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ASymptotic Distribution of Sample Size
for Certain Methods of Sequential Sampling
.L. L. Lasman a.nd E. J. Williams
!
~
...
l'
f.
Instiillte of Sta:ti.stics,.i
Mimeo Series No. 203
June, 1958
k·
•
, t
4
_
TABLE OF CONTENTS
Ohapter
I
II
...
INTRODUC TION • • • • •
LITERATURE REVIDl
o
•
•
•
•
•
1.
2.
Acceptance Sampling • • • • • • • • • • • •
Estimation of p in a Binomial Population ••
3 • Es timation of f.l. in a Normal Population •
4. Asymptotic'1)leory. •
• • • • •
, • A Recent Application. • • • • •
• • •
nI
SEQUENTIAL STOP RULE9. • • • • •
Rectangular Stop Rule •
2 • Linear Stop Rule. • • • • •
3 • Number of Paths to a Line •
4• Hyperbolic Stop Rule. • •
IV
•
oJ
. .. .. .. .. . .
.......
..
EXTENSION OF RFSULTS • • • • • • • •
0
•
•
•
•
•
e
1. 'General Function in, Two Dimensions.
2. Functions in s Dimensions • • • •
v
Sl.1MMARY • • •
BIBLIOGRAPHY •
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••••
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•••••
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12
14
18
20
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46
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,4
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61
.
Chapter I
INTRODUCTION
World War II stimulated the imagination of many great people and
this resulted in new ideas that were destined to change the ways of the
world.
One of the more important statistical ideas was in connection
with sequen tial theory.
To say that sequential analysis has become
important as a statistical tool would be to understate the obvious.
Almost every practicing statistician has used sequential methods in one
form or another.
A dictionary
definition of the word Itsequentialtt is: IlSucceeding
or follow::ing in order. It
of in statistics.
And exactly in this sense should it be thought
Not every sequential experiment was planned with
sequential theory in mind, but any experiment whose observations are
dependent on the previous observations, are taken one at a time, or
taken in more than one group is a sequential experiment.
Many people have been using sequential methods all the time withou t knowing it.
These were common-sense procedures such as engineer-
ing inspectors used when a first sample drawn was not decisive.
Sequential theory makes precise these common-sense procedures.
E:xamples of sequential sampling can be found in many fields of
application.
Industrial concerns use sequential methods extensively,
and people in agriculture may soon find uses for the method.
The
first important departure from the fixed-sample-size approach came in
the field of industrial quality control, where double-sampling methods
were used.
Double-sampling schemes have been developed to solve many
•
2
industrial problems, and the extension to sequential sampling as it is
known tDday is essentially an outgrowth of these schemes.
But even in industry, sequential sampling is not confined to its
role in quality control inspection.
situations.
It is used in many
decision-m~ing
The continual collection and ana:qsis of data gradually
remove doubts existing in many problems.
This leads one into the field
of sequential experimentation in which one experiment helps the experimenter decide on the procedures for the next.
Work in developing design
of sequential experiments has only recently begun (Fisher, 1952, and Box,
1957) •
However, sequential. techniques are not always practicable.
If the
time between successive samples is long, and conditions have changed
drastically, the initial results may be unusable.
difficulty of compll ta tion may be too great.
In other cases, the
Or yet, to use an often
quoted situation, we may be experimenting with the growth of trees, and
it may take years for a tree to mature.
In this case it would be
ridiculous tD grow one tree first, see what happened, plant another one,
etc.
There are, of course, some instances where sequential sampling is
necessary"
For example, in' obtaining data on a certain illness Which
is not prevalent, the experimenter must wait until the patients go to a
hospital for trea:tment, one by one, to collect these data.
A main advantage of' sequential procedures is that when they are
applicable, they require, in general, fewer observations than the more
standard approa.ch in obtaining a desired result.
Thus the actual cost
..
3
of experimentation can be greatly reduced and much of the exper;1menters I time saved
0
Most of what has been done up to now concerns seqaential methods
as applied to the general' area of hypothesis testing
0
In more recent
;rears, a new look has been given at these methods, and the problem of
estimation by means of sequential sampling has been approached. Much of
this work has dealt with the estimation of p in a binomial population,
and there is some on estimating the mean of a normal population and the
difference between two means.
This dissertation is concerned mainly with the problem of estimation using sequential sampling.
One of the examples' of the theory
which is developed concerns the estimation of the difference between
two means, so that this estimate will have a specified degree of
precision
0
More generally, this dissertation is concerned with finding the
asymptotic distribution of the sample size required to reach a certain
fixed boundary when the sampling is of a sequential nature.
The
general theory will consider s populations mixed in unknown proportions.
Observations from this mixture will be drawn one at a time, and can be
classified correctly 'by population at that time.
When two populations
are concerned, we get the usual binomial situation, and most of the
work of this thesis will deal with this case.
"The asymptotic distribu-
tion of the sample size is obtained when the boundary is of the form
f(:x:t,
42'
000,
X
s ) ... 0
where x.J. is the number of observations from the ith population and
4
where f has derivatives with respec t to
darivativas are of the same sign
~,
••• , x s ' and all the
0
It is shown that the problem of estimating the difference between
two means with a specified degree of precision yields the boundary,
1
11
-+--.,.
x
c
,
2
a rectangular hyperbola having the desired properties, and the distribu tion of the sample size for this case is derived.
Another example of a boundary is a negatively sloping line in tne
~,x2-p1ane
and again the asymptotic distribution of the sample size is given.
When the boundary does not have the required properties, the
gener~l
formulae givan cannot be applied.
For the case where the
boundary is formed by the two straight lines
exact formuJ.ae for the mean and variance of the sample size are
obtained.
•
Ohapter II
LITERATURE
REVIM
In this review, much of the work in sequential theory will be
discussed, and the cases where the expected value and variance of the
sample size was derived will be pointed. out.
In a few cases the asymp-
totic distribution of the sample size was obtained, and the conditions
will be given for these particular cases.
Work in sequential methods,
whether it be hypothesis testing or estimation, is of little value until
there is some idea of what the sample size will be.
n
.1.
Acceptance Sampling
In Ohapter I it was stated that sequential analysis was an outgrowth of World War
n'.
This is true but for one notable exception.
Dodge and Romig (1929) really had the first sequential test procedure
when they constructed a double-sampling procedure.
Their idea was that
the decision whether or not a second sample should be drawn depends on
the outcome of the obserwtions in the first
s~ple.
The advantage of this and other schemes that were to follow is a
reduction in inspection costs
j
for the expected value of the number of
observations where samples were taken in more than one group was less
than the number necessary in a, single-sample inspec tion plan.
In another attempt to reduce costs, Ba.rtky (1943) applied a multipIe sampling technique to estimate the fraction p of defectives in an
infinite population.
*
His procedure is to take an initial sample of nO
An excellent review of the literature together with a. rather
complete bibliography is found in Anscombe (1953). This chapter is
based to a. large extent on that pa.per.
6
units and all additional samples are to be of size n.
If the number of
defectives in the initial sample is at most c, the lot is accepted.
this number exceeds c+k, the lot is rejected.
is taken.
In general, if, after taking
I'
If
Otherwise the second sample
additional samples of size n,
the total number of defectives equals e+r, the lot is accepted.
If this
total exceeds c+r+k, the lot is rejected.
Bartky proved that the process terminates with probability one,
gave a general formula for the probability of acceptance and for the "
expected sample size required, and gave these values more explicitly
under 10 different conditions on his sampling scheme.
'What is known today as Sequential AnalYsis was begun about the
same time in the United States as it was in England.
The pioneers of
this work were Wald in this country and Barnard in England •
Because
of the secrecy connected with this classified work, it was only near
the end of the war that any theory was published.
However, W'ald did
submit a restricted report to the National Defense Research Oommittee
in September, 194.3.
In this report, W'ald devised his sequential prob-
ability ratio test.
Somewhat later 'Wald (1944) published a general
theory of cumulative
S'UnlS
which gave the operating characteristic
curve of any sequential probability ratio test.
In May of 1945, after the restricted classif1cai!;Lon was removed,
Wald published the material of the report together
lft-eli'the
new ad..
--
vances he had made since 1943, in the article, taBequenti~Tests of
\
Statistical Hypotheses
ti
,
(1945).
book, Sequential ,Analysis (1947).
"
This paper was the basis of Wald's
By use of his now famous identity,
\
•
7
W'ald gave the expected value of the sample size under his sequentia.l
approach, and for this special case, derived the asymptotic distribution of the sample size.
In each of the non-single-sampling plans in
acceptance sampling, it is the mean and variance of the sample size
that are important when the problem is one of reducing costs.
Meanwhile, in England, Barnard (1946) was working on similar problems.
He considered first a simple inspection problem, then sequential
tests in general and finaJ.ly general inspection problems, and on this
last point gave the distribution of the sample size as a function of
the particular process curve, which is obtained by plotting fractions
defective against frequency of batches having this fraction defective.
Barnard also gave formulae for the mean sample size and the variance of
the sample size. Working with Barnard, Burman (1946) developed workable formulae for the operating characteristic and average sample size
when considering a binom:i.a1 population.
11.2.
Estimation of p in a Binomial Population
As far as testing hypotheses was concerned, Wald as book covered
the most general of situations.
these results.
There was little to be done to extend
Next, attention was turned to the problem of estimation,
and many authors considetoed the problem of estimating p in a binomial
population.
About the time of Wald's 1945 paper, Haldane (1945) first considered this problem.
He considered a population of which a certain
proportion p possessed a particular attribute.
If' one samples se-
quentially until a number c of the observations were found having
.
8
this attribute, and if N is the total number of observations taken,
then an unbiased estimate of p is
1\
P
0-1
N-I '
a-
and
where q
llll
I-p
0
Haldane pointed out that if' p is sm.aJ.l, it is often desirable to
have the standard error of p roughly proportional to p rather than
proportional to
<1
p
fP 0
For example, when n ... 1000, if' p == 001, then
.0031, so that we could not distinguish between populations where
go
p was 001 and 0005.
for c
~
The procedure described above accomplishes this
30 .
An unbiased estimate of v(~) was shown by Finney (1949) to be
1'.
AI.
1\)
!llL
v(p "" N-2
•
Tweedie (1945)· considered a method of finding the relative frequencies of one type' of blood cell, in which counting is stopped when
a fixed number of this type has been recorded.
This is the procedure
described above, and Tweedie gave it the name Itinverse binomial
sampling
0
It
The name stems from the fact that the estimate of p
obtained by direct and by inverse sampling have cumulant generating
functions which are inverses of each othero
An extension of inverse sampling to multinomial variables was
also given by Tweedie (1952) 0 Here he considered sampling until some
linear function of the variables reaches a predetermined value.
With
*
9
N groups in a population, let Pi be the probability that any one observation will fall into group 1.
Then for the simple stop-rule in which
sampling ceases when the number of observations in a certain set of
g;.roups (i
fill
1, •• 0, k, say) becomes equal to some previously fixed
number, c, Tweedie showed that the expectation of the sample size is
c
k
2p·
1
1.
and its variance is
This paper is discussed further in section II.4.
Girshick, Mosteller anti Savage (1946) presented some theorems
concerning the unbiased estimation of p for samples drawn from a
binOmial population using arbitrary stop rules.
They gave applica-
tions to single sampling, curtailed single sampling -- with some
rejection number, c--curtailed double sampling, and the sequential
probability ratio test.
The most important of these theorems for
our purposes concerns the estimation of p from the ttsample path. lt
Consider the positiva quadran t in the x, y-plane ..
Let x be the .
total number of observations taken ani let y be the number of observa-
tions so far in the sample which possess the attribute under study.
Each observation can then be regarded as a lattice point -- that is, a
point with integral coordinates -- on the diagram.
The sequence of
10
points is called a ttsample path."
;rixed
bound~,
When this sample path reaches a
sampling ceases and the estimation is made.
The
authors show that an unbiased estimate of p is
$ (:xl'Yl)
,
k(J\].'Y1 )
where $(J\].'Yl) is the munber of possible paths from the point (1,1) to
the point (J\].'Yl ) on the boundary, and k(J\].'Y1) is the number of
possible paths from the origin to this poin t.
This theorem is readily generalized to the following situation as
stated by Anscombe (1953)g
-
-
Let (t+u, t) be any ua.ccessib1e" point in the diagram,
Leo, one tha.t, can be reached by a possible sample path
from the origin which does not previously meet the boundary
Then i f when sampling ceases the sample path has
reached the point (N,R), an unbiased estimate of pt(l-p)U
is
0
$~N,R~
k N,R
'
where k(N,R) is the number of possible sample paths from
the origin to (N,R), while $(N,R) is the number of
possible sample paths from (t+u, t) to (N,R).
Relating this work to the problem of inverse binomial sampling,
'we have for the fixed boundary x
origin to (N,c) is
=r
The number of paths from the
0
(::i) , and from (1,1)
to (N,c) is
(::~).
Their
ratio is
c-l
N-1
the result of Haldane
,
0
For estimation, the number of paths to points on various boundaries becomes important
0
Plackett (1948) presented a method for
11
determining the number of admissible paths from the origin to a point on
a boundary
0
This boundary is charac terized by the fae ts that no more
than n observations can be taken, and the boundary is to have n+l points
where a path may cross ito
Plackett showed that if N(x,y) is the number
of paths from the origin to the point (x,y) on the boundary, B, then
where x and yare the numbers of successes and failures, and where p
corresponds to events causing the x coo rdinate to increase and q ... l=p.
By equating coefficients of pi, the values of N(x,y) can be obtained for
various boundaries with the given properties
0
He also stated that the
average sample size can be given in terms of the quantities, N(x,y),
which can be done by evaluating
Wolfowitz
Savage
0
2 (xty) N(x,y)
pX
cr.
(1946) generalized the work of Girshick, Mosteller and
He showed that their estimate is the only proper unbiased
estimate for sequential tests defined by doubly simple regions, where
a proper estimate p(c<) of p is one for which 0 ~ p(o<) ~ 1, for all ..(
that are points on the boundary, and where a doubly simple region is
f (x) and x ... f (y), where f (x) is non1
2
l
decreasing for all x~ 0 and f (y) is non=decreasing for all y ~ 0,
2
one between two curves y
lOS
and fl(O), f 2 (0) > 00
Further generalizations have been given by Blackwell
Blackwell and Girshick
(1947), and
(1947), where a lower bound was obtained for the
variance of an unbiased sequential estimate
12
of a parameter 8, where :lC.l is the first observation and W = :lC.l+x2+·' .+xn
is a sufficient statistic for
e.
This was shown to be
The lower bound is attained in the classical case of samples of
constant size N, since
and this is the only case where the lower bound is attained •
.A recent article by Lindley (1957) gave methods of sampling a
binomial population to estimate p with a prescribed accuracy.
The
basis of his approach is the amount of information about p provided
from a sample.
The rule he proposed is to continue sampling until some
measure of this amount of information reaches some prescribed value.
In general, a sampling scheme ad,opted to obtain a prescribed amount of
.
information about a monotone function of p will be different from a
scheme regarding the information for p.
Lindley applied his results to p and to two , functions of p:
I
:=
2- arcsin
fP and 't' = In 1 ; ' These three cases yield approximate
equations for three different stop ru1es.o
IIi a previous article (1956),
Lindley introduced these ideas as a consequence of the concept of information due to Shannon (1948).
In neither article did Lindley make
an attempt to estimate the sample size required for information to
reach the desired leyel.
II.3.
Estimation of \J: in a Normal Popu1ation
s
While work was being done on inverse binomial sampling, Stein
13
(194.5) .c0Ilsidered estimation from the standpoint of double sampling.
By
a two sample test he. solved the problem of determining
confidence
."
..
intervals of prescribed length and confidence coefficient for the mean,
11, of a normal distribltion with unknown variance
(i.
SteinVs method
corsisted of drawing a sample of size no' and then a further sample of
N-n
o
observations.
N is determined by
N
lOl
max
{[S2/z1
+ 1, no]
where s2 denotes the usual estimated variance from the first sample of
no observations and z is defined by
t
iii
t.,(.JZ ,
.e is the desired length of the confidence interval, t.,( is such that
P ( -t.,( ~ x ~ t)
:0
1-.,( ,
where x has StudentVs distribution with n -1 degrees of freedom, and
o
1-.,( is the desired confidence coefficient. [s2/z ] denotes the greatest
integer less than s2/z •
=
After all N observations are taken, the mean Y of these observations
is computed.
Then
=
(Y -
21 t
1
~11 ~ Y + 2 t)
is the required confidence interval, and the confidence coefficient is
~ ~ l-..{. ~ will be close
to 1-..< if no
on the expected sample size.
< ci/z.
Stein obtained bounds
These bounds are a function of
cl
and can
be evaluated from tables of the incomplete Gamma function, but using
2
E(n) ~ ;- is a fair approximation provided
(i > zno •
Chapman (19.50)
extended Stein IS method and tested the ratio of the means of two normal
14
populations.
Wald
(1947) considered this problem of est:i:mating the mean of a
normal population with unknown variance by means of a confidence interval of given length and confidence coefficient, but he did not obtain a
useful solution to the .problem.
Stein and Wald
(1947) obtained confidence intervals of prescribed
length and confidence coefficient for the mean of a normal population
wi th known variance by use of sequential procedures.
They showed that
in order to have the expected number of observations as small as
possible,' a fixed sample size should be used.
n .4.
Asymptotic Theory
tlLarge Sample Theory of Sequential EstimationI' is the title of two
papers by Anscombe
(1949), (1952), who seems to be one of the first to
attempt to apply asymptotic theory to sequential estimation.
In his first paper, Anscombe stated that the general problem of
sequential estimation is formulating a rule of sampling such that an
unknown parameter can be estimated with specified accuracy and with
He considered estimating a single
minimum expected sample size.
unknown parameter 9 using only statistics Z which are the sum of m
m
observations z1' where E(zi)
:=
e
and V(zi)
:=
v(~) = v~e}
v(e).
Then
,
for large n, whether n is fixed or determined by a sequential procedure,
i.e 0' n is the smalles t value of m for which an inequality of the form
zm ~ k(m)
or Z ~ k(m) is satisfied.
m
15'
If the boundary satisfies the conditions
It is approximately linear in the neighborhood of n , where
o
nou is the abscissa of the point where the mean path intersects
the boundary,
(ii)
The possibility that it can be crossed elsewhere than in
the neighborhood of no can be ignored,
(iii)
It crosses the mean path at a non~zero angle,
then Anscombe showed heuristically that the sample size is asymptotically normally distributed with mean n
and variance n vee) [8-k'(n ) f2,
0 0 0
A
and hence that
a
.
is asymptotically distributed with mean
a
and variance
vea}
no'
In the 195'2 paper Anscombe showed that under certain conditions
fixed-sample-size formulae are valid for sequential sampling, provided
the sample size is large
0
Oox (195'2a) gave a general treatment of double sampling using
asymptotic methods
0
He desired to estimate a parameter
accuracy using as few observations as possible
0
a
with assigned
Applying large sample
theory, he obtained an estimate with bias O(n~2), and gave applications
to the normal distribution with variance known and unknown, and to the
binomial distribution
0
He considered the problem of estimating with
given standard error the difference between two means of normal populations.
Cox showed what the expected sample size would be using his
method, gave an inequality for the expected sample size if the procedure
had been done sequentially, and conclu4ed that,
16
except where the preliminary sample size is small,
the best double-sampling procedure has an expected
sample size only slightly greatar than for the
best sequential procedure
0
Tweedie (1952) used an extension of Waldls (1945) identity and
thereby gave the asymptotic mean and variance of the sample size under
the rule that sampling stops when some linear function of the observation ceases to lie between two specified limits.
Cox (1952b) derived the bias of Anscombe IS 1949 estimate to the
term in n-1 when the unknown parameter is the population mean
0
He used
the method of plotting the cumulative sum of the observations against
the number of observations until a boundary is reached.
Cox presented
a new estimate
n
2zi
where to'"
1
n
and b l
n
is the slope of the boundary at the point at
which it is crossed, and he showed that the bias is O(n-2)
0
'Wald (1945), as mentioned previously, derived the asymptotic
distribution of the sample size for his particular cases.
In another paper, Anscombe (1953) made further strides in the
asymptotic theory of sequential estimation and obtained the secondorder asymptotic distribution of the sample size fo,r a certain general
type of sampling procedure
0
Second order asymptotic theory is devel-
oped by adding a correction term to the asymptotic distribution. of the
estimate, so that the error is of lower order when n is large
0
He set
confidence limits on' the mean, ~, of a normal populatic n with unknown
17
variance by first making the transformatiom
,
f xjJ
i
1\ • id+1) [ i Xi +1 -
'
2'
i . 1, 2, •
0 0'
11-1 ·
His first order stopping rule then becomes8 n is the least N for which
where
e is
small, and
the desired length of the confidence interval, supposed
1-~
is the desired confidence coefficient.
It 'is shown that the second-order asymptotic procedure for this
problem modifies 'the rule tog n is the least N greater than 3 for which
N-1
e2
z
2
~
u. ~-2 N(N - 2.676 - -2,)
J.
4'Z ..(
~
•
J.=
The right hand side is not unique, in that the multiplier of
e2
~
may
4z..(
be modified by a.dding a function of N which is O( (N) for N large.
For
this rule the expected sample size is approximately
The first term on the right hand side is the number of observations
2
'
that would have been needed if (] were known, and the rest is the
,
.
average cost in observations of not knowing (J2.
knscombe found the asymptotic distribution of n by considering the
sum of a. sequence of independent ·random variables
also applied to
betwe~
So
0
These results are
simple birth-death process, so that the difference
the birth rate and the death rate is estimated with given
18
standard error.
Anscombe investigated numerically the accuracy of his
approximations.
A follow-up
of some of the results of Anscombe is given in a recent
article by Ray (1957), who modified the procedures slightly and attempted
some results on small sample theory.
t instead of the normal deviate.
one
change is the use of Studen t' s
His sampling procedure is first to take
three observations and then sample in pairs.
His results are applied
to the estimation of a mean with given standard error and estimation of
the difference' between two means with given standard error.
Ray gave
the dis·tribtltion of the sample size as the probability of stopping at
the (2rn+1Jst observation and thus the mean and variance of the sample
size in-terms of an infinite series, none of which seems to be of much
use in practice.
II. 2• A Recent Application
. The applications of sequential methods may profitably be applied
to another field of statistical theory, that of discrimination, although
to this writer's know1e4ge nothing yet has been published in this field.
~
In an unpublished paper, Williams (1957) did consider the discrimination of two groups where he assumed that members are drawn one by one
at random from the popu1ati,on and then classified into one of two
groups.
Sampling is continued until a sample with the required prop-
erties is obtained.
Two such properties might be the following (both
of which are considered in the text of this thesis):
1.
Sample until at least n members from each group are obtained,
in which case
19
E(n) ~
n
.
.min(p,q)'
where p and q are the proportions of groups I and II in the
populations.
2.
Sample until the difference of the means for the two groups is
d~termine<i with
a given accuracy.
ference is proportional to
1- + 1-, sampling is continued until
. nl _ n 2
the condition
is satisfied.
Since the variance of the dif-
Chapter III
SECJTENTIAL-STOP RULES
In this chapter and the next various sequential stop rules as
~efined
earlier in Chapter II are considered, and the mean and variance
of the sample size under each rule is obtained.
The most general rule
is:
Given.s populations mixed in unknown proportions and
a function of s variables f(~, ••• , x s }' in the nonnegative part of the s-space, sample observations one
at a time, evaluate the function after each observation is drawn, and cease sampling when the func tion
first changes sign.
This
rule is equivalent to plotting the function and the sample
.
path and. continuing sampling until this path crosses the curve of the
func tion.
~he
sample path is a sequence of points with integral co-
ordinates, the coordinate x. at any time being the numbers of observa.-
J
tions f~om population j thus far in the sample.
The main type of function considered is one whose derivatives with
respect to:x:t.' •• 0,
X
s
are all of the same sign, so that once the boun-
dary is crossed, it cannot be crossed again.
TIlo1.
Rectangular Stop Rule
The first situation to be considered is for the simple binomial
case; 8=2, and for -a func tion which will be called the rec tangula.r
stop rule.
~re
sampling continues until at least
~
observations from
the first and a.t least m2 from the second population are obtained.
A
21
graphic representation of this situation appears in Figure 1.
Figure 1
Rectangular Stop Rule
x2
/
/
//
" - ; - - - - - - - - - - x..t
(~,O)
With the exception of the point (~,m2)'
points on the line ~=Inl can be approached
only from the left, while points on the
line
~=m2
can be approached only from below.
Let the proportion of population I in the mixture be p and that of
population II be q=l-p.
The probability of stopping at the nth trial is the probability
that at the (n-l) th trial we have u ~ ~ successes (Le. observations
from population I) and v
at the nth trial .QI. u
=:
:=
m -l fa.ilures (population II) and a failure
2
~ -1
successes and v
~
m2 failures and a
success on the nth trial.
If there are"m 2-1 failures, then there are n-m successes in n-l
2
trials.
If ml-l successes then n-ml failures.
Hence the probability of stopping in n trials is
22
since the probability of
1. m2=1 fai,lur.es in n-l trials is
Cn-f)!
. n-m 2 m2-l
. (n-m2~! m-l)! p , q
.,
2
2 • ~-1 successes in n-l trials is'
3 • failure at nth trial is q,
4. success at nth trial is p.
NoW,
since
~+m2
is the minimum possible sample size.
Thus
If the sUIllD'lation were from zero to infinity, 'this expectation
would be
~
+~ ,
q
P.
but with the
term must be subtracted.
~ange' m.. +m2
.L
to infinity.a correc tion
This' is the pair. of sums up to ~+m2.
Hence
23
+" ~.... ~
p
~p
~
n~
+m
-1
In the first su.mmation let n ... s+m
Then the sums are for s ... 0, 1, ••• ,
n!
. n-zrt:t
(n~)!~! q
2
2
•
and in the second n ...
~ -1,
and s
= 0,
1, ••• ,
respectively.
The first two terms of E(n) become
To evaluate this quantity it is convenient to consider
M
s+~.
-l
2
24
Summing this last expression for i
out except the last for i ...
l!lIl
0,1, ••• , ntr-l, all terms cancel
~-1-
Therefore
and
where B(u, v} refers to the Beta distribution with parameters u and v.
Hence the first two terms reduce to
Similar results hold for the remainder of the expression, so that
!l!l
B
~
m
J
[ l. (puI!1.-1 (l-u) In.2 + 1 }l u~ (l-u) m.-l
2 du
du
q <5
~,m2
p p
In a similar marmer, an expression for E(n 2) may be obtained and
hence V(n).
The first of these 'tnrns out to be
25
which reduces to
The expected value and. variance of n can be obtained numerically,
bu~ ~o~
large values of
involved
IIJ.
2 the calculus becomes tedious and
Various special cases are next considered and the results
0
and
II1
summ.a.rizedo
bo
For
~Il!l
Co
For
~
2,
... 3,
For the same oases, i f p
:=
q
=
t
26
V(n~ ... 2m2 [ 1 +
E(n)
lllI
m
<t> mJ
2 + 6(t> .2 -
1'm
2m2 + (2) 2 [4+m2]
4(t)
2m
2
0
0
yen) .. 2m + (t)m2 [20-2m22-3m2] - (t)2m2
2
(4+~)2
0
•
t)
In all cases, when pDq, as m increases,
2
E(n}
----> 2m2+
yen) decreases until m2
lllI
In..,
.L
for which yen)
m2 •
~
• V(:Il)
m2 •
1
~-
'Whim m .. 0, we have the inverse binomial s~ling situation men2
tioned in Chapter II, and the results obtained here agree with those
obtained by applying that method directly.
Another special case considered is for m1
= m2
• m and p • q •
t
0
o.
•
27
The sample size is then given by n = m+t, t
~
(Wt-IP
2
~
m.
(~-1)1
Then
2
~l tIm-I! tJl'"t + m! m-l! 22m =
. where the first term is for t
> m and
1
the second for t = m.
The factor
of 2 is introduced since the sample path can cross the boundary from
below or from the left.
Replacing m by m+l in (1),
2
22m+2
•
= 1
,
hence
(2m+l) !
(m-l)! (m+l)!
2
..
22m+l
2m
,
so that
E(n)
or
E(n) = 2m +
(2m)!
mHin-l}t
1
22m-l
'V
2m +
4~ l!!..211
'
for by Stirlingls approximation we can show that
This result can be Verified in the special cases given previously.
•
28
ill.2.
Linear Stop Rule
An example of a negatively sloping function is, for s=2,
k:x:t+~ .. m,
k
> 0,
as in Figure 2.
Figure 2
Linear Stop Rule: k:x:t+x2
=m
~
(O,m)
.(m/k,O)
'.XJ.
Lattioe points on the -boundary can be
approached either from the left or from
below, while points which are not lattice
points must be approached from the left.
Let p and q be defined as in the last section •
Then. the proba-
bi1i ty of the sample path reaching a particular lattice point, (ll, v)
say, in the
~, x
2
-plane is
(u+v) ~ u v
~ u p
q
u.v.
-
For the present, let k be an integer,then for c!ach integral value of
:IJ.,
there are k integral values of x 2 where the line may be crossed by
a sample path.
Let PO(r) be the number of paths which e~d by crossing
29
the line at JI]. = r,
PI (r) be the number of paths which end by crossing the line at
:x:L=r-
1
k,
Pk - l (r) be the number of paths which end by crossing the line at
k-l
'
x..
=r-. .L
k" " r - l + ; a
k •
Then
is the total number of paths crossing the line for r-l
<x
1
~
r.
The
probability of reaching the point (r, m-kr) is then
Similarly, the probability of crossing the line at the
0
ther k-l
values of x are
2
= (~-(k-l)r)
r-l
,
]\-1 ( r ) p r qm-kr+k-l
,
P
k-l
(r)'" (m-<k-l) r+k-2)
r-l
since the number of paths crossing the line at a point which is not
30
a lattice point is equal to the number of paths reaching the lattice
point just to the left of the line, however, one more step in the
~
direction is needed to cross the line.
Hence the probability that the sample path crosses the line in the
< Xl ~. r
unit interval r-1
is
= Lk ,
say.
For k=l,
L
1
III
r!
<:!r)! prqm-r,
the ordinary binomial result.
Similarly
r m-2r
L2 = P q
In order to obtain the expected value and variance of the sample
size we replace L. by a continuous approximation using Stir1ing i s
J.
formula and the general method of approximation to a binomial.
To find the appropriate transformation, we consider the Itmodal lt
line, Le. the line around which the sample paths will vary.
'modal line is one from the origin to a point on the boundary.
point is
The
This
31
or
This is a form of an average of r.
In order to transform to a
standardized variable z, we use the transformation
r
I:
mp!'zm
qfokp
,
that is, we expect the variation about this line to be o(
assumption which is later verified
fi),
an
0
since r takes unit steps, and of course
~ r..J
-
1
~ 2npq m
e
2
-z 12pq
.1
-z
dr =.~. e
v 2npq
the usual normal appro:x:i.mation to the binomial.
2
12pa •
'UZ ,
Hence z is approximately
normally distributed with zero mean and variance pq.
Now,
Taking logarithms, applying StirlingVs formula, and collecting terms,
-
log L 2 rv
(m-2r ),log [ (m-r)gl
m-2r
J
.
...+ r
log [(m-r)p]
r
+ log [21fI'1;:2r
Using the transformation
1
>J. 2 + log (1 ... m-~;+l) •
32
r
==
mp+zu
. 2
qt'p
we have
log
L
mg-2z rm
(mg-zg .fm,
2,v qt2p
log mq-2z f1il'J
The firs t two terms of 10 g L
2
can be approxima. ted . by
2
+ mptzu
log
[1
(
.....
2p)(..L.
,",..L
qt2p .
':\"
{iiiP.r; 2
" mp
)J.
~ -mg-2z
rm; r(.q+.2) I ...lL + ..i.) _ (qt2 ).2 (..L + ~) 2J
q+2p L:. p. \ tmq .fijiq2
p.
'-/1fiq rmq2
(mptz fi) [(qt2P)
q+2p
(..L
- -l)
(Mp
2
+ (q+2p) 2 (..L. -
..:1.) J.
fIiip .r.:
vmp 2
.tw
vmp
so that
1
g+2p ) 2
log L ~ log ( 2nmpq
2
2
)
- z 2(0+2p)
+ log (
qt2p
pq
or
(q+2p) •
33
To obtain the differential element we note that
dr
~
q+2p
Ill!
•
Hence
!
=
( ~.)2
2npq
i 2 (g+2p)
2pq
e
dz.
That is, z is normally distribu.ted with zero mean and variance
.
and
E(r) • -!!m...
VCr) ""
q,+2p ,
!92'
~.p
,
,
mV{z)
(q+2p) 2
so that r is normally distribu.ted with mean and variance given above.
Now n
Ill!
m-r, hence
tV
n
(
)
N.l!!mpg
q+2p , (q+2p) 3'
•
This method is readily extendable to the general k, where
n '"' m-(k-l) r and where \: can be written as
2
1 + . rg + r(m-{k=1}r+llg
~~.-:;w.;~ [
m-kr+l
Cm-kr+l) \m=kr+2)
+ r[m-(k-l)r+l )[m-(k-l)r+2 }s!3
(m-kr+l) (m-1a*2) (m-kr+3J.
34
An argument similar to the one used for L is used on the factors
2
outside of the braces, with m"';kr in place of m-2r and m-(k-l)r in place
of m=r.
Using StirlingUs formula and expanding the logs this first part
becomes
1 - z2(g+kp)
(r:~) 2
e
2pq
mg-kz iii
z {i
Replacing r, m=kr and m- (k-l) r by mp+
o/kp
, qtkp
m-(k-l)z
o/kp
so tba t in the limit
and
rm
The general term of the series in braces is then approximately
... p ,
I,
so'~'that
the factor in braces is
l+p+.oo+P ... l+(k-l)p ... qtkp.
Again
vn
dr "'" dz
q+kp
and
3S
and the distribution of n j;s then asymptotically normal with mean :kp_
- ' 2
and variance
ma Ck- l )
':l'
-
0
(q+kp)3
The above analysis has been done With the assumption that k is an
integer; however, it is.clear that the result holds for any k
proof of this statement follows,
* when
> 00
A
in fact the line in question is
of the form
tet the probability of obtaining an observation from population I be
p and from population II be q, and let n
l
and n
2
be the number of
observations from the two populations in a final sample of size
n ... n +n o
l 2
From elementary theory we know that n
l
and n
2
have a bivaria.te
distribution with means np and nq, variance npq and covariance -npqo
Let
Y1 and y2 have zero means and variances pq.
The line can now be written as
or since Yi+ Y2
*
l!lil
0, this is
I am indebted to Dr
this proofo
0
•
Go S. Watson for some of the features of
\
36
Hence,
where the final term on the right is to allow for discontinuities at
the boundary.
Taking e:xpec tations ,
and
,
where n is replaced by its expected value.
obtained previously when f
l
0=
k and f
2
:=
These are the results
1-
This proof is of a heuristic nature, in that it treats n as if
it were fixed and not a random variable.
proof follows.
A mathematically rigorous
This proof makes use of an identity due toWald (1945).
For the present purposes, it can be written as
{
l!(~,t2) ]
-n •
~nl+t2"2}. 1
,
where ~(\, t ) is the moment generating function of the binomial dis2
tri bl tion, that is
The boundary restriction is
37
and by taking t, proportiona.l tof1.' , i.e.
1.
the identity becomes
E
S ~e
r. tfl+qe
L
tf'J
-n }
E
[
e -tc
,
2
= e -tc
,
or equivalently
. e·~KJ
"Ill
where K is the cumulant function, i.e. log ~
= K.
By denoting the moment generating function of n by 'fl(t), the
a.bove equation becomes
\fI(-K) .. e
-tc
,
•
and we must solve for t in terms of K.
...
Also...
2
sK
Then, to a first approximation
and
= l+K+ 1L
+
2
...
38
Now,
or
K2pq(f -f ) 2
2 l
2(f P+f q) 3
2
l
Substituting this value of t on the right side of the identity and
replacing -K by K
,
as the moment generating function of the sample size, and hence the
desired results are obtained.
If the approx::i..mation is carried further, it can be shoWl'l that the
skewness coefficient goes to zero, for then
e
K
2
t
=.1+t(fl p+f2q)+ '2
+
t-
(f 3P+f 3q) +
1
2
= l+K+ -K2
2
K3
+ -
6 +
and the third order approx::i..mation becomes
(f
2
l
0.0
p+f
2
2
q)
39
e.
so that the third cumula.:nt will also be of order c.
~ and will
The skewness is
then be proportional to c;/2 = c-1/2 and thus will tend to
zero as c increases.
nI.J.
Number of Paths to a Line'
There exists some interesting:m.athematics in connection with
~ind~ng th~
number of pa ths from the origin to the line considered in
the last section
This becomes useful when the probability of any path
0
finally crossing the line is the same for all paths.
Let P(k,m) be the tot;al number of paths to the line kXJ.+x2:~ m.
Wh~n
m
~-
P(l,m)
~
k=l, we again have the usual binomial situation, and
llO
2 •
'!'he ratio between the successive number of paths as m is
.
~
increased in steps of one is 2.
For k=2, by direct enumeration we have
P(2,O) = 1
P(2,1) = 2
.
P(2,2)
l!!l
3
P(2,3)
l!!l
5
_.
~
P(2,4) "'" B
P(2,5)
=13
40
P(2,6) == 21, etc.
These 'numbers form the ordinary Fibonacci sequence, for which the recurrence relation is
P(2,m) == P(2,m-l)+P(2,m-2), m
> 1.
The ratio of the succeeding terms approaches a constant
~
~
::::::; 1.618.
is such that
for then the recurrence relation for the sequence is
which reduces to the above equation.
Note that the roots of the equa-
tion are t.. and _~ -1.
Hence we obtain
Hardy and Wright (1938), i.e. the number of paths is a linear combination of a power of the roots of the basic equation.
For large m,
(_t..)-(m+l) approaches zero, so that t.., the root greater than unity,
becomes the only one of importance.
Considering the case for k=3, (Figure 3), let P (3,m) be the
o
number of paths from the origin which cross the boundary at a lattice
point, P (3,m) be the number of paths crossing the boundary one unit
l
from a lattice point in the ~ direction, and P2(3,m), two units in
the x 2 direction.
Then P (3,m)
o
0:
P (3,m-l) + P (3,m-l)
o
l
41
Pl{3,m}
P2(3,m~1)
P (3,m}
2
P (3,m-l)
O
2
.
P (3,m) ... fP i (3,m)
=PO(3,m-l)
~
+ P(3,m-l)
~
... P (3,m-2) + P (3,m-2) + P(3,m~1)
O
l
lOll Po (3,m-3)+ P
(3,m-3)
+ P2 (3,m-3) + P(3,m-l)
l
:= P(3,m-l)
+ P(3,m-3) •
Figure 3
Proof of the Recurrence Relation for k=3
x2
I
I
(0, m)
(O;m-1) ~
1\\
\\
\ ~~
~
~\~
!!\\ e:
&'\ !\
~
1\\
-
\'
\\
For r-l < XJ. ~ r, "ijle line 3XJ.+x2 I: m can be
crossed at points ~, ll, and.Q.. Let the number
of paths to these points be Po(3,m), P1 (3,m)
and P2(3 ,m) respectively. The number of paths
to both points 11 and .Q. is the same as to the
lattice point just to their left. The result
then follows.
•
Similarly it may be shown that for general k
P(k,m)
l!!l
P(k,m-l) + P(k,m-k) •
Assuming then that P(k,m) can be written as a linear combination of
the roth power of k values \ , ••• , \:,
i •e •
k
m
P(k,m) ....~aiti '
i~
we then find that \ , ••• , \: are the k roots of
tk_t\c-l
:0:
1.
This is easily shown, for we have
~' a.1 to1 m =
~ a oto m-l + ~ a. t m-k
111
i
,
or
so that the to are roots of an equation of the form given above.
1
Without loss of generality, we may assume that t.
1
and we know that ~ t
i
!!il
t. for i
J
will be the dominating term
is the only root greater than unity.
This can be shown by
considering the variation of the function tk_tk-l_l with t.
monotonic increasing function for t
values of t.
~
1 and is positive for large
> 1.
Thus P(k,m) is approximately a geometric
progression for increasing integral values of m,
P(k,m)
~
This is a
For "l:Fl, it is negative, showing that one and only one
root exists for t
where
<j
1.
In the expression for P(k,m), ~ \
since \
~
is as yet undetermined.
rV
al\m
,
43
To obtain al' consider the system of equations
TA=,2
where
1
T
=
~
~2
~
Ja.
1
t
2
0
•
00
1
•
\:
t 2
2
k-l
t2
k-l
AU .,. (~
a
:e,t
2
0=
o
(1
2
\:2
k-l
\:
•• 0
~)
••• k )
is the vector whose elements are the number of paths for
m'" 0, 1, 2, •• 0' k-l respectively.
To find A, we must invert T; however, since' only
~
is needed, we
need find the cofactors of the first column only, i.e., the elements of
1
1·
the firs t row of T-.
Call these c J.
Then
T is called the matrix of Vandermonde, and its determinantal value
is
k(k-l)
7T (t.-t.)
i(j
\
Ie
I
t
I
I.
J
~.
a
(-1)
2
7T (t.-t j ).
i<j
~
..
44
Using this value and a theorem of elementary symmetric functions,*
we find
C
lp
:=
~p
1
7T (t -t.)
j>l
1
,
p
<k
J
and
Applying the theorem mentioned above, we find
7T
\k + k-l
(\-t.)
j>l
lIB
J.
~t-1
Therefore
* thm
If
~, x
2
' •• c, .x are roots of the equation
n
.&
k k-l
Hence for the equation t -t -1 -= 0,
so that ~ t
i
"" 1,
the roots are zero.
1Tti
lIB
(_l)k-l and all other sums of products of
45
t k-2(t k + k-l)
1
1
so that in order to find al' only one root, namely t , need be obtained.
l
a can be further simplified by multiplying numerator and denomil
nator by (1 -
1 2
r)
Many of the terms in the numerator cancel out and
0
1
we obtain
t k
~ = k(\~l)+l
'.An asymptotic expression for t
l
•
can be fotmd, which is valid for
~
large k, so that the equation
need not be solved explicitly.
If a closer approx:i.ma.tion is desired,
successive approximations can be obtained satisfying the equation to
the required degree of accuracy.
Use of high speed computing is well
suited to this need.
Now if \
is the root greater than unity and if we let t
l
=1
+
then the equation becomes
taking logari thIns
k-l
Therefore
Substituting for
r:l ,..J
t'
~
lIll
_
lQg S
log(l+~)
/V
_
log
~
@
-log f3
k-l
on the right, and ignoring log log terms, we have
~,
46
On keeping the log log terms in the approximation, we find similarly
t:L rV
k . l/k
(log k~
,
hence for large k, we will have
It is interesting to note that the bounds on t
for (io g
k) -11k
are not very wide,
l
approaches unity quite rapidly.
~
The following table gives the values of
and
4,
~
and
~
for k == 2, 3
the actual number of paths, P(k,m) and the estimated number of
"'-
paths, P(k,m}, using the values of t l and aI' for various values of
m.
TIl .40
Hyperbolic. Stop Rule
It is often desirable to sample until a specified accuracy is
obtained.
This accuracy might be given in terms of the standard
error of the statistic used; for example, we might want the standard
error of the difference between two means to be a given small number,
sa:v:rt.
Assuming independence and a common variance,
'P'o
expression is
or
ci,
this
e
e
e
kl!O!.~
k
1.61803
t1
III
~
... 1.17082
m
P(2,m)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
1
·2
3
5
8
13
21
34
55
89
144
2.33
.377
610
987
1597·
2584
4181
6765
1\
~
liII
'3
k-ij
... 1.46557
~
8.1 • 1.31342
... 1.38028
a1 - 1'.43970
,1\
;'\
P(2,m)
P(3,m)
P(3,m)
P(4,m)
P(4,m)
1.17
1.89
3.07
4.96
8.02
12.98
21.01
33.99
55.00
89.00
l44.0
233.0
377.0
610.0
987.0
1597.
2584.
4181.
6765.
1
2
'3
4
6
9
13
19
28
1.31
1.92
2.82
4.13
6.06
8.88
13.01
19.07
27.95
40.97
60.04
88.00
129.0
189.0
. 277.0
406.0
595.0
872.0
1278.
1
2
3
4
5
7
10
14
19
26
36
50
69
95
131
181
250
345
476
1.44
1.99
2.74
3.79
5.23
7.21
9.96
13.74
18.97
26.18
36.14
49.88
68.85
95.03
131.16
181.0
249.9
345.0
476.1
hI
60
88
129
189
. 277
406
595
872
1278
-F
-.:J
,-
48
wheren l is the number of o·bserva.tions from population I and n
population
2
from
:a.
The stop rule then becomesg selec·t observations until the above
inequality is satisfied.
A graph of the boundary appears in Figure
Figure
-
-.
4.
4
Hyperbolic Stop Rule
x2
(0 ,c) . -_ _- L
--=~~_
&..-_------(c,O)
~
To be consistent with the previous sections, replace n. by x.,
~
~
since
the n, can assume only integral values; then (2) is a rec- - -
~
taIlgular hyperbola in the
~,x2""Plane.
We are interested only in the
positive branch of the hyperbola., which has asympotes at:x.t .. c and
x
2
:e
c, since (2) is equivalent to
... c 2
The slope of this curve is
c
2
, 2
(~-c)
=0
-k
,
using the notation of III.2 for the slope.
F'or a gi van va1ue
0 f ~,
cr
r say, x 2 ... H
'
an(i the slope
a._
t the
point is
49
2
=
--:c~_
(r_c)2
The equation of a straight line passing through
0
the point (r,cL) with this slope is
r=c
We may now consider this as a special case of the straight line of
Section II102 for in the neighborhood of
reg~ded
as linear
departures from
XI
==
:~
,
the curve may be
As in the case of the straight line, we expect the
0
XJ. ""
:~ to be 0 ( {M)
0
For the straight line of slope -k we had
co
z2(q+kp)
2pq
dz
In· iths case, where m ~
2
2 and k
(r-c)
cr
=
c
2
(r-c)
that
The distribution of z becomes
=
~dz ~
_.
~
e
...L
22
p
dz,
2np
.-
and the distribution of the sample size n, where
2
cr
r
n""r+-lOllr-c
r-c
'
is
In the case when p
m
q, the modal line intersects the hyperbola
2 '
50
a t the point where the slope is 1, and the variance vanishes.
The
fact is that now the variance does not go to infinity with c, but tends
to a finite limit.
In practice these results may not be useful because the variance is
usually unknown.
For this case the variance may be sequentially esti-
mated, as each observation is drawn, by the method siJuilar to the' one
due to Anscombe referred to in Chapter II o
Since common variance is
assumed, the estimate can be based on the pooled sums of squares from
the two populations.
After a sample is taken, make the transformation
i f the observation is from population I, where w. is the value of the
J
random variable on the jth observation from this population, or
Vi ~ id+1) {i "1+1 i f the
0
~"d 2
bservation is from population II and wj is the value of the
-
-
random variable on the jth observation from population II.
Then
where there are nl and n observations taken from populations I and II
2
respectively.
In this case c
1\2
l!!l
co(J , which is a function ofn, so that the curve
changes as the sample size increases.
51
Rea.rran ging the inequa.1i ty, we see that the criterion for stopping
d
becomes~
continue sampling until the following inequality is satisfied
A short hypothetical numerical example is given for 11.1ustrative
purposes.
Suppose the variance of the mea.n difference is desired to be less
than or equal to 1/2, Le., c o = 2, and suppose the observations were
taken in the following order, with the values of the random variable
given in parenthesesg
Then the analysis can be set up as follows:
w!~
Vi
~Vi
~Ui+2Vi
W(nl'n 2)
.50
2
2.00
2.00
017
067
4
0.00
2.00
2.67
4.75
4.75
4.80
1.80(3,2)
2.67(4,2)
4.29(4,3)
5.62(5,3)
3
2008
2 75
4
5
005
2080
5
4
~Ui
i
wi
U
i
1
3
050
2
4
3
0
Hence sampling stops when n • 8, n
A2
(J
·e
.
...
4·6 80
5'
I..
!Ol
0
•
s..
"i:'
Jlli~larly p ... t
l
8!Il
5, n 2
'"
and q =
III
8'~ .
30
(J2
is estimated by
...
Chapter IV
EXTENSION OF RESULTS
IV 010
General Function in Two Dimensions
The first generalization that will be considered is the na.tural
~xtension
to a general negatively sloping function in two dimensions
defining the stop rule 0 Let this function be
an.ci
let Pl and P2 be the proportions of populations I and II in the
mixture.
a f(:x:t,x2)
fl
go
@
~
\ a f(3.'X2)
f2 =
@~
where both f
l
and f
2
are of the same sign.
I
:x:t
= API
x 2 ... AP2
It is reasonable to expect
that Xi will be proportional to Pi' i.e. equal to APi' so that
f(A. Pl,A.P 2)
=0
will give the value of A. at the curve, and hence define where the
modal line crosses the curve.
Assuming that this curve will almost certainly be crossed in the
ne:ighborhood of this intersection, the results for the straight line
may be used in the appro:x::i.mation •
As before, Ie t
53
e-
n 2- nP 2
Y2""
rn
Then the transformation of Watson is equivalent to
where
Again the final term on the right is to allow for discontinuity
effec ts at the boundary.
Now
so
n "" A+O({1)
,
rn vt+ocJ5
,
m
.
and we may write
,
then
E(n) "" 1.+0(1)
V(nl "" ~~2V(Yl)+0(1)
,
"" A.PlP2~ 2+0(1)
To the desired degree of approximation, the bounded terms may be
ignored
0
Again this argument is heuristic and the mathematically
--
rigorous proof will be given for the more general function in the
next section.
An application to the hyperbolic stop rule will indicate how the
resul t holds for non-linear stop rules.
54
The basic equation is
1-+
1-.1
.
x2
~
c
0
Replacing x. by A.p., we have
1.
1.
Hence
Now x.
dPV
'1.p.
.1.1.
so that
f i ;-J -
1
2 2
'
A. p.
1.
..
and
A.(f2~fl)2PlP2
2!lll
,
(fl PI+f 2P2)
A.(P2-Pl)2 PlP2
2 2
PI P2
c(P2-PI)
~ . 2 2
PI P2
2
= yen) ,
which are the values found in Chapter III for this specia.l case.
IV 02 0 Functions in s Dimensions
For the general function in more than two dimensions, a generalization of the heuristic method and results of the preceding. section is
first used.
Let the func tion be represen ted by
•• 0,
x )
s
s
and let the proportion of popula.tion i in the mixture be Pi· fPi!lll L
55
Define s quantities y.]. by
n.-np.
].
].
Yi
rn
==
where CIj. == I-Pi' and s partial derivatives by
@f(:x:t, "', x )
f.
].
which all have the same sign
s
= ---=-----C> xi
Xj == 'A.Pj
j == 1, 2, ..• , s
0
Then
will define the modal line to the surface and give the value of 'A. at
the surface.
The s dimensional generalization of the transformation then becomes
~
_ .
:2 f ]..y.]. + 0 (1)
2'fi Pi
E(n)
==
V(n)
==
'A.+O (1)
'A.
I"':' f. P .)
~
].
2 V(2 f .Yi)+O(l)
].
I
].
Now the problem reduces to one of find in g the variance of .L.
~ f ].. y ]..•
V~f.y.)
~ ].].
==
2
""f.
p.q. - .L.].
~f.f.p.p.
.L.].].].
J ]. J
56
The generalization of the valid proof of Chapter III considers
where s/J is the moment generating function of the multinomial
distribution.
Setting t.~. equal to tf.~ and using the restriction ~
~ Ln.
= c,
~ ~
the identity becomes
e
-tc
as before, where K ... log ,;.
Soiving for t in terms of K and taking the terms in the expansion
.
2
up to the term in K as was done in Chapter III, the results of the
heuristic proof then follow.
In attempting to apply these results to the hyperbolic stop rule
when more than two populations are concerned, we consider the between
population component of variance, which is the variance among population means and is defined by
2
O'b
-
13=1
where we have s populations and n
i th population.
,
observations are taken from the
i
For sg)2, the term in parentheses is
57
and i f set equal to a constant gives the hyperbolic stop rule.
Hence
making the standard error of the difference between two means a given
popul~tion
small n1.Ul)-ber is equivalent to making the
ance
Q"
component of vari-
constant.
Then for the
13
'populations, the direct generalization
2ninj
C
=0
2;ni
or equivalen tJ.y
defining a rectangular hyperboloid in
tionship involved in the stop rule.
the sample size n ...
:2 ni
dimensions, becomes the rela-
13
To find the mean and variance of
to attain the boundary, we use the results
of the general case.
The mean A. is found by setting
~A.PiA.Pj
,
~A.Pi
>=
c
go
c
,
then
A. ~p.P.
~ J. J
so that
A.
:= _
....c"'-_
2 Pi P j
To derive the var;i.ance, we replace n
i
by A.P , then
i
58
~Pifi
Hence
becomes
2PiPj and f:i..-fj = PfP i '
Chapter V
This dissertation was concerned with finding the asymptotic distribution of the sample size required to reach a certain fixed boundary
when the sampling is of a sequential nature.
For the most general case
consider s populations mixed together in the proportions PI' P2' • 0., ps.
Observations are drawn one by one and then can be correctly classified
by populations.
Let n, be the number of observations taken from the
J.
1 -t'J.on; then
J..th popuua
1'1 =:
:2 ni
is the total sample size
0
A sample path is defined by a sequence of points in s dimensions,
each point having integral coordinates, the coordinate Xi at any given
time being the number of observations n i thus far in the sample.
Sampling ceases when the sample path crosses a fixed boundary.
If this boundary is of the form
(A)
where f has derivatives with respect to
~,
x 2'
000,
X and has negas
tive slope
everywhere, or equivalently if all the derivatives are of
the same sign, then it is shown that the sample size,n is asymptotically
~orrnally
distributed with mean 'X. and variance
,
where
X ' == 'X.p,
J
J
j == 1, 2, ••. , s
60
and '1. is defined by
Special cases of this general result are given for s=2, where the
-
boundaries are a rectangular hyperbola
I - +I - = I
-
:lCJ.
~
c
and a negatively sloping straight line
Results for a rectangular hyperboloid in s dimensions are also given.
One case where the boundary is not of the form (A) is also considered •
This boundary is :lCJ. =
111.
and x2 = m2 and the mean and vari-
ance of the sample size are given in integral form.
If the boundary is a straight line of the form
kXI + x 2
= m,
k a positive integer,
and if the probability of any path finally crossing the line is the
same for all paths, then
where
P(k,m,r) is the number of paths crossing the line for (r-l
<~
and P(k,m} is the total number of paths to the line.
An
approximation to P(k,m) is considered and it is shown for
-
increasing integral values of m, that the sequence so formed is
approximately a geometric progression.
P(k,m}
I'V
That is
atm ,
and the values of a and t can readily be obtained.
~ r)
61
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