (a' I b f
) •
1-1
in a sequence with decreasing a+b,
Hith the help of the criterion developed in
Theorem 2.2 we may then search for
(ao,bo), as described already,
along this sequence.
special case:
(
~
j
\"
constant cost
Po
= ra
ra
+ r
b
'
as defined in (2.42)
In this special situation we do not have to go beyond the point
(a~,bl) say, along this sequence, where
k
°
being the largest positive integer k
for which
P (l-p )
>
A(r k, rbk);:
•
a
era+rb ) k+l
In fact, it may be shown in this special situation that the
°
point of truncation,
°
(ao,bo) is one of the (ra + r b - 1) points of
55
,-1
~
°
°
between the two successive points (ra(k +1) , rb(k +1»
(ra kO, rbko)
of
'{-19 including the first. This follows since
from the definition (2.58) of kO, (r kO, r kO)
b
a
point'VTbile
all points
and
(ra(ko+l), rb(ko+l»
is a continuation
is a stopping point.
Moreover,
(a,b) with
a + b
> (ra + rb)(k O + 1) + 1
-
are stopping points.
This follows from (2.4;) since for any such
point (a, b) ,
"'(a,b)
<
2A Po(l-Po)
by lennna 2.
a+b
<
2A po(l-Po)
by
(2.59)
(r +rb)(ko+l)+l
a
<
1
by (2.4;), (ra(ko+l), rb(ko+l»
being a stopping point.
Finally, it may be shown just like in corollary 2.2 that the
points
(r k, r k) with
a
b
k ~ k
O
are all continuation points.
We
cannot assert, however, that all the points of i l l and i 12 for
O
O
which a+b < a + b
are also continuation points, since ~(a,b) is
not necessarily monotonically decreasing in
i
= 1,2.
a+b
for
(a, b) ~
il
1
This is the reason for the particular definition of the
point of truncation that we chose in the beginning of this section.
He expect, however, that most of them will be continuation points
because by Lemma 2.7
"'(a,b) is closely bounded above by a function
,
56
that is monotonically decreasing in a+b.
2.5 Procedure for obtaining the optimum boundary in the binomial
;problem.
In this section, we describe how to obtain the optimum boundary
constructively once the point of truncation is known.
we first consider the case p
Special case:
po
o
For simplicity
= 1/2.
= 1/2 •
The neutral boundary,' ~(-)
, consists of points (a,a) in this
-0
case., Because of the symmetry of the problem, the optimum boundary
will be symmetric with respect to the line a
= b.
any loss of generality, we consider only a
b.
Let
(ao, a O)
be the pomnt of truncation obtained as indicated
in the previous section.
are stopping points.
~
Hence, without
By Lemma 2.6, all the points on
Also, fram Corollary 2.2, all the points (a,a)
With a < a o are continuation points.
Next, the recursion relation
(2.16) reduces to
where
(2.61)
and
(2.62)
C(a,b)
=\
J
1
, all (a, b )
l EL /p- ~ la,E7·
r
, for constant cost ;
, for absolute deviation
cost.
57
For stopping points p
points
=p
, a known function, and for continuation
°
..
p < P , as follows from Lemma 2.2.
°
classified as a
Thus a point may be
stopping point or not according as
p
= po
or
p < Po at this point.
Consider the points on the line
a
=aO -
(ao-l, aO.l) which is a continuation point.
tained with the help of
(2.60)
since
I, starting from
p(ao.l,ao-l)
(ao, aO-l)
may be ob-
and (ao-l,ao) are
stopping points as noted earlier and hence have known p-values.
this
Us ing
p(aO_l,ao_l), and the known p-values of the stopping points on
the line a
= a O,
the p-values of the points on the line a
=aO
-
1
may be calculated with the help of (2.60) successively with increasing
a . co-ordinates.
Thus, these points may be examined for a stopping
point or a continuation point one after another.
\tie
need, however,
only continue on this line till a stopping point is reached for the
first time.
This is due to the fact that all further points on this
line are stopping points as can easily be shown using lemma 2.5.
The procedure described in the previous paragraph determines
the boundary point (i.e., the first stopping point encountered) on
the line a
next line a
= a°- I .
=a O -
We then consider finding this point on the
2, in the same way, by examining the points on
it for a stopping point till we reach one. This time we start with
o
the point (a - 2, a O • 2) whose p-values can be obtained from (2.60)
by those of (ao.l, aO-2) and (a o_2, aO-l) already obtained. This
process is then repeated to find the boundary points on the lines
a
= a ° . k, successively for k
=
3,4, etc •• ~he boundary points on
58
the part a
b are obtained fran those on a > b by symmetry.
~
General ca.se:
0< p
°
< 1
As before for each point (a,b) we associate the Bayes risk
p(a,b) which satisfies the same recursion relation as (2.60) where
nOiv (2.61) is replaced by
(2.6:; )
= A E ~rp-polla'E7
po(a,b)
-
A
I a:b
-
Po r_7
and (2.62) by
for constant costj
a,b,
(2.64)
C(a,b)
for absolute deviation cost.
In the general case, the optimum boundary may not be symmetric
1-10 :
with respect to
a = P po/(l-po). vIe have to consider the
boundary both above and below this line.
of truncation on
1-1 ,
1-12
(1):
(ao,bo)
(ao, bO) belongs to
€
.rlo
By Lemma 2.6 the points on the lines
a = a
stopping points, hence their p-values are known.
(ao, bO-l)
1-1,
and
.{-10
'
•
(ao, bO)
next point on
be the point
obtained as indicated in the previous section.
Three different cases arise according as
~1l or
Let
°
and b = b
°
are
The p-value of the
i.e., (ao-l, bO-l) is obtained fram those of
(ao-l, bO).
of the points on the line a
points on the line a
= aO-l
Using this
p(ao-l,bo-l) and the p-values
= a O, in the equation (2.60), those of the
may be obtained successively fram the
59
point (ao.l, bO-l) till we reach the first stopping point on this
line. Similarly using pea 0 _ 1, b O-1) and the p-values of the points
O
on the line b = b O, those of the points on the line b = b - 1
Consider the next point on .C1 which
may be obtained successively.
may be either on the line a
=aO
•
2
or on the line b
= bO
-
2.
In either case, it should be sufficiently apparent by now how to
proceed.
(2):
B,y
(ao, bO)
e
i-1l •
Lemma 2.6, the points on the lines a
=a O
and b
= bO +
1
are stopping points in this case; hence their p-values are known.
Proceeding as indicated above, the boundary point on the line b
is obtained first.
b = b
°•
1
(3):
°
(ao, bO)
= bO -
e
i-12
1,
•
2.6, again, the points on the line a
=aO +
1, and
°
are stopping points.
is obtained first.
b
= a° -
and so on.
B,y Lemma
b = b
Next the boundary points on the lines a
= bO
The boundary point on the line a = a
O
Next the process is repeated with lines a = a - k,
k, successively with increasing k.
2.6 Some points useful for computation of the optimal boundary.
Since the optimum (Bayes) boundary does not depend on the absolute magnitudes of the Bayes risk
p, but rather on the relative
magnitude of the two quantities on the right hand side of (2.60),
we may, for convenience of computation, consider the following translated quantity
60
which may be called the gain function associated with
(a,b).
Using
the fact that
the equation (2.60) may be vTritten in terms of (2.65) as
(2.67) V(a,b)
rV (a,b),+aa b V(a+l,b) + a+bb V(a,b+l) - C(a,b)7
-
= max -
0
't'There
Following Moriguti and Robbins ["39,7, for actual computation, we
may still consider anothe.r quantity defined as
M(a,b)
=
V(a,b)
-
V (a,b)
o
which may be called the net gain function associated with
can be shown from (2.67)
and
(a, b).
It
(2.6 ~ that the optimum boundary may be
characterized in terms of M(a, b) as follovTs,
(2.70) M(a,b)
= max~O,
a:b M(a+l,b) +
a~b M(a,b+l) + A(a,b)-C(a,b17
where
(2A •ab/f(a+b)2(a+b+1).7,
) 2A • b/(a+b) •
A(a.,bl.)= {-
_. '
'\2/\0 · a/(a+b}
if
(a,b)
€
rp~;';.. ''iJ.j(a,+b+i.17, :if
..__
110
(a,b)
'
€
{]l '
• f(a+l)/(a+b+l)-po-]' if (a,b)
€
1-12 '
, other;vise •
61
We give below a closed form expression for
C(a,b) for the abso-
lute deviation cost in terms of tabulated functions:
C(a,b)
= Er /p-p
-
1
0
Ila, -b7
S'
=
lp-p0 Ids a, b(P)
o
-I
r2 I Po(a,b)-17
+a
r2
a b -
p0 -
=
where
Ix(P,q)
p
0
(a+l"b) -17
-
is the incomplete beta integral
J
x
1
B{p,q)
t p- l (l_t)q-l dt ,
p,q > 0
0
tabulated by K. Pearson
L3g7,
For the special case p
o
lemma another expression for
= 1/2, we give below in the form of a
C(a,b)
that might
be more convenient
for computation:
Lemma 2.8
C(a, b) -;
EL Ip
-
~I la'E7
=
max (a, b)
a+ b
where
(2.74)
g(a,b) =
b1(a,b;~)
r Ja -bl 7
-·2
-
Ia-b/
( 2j
1::
j=o
and
and L~7 denotes the largest integer < x •
.)
g(a,b)
2.7 Trinomial model.
In this section we write down without much elaboration the
analogous results for the trinomial problem stated below.
These
results can be obtained similarly as in the binomial problem with
(2.76)
f
(i)
~1'~2
= ~i'
= 0,1,2;
i
0 ~ ~i ~ 1, ~o + ~l + ~2
is preferable if
~l
>
~2
'
a 2 if preferable if
~l
<
~2
•
al
L(~1'~2; a l )
=
j 2A(~2 L
~l)' if
1t l
~ ~2 '
, if "ttl > tt 2
0
•
It follows from the above specializations that
'l gf
rl
=
f2
=
tI
1-)-0
=
19
g
Eg ttl
> Eg tt2 ~ ,
~l
< Eg 1t2 ~ ,
Eg
, E 1t
g l
=
Eg 1t2
"'\
S
and
Also, (2.1) reduces to
(2.80)
e
¢g,j
=
{~
,
,
,
if Ego ttl > Eg
j
J
if
" =
if
" <
j = 0,1,2, ••• ,n; n = 0,1,2, •••
1t
2
"
,
,
" ,.
= 1.
63
and (2.2) reduces to
where
Corresponding to Theorem 2.1 we have in this case the following
theorem which can be proved analogously.
Theorem 2.3.
5
s
A sufficient condition for a Bayes sequential procedure
for the trinomial Pfoblem to be truncated with probability one is:
(2.83)
t
(a)
5
(b)
lim
n ->
C(1t l ,1t 2 ) > 0
~
=
1
and
(2.84)
Var (1t l -
1t
2 / Sn)/
C(Sn)
=
0 a.e.,
(Xl
uniformly in n •
takes at most no observations, where no is the
5
smallest positive integer such that for every n ~ n - 1,
o
Moreover,
8
It is possible to show as in the
r~
(2) after Theorem 2.1
that it is only necessary to show that condition (b) holds for
those
5n E ~-o
(-)
•
64
2.8 Dirichlet a priori distribution.
From now onwards we consider an a priori distribution belonging
to the bivariate Dirichlet distribution family (1.5) some of whose
properties have already been considered in subsection 1.5.1.
From
the property of closedness of this family under sampling" all the
considerations of the binomial problem hold equally well.
The only
Change in this case is that now we have to consider all the lattice
points of the positive octant in the three-dimensional space.
various sets
A 1"
1\
2"
/\"
f 1"
f 2"
"{-1 0
The
are considered as
before, with respect to the fixed frame" of reference with (0,,0,,0)
as the origin in the '-space.
From (1.5 a"b)
and (2 •. ')we may
write
(2.86)
,.
f l =ia"b,c
The optimum boundary may be defined in this case as the set of
stopping points (a,b,c) such that
(a-l"b"c), (a,b-l,c) ~ (a"b,c-l)
is a continuation point.
As before, by dSn(~1"~2) we shall mean
a +b +
C
ds(a"b,c) where
= n.
Analogous to corollary 2.1, we have the following corollary:
Corollary 2.3.
A Bayes sequential procedure for the trinomial
problem vdth a Dirichlet prior distribution is truncated with probability one.
65
Outline of proof:
Condition (a) of Theorem 2.3 may be shown
to be satisfied for the special cases of cost and a priori distributions we are concerned '\-lith here.
Now" for
S {(-)
"it follows from (1.6)" putting a = b
.1 -0
Also" it can be easily verified that
;f
1
_
'
) Ef l7t l - '1t 2 J la,a,;:/ "
for constant
case
for absolute
cost: case
costl
(i)
deviation
(ii)
l= 4a/f(2a+c)bi(a"a17
where bi(a"a)
with parameters
is the central term in the bino.rrAal distribution
(2a, 1/2).
diately for constant cost.
(2.91)
From these results, (2.
4)
follows imIDe-
For the other cost, we have
Var(7t l - '1t2Ia,a,c)/C(a,a,c)
= f2(2a
V"l
+ c + 1)bi(a,a17 -1
(7ta)1/2/2(2a + c + 1) -> 0
as
2a + c -->
ro •
2.9 Determination of the points of truncation in the trinomial problem.
The results of the previous scction ensure that a Bayes procedure
for the trinomial problem is truncated.
This means that for a suffi-
ciently large value of n, all the lattice points of the plane, say
Ln : a+b+c=n are stopping points.
In other words the continuation set,
66
1\ ,
lies below this plane.
A to
narrows down
Just like Lermna 2.6, the lemma below
s~e subset of a+b+c < n.
Let the two planes
Land
,-)
n
.l -0
intersect each other (which
they do perpendicularly) along the line, say,
f:
n
eonsider a plane P~
~n at 45 to the plane
,-
.l
2
0
.
1,2.
J. =
'
r1
in
Obviously,
passing through
n
a::: (n-c)/2.
0
n
P1 and P2 JJleet each· other perpendicu-
fn .
larly at
.
Lemma 2.9.
cated at
If a Bayes procedure for
th~
trin?mial problem is
tr~
A
is contained within the
n
n
n
region R . bounded above by the two p~es P ~ P2 •
l
Proof:
that
/\
by a
n, then the continuation set
a
O
= (n-co)/2
c -plane, where
c
°
is such
is an integer (positive), the continuation set,
(co), in that plane is contained within the sq~e bounded above
= a°
and bounded on the right by b
(ao, aO-l, co)
in
rl •
(ao, aO..l, cO+l)
(aO, aO..l, co)
all in
is also in
Consider the point
(a °, a °, c 0) (a °+1, a 0-1, c 0) and
/''1 (being in Ln ) and hence by Lermna
:I
/\ 1.
(ao-l,ao,co)
belongs to "'2.
of the plane
L ..
n 1
on the
2.5
It can similarly be ShovlIl that
(ao, aO-l,co)am'({;l,°l'il'-l,aP;;co), are two points
co-plane.
that all the points of the plane
stopping points.
= a °.
It can be represented as a convex linear
combination of the points
plane
°
The lemma asserts that for each
L _
n l
It can be shown successively
except on
,,{-10
'
if any, are
UtiliZing this fact again, all the points on the
n
L 2 and on or outside R may be shown to be stopping points.
n-
ContinuilJlg in this way successively we arrive at the above lemma.
Formal definition of a point of
truncati~
For a fixed c, we may formally define the point of truncation in
O
the c-plane just as in section 2.4, as that ~0int (ao, a , c) of 1-1,
such that a O is the smallest integer such that no poiD~ (a,a,c) ~
..{ 20
~~~~plane
with a
~
a
a
is a continuation point.
Because
of the symmetry of the problem, it will turn out that just as in the
symmetric binomial problem (i.e., "lith p
cation on the
stopping point
°
= 1/2)
the point of trun..
c-plane might alternatively be defined as the "first"
(ao, a O, c) in (-)
-
meaning with smallest a 0 •
-0
on this c-plane, "first fl here
It may be noted here that unlike the
symmetric binomial case, the points of trunca.tion defined as such
do belong to the optimum bouncla.ry as defined in the previous section.
In fact, it may easily be seen from the above lemma that they cannot
be reached from any point in the same c-plane, but may be reached from
the next lower c-plane.
We now state how to determine these points of truncation.
Just as
in the binomial problem, the knOWledge of the:t.;r exact location enables
us to obtain the optimum bouncla.ry, as will be described in section 2.10,
by working backwards from these points with the belp of the recursion
formula (2.81).
To find the points of truncation for a Bayes procedure knovffi to
be truncated at
n, say, we need, because of Lemma 2.9, only the set
n
L _
1-20 to determine whether they are stopping points
n l
or not. A point belonging to Ln _ n 1-20 is of the form (a,a,c)
l
where 2a + c = n-l. Notice that these points are on c-planes with
of points
68
odd (even)
c
if
n
is even (odd).
A further obs ervation at
this point leads to either (a+l,a,c), (a,a+l,c) or (a,a,c+l), each
of which is a stopping point.
The next theorem gives an easily appli-
cable criterion to deter.mine whether or not such a point (a,a,c) is
a stopping point.
Theorem 2.4.
such that
then
If
(a,a,c)
---=--.-;.;.---------------C)
is a point on the neutral boundary
-0
(a+l,a,c), (a,a+l,c) and (a,a,c+l) are stopping points,
(a,a,c)
is a stopping point or a continuation point, according
as
(2.9 2 )
~(a,a,c)
5
C(a, a,c)
or ~(a,a,c)
> C(a,a,c)
where
and
The proof of this theorem is analogous to that of Theorem 2.2
and hence omitted.
If all the points of L I n
n-
C)--0 are thus found to be
-
stopping points, we then restrict our attention to the next smaller
n
region R - l • By Lenma 2.9, again, we need examine only the points
of Ln_2 () ~lo for stopping or continuation points, and this is
done with the help of the above theorem as before. We proceed in
the same way till we reach a stage when not all the points,of, say,
n (-)
Ln- k .
..l -0
are stopping points.
It turns out that some will be
stopping and some will be continuation points depending on the value
of c.
Suppose for a fixed value
(n-k-co)/2" co)
CO of c, the point «n-k-co)/2,
is found to be a continuation point.
Now, using the
monotonically decreasing property of ~(a,a,co)/C(o,a,co) with respect
to a for the two types of cost C that we are considering, it is
possible to show just as in Corollary 2.2, that all the points
with a < (n-k-co)/2 are continuation points.
(a,a,co)
Hence, by definition,
the point
is the point of truncation on the
cO_plane.
If, for other values of
c, all the points of L _ n 1-1 are still stopping points, we move
n k
0
on the next set of points L _ _ () 1-1 and so on. It is now
0
n k 1
evident how the points of truncation may be obtained for any value of
c.
Using (2.92) and (2.87) we are thus led to conclude for the case
of constant cost that the set of boundary points for the Bayes procedure consists of the lattice points of a curve, having the following
parametric representation:
where aO(c)
is the smallest positive integer such that
(2a+c)
for a fixed c.
Obviously a 0
lar notation aO(c).
aO(c)
2a
(2a+c+1)
<
1
A
depends on
c and hence the particu-
In fact, it is easily seen from (2.97) that
is decreasing in c. This means that the point of truncation
moves to the side of the origin as
c
increases.
The set of truncation points for the case of absolute deviation
cost is also given by the same type of curve as (2.96) where (using
(2.87) and (2.90»
aO(c)
is nov defined to be the smallest positive
integer for which
1
2 (2a +
for a fixed c.
C
+ 1)
1
•
For large A,
<
a O( c)
1
A
is given approximately by
1
A •
As
a numerical illustration, we give below for A
= 10,
same of the
points of truncation (2.96).
·e
constant cost:
c
a o(c)
1
2
4
1
absolute deviation cost:
c
1
2
3
4
5
6
•
f1.
aO(c)
20
17
16
15
14
13
Finally, we note that if the point (2.95) is a truncation point,
then the point «n-k-co)/2, (n-k-co)/2, eO + 1) belongs to L
n-kfl
and hence, is a stopping point from the very definition of L k' We
nmay formally put this observation in the form of the following lemma
which may otherwise be proved directly from the definitions of aO(e)
from (2.97) and (2.98).
Lemma 2.10.
ROint, then
1£
(ao(c), aO(e), c)
(ao(c)-l, aO(c)-l, c+l)
represents a truncation
belongs to the stopping set.
71
We shall use this lemma in the next section.
2.10.
Procedure for obtaining the optimum boundary in the trinomial
problem.
We describe in this section how to obtain the optimum boundary
systematically, once the points of truncation are known,
~sing
some
recursion relations as well as some simple facts about the optimum
stopping and continuation sets as embodied in Lemmas 2.5, 2.9, and
2.10.
Because of the symmetry of the problem, the optimum boundary
is symmetric With respect to the plane a=b.
generality, we may thus consider only a
Without any loss of
>b
•
Suppose the points of truncation (aO(c), aO(c), c)
-e
as indicated in the previous section.
are obtained
Consider any particular c..
plane, in which, omitting the fixed co-ordinate for the moment,
(ao(c), aO(c»
is the point of truncation.
points on the line a = a
0
By LeInma 2.9,
are stopping points.
all the
Moreover, as noted
earlier in the preVious section all the points on the diagonal a-b
With a
< a 0 are continuation points. Finally, the recursion rela-
tion (2. 1) reduces to
+ a+~c
p(a,b,c+l) + C(a,b,c17
where
(2.101)
and
po(a,b,c)
= A EL:lnl
..
n21Ia,b,~7
.. A
~:~J
'
72
(2.102)
(a,b,c), for constant cost
, for absolute deviation cost.
For stopping points
points
p
= pa ,
a known function and for continuation
..
p <p
. a , as follows from Lemma. 2.2. Thus a point may be
examined as a stopping point or not, according as
p
= p0
or 0
p < Pat
this point.
o
Consider the points on the line a = a • 1, starting from
(ao.l, aO.l). (ao,ao_l,c), (ao_l, a O, c) are stopping points, as
noted earlier, by Lemma 2.9.
Hence,
So is (a 0_ 1, a 0_ 1, c+l) by Lema 2.10.
p(ao.l,ao.l,c) may be obtained from (2.100).
Also, since
(ao.l, aO_l, c+l) is a stopping point, by Lemma 2.9 again, all the
points on the line a
= aO.l
on the (c+l)-plane are stopping points.
Using this fact, the p-values associated with the other points on
the line a
= aO_l
on the
of (2.100) successively fram
c-plane may be calculated with the help
(ao_l,ao.l,c)
onwards and hence may
be determined to be stopping points or not.
One needs to proceed
a
till one encounters a stopping point on this line a = a
- 1 for
the first time.
All further points on this line can easily be shown
to be stopping points with the help of Lemma 2.5.
The procedure described in the above
parag~ph
determines the
boundary point (the first stopping point encountered) on the line
a
= a o( c)
- 1
on the c-plane. This is nO"W carried out for all poss i-
ble values of c,
before one considers finding out the boundary point
on the next line a = aO(c) • 2. For a fixed c, we start fram the
o
O
point (a _2, a _2,c) and proceed to the left on the same c-plane
e
~
as before. It is to be noticed that the p-values of the points
(a o_2, a O_2_k, c+l), k ~ 0, that are needed in this connection,
must have already been obtained in the procedure described so far
(if they are not already stopping points, of course).
carried out for all possible values of c.
point on the lines
a
= aO(c)
This is now
In this way the boundary
-k are obtained for k =
3, 4, etc.,
successively.
The boundary points on the part a
on a
~
b, by symmetry.
~
b are obtained from those
The description of the procedure is thus
complete.
2.11 Some points useful for computation of the optimum boundary.
-e
For convenience of' computation we may replace (2.100)" as in
the binomial problem.1 by an analogous relation in terms of the socalled gain function V(a,b,c) defined as
(2.103)
Using the relation
(2.100) becomes in terms of V:
+
a+~c V(a,b+l,c) + a+~c V(a,b,c+l)
.. C(a,b,c17
where
(2.106)
=
Va (a,b,c)
A ~b+"b
a+ c
•
Alternatively, (2.105) may be put in terms of the so-called net
gain function,
(2.107)
M(a,b,c)
M(a,b,c)
=
defined as
V(a,b,c) - Va (a,b,c) ,
as follows:
(2.1C8) M(a, b,c) = max£o,
.
a+~ c M(a+l, b,c) + a+ ~ c M(a, b+l,c)
for a
t
b.
For a
=b
,
(2.109) M(a,a,c) = V(a/a,c) = max £0, A(a,a,c) .. C(a,a,c)+ ~c M(a+l,a,c17
We give an expression for C(a,b,c)
cost that might be useful for computation:
for the absolute deviation
75
Lemma 2.11
(2.110)
C(a"b"c)
E£
=
where
/7t1-7t2 f fa" b,,2,7
4 max (a,b)
a + b + c
g{a,b) is defined in (2.74).
•
g(a,b)
CHAPI'ER III
BAYES PROCEDURES WITH MODIFIED LOSS STRUCTURE
3.0 Introduction and s,ummary
The modified loss structures as defined in (1.3) and (1.10)
are only some special cases of the general situation when loss incurred at any stage depends on the observed outcome.
One important
feature of these special cases is, however, that any optimum sequential procedure for a decision problem involving this type of loss
must be truncated at A.
This follows intuitively from the fact
that any terminal decision after A observations must be correct
in the sense that no loss is involved and the situation cannot be
improved upon by taking further observations whereas any further
observation is going to cost us more for any type of positive cost
function.
This argument, may, however, be put in formal mathematical
terminologies by developing an analogous version of lemma 2.4 for
the modified loss structure.
It is well-known
£6, 27
how to obtain the Bayes sequential
procedure within the class of procedures truncated at a fixed stage,
by the working backwards technique, for loss and cost depending on
the observed outcome in general.
The existence of the Bayes pro-
cedures for the various decision problems involving the modified loss
structure are thus assured.
The problem is then to give a simpler
77
characterization of the Bayes procedures that might be applicable
readily in actual computation.
Section
,.1
contains such a charac-
terization for the general model that is analogous to (although not
exactly the same as) that given in
£27 for
the traditional loss, con-
stant cost, and independent and identically distributed sequence of
observations.
A further simplification in the characterization results as
shown in section
,.2 when
one considers the appropriate absolute
deviation type of cost corresponding to a modified loss.
As noted
already in section 1.4, it follows from physical considerations that
the modified loss structure fits more naturally with the absolute
deviation cost.
The results of the first two sections are then applied in
section ,., and
tively.
,.4
to the binomial and the trinomial model respec-
It is in section ,., that a rather undesirable feature of
the Bayes procedures with the modified loss structure compared to those
with linear loss structure is illustrated in details with the binomial
model.
,.1
Characterization of the Bayes procedure with a 6eneral model.
Extending the results in
£2::'7
in the desired direction we
arrive at the intended characterization stated as a theorem below.
Apart from the assumptions of independent observations and constant
(but may depend on the unknown parameter) cost per observation, we
introduce the assumptions regarding the particular structure of the
loss in the appropriate places.
We shall retain the notation of
section 2.1 as far as possible changing only the relevant ones.
78
Let
Notations:
be the loss incurred due to action a
when
is the true :parameter.
Q
after observing xl, ••• ,x ,
j
i
For consistency of notation we de-
note by
i = 1 1 2;
the loss incurred when a
~
and when
is true.
(3.0)
Let
We assume the usual [" 9
min
i=1,2
ljr0
5~ _
(ljr gI
Let
ail a l(g), j
= O,1,2, ••• ,nJ i = 1,2.
Lj(g,a ),
i
j = 0,1,2, ••• ,n.
be an arbitrary (but measurable) sequential decision
:procedure truncated at
Let
.~Lj(g,
Lj{Q, a ).
i
_7 measurability and
o
=
Aj(g)
5;: (ljr, ¢)
=
e@
is taken without taking any observation
i
integrability conditions for
Lj(l,a i )
Q
nl i.e,"
+ 1jr1+ ••• + ljrn ~ 1
¢g)
a. e.
for all
denote the Bayes procedure for
Q €
s•
(0 .
Also let
j = 0,1,2, ••• , n
be the conditional :probability of stop];ling with
that
j
or more observations are taken.
quely fram
~g,j'S
and vice versa.
ljr~lj 's
j
observations given
may be obtained uni-
79
With these added notations to those of section 2.1, we may
write the following results.
The risk
r(Q, 5) associated with 5 when
Q is true is given
by
=
Expected loss + Expected cost
j~O I'"J
=
{Lj (Q'&2)¢j + Lj(Q'&l)(l-¢j)
1
"
:'£'"
The average risk associated with 0
(:j.2)
r(g,5)
Ir(Q 5) ds (Q)
=
J
is given by
•
GD,
Substituting (3.1) in (3.2) and changing the order of integration
!:
n
j=o
f
,
. j
At
(3.4)
>
+ j
(3.6)
=
~
j=o
C
!,.rAj(S.)+jc(s.)7fe
J -
y)'
J
J-
(g j).7 fg,j dj..Lj
.df.1j
s,J
•
80
The equality is obtained
, i f ~j(Sj)
o
j
= 0,l,2, ••• ,n
where
().5) when ¢ = ¢g defined as
, if
It
,if
"
= Lj(Sj,a l )
=
"
< Lj (Sj,a 2 ) ,
=
"
•
u is any arbitrary number,
0 <u <1 •
Assumption (L ): The loss depends on the observations in a
l
rather special way in the sense that it does not depend on the
actual magnitude of the observations.
Specifically,
Further notations:
(3.9)
*o +
Ijr
/)
*
*
... +'!rn- 1
=
1
*}
* 1; ¢*'0 , ..., ¢n- 1
n-
• •• , Ijr
vJe shall show below that with the restriction (3.8), we may write
(3.10)
81
Writing the first term in the summation in (3.3) separately and
using (3.8) we have
(3.11)
(3012)
j~2 f "';-1
+ (1-'"0)
j ["
(n-l - j-l) { L( 9,a2 )
~-1
G)
J:.i
+ L(Q"al)(l-¢;_l)}+ (j-l)c(Q17d5j(Q) fs"jdll
where
(3"01,3) (1-'"o)Al =
J
"'1
X
i.e."
f (n-l)
0
j
,
82
NOlii'
we may write
Substituting (3.15) in the sum,
(3. 1 7)
=
A1 + A2
(
) f g(X1 )
A say, in (3.12), we have
2
. C4.t(x1 ) •
(Xl)
r(sl
')!
thus arriving at (3.10).
p
n
Now remembering the notation
(g) =
inf
5 t:.
r{ g, 5)
/::,n
we have
(3.18)
J
.
J"
(xl)
r{sl
*
' 5 ) fg{Xl)d~(Xl) ~
.
*
,5),
Hence it 'follows from (3.7), and (3.10), that
(3.19)
inf
{, ( ~n
'"o fixed
+0
(sL7
and hence
(3.21)
,
if
It
=
If
•
It may similarly be shown 'that
j
and hence
';'There by
(3.0) and. (3.8)
('.24)
~j(E)
=
(n-1)
min
i=1,2
L (e,a i )
,
= 1,2, •••,0-1
;
84
X(s)
(n-j)
, say.
and Po (s) is defined to be identically
°
for all
S • We may thus
state the results in the form of the following theorem:
Theorem 3.1.
Under the assumptions of identically and independently distributed
sequence of observations and with constant cost per observation, the
Bayes terminal decision rule in the class
6
is given by (3.7).
n
If,
(Ll ), i.e., (3.8),
can be characterized as:
moreover, the loss function satisfies the assumption
then the Bayes sampling rule in the class
(3.26)
"g,j
=
6
n
{a
,
if Pn_j(Sj) < ~j(Sj) ,
I\ .. 1
,
if Pn_j(Sj) = ~j(Sj);
j
= O,l, ••• ,n-l
where
(3.27) Pn_/s) = min
.L~j(£)'Jpn-j-l(siY»fg(y)djJ(y)
:{
j
~j(S)
+ c(sL7
= O,l, ••• ,n-l
;
being given by (3.25) •
~ve
may express the Bayes sampling rule in words as follows:
After any stage of sampling, j, say, j
take another observation according as
= O,l, ••• ,n-l, we stop
Pn_j(Sj)
= (n-j)X(g)
or
or
Pn_j(Sj) < (n-j) Xes) •
We may alternatively define the Bayes sampling rule in terms
of certain sets in
~~
Black'Vlell and Girshick
n
• For this, let us denote, following
.L27 ,
,
(n-j)
~
(s)} ,
j
= 0,1, ••• ,n-1
.
It may be verified that
where
r-·' /
\~ois
Now let
S*
jn'
(;.;0)
j
the set of all g's.
= 0,1, ••• ,n.
Then,
(;.31)
S*,
n
=
-
(S*' ,
on
*'
*'
Sln' ••• , Snn )
forms the required partition of the sample space 3C n
Bayes procedure takes
j
observations if and only if
such that the
*,
xES.
-n
In
It may also be verified that
(3.32)
*'
San
* '
C Sjn
j
= 0,1, ... ,n
1-There S* = (S* , ••• , S* ) is the partition of the sample space 3e n
n on
n,n
corresponding to the truncated Bayes procedure for the decision problem vTith the traditional loss function
(;.3;)
i
= 1,2,
This means in other words that if the truncated Bayes procedure with
the loss (;.;;) tells one to stop then so will also be told by the
Bayes procedure with the modified loss
from physical considerations.
(3.8), which is rather obvious
86
3.2 Modified loss structure with absolute deviation cost.
He now consider the fol101n.ng special situations:
I.
Assumption (L ):
2
L
(~, a )
l
L (Q,
a2)
=
{2
=
J
oQ
, if Q<O
,
if 0.>0
, if Q < 0
0
L2Q
, if Q>O.
Then,
~ (~)
= Es fQI - fEs Qf
.
..
.
From (3.3'5), it follows that the various sets
ponding to
where k
r l , r 2 corres-
(2.7) vTith this modified loss may be defined analogously
as in (2.12) or
II.
110 ,
(2.78) in terms Eg
Q •
Assumption (C ):
is a positive integer.
It may easily be seen from
(3.35),
(3.25), (3.27) and (3.26) that with these specializations the Bayes
procedure is truncated at
n-k.
With k=l, i.e., for the absolute
deviation type of cost we may thus state the result in the form of
the following corollary:
Corollary 3.1.
The Bayes procedure for the decision problem with
the modified loss structure, viz.,
e
87
L(Q, al;j)
=j
l
2(n-j) IQ
!
, if Q ~ 0,
j ~
n ,
0
_ n,
, if Q ~ 0, J.>
0
, if Q > 0, j = 0,1,2, ••••
0
, if Q ~ 0, j = 0,1,2, ....
0
, if Q > 0, j
(3.36)
L(Q, a 2;j)
=
2(n-j)Q
, if Q > 0,
~
n ,
j ~
n ,
and with the absolute deviation cost, viz.,
is truncated at
n-l.
From now on we shall only consider these two specializations,
(3.36) and (3.}1) which lead to the simpler characterization of the
Bayes sampling rule in terms of the following quantity,
Vj(S)
= {(n-j)E$IQI- Pn_j(S); j =0,1,2... , n,
o
,
which may be called the gain function.
j~n
;
It follows from
(3.~7),
(3.25) and (3~35), that
Vj(S)
= max f(n-j)
Es /QI - (n-j) {EgIQI -
~EsQ~}
1
(n-j) Es/QI - Jpn-j-l(slY»fs(Y) dil(Y)
;)?
- E
Now, using the fact that
.
s /gL7,
j
= 0,1, ••• ,n-I.
(3.38),
88
it fo100ws from (3.39) that
Vj(S) = max f(n-j) 'EgQI ,
f
Vj+1(siY»fg(y)dll(y17,
)e
j :: 0,1" ... "n-1.
The Bayes sampling rule may thus be stated as follows:
stage of sampling" say,
j
= 0,1, ••• ,n-1,
after any
we stop or take another
observation according as
It may be noted that it follows from the definition (3.38) of Vj
that
thus corroborating with the corollary 3.1 that the Bayes procedure
is truncated at n-1.
3.3 Binomial model.
To obtain the optimum (Bayes) sequential procedure for the
special problems we have been considering, the results of the previous section are specialized in the usual way, replacing the
there by A, and
as in (2.36).
j
there by n"
there by p-p.
o
,-)
.1-0
is defined
r 1 and r 2 are defined in an obvious way. Equation
(3.41) reduces in this case to ,
(3.44)
Q
n
89
where
vn,o (a"b)
From
(3.43)
(3.46)
= (A-n)
f·a+ab
- p0
r"
n = 0"1,, ••• ,A-l .
it follows that
VA- l(a"b) =VA- 1 ,0(a"b)
= Ia+a b
- p0
I
for all (a"b).
With the help of (3.44) it is possible" at least in principle"
to calculate V (a"b) for all possible values of (a"b) and n
n
.
starting from the known values (3.46) for n = A-l. Once these values
are known, the optimum sampling rule becomes specified.
We now compare this procedure with that ob-roined in the previous
chapter with linear loss structure. We first note that the subscript
n
in Vn(a,b) 4enotes the number of observations already obtained.
If we start with an a priori distribution
dg(a"b), then the releo
0
vant points (a,b) for which V (a,b) has to be defined are those for
n
which
a+b = (a o + b 0 ) + n
•
The dependence of V on the stage of sampling, n, is thus" essentially"
a dependence on the particular a priori distribution ds(a o' b o )
through a + b. The prior distribution being fixed for a particular
o
0
decision problem" we roo.y, alternatively" write
(3.44)"
omitting the
dependence on n" as thus ,
(3.4)
where
V(a"b) = maxrV
(a"b), a+ab V(a+l,b) + a+bb V(a,b+l)7
0
- ,
90
(3.48 )
vo(a" b) = (A + a 0 + b 0 - a - b)
I...!:...
a+b -
p0 ,
(which is not to be confused with the earlier V (a"b)
.
The difference of
0
in
(3.44».
(3.47) and (2.67) lies in the difference between
the two different definitions
(3.47) and (2.67) of Vo(a"b). It may
be noted that the nature of a point (a"b) as to whether it is a
stopping point or a continuation point as determined with the help
of
(2.67) did not depend on the particular a priori distribution
one starts with" whereas this is the situation ,in the present case
with
(3.47). This is an undesirable feature of the procedures con-
sidered in this chapter in comparison to those of Chapter II in the
sense that one has to compute the optimal boundary separately for
each prior distribution" whereas" as noted earlier in section 2"
for the procedures of that section" we need compute the optimal
boundary only for the particular a priori distribution dg(l"l)
and the optimal boundary for any other a priori distribution may
be obtained by a simple change in the frame of reference.
It is possible to prove the convexity of the two stopping
regions in this case (for the general model) just as in the proof
of lemma 2.5 for the traditional loss.
This result may analogously
be used along with the fact that all the points (a"b) with a+b
a o + b0 + A - I
=
are stopping points" as in the traditional case"
to arrive at a result corresponding to lemma
region of sampling.
2.6 , delimiting the
We have" however" preferred to show this directly
91
using (3.44).
For simplicity we only consider the symmetric case,
Le., with p = 1/2.
o
LeIllID.a
3.1
Vn (a,a) > Vn,o (a,a) = 0,
V (a,b)
n
= Vn,o (a,b)
n < A-I, all a,
for Ja-bl ~ A-n, n < A-I.
Proof:
(a)
Putting a
=b
= max;-O,
-
Vn(a,a)
in (3.44). and (3.45) with Po
= 1/2
,
1
1
-2 (Vn+ l(a+l,a) + -2 Vn
l(a,a+l)
+ -7
Now, because of symmetry of the problem,
Vn(a,b)
= Vn(b,a)
all (a,b), all n;
and hence in particular,
'{tie,
thus, have from (3.51)
V (a,a)
n
= max -rO,
Vn+ 1(a+l,a)7
-
Now from the definition (3.44) of V, it follows that
Vn+ l(a+l,a) > Vn+ 1, o(a+l,a)
=
A - n - 1
>
o
2(2a + 1)
for
n < A-I.
Hence
Vn(a,a)
= Vn+ l(a+l,a) > 0 = Vn,o (a,a)
for all a, n < A-I •
92
(b)
(3.51) for
Because of symmetry, it is sufficient to prove
a > b only, i.e., to show
Vn (a+k,a)
We prove this
n
= m,
for k ~ A-n, n ~ A-l •
(3.56) by induction on n. Suppose (3.56) is true for
we show that
From
= Vn,o (a+k,a),
(3.56) is also true n = m-l.
(3.44) and (3.45) it follows that
-:--'~ V
(a+k+l,a) +
m
a
+ 2a+k Vm(a+k,a+l)- 7
From induction hypothesis, we get
(3.58)
., (a+k+l,a) = V (a+k+l,a) =
m
m,o
and
(3.59)
Hence,
Vm(a+k,a+l) = Vm,o (a+k,a+l) = (A-m)(k-l)
2(2a+k+l)
(3.57) reduces to
= V 1
m-
Thus
by
-
• k > A-m+l •
'
,0
(a+k,a),
k
>
A-m:I
k
>
A-m:I •
(3.56) is true for n = m-l. But (3.56) is true for n = A-l,
(3.46). Hence (3.56) follows by induction and the proof of the
lemma is complete.
When interpreted in words, part (a) of the above lemma means
that the points (a,a) on the neutral boundary that may be reached
93
at any stage of sampling up to A-2 are continuation points. We
stop anyhow after stage A-l, whatever the point (a,b) is reached.
Hence the point of truncation, may be defined as the point (ao,a o )
where a
a
O
O
= (a o+b 0 +A-l)/2,
= (ao+b 0+A-l)/2
in case a +b +A-l is even or otherwise
00-
+ 1/2 in case a0
+b +A-l is 0
odd. Part (b)
of
'
the above lemma then asserts that the optimal region of sampling lies
o
0
within the square bounded above by a = a
and on the right by b = a •
A possible method for obtaining the optimum boundary numerically
may be described analogously as in section 2.5 with the help of
(3.47), starting from the point of truncation and working back'Wards.
3.4 Trinomial model.
For this specialization, we need put
1-10 , r l' r 2
of. section 3.1.
(2.86).
(3.61)
~
= ~l
-
~2
in the results
are defined in the same way as in
Equation (3.41) reduces in this case to
Vn(a,b,c)
= max;-V
- n,o (a,b,c), a+b: c
Vn+ l(a+l,b,c)
where
V
n,o
(a,b,c) = (A-n)
~bb:
a ++ c
•
Again, it follows from (3.43) that
VA l(a,b,c)
-
= VA- 1 ,0 (a,b,c) = ~bb
,
a+ + c
for all (a,b,c)
~
With the help of (3.61) it is possible, at least in principle,
to calculate V (a,b,c) for all possible values of (a,b,c) and n
n
starting from the kno'Vm values
(3.63) for n = A-l. Once these
values are kn01m, the optimum sampling rule becomes specified.
As in the binomial problem, vTe may alternatively write
(3.61)
as
(3.64) V(a,b,c) = maxrVo(a,b,c),
+ a+ ~V(a,b+l,c)
:L
a+~ c V(a+l,b,c)
.
c
+ a+ ~ c V(a,b,c+117
,
'VThere
( 3. 6)
5
and
V o( a,b,c )
= (A +
'a_b I
a o + b o + Co - a - b - c ) ~
(a,
b , c ) represents the starting point.
000
The same remarks as
in ~he binomial case apply in comparing (3.64) with (2.105).
We state below without proof a lemma corresponding to Lemma 3.1
for the binomial problem.
Lemma 3.2
(a)
V (a,a,e) > V (a,a,c) = 0, n < A-l, all a,c
n
n,o
(b) V n(a,b,c)
, for ja-bl ~ A-n, n
= Vn,o (a,b,c)
5 A-l,
all
c.
The proof is analogous to that of Lemma 3.1. Just like Lemzna 2.9,
this one also delimits the region of sampling Within a certain region
defined as in Lemma 2.9, where
n is now replaced by
a + b + c + A-l •
°
°
°
Using the lemma above and the recursion relation (3.64), a possi-
ble method for obtaining the optimum boundary systematically may be
described analogously as in section 2.10 starting from the points of
truncation (ao(c), aO(c),c)
satisfying 2a~(c)+c
where
aO(c) is the smallest positive integer
> a + b + c + A-l,
°
°
°
c > c
- °
95
It follows from the ~efinition (;,66) of aO(c) that
non increasing in
aO{c)
is
c.
Suppose we start With a unifoT;m prior i.e., With
a0
=0
b =
c
0
Then for
A
= 10,
=1
the points of truncation are obtained from (;.66)
as
c
1
~
6
5
5
4
5
4
4
6
7
These may be compared With those values obtained in page 70.
CHAPTER IV
ASYMPTeJrIC BEHAVIOUR OF OPTIMAL SOWTION
4.0 Introduction and summary
In this chapter we study the limiting behaviour of the optimum
boundaries of the various sequential decision problems considered in
Chapter II under limiting conditions which tend to require large
samples. As sample size is not one of the given parameters of a
sequential problem., a natural approach to the large sample theory
is to make the cost of an observation tend to zero (when the loss is
held fixed) as considered by Vlald
f39!,7. Since it is the relative
magnitude of the two types of cost., namely., the cost of an observation and the cost of making a wrong decision., that matters in 0btaining the optimum boundaries., an equivalent approach is to make
the loss tend to infinity keeping the cost fixed.
We consider this
alternative approach as it is particularly suited for our sequential
decision problems.
The limiting condition suggested can then be ob-
tained by letting the parameter A in the various loss functions
approach infinity.
From physical considerations it is apparent that as
A becomes
larger, more and more sampling is needed to reach the boundary. Thus,
in the limiting case, any point with finite co-ordinates (a,b) or
(a,b,c), whichever the case may be, will be Within the continuation
97
set.
The boundary wlll consist of points With their co-ordinates also
approaching infinity.
The limiting behvaiour of the boundary" if any"
has to be specified only in terms of suitably normalized co-ordinates.
The first point to be resolved in this study" then" is what
constitutes a suitable normalization of the co-ordinates in each case"
in order to Jrield non-degenerate limits.
It turns out that the appro-
priate normalization for a given case depends on the cost tunct10n •
Throughout this chapter tbebinamial problem is considered for
= 1/2. The asymptotic behaviour of the optimal
o
solution has been studied by Morigut1 and Robbins £39,.7 for this
simplicity with l'
problem With constant cost.
Following them closely the problem of
finding the asymptotic optimal solution for the binomial problem
With absolute deviation cost is reduced in section
4.1
to that of
finding the solution of a partial differential equation free boundary
value problem.
In section
4.2
certain series representations of the
latter solution are obtained and compared with those given by Moriguti
and Robbins for the corresponding case with constant cost.
Sections
4.3
and
4.4
correspond to sections
tively for the trinomial problem.
4.1
and
4.2
respec-
The m-eIJ representation of the
optimal solution is obtained in this case for both types of cost.
The formal expansions of the asymptotic optimal boundaries for
the various decision problems" obtained in this chapter" are useful
in giving us some idea of the asymptotic shapes of the Bayes sequen-
tial testing regions" a concept developed by Schwarz
what similar context.
£3!J:.7
in a some-
We hope that the exact boundary for large A
can be computed from the asymptotic results involVing no appreciable
error.
4.1
Binomial problem.
As
shown in Chapter III (2.67)1 the optimum boundary can be
cha.ra.cterized in terms of the gain function V(a"b) which assumes"
on the boundary and outside it" the value
(4.2)
V(a"b)
=
a:b v(a+l"b)
+
a~b
V(a"b+l) - C(a,b)
where
C(a"b)
I
for constant cost,
" for absolute deviation cost.
It is possible to give" as in (2.70)" a similar (With obvious modifications) cha.ra.cter1z~dion of the optimum boundary in terms of the
~ ~
(4.4)
function"
M(a"b) = V(a"b)
-
Vo(a,b) •
We now consider what kind of limiting conditions are appropriate
if the boundary :L$ to be non-degenerate.
Without going into details
we state that so long as Po is a fixed positive number bounded
away from 0 and 1" a
and b must approach infinity at the same
rate in order tbat the asymptotic boundary be non-degenerate.
this limiting condition" the problem of testing p
p
In
< p 0 against
-
> Po reduces after proper standardization to that of testing p
~
J.1
0
99
against
I.l.
> I.l.o , where
Gaussian process.
is the mean rate of increase in a standard
I.l.
With p
o
= 1/2, I.l.
0
= O.
In this symmetric case,
it is the limiting tendencies of the two pertinent parameters a+b;: u
and a-b: v
that determines whether or not the optimal boundary is
non-degenerate asymptotically.
11!J:.7
It is possible -to show from Chernoff
that the appropriate normalization of
u, v
and V or M for
a non-degenerate boundary depends on the tY.l?e of cost and can, in
fact, be determined as given below.
The appropriate nor.malization consists of:
Case (i):
-
constant cost:
..;..;.;..;....-.;",.;,;;~
f
Case (ii):
I
(4.6)
2
(a+b)/A / 3
,
(a_b)/Al / 3
,
absolute deviation cost:
-
x
= u/A =
y
= V/Al / 2 = (a_b)/Al / 2
V(x,y)
(a+b)/A
= V(a,b)/Al / 2
Moriguti and Robbins
["39.7
,
,
•
studied the limiting behaviour of the op-
timum boundary under normalization (4.5).
this chapter under nor.malization
(4.6).
L the limiting tendency in which a,b
x,y
in
(4.6)
remain finite.
We shall do the same in
For convenience we denote by
~
00
along with A such -that
100
Lemma 4.1
say ,
where
and ..f
¢
are the standard normal density and disttibuticn
function respectively.
Proof:
It is we1l-known~15, p.25~7 that the beta density of p
converges to the normal density under L.
The standardized variable
is
t
=
,
P - E(p)
JV{p)
where
E(p)
=
=
1
1
'2 + A1/2
•
..z...
2x
2
and
1
V(p)
ab
= (a+b)2(a+b+l)
=
~
_...:L.
4x2
A(x +
Hence,
p
=
=
so that
(4.6)
E(p)
+
1:+...:L.
2
2x
t
JVGiJ
1
• A1/ 2
+
t
2-(xA-)=-lj'=2
.
• 1
A
j)
101
NO'tv if t
fixed a l
f
is
N( 0,1) i it follovTS from Theorem 2 of ["1'L7 that for
and b l ,
which implies that EJtl
is bounded and hence the last term in (4.6)
converges to zero in mean as A"":>
co.
Thus
1 (y
1
Iyl
}
r¢(- ..LU) 7..
JX- 2 - jXIy I
= -"'\¢(-) + .L!J.
/Xl IX
The following development parallels very closely the corresponding
development sketched very briefly by Moriguti and Robbins ["327 for
the case of constant cost.
It follows from (4.2) that within the go-region V(x,y) satisfies
the following
relation~
(4.7) Al / 2[V(x,y) - V(x+A-\Y17
=:
~
Al/2[V(X+A-l,Y+A-l/2_'8(X+A-\Y)
+ V(X+A- l ,y_A- l / 217
+ (y/2x)A- l / 2 Al / 2[V(X+A-\Y+A- l / 2 ) _
V(X+A-l,y_A-l/~7
.. c(
~Ax + ~Al/2y, ~Ax- ~
Multiplying both sides of (4.7) by Al / 2 , letting A ->
co
Al / 2 y) •
and using
lemma 4.1, we see that V satisfies, under L, the following partial
differential equation:
102
21 V
1 219;
- --=
--" x
2 21 y2
(4.8)
y
+ x
"V
o
y - -(
C x,y ) •
On or outside the boundary,V(x,y) is given by
'lvhere
(4.10)
_ 1/2.Al / 2y)/Al / 2 = JyJ/(2x).
If follows from
(4.10) that
Vo
satisfies, in the region y >
0,
the following homogeneous partial differential equation:
(4.11)
1
"2
2
o Vo
y d Vo
+ -- + 0 y2
x
" x
0
V0
"y
=
0
•
Thus, the standardized net gain function,
(4.12)
w(x,y)
~
2
M(a,b)/Al /
=.LrV(a,b).
l/
- V0 (a,b)7/A
-
2
= V(x,Y)-V 0 (x,y)
satisfies, under L, the partial differential equation,
21 WoW
--o y + --0 x
(4.1;)
=
-(
C x,y)
in the region between the straight line y=O and the boundary curve
(4.14)
say.
At the boundary, y = 0, synunetry implies the condition:
(4.15)
and this, together with
21 V \ .
(4.16)
o
ay I
y=O
=
1
2"X
103
yields the boundary condition
~I
oyI
At
=
y=o
1
-2x
Y = Yl (x), a continuity requirement yields the boundary condition
I
(4.18)
=0
WI
IY=Yl ( x)
Another boundary condition is obtained by the same arguments as
in Moriguti and Robbins.
boundary,
region.
Suppose that the point (u,v) is on the
(u+l,v+l) being outside, (u+l,v-l) being inside the goThen
=
V I ( U, V )
and
VI ( U, v)
o
,
VI(u+l,V+l) = VI(u+l,v+l);
o
where
and
= Vo«u+v)/2,
V~(u,V)
Let us further assume that v-l >
VI(U,V)
= VI(U,V)
= AV/2u,
o
(u-v)/2) •
o. Then substitution of
and VI(u+l,v+l)
= VI(u+l,v+l)
= A(V+l)/2(u+l)
0
into
V'(u,V)
= '2~ fVI(u+l,v+l)
+ V'(u+l,v- 117
rv' (u+l,v+l ) .. VI (u+l,v-l_)7 - C(u+v
2 ' 2u-v)
v_
+ 2u
obtained from (4.2), yields
104
Av
2u =
I(
)7
'21 _r A(v+1)
2(u+1) + v u+1.. v-1_
r
v
+ -2u
i.e ...
vr(u+1..v-l)
A v+1) - VI( u+1 ..v- 1)7 - c(u+v
u-v)
2 u+1
_ .- 2 .. - 2
=~f~:i~
2u
+ u - v
c(u+v
..
u-v)
2"2 ·
But
V~(u+1.. V-1)
=
A(v-1)
2(u+iJ-
•
Thus
2u c(u+v .. E.:! )
VI (u+1.. v-1) - V~( u+1..v-l) =
2
2
u-v
Le.,
W(X+A-l,Y_A- 1/ 2 )
1/~
=
x-y
A
c(~
Ax + ¥1/2y ..
~Ax
_
~
A1/ 2 y) •
Expanding the- left hand side and using Lem:ma 4.1.. we get
But W(x..y)
=
0.. since (x.. y)
is on the boundary.
Hence.. (4.19)
implies that
(4.20)
Therefore.. in the limit, we should expect that
(4.21)
oW
oY
=
0
Y=Y1(x)
The problem of finding the optimum boundary along with the optimum
net gain function thus reduces to the following
105
FBEE.BOUllJDARY PROBLEM:
To solve for the unknown W(x"y) and the unknown boundary Y (x)"
l
given that W satisfies the partial differential equation
(4.22)
1
o2w
+ I.
2 Oy2
£..!!
.£.1!
+
x 0 y
0 y
+ 0 W
0 x
=
C(x"y)
where
= -
C(x"y)
Y
2x
1
(
Y
+ - ) ¢ (-)
)Xl
JX
y
.-
jX
(_ jX
..L)} .
in the region
(4.24)
and where W(x"y)
(4.25)
U! ,
o y
satisfies the boundary condition
= - 2~
"for
y=O·· .
0
<x <
for
0
< x < co
for
0 <x <
co "
,
and the free boundary conditions
(4.26)
Y1(x)
W\ Y = Y ( x )
l
--
0
o wi
o Y Y = Yl(x) -
0
"
,
gives the upper boundary.
With respect to Y"
and W(x"y)
(Xl
"
•
The problem being symmetrical
= W(x,-y)
, the lower boundary is
given simply by -Y (x). Thus it is sufficient to consider only Y > O.
l
We now make a transformation to reduce (4.22) to the heat
equation; we conclude" omitting the proof:
106
Lemma 4.2.
The non-homogeneous partial differential equation
,
where
f(x,y)
satisfies the corresponding homogeneous partial dif-
ferential equation, is reduced to the heat equation, viz.,
-"
Fl
=
0
,
" t
by the following transformation :
(4.;0)
--1
2x
t
=
z
= X.
x
,
,
It is easily verified that C(x,y)
satisfies the homogeneous
partial differential equation:
1
•
2
Hence, by the lemma above, if we define
(4.;2)
where t
Wl(t,z)
=
w(x,y)
•
X
C(x,y)
and z are defined by (4.;0), then the free-boundary
problem stated above reduces to the following
TRANSFOBMED FREE·BOUNDARY PROBLEM:
To solve for the unknown WI(t,z)
given that WI
and the unknown boundary zl(t),
satisfies the following partial differential equation:
107
=
(4.33)
) 0 < Z ~ Zl(t)
0 in
'\
/O<t<co
,.
,
and satisfies the following boundary conditions:
(4.34)
oWl \
oZI
- i
0 Cl i
1
=
-
az i z=O
'2"
(4.36)
!
1
""2 '
Z=O
,
I
Wl
=
Z=Zl(t)
l!
-1
o Z I Z=Zl(t)
= .. Cl(t'Zl(t»
-
(;) W
=
-
I
0 Clt
.
,!
~l
Z=Zl(t)
.
/I
~--
where
Cl(toZ)
,
4.2
=
12t
C( ~
Z)
2t'"2t
•
Series expansion for the optimal solution for large x (binomial
proglem)
As it is very difficult to find the solution of the transformed
free-boundary problem, our objective in the following analysis is
limited to finding a formal series expansion of its solution for small
t (as in Moriguti and Robbins ["39,7).
Thus, assuming a series expansion
for the unknown boundary as
(4.38)
for small t, where the range of the summation index m will be
settled presently} we are led to consider solutions of the heat equation
(4.33) involving some powers of t. It may be verified that the
function
108
where
(4.40)
denotes the confluent hypergeametric function and n,
satisfies the heat equation
(4.33).
The most general form of the solution of
this family and satisfies
A are constants,
n
(4.33) which belongs to
(4.34) is, thus, given by
(4.41)
where the range of the summation index n is detennined along with that
of m in
(4.38) from the two free boundary conditions (4.35) and (4.36).
It will be seen below that these two conditions are sufficient to
determine the unknown constants
From
(4.41) and (4.40),
C
and A
•
m
n
(4.35) and (4.36) reduce to:
109
To facilitate the later analysis we express the ¢ and
tions in terms of confluent hypergeometric functions.
f
func-
To this end we
note the fOlloWing results (Erdelyi £117)
2
¢(x) = ....!.- IF1(a, a; _x /2),
j2ic
-
co
<
X
< co
J
a:j
°.
(4.44)
and in particular
2y IF (1/2, ;/2; y) = lF (1/2, 1/2; y) - lF (-1/2, 1/2; y).
1
l
Therefore, putting y
= _x2/2
l
,
(4.45)
=....!.-
fii
2
lFl (-1/2,1/2;-X /2).
utilizing (4.44) and (4.45), (4.42) and (4.4;) can be expressed as:
~ A t n lFl (-n,l/2;-zi(t)/4t)
n
=~
Zl(t)-....!.-
-J_
j2tj21r
lFl (-1/2,1/2;-zi(t)/4t)
n Zl(t)
2
1
1 1 Zl(t)
.2
t
lFl (-n+l,;/2;-Zl(t)/4t) = '2 - 2t IF1.(l/2,}'2;'-Zl(t)/4t)
n
fiifii
.
I: nA t
We now consider the range of m and n in (4.38) and (4.41)
respectively.
vie first note that in (4.;8) m must be non-negative,
otherWise the series might diverge for sufficiently small t.
Next,
it has been shown in Chernoff £1!!.7 that Yl(x)//X is decreasing in
110
x for the case of constant cost.
His argument remains valid for
absolute deviation cost too. This means, in terms of the transformed variables, that zl(t)/j2t is increasing in t, i.e.,
(4.46)
This rules out the possibility that m <
1/2, because, otherwise,
for sufficiently small t .. the term involving a negative coefficient
in
(4.46) might be dominant making (4.46) nonpositive contrary to
hypothesis.
Thus if
(4.47)
zl(t)
=
I:
m~ 1/2
c tm
m
..
then
~e
zi(t)
4t
(4.4B)
say. Substituting
hand side of
=
1:
m>o
dm t m
,
(4.48) in (4.42), we notice that in the right
(4.42) the minimum power of t is -1/2. Hence the
series on the left hand side should start with n
what steps
= -1/'2.
Now by
n should increase in the left hand side of (4.4'2) de-
pends, of course, by what steps m increases in
plicity, we take the step in
(4.38) as
(4.38). For sim-
1/2, any smaller steps only
increasing the number of unknown coefficients to be determined. We
are thus finally led to consider both m and n as half-integers..
m starting from
1/2 and n starting from -1/2 in (4.38) and
(4.41) respectively.
If we now substitute
111
in (4.42) and (4.43) and equate the coefficients of equal powers of
t
from both sides of the equations we are led to the following results.
Equating coefficients of
t- l / 2 from both sides of (4'.42), vle
get
(4.50)
..l
_..l
j2j21r.
lFl (-1/2, 1/2;-ci/4) = A_ l / 2 lFl (1/2,l/2; ..ci/4 ) ,
while equating coefficients of
(4.51)
1
- "2
l
t- . from both sides of (4.43), we get
1
fi1r
for all
a contradiction.
Hence
Cl = 0 and hence from (4.51), A_ / = -1/2./1c.
l 2
Substituting these values for the constants
subsequent equations, the next two coefficients
C , A_ /
l
l 2
C , A are determined.
2
o
These are then substituted in the subsequent equations.
is repeated, giving us the follmv1ng result:
C~
- C4n_l
:J:.
0,
n = 1,2,3,. .. ,
in the
This process
112
and all other coefficients are zero.
The first few values are found
to be:
,
C'
= '"1/2
C'2
- 12
C'
3
197 ",5/2
= TI)O
1
_- l
~/2
A'0
,
,
We thus arrive at the
=
1
- '2 '"
,
1
",1/2
,
1
",3/2
,
Ai=
"4
A'2 --
b
,
A3
529
= 14~O
follo~ng
-1/2
'"
5/2'
•
series expansion of the solution
of the transformed free boundary problem:
and
• • -e
•
In terms of the original variables
x and y (see
(4.6)) we get the
following series expansions, for large x, of the optimum boundary as
113
_ J?ic
• -lr1
4
jX -
-
_ 48.
~2 +
.
X
9 J£~
1 7'
1 o,a
•
-±r,
- · ._.7
x
"l-
and of the optimum net gain function (4.12) as
..
... .
In particular,
1
• U(2 - •••
x
- -·..__7
For the case of constant cost, Moriguti and Robbins
the
fo11oWing~
1
Y1(X) = ¥x
-
1
48x4
- ...
•
132.7 found
114
_2 _
F
3d6x5 1 1
•
••
(-5, 1:; _ L
2
2
)
2x
I
..
5
where x and yare defined in
(4.5)
...
,
and
•
4.3 !!:!.!~omJ:.al llr:obl~.
As shown in Chapter II, (2.105), the optimum boundary can be
characterized in terms of the gain function V(a,b,c) which assumes,
on and outside the boundary the value
,
and satisfies in the go-region the recursion relation
(4.58)
V(a,b,c)
= a+ ~ c V(a+l,b,c) + a+D+C
? V(a,b+l,c) + ~b
a+ +cV(a,b,c+l)
-
C(a,b,c)
where
(4.59)
C(a,b,c)
=5
'I E£ /1t
for constant cost ,
1
1 -1C 2
i fa, b,!:.7
1
for absolute deviation cost
115
It is possible to give" as in (2.108)" a similar (1-lith obvious
modifications) characterization of the optimum boundary in terms of
the net gain function ,
(4.60)
M(a"b,c)
= V(a"b"c)
-
V (a,b,c)
o
As in the binomial problem, the appropriate limiting conditions
for the asymptotic boundary to be non-degenerate depend on the type
of cost per observation.
a+b = u
nent parameters
In this problem" apart from the two perti-
and a-b = v, we have another" c.
clear that if the order of magnitude of
a+b
c
It is
is less than that of
as these parameters tend to infinity" the trinomial problem
essentially reduces asym,ptotically to the binomial problem considered
in the last two sections.
To retain the essential feature of the tri-
nomial problem) then" we keep the order of
c
The two different kinds of approaches of u
the same as that of
and v
a+b ..
to infinity in
this problem for the boundaries to be non-degenerate are the same as
those in the binomial problem for the corresponding two types of cost
function.
Specifically they are as follows:
The appropriate normalization consists of:
case (1):
constant cost:
x>
(4.6l)
a-b
Y
=
Al !}
z
=
a+b+c
A2!3
•
,
'
V(a,b"c)
A2 }
z
..
>x
j
0
j
116
absolute deviation cost:
case (ii):
I
(
a+b
,
=
a-b
Al !2
;
=
a+b+c
A
,
x
=A
y
z
x> 0 ;
1
)
/
(4.62) \
\t
\,...
V(x,y,z)=
V(a,b,c)
Al !2
z > x ;
•
We denote, for convenience, the two types of limiting tendencies
specified in the above two normalizati0ns by L and L2 respectively.
l
He note that if the order of magnitude of c is less than that of
a+b, then the corresponding solutions can be obtained as a special
case with z = x.
We now treat the two cases separately.
case (i).
(4.63 )
+ J...
2z
-
1.
It follows fram
(4.58) that
117
Letting A -->
we see that V satisfies, under L , the following
1
partial differential equation:
(4.64)
.
00
oV
o z
2
0 V + x
~
z
oy
x
-
= 2Z
oV
ox
On or outside the boundary , V(X'Yi Z)
oV
+ l.
z oy
-
1
•
is given by
(4.65)
where
(4.66)
Vo(x,y,z)
= Vo(a,b,c)/A2/ 3
= Vo(~A2/3x+
~Al/3y,~A2/3x_ ~Al/3y,
A2/3Z.A2/3x)/A2/3
=
fyl /z ,
which satisfies, in the region y > 0, the homogeneous partial differential equation:
(4.67)
x
2z
x
+
+ -Z
Thus the normalized net gain function
. W(x,y,z)
( 4.68)
oV
oz
----.£ =
°.
,
(~
= M(a,b,c) / A2/3 =-;-V(a,b,c)-V0
a,b,c),A2/3 =
-
=
V(x,y,z) - Vo (x,y,z)
satisfies, under L , the partial differential equation:
l
x 2l2w
x 0 W yoW
0 W
--+--+--+-=1.
2z Oy2
z 0 x
z 0 y
0 y
We state below a lemma, Without proof, that will be used in the
second case as well as in Chapter V.
118
Lemma 4.3
!f the parameters
a,b,c
of the Dirichlet distribution ;(a,b,c)
tend to infinity along With A in such a way that
i
i
Ij
,
(4.70)
a+b
A2a
=
x
a-b
ACt
=
Y
a+b+c
=
z
!
/
'\
"\
j
l
A
2Ct
Ct>O
remain finite, then
C(x,y,z), say.
case (ii):
Just as in case (i), it follows from
r -(
A1/2_ V x,y,z
= - zx2 A1/2-r
) - -(
VXl y,
Z+A-1
(4.58) in this case
L7
-(
-1
-1/2 ,z+A-1) -2V
-( x+A-1,y,z+A-1)
V x+A ,y+A
-1)7
+ -(
V x+A -1 ly-A-1/2 ,z+A_
+ -x A1/2
z
r -(V x+A-1,y,z+A-1) - -(V x"y,z+A-1)- 7
-
-1) 7
- -(
V x+A -1"y-A-1/2,z+A_
_
C(~ Ax
+
¥1/2YI~
Ax _
~
2
Al / y, 4z - Ax) •
119
MUltiplying both sides of (4.72) by Al / 2, letting A ->
and using Lemma 4.; with a = 1/2, we see that
L~1
c;
co ,
V satisfies, under
the following partial differential equation:
-
oV = --x o2v
x 0 V
y 0 V -(
)
--o z 2z --0 y2 + -z --0 x + -z --0 Y - C x,y,z •
It turns out that
Vo(x,y,z)
l 2
/ = jYI/z , as in case (i)
= V0
(a,b,c)/A
,
and hence satisfies the equation (4.67).
Hence the normalized net gain
function,
( 4.7;a)
\v(x,y,z) = M( a,b,c) / A1/2 ,
satisfies, under L2 , the same differential equation (4.7;).
Combining the two cases, ive may state that under the appropriate
limiting conditions W satisfies for both types of costs per observation, the partial differential equation:
x 'O~l
x
2'Z" ""2 + Z
oY
0 W +
oX
oWy
X o
Z
oW
+ '0 z
-
= C(x,y,z) ,
where
C(x,y,z)
(
=)
<
1
'.-
, for constant cost
1
?If 1/X
,S¢(L) + .w f~ - fi(- .w)7J~ for absolute
IX
IX 'deviation cost
in the three dimensional region between the plane y=o and the
boundary surface y = Y (x,z)" say. At the boundary surface y=O,
l
symmetry gives the condition '0 VIIloyl y=o =
in both cases. This,
°
together With 0
Vo~ yl y=o = liz, gives the boundary condition satisfied
e
120
by W, in both cases, as
o wI
GI
(4.76)
y=o =
-
1
z
•
As before, from continuity, we have
i
(4.77)
i{l
= 0
\Y=Y1(x,z)
for both cases.
,
The other free boundary condition can also be
derived as in the binomial problem; it turns out to be
\ up to 0(A- 1 / 3 ) for case (i)
1 2
; up to 0(A- / ) for case (ii)
(4.78)
-e
Again, due to symnetry,
ll(x,y,z) = vl(x"-y,,z)
and the optimum
is sJ~etric with respect to the plane y=O.
boundary Y1(x"z)
Thus it is sufficient to consider only y > 0 •
The problem of finding the optimum boundary
along with the optimum net gain function is thus reduced to
a free-boundary problem which can be stated as follows after the
following transformation of variables:
r
=
zx ,
1<
s
=
1.
o < s < (Xl
t
= 2x
W1 (r,s,t)
X
1
"
J
r
,
< (Xl
O<t<(Xl
J
,
=
w(x,y,z) - x 5(x,y"z)
=
,1
s
r)
W\~, 2~" 2t
-
-(
)
C1 r,s,t , say •
121
TRANSFOBMED FREE-BOUNDARY PROBLEM:
To solve for the unknown boundary
\'T (r,s,t) where
l
(r,t) and the unknown
1
satisfies the partial differential equation:
W
l
6
i
o vTl
in
0
at=
;
\
"\
\-
o< s
,
,
~ sl(r , t)
l<r<
co
O<t<
co
with the following boundary conditions:
o Wli
(4.80)
d sI\ 8=0
=
-
1
-r
0 eli
-0 Is
-
s=o
,
(4.81)
o Cl'
(4.82)
=- o s
l
Now
=ii
I
.t:'\
r
1
1"\.1-
c:.u
for case (i)
.
1 - ! J,(
)\ -1 . /r;:;z
2t.LT¢(- s) + -s ' ;\p I -S)1
- I 7, for case (ii)
:.t"\
j2t
j2t . 2
,j2t -
the latter formula reducing to
-t2
using
(4.84)
(4.45).
o C1
.
Thus
, for case (i)
as
1
for case (ii).
122
Hence a
clIo
sl·
s=o
is
0 for both cases.
4.4 Series expansion for the optimal solution for large
x (trinomial
problem)
Let
(4.85)
•
He no'W consider the two cases separately.
case (ih
constant constant: .
sl(r,t) and W2(r,s,t)
satisfies the partial differential equation:
To solve for the unknown
(4.86)
'Where W2
0~1
~ -
as
'With the boundary conditions:
s=o
(4.88)
=
I
W2 1 s=sl(r,t)
a W2 \
~
0
,
= 1
s=sl(r,t) =
r
r
,
+
2t
•
As we are interested in solving this free boundary problem only
for small t,
and Robbins
we may assume, as in the free-boundary problem of Moriguti
1 32.7,
a series of ex,pansion in t
of the unknown boundary
surface as
(4.90)
sl(r,t) = co(r) + Cl(r)t + C2 (r) t
and, hence, a solution of
form
2
+ •••
(4.86) with boundary condition (4.87) of the
123
w2(r l s , t) = E An(r)
(4.91)
t
n
l Fl(-n,l/2;-s2/4t )
1
n
•
where, unlike the previous situation, the coefficients C (r)
m
An(r)
and
It is possib~e to show SoBin Meirigutj,'and
may now depend on r.
Robbins' problem, that the sUlIlIllation in (4.91) is over integral values
of n starting With n
= -1.
Moreover, the two free boundary conditions
(4.88) and (4.89) are sufficient to determine the arb:ttr&J'Y functions
An(r)
and Cm(r)
completely.
In fact, they may be obtained successive-
ly just as before by equating the coefficients of equal powers of t
from both sides of each of the following equations obtained by substituting (4.90) and (4.91) in (4.88) and (4.89);
~
(4.92)
00
n
E A (r)t F (-n,1/2;-E E Cm(r)tm_72/4t)
n
l l
n=-l
m=o
~
(4.93)
E 2nA (r)t
n
n=-l
=
n
1
r
~
E
00
=
00
E Cm(r) tr:J1Fl(-n+l,3/2;-EE Cm(r)tm_72/4t)
~o
~o
•
The results are:
and all other ct s
and At s
vanish.
The first few coefficients are:
124
CI(r)
C (r)
2
,
16 -6
=- 3 r ,
2r- 2
=
AO(r)
= - '21r
Ai(r)
= r -;
.
896 -10.
15 r
,
c3(r) =
A (r)
2
•
•
,
,
= _ l~
r- 7
,
148 -11
r
~(r.> =
l5
•
•
Thus
a. series expansion in
t
of the optimum boundary sl(r,t)
for small t may be written as:
(4.94)
sl(r,t)
t 2,
-
= 2(-)
r
16 t 5
896 t 8
-; (-) + - ' - (-) - ....
r r
l5r2 r
•
which may be :put in terms of the original variables (see (4.61»
.
(4.95) Yl(x,z)=x
,
sl(~'
x
-k)
~
-
=nl
~z
(~)fl - .1:..-; (~) + .J-,... (~)2 - •••_7
z
;z
=...l....(~)2 rlz -
2x
whichever form is convenient.
(4.96)
Yl ( x,x)
As
z
l5zo
z
8
~
(~)
+ .J-,...(~) 3x' z
l5x z
4
1
...:..7
0
z-> x, both, however, reduce to
- ...
11
- 6x4
= 2x
A more :physically meaningful
as:
re~rametrization of
the boundary surface
may be given in terms of the :parameter
'fJ =
.!r
= ~
z
,
0
<
-ui -<1
(the :pro:portion of untied observations, in terms of the original
double dichotomies :problem) as:
125
- ••• (for large
z)
may be written as
2
5x lF ( 5,,1I·
-48z
--rr
2,-y2/)
2x + 37~
11 lFl (8 ,,1I·
2,-y212x )'"
- •••
l
2880z
In this case W satisfies (4.86) with the boundary conditi~n
2
(4.87) same as in case (i). Only the free-boundary conditions (4.88)
and (4.89) are changed to:
) 2\~ 8=sl(r"t
-
(4.102) 1{
-
2
-t •
...L
~
IFl(-1/2,1/2i-s12(r,t)/4t)+sl(r"t)/r"
+ l/r
respectively.
may be taken as
For small t, a series expansion in t
,
of the solution
•
126
(4.104)
(4.105)
sl(r,t)
=
w2(r,s,t) =
E Cm(r) t
m
m
E An(r) t n 1F1(-n,l/2;-s2/4t)
n
where m and n are half integers and the summations starting from
m=3/2 in (4.104) and n
.
binomial situation.
= -1/2
in (4.105), just as in the corresponding
.
Substituting (4.104) and (4.105) in (4.102) and
(4.103)" and equating coefficients of eqt:Ial poWers ot 't':i'rom"both 'sidles of
the resulting equations, the unknown coefficients Cm(r)'s and An(r)'s
may be determined successively. It is found that
and all other Cn •s and An •s are zero. The first fevT non-zero
coefficients are:
=
j1c
r
A'(r)
o
=
Ai(r)
=
3
= - ..:L(fiC) ,
12
r
- -1
,
.fiC
..l.. (/iC )
2/iC r
2
,
_ ..l.. (~)4,
;/it
105§
144oj1(
Thus a series expansion in t
-
for small t
may be written as:
r
(fiC)
6
•
r
of the optimum boundary sl(r,t)
127
which may be put in ter-ms of the original variables (see (4.62»
(
I.
)
I
)
(X)1/2' ~
'1-.107 Yl \x,Z = Z
•
1
vZ
r
r.:- J.. 1 -
7'Jt
1
Z
48 2
+
as:
1971(2 1
7
160'.16'" - ••~ ,
Z
or, alternatively, in ter-ms of 'W"' , the proportion of untied observationa, and z, the total amount of sampling, as
( 4.10 8)
r
*(
J)
J2'Jt-Gr
1
7'Jt
1
197' 'Jt2 1
Y1 z,1J =
4 - _ 1 -"4tr 2 + 1bO..16 ..,-
JZ
z
or, still, alternatively, in ter-ms of If;f and
z
x, as
~4
T
x
All three for-ms coincide as
z
->
x
i.e., as 1:jj
- ...
-> 1 to
The net gain function turns out as
(4.111)
+
2(2'Jt)5/2 529 (x)6 1
F1(-11/2,1/2;-Y2/2x)
4
~2
r
1
1440 _. 4
z x .J./ c:
- ...
.
_7 •
128
Thus
- ...
In tbe special easel
Z = x
I
1
1
~/2
-
-;;rT2
2(21r)?/2 529
+ 1440'" 44
1
1172 - •••
x
•
It may be noted that (4.110) is the same as (4.54)1 the ex,pansion
for the boundary Y1(x)
viation cost. A1so1
in the binomial problem 'Wi~h absolute de-
W(x,y"x) and W(x"O"x) in (4.113)
are found to
be double the corresponding eXJ?ressions (4.55) and (4.56) for the later
situation.
CRAFTER V
BOUNDS ON ASYMPTCYrIC SOWTIONS
5.0 Introduction and summary.
In Chapter IV we obtained a formal expansion in negative powers
of x (for large x) of the (normalized) optiIllal (Bayes) boundary,
Yl and that of the corresponding (normalized) net gain function W
as A ->
CIO
for the various sequential decision problems considered
in Chapter II.
It is, however, very difficult to justify these
formal expansions.
132,7
The expansion obtained by Moriguti and Robbins
for their problem (involving binomial model with constant cost)
has been shown by Breakwell and Chernoff fl?;7 to be an asymptotic one
as
x--->
term.
CIO
in the sense that the error is small compared to the last
Their proof is rather involved.
We shall consider in this
chapter a less ambitious objective of obtaining some bounds on the
true boundary using Bather's f§.7 method.
for all x,
large and small.
boundary when x ->
CIO
These bounds are valid
In fact, they converge to the true
as well as when x ->
o.
In as much as
these bounds provide even approximately the true boundary for intermediate values of x, they are useful in themselves.
Besides, in so
far as the expansions of these bounds for large x are consistent with
the expansions obtained :i.l'l Chapter IV, they provide an independent check
on them.
130
As we are going to follow Bather's methods rather closely, we
try to keep matters simpler by retaining his notations as far as
·possible consistent with our previous ones.
Chernoff's
f1!J:.7,
Bather's results, like
are concerned with a Gaussian process and deal
with the risk function rather than the gain function approach of
Moriguti and Robbins
sampling rule.
["39.,7
to characterize the optimum (Bayes)
Since the optimum boundary remains the same for
either approach, our choice in the various studies has been prompted
by the relative advantages one offers over the other.
In Chapters
II and III we concerned ourselves with the risk function as far as
the general theory developed with respect to it
-e
£9,
2!J:.7 could take
us and then changed to the gain function for some obvious reasons.
The recursion relations developed in Chapter III could only be obtained in terms of the gain function.
In Chapter II, the computations
of the optimum boundary are easier with the recursion relations involving the gain function rather than with those involving the risk
function.
In Chapter IV, we retained the gain function in deriving
the partial differential equation and the various boundary conditions
that it satisfies just to be consistent with Moriguti and Robbins, but
this portion could equally well have been carried out in terms of the
risk function With slight modifications of the resulting partial differential equations (viz., changing the sign of
boundary conditions.
thus reversible.
TI,
etc.) and the
For the asymptotic theory the two approaches are
In order to adopt Bather's technique, it is,
ho~~ver,
imperative that one should consider the risk function rather than the
131
gain function, because same of his arguments hinge critically on the
In the following analysis, therefore, we adapt the derivation of the
various differential equations along with the boundary conditions in
terms of the noraalized risk function.
These could, however, be easily
anticipated for thebinamial problem with absolute deviation cost, from
Chernoff's results for the Gaussian process problem with constant cost.
Under the particular type of limiting tendencies that we are considering,
the trinomial problem would have reduced to a certain (rather specialized)
bivariate Gaussian process problem, just as the binomial one reduces to
the univariate Gaussian one.
·e
We have, however, no special reasons for
considering this bivariate Gaussian process problem in order to derive
the partial differential equations and the boundary conditions satisfied by the standardized risk function for the trinomial problem.
They
are simply obtained by modifying, as in the binomial problem, the corresponding expressions in terms of the gain function that
~
are derived
in Chapter IV.
In section 5.1, we first consider the binomial problem with absolute deviation cost.
The associated free-boundary problem (FBP) is
stated in terms of the normalized risk,
variables, the
~(t)
FBP
is put in the appropriate form,
for the true boundary aCt)
obtained.
p(x,y'f.Transforming the
The upper bound
and. its expansion for large
tare
A similar development is given for the lower bound p(t
Section 5.2
f.
is concerned with the corresponding developments for the
trinomial problem with both types of cost.
4JlIhe ambiguous use of the same symbol p for two different thltufigs may
be resolved by referring to the arguments.
5.1 Binomial ;problem 'With absolute deviation cost.
We have seen in Chapter II, (a.60) that the optimum boundary
can be characterized by the Bayes risk
p{a,b)
which satisfies in the
go-region the recursion relation
p(a , b)
=
a:b
C(a,b)
=
E~lp -
,
a+bb p(a b+l) + C{a,b)
p(a+l,b) +
where
1/21
la,E7 ;
and assumes on and outside the boundary the value
po{a,b)
·e
=
A Effp -
1/21 'a,E7
=
A C(a,b)
-
A
~
Using the appropriate normalization
~
•
(4.62) for this case, namely,
a+b
x
=T
y
=
a-b
Al / 2
=
p(a,b)
Al !2
p{x,y)
- A
,•
. and modifying the results of section 4.1, we see that, under L, the
normalized risk p(x,y)
satisfies the partial differential equation
OP
oy
+
0;';
0 x
=
- C(x,y)
,
in the symmetric region
o<x<
co
,
e
m
where Y1{x)
is the unknown boundary on which bounds are desired;
the boundary conditions are
-
iLQ
o y :: 0
on y = 0
-p
on the unknown boundary
:Le
oy
where
po (x,y)
.
=
p
0
(a,b)/A1 / 2
•
Using (5.1), (5.2), (5.3) , (5.4) and Lemma 4.1, we may write
-e
= 2. ,~ ¢(..L)
Jx L·jX
_
M
jX
§ ( _ Jxl
IX
)} .
j
Considering only the upper boundary, (5.4) and (5.5) become
where the prime denotes differentiation with respect to the second
arguJ:DeZ1,t.
Let us
001'1
put the above facts in Bather's notation.
s
= ..1L
t
=
x
=
=
p(x,y)
f{s,t)
K{s,Yt)
IX
po{x,y)
,
,
,
,
Let
tt
~4
=
c(s,t)
Then
f(s,t)
C(x,y)
,
satisfies the partial differential. equation:
of'
as
s
of7
at
_ =
+ 2t _r (
c s,t) +
0
in
Is I
<
(;: (t)
with the following boundary conditions:
oo sf =
(5.8)
...
due to symmetry
-e
f(s,t)
on
0
s
=0
,
=
.~
X(s,t)
= ; . {¢(s) -lsI ¢ (-lsI>} _ t-1/,,(s),
say
whenever
Is I
o<t<
> (;:( t) ,
cg
,
and finally
~ ~ (iT( t
), t )
=
~ ~ «(;:( t ), t )
= ...l... -1(Jt
(;:(t» •
Due to symmetry again, we need consider only
s >
o.
Upper bound:
. Following Bather, we consider the following family of separable
solutions Of
(5.7)!
f{s,t)
=
t
r
g(s)
,
r
any real constant.
135
It follows from (5.9) that
= -1/2.
Hence
=_ 2t3 / 2
c(s,t)
r
g(s)
must satisfy the
ordinary differential equation
g"(s) +
S
gl(S) _ g(s)
= - 2t { ¢( s) + s
r .! _
rl (- s)
-2.:e
in order that
t- l / 2g(s)
7~
-)
is a solution of (5.7) * Now (5.11) does
not make sense as the right hand side involves
t.
In order to make
sense we consider for the present an auxiliary problem where everything remains the same as in the original problem except that the
cost function is now given by
where a
is some positive constant.
The optimal continuation region
for this auxiliary problem then is bounded by straight lines, viz.,
tel
=
{(s,t);
-
The unkno'Wn constant ~ and g( s )
g(s)
X<
~l
<
X} '.
are found from the following facts:
satisfies the differential equation
g" + sgl - g = - 2a {¢(s) + sf
in
s
~ -1 (-s>-7}
With the boundary conditions:
(5.13)
g'(O)
=
0
(5.14)
g(X)
=
L(X)
(5.15)
g'(X)
=
L'("-) •
,
-
Now the general symnetrical solution of the homogeneous equation
g" + sg' - g
=0
may be written as
•
where y
is an arbitrary constant.
A particular solution of (5.12)
with (5.13) is given by
where
u
2
1
• "U
2u
/2 du
f
2
2
y .y /2ijy
o
i
v
.
¢(w) dw
0
Let
be the general solution~ Then the two constants y
determined from (5.14) and (5.15), i.e., fram
hex)
Writing
(5.17)
and
=
(5.18)
Lex)
,
explicitly, we have
and
Eliminating y
from (5.19) and (5.20), we get
and
X
are
•
137
which
the critical level
dete~ines
~
for each a > 0 uniquely.
Uniqueness follows from the positivity of the left hand side of
(~;'21)
which is a consequence of the definitions of the various functions
involved therein.
If we now replace the constant a by the running variable t,
then the resulting solution ;;:(t)
It may oe verified" moreover" that ~(t)
an upper bound on O:{t).
is decreasing in t.
will const:l.tute, following Bather,
To show this it is sufficient to show that the
left hand side of (5.21) is increasing in
X.
This can be proved in
its turn by direct differentiation and by utilization of the differen-
'.
tial equation that
-e
vi
Vl(s)
satisfies" namely"
s vies) - Vl (s)
(s) +
= s
1'21 - i(-st7
It aan also be seen from (5~2l) that '5::(t) -->
'5::(t) -> 0 as
t _>
co
co
•
and t -> 0 and
•
It is thus possible" in principle" to obtain the actual values of
'5::(t) by solving (5.21) for '5:: after substituting a = t.
In practice,
however" difficulty arises because of the presence of texms involving
and V~.
the untabulated functions
"'1
the definition (5.16) of
Vl that
11r
t
Yl
('5::»
-
Now it follows directly from
0 ,
and
It can be seen from (5.21) that if some positive terms are omitted
from its left hand side" the resulting solution" ~(t)" say" can only
exceed '5::( t ) • Thus an upper bound. to the upper bound '5::(t) may be
obtained by solVing the equation:
t(t)" unlike >::(t)" is amenable to computation using the normal integral
It may also be verified from (5.22) that ~(t) is decreasing
tables.
in
and that 1(t) ->
t
>
t _
00
t -> 0 and ~(t) -> 0 as
as
00
•
ASymptotic form of upper bound for large t.
Lemma 5.1
This follows readily from the definition (5.16) of V (s)
1
following transf~tion:
-e
w'
=-vw ,
V
= -"
u
v'
u
'
u
= S ' o -< w' ,
v'
, u'
<
using the
1
In fact" one finds that
(5.23)
J21c
111
I2
I2
I2
'l(S) = s4 (
(U 12 v. 3 e- s2 /2£u {1-v (l_w )17
S
0 ~
)0
M4 s 4 + M6 s 6 + •••
=
dw ' dv ' du I
•
•
where the coefficients M4" M6" ••• may be determined by expanding
the integrand in (5.23) and integrating term by term.
Using Lemma 5.1" it can be shown that for large t"
-(
A. t)
=,J2ic
-r.~
1
t
-
-
1
2
-
(4~ ) 3
1 +
t3
-
0
(- 1 )
t3
,
139
and
'R( t)
=
If
-tl
-
1 (
J21i?
b4
1
t
+
3
0
(
1".2:)
•
t,.l
For large t, we do not lose much by using the computationally more
convenient form 'X(t) instead of ~(t) •
Lower bound:
Without going into the details of the arguments that lead to the
inner bound, for which vTe refer to Bather, we just indicate the pertinent steps relevant for our case only.
Following Bather" we consider
here the following family of solutions satisfying (5.7) and (5.e)
•
Eliminating the parameter a
from the two equations below obtained
from the boundary conditions (5.9) and (5.10) ,
F~(p(t)"t)
the inner bound,
p(t)
p(t)¢(p(t»
=
KI(p(t),t),
is given by the equation:
+
.ri -J(..p(tlt7["1 ... p2(tt7
2rl + p2(t)7¢(p(t»r}(-p(t»
-
-
- ¢(P(t»
The existence and positivity of pet)
_ pet) 7
1+p2(tf
=
1
2t
•
folloWS from the positivity of
the left hand side of (5.26) that is obtained by using Gordon's inequality ["29.7 on Mill's ratio" viz.,
140
>
equality obtaining as
x _>
It may be verified fram
x
2
1+x
x> 0
co •
(5.26) that pet) is decreasing in t
by showing that the numerator and the denominator of the left hand
side of
(5.26) are, respectively, increasing and decreasing in p •
It may also be verified that pet) ->
as
t
->
co.
Fram
co
t -> 0 and p(t)-> 0
as
'5.26) again the asymptotic form of p( t) for
large t. turns out as follows:
..
•
Now writing
(5.24) and (5.27) in terms of the original variables
y
(5.2) and (5.3) respectively, the bounds on the optimum
given by
x
upper boundary for large x may be written as
..l-+
2
x
and
Yl(x)
- .
=
if
..l-
rl -
jX' i.
whereas the exact upper boundary
(4.54) from Chapter IV is
r
Yl ()
x = J21c
T
-1 _ 1 ~
.!x + o( xc.
lrJ _7
71C 21
'lm
.
x
+
0(T1 ) _7
x
It is to be noted that the first order term is the same in all three
expansions above and that Yl(x)
for sufficiently large x.
lies between
Yl (x)
and Xl(x)
and
141
5.2 Trinomial problem
As already noted in Chapter II, (2.100)" the optimum boundary
.
can be characterized with the help of the Bayes risk
p(a,b"c) which
satisfies in the go-region the recursion relation
p(a"b"c)
c
p(a+l,b"c) + a+~ c p(a,b+l,c) + ---b
a+ -+c p(a"b"c+l)
= a+~ c
+ C(a,b,c)
,
where
C(a,b,c)
=)
for constant cost ,
1
~
l EL ,~"1-"2 Ila" b".£7
·e
,
for absolute deviation
cost ;
and assumes on and ou1\lside the boundary the value
•
Considering the two different types of normalizations for the two
types of cost" viz."
x
y
z
p(x"y"z)
=
(a+b)!A2!3 ,
= (a~b)!Al!3 ,
2
= (a+b+c)!A !3
=
p(a" b,C)!A2 !3
;
,
2
po(x"y"z) = po(a'.b'.c)!A !3 ;
for
(i)
constant cost" and
142
x =
Y
=
(a+b)/A,
(a_b)/Al / 2 ,
z
=
(a+b+c)/A
;
= 1/3
for (ii) absolute deviation cost, using LeIIJIIla 4.3 with a
case (i) and with a
= 1/2
for
for case (ii), and re-expressing the
results of section 4.3 inter.ms of the normalized Bayes risk
P(X,y,Z), we have the following~
Under the appropriate limiting tendencies for the two types of
cost,
p(x,y,z)
satisfies the partial differentiation equation
2
0 P + 1. 0 P +!
O2y Z oY z
a p + 0~ P __
oX
oz
-
-(
c x,y,z )
in the region (symmetric with respect to yeo):
O<X<z<
Yl(x,z)
;
being the unknown upper boundary surface on which bounds
are desired; and
,
C(X y"Z)
~here
= {. Jrl +
Z
Further 1
00
P(X'YIZ)
12,/i
Z
{¢(M) _M
IX
IX
satisfies the following boundary conditions:
(5.29)
~
(5.30)
-p
(5.31)
a
oy
oY
on yeO due to symmetry ,
0
=
j
'\
=
e_
-
Po
o "0
oY
on the unknown boundary surface ;
where
-(
po x,y,z ) -_?fX
z
•
Consider:tng only the upper boundary yl(x,z), (5.30) and (5.31) may
be written explicitly as
(5.33)
where the prime denotes differentiatIon with respect to the second
argument. We now make the following transformation
s
=-Y
t
=x ,
f(r,s,t)
(5.34)
,
jX
.,;:.
K(r,s,t) ""
c(r,s,t)
:;:
_00
< s < 00
O<t<oo
-,p,x,y,z )
p0 (x,y,z) =
!x C(x>y,z)
-r[r"2
{ ¢(s) - /sl
~(-lsf)}
=j
, for case
(i)
r
t';' {¢(s)
+ fs
'f~ - i(-/s' L7},
for case (ii)
•
It follows from
(5.28), (5.29), (5.30) and (5.31) that f(r,s,t)
satisfies the partial differential equation:
:2)
s
+
~
+ 2tfc(r,s,t) +
;
~ ~3
0
=
in
fs J < 0- (r,t) ,
-e
1 ~ r < G9,
0 <t <
0)
,
with the following boundary conditions:
oa fs =
0
on
s
=0
due to symnetry ,
f{r,s,t) = K{r,s,t) whenever fs f > o-(r,t) ,
and
~ ; (r, rr(r,t),t)
=
~ ~ (r, rr(r,t),t)
= 2...
r/t
Again, due to symmetry,
to consider only s
case (i):
~
f(r,s,t)
- 1(-
rr{r,t»
= f(r,-s,t),
by
(5.34) •
and it is sufficient
0 •
constant cost:
Upper bound:
Because of the particular separable form
we seek solutions of the form
(5.34) of K-(r,s,t)
= -1-
f(r,s,t)
Substituting
g(s)
rjt
(5.38) in (5.36), we see that g(s) should satisfy the
ordinary differential equation
g il + sg' - g
=-
2.rt3/ 2
c ( r,s, t)
Remembering that
for this case, we are led to consider the auxiliary problem with the
new cost function as
where a
is an arbitrary positive constant.
Now proceeding as in
the binomial problem, an upper bound X(r,t)
on 'ir(r,t)
from the following equation:
From (5.40) one can easily deduce the following facts:
1. X is decreasing in r
2..
as well as in t.
For a fixed r 0 >
1 ,
~(ro,t)
-> 0 as t -> co
~(ro,t) ~ ~
as
t ---> 0
,
•
is obtained
146
.3.
For any two fixed 1
~
rl:S r 2
;::(rl,t) ~ ;::(r2 "t},
0
'
<t <
co
,
and hence
~(r"t}
4.
(5.44)
~(l, t) ,
<
r ~ 1, 0
< t < co
For large tr, an asymptotic form of ;::
•
is given by
=
;::(r,t}
Lower bound:
We consider the following family of curves satisfYing (5 •.36) and
"
-.
(5.45)
Fa(r,s"t)
= ..
rt *(s) + a(r)
J€
e..
s2 2
/
,
where the :parameter a is now allowed to de:pendon
2
*(s)
=
s
e" s /2
s
S.
2
eX /2 dx
r,
and
.
•
o
The boundary conditions (5 •.32) and (5 •.3.3) for the family (5.45) involve
the inner bound p{r,t)' as can be shown similarly to the binomial
:problem, using the arguments of Bather, as follows:
F~ (r, p{r,t),t)
=
K'(r,p(r,t),t)
Eliminating the :parameter a(r)
and the relation
•
from (5.46) and (5.47), using (5 •.34)
147
'P(r,t)
is given by the following equation
(1 +
i Q) !(- 'ii)
i) + '!r(p)/
The facts
p,
- 'ii ¢f;;)
=
P
(5.41), (5.42), (5.43) stated above for
as can be verified from
~
also hold for
(5.48) utilizing some facts about
~
given by Bather. Also, using the asymptotic form of W(s) . given by
Bather, an asymptotic expansion of
p
for large rt
1
. 2
1
2 372 £1 - r;r:.( 2 372)
/2.r t
v2n
r t
+
is given by
1
2
0 ( 2 372) _7 ·
r t
The asymptotic expansion of the exact boundary Yl(x,z)
in
obtained
(4.95) for large x and hence for lage z, when written in the
notation of this chapter, is given by
A comparison of (5.50) with (5.49) and (5.44) gives a verification
of
(4.95).
case (ii):
Upper bound:
absolute deviation cost:
Equations
Remembering from
(5.38) and (5.39) are obtained as in case (i).
(5.35) the expression for c(r,s,t) in this case,
we are led, as before, to consider the auxiliary problem With the new
cost function as
•
148
Proceeding as usual we arrive at the same equation (5.21) defining
';;: for a fixed a
(5.21)
in the auxiliary problem.
Replacing a
in
by rt" the resulting equation gives ';;:(r"t)" an upper bound
on ;(r"t). We may" as in the binomial problem" obtain an upper bound
~(r"t)
to X(r,t)
defined by an equation like
in its right hand side replaced by rt.
(5.42)and (5.43)
case.
Fi~ally,
(5.22) with the
The same three f8.~ts
also hold for either ';;:(r,t)
t
(5,,47),
or ~(rJt) in this
for large rt" asymptotic expansions for them may be
obtained as follows:
(5.51)
);:(r,t)
2
2
/21c ..l r l .1: (./2ir
23t) (2:_)
= ;--If'
rt 2 T
~rt,
1(r)t)
1 rl
= J21r
rt-"b1
T
Lower bound:
(J2;.)
2
~~
Cr ;
1 3
+ 0(--) 7
rt -
,
3
.
2
)
+ 0 (~
) _7.
As i:1 the binomial problem (Sec (5.25)}" we consid.er the
following family of solutions to (5.36) a~d. (5.37) ,
where the parameter a may now depend, as in case (i), on
Eliminating a(r)
r.
from the following two equations, resulting
from the boundar"J conditions (5.32) and (5.33) "
Fa(~~ ~(r,t),t)
=
F~(::·.o -~(r,t)"t)
= K'(r, p{r,t),t)
the inner bound p(r"t)
K(r, p(r,t),t)
,
,
is given by the same equation as (5.26) inth
it~ right hand, side replaced by 2(1 +
i.:k·).
(5.42)
and (5.43) also hold for p(r,t).
p(r,t)
for large tr is given by
The same three facts
(5.41),
An asymptotic expansion of
-(
) ,= .}21c
p r"t
T
1 2
1 3 .
-r1t -r1 - (-)
+ 0(-)
rt
r t -7
.
Expressing the boundary Y1(X"Z)" obtained for this case in (4.107)
of Chapter IV" in the notations of this chapter gives
(5.53)
a(r"t)
= J2ic
T
1
rt
11 -
7'1C (1 2
(1)4 7
4E
rr-) +0 rt
_.
Comparing (5.53) with (5.51) and (5.52) gives an independent check on
.
.
(4.107).
150
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