Download Achievable region of reliabilities in multiple hypotheses two

International Journal of Applied Mathematical Research, 3 (3) (2014) 407-415
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°Science
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doi: 10.14419/ijamr.v3i4.3060
Research Paper
Achievable region of reliabilities in multiple hypotheses
two-stage testing for the pair of families of distributions
Farshin Hormozi-Nejad
Department of Mathematics and Statistics, Ahvaz Branch, Islamic Azad University, Ahvaz, Iran
Email: [email protected]
c
Copyright °2014
Farshin Hormozi-Nejad. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Abstract
The achievable region of reliabilities in the model with several possible hypothetical probability distributions partitioned into the pair of families is considered. The achievable region for many hypotheses testing was examined
by Tuncel. Decisions concerning realized probability distribution of the object must be made on the base of the
samples which are received in each stage of the two-stage test. It is proved that the defined region for the vectors
of reliabilities in the two-stage test characterizes the set of all achievable vectors and advantages of the two-stage
testing are revealed.
Keywords: LAO test, method of types, multiple hypotheses testing, reliability, two-stage test.
1.
Introduction
In present paper, the approach of Tuncel [11] for the achievable region of reliabilities of testing of the model that
follows a probability distribution (PD) from one of a pair of families is developed. The list of S hypothetical PDs
is given. The statistical problem is to detect actual PD from this list and to find the achievable region using a
sample of N experiments outcomes. The overwhelming majority of published works are dedicated to the case of
two hypotheses [10]. The Logarithmically asymptotically optimal (LAO) testing of two hypotheses is the procedure
ensuring the best exponential decrease with growing N of the error probability of one hypothesis subject to the
given exponent of the error probability of the other hypotheses. Hoeffding [9] and later Tusnady [12] and others
investigated LAO tests in the case of two hypotheses. The LAO testing for multiple hypotheses first was considered
by Haroutunian [6]. In [1, 5, 6] some results on multiple hypotheses testing and identification are presented. The
two-stage tests have become popular in applications, especially in the field of clinical trials and genomics [5].
Ahlswede and Haroutunian [1] proposed the problem of statistical hypotheses optimal testing and identification
for many objects. Haroutunian et al. [7] investigated reliability criteria in information theory and in statistical
hypothesis testing. Yessayan et al. [13] solved achievable region of reliabilities in testing of many hypotheses for
two independent objects. The model of the two-stage LAO testing of multiple hypotheses for the pair of families of
distributions is investigated in [8].
The map of this correspondence is as follows. In Section 2, briefly the problem statement and some needed
definitions are declared. In Section 3, achievable region in the one-stage test considered by E. Tuncel [11] is
introduced. In Section 4, the analysis of achievable region at the first stage of two-stage test is investigated. In
Section 5, achievable region at the second stage of two-stage test is studied. In Section 6, achievable region at the
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International Journal of Applied Mathematical Research
two-stage test composed of the first and the second stages of test is declared. Finally in last Section, a conclusion
for achievable region at two-stage test in the pair of families of PDs is disclosed.
2.
Problem statement and definitions
Let X be a random variable (RV) taking values in the finite set X and P(X ) be the space of all possible PDs on
X . Suppose S known hypothetical PDs from P(X ) are given and they are partitioned into two disjoint families of
PDs. The first family includes R PDs Ps = {Ps (x), x ∈ X }, s = 1, R, and the second family includes S − R PDs
Ps = {Ps (x), x ∈ X }, s = R + 1, S such that the considered object characterized by RV X follows to one of this S
PDs.
Let x = (x1 , x2 , . . . , xN ) be a vector of results of N independent observations of the RV X and N = N1 + N2 ,
such that:
N1 = dψN e, N2 = [(1 − ψ)N ], 0 < ψ < 1,
x1 = (x1 , x2 , . . . , xN1 ), x1 ∈ X N1 ,
x2 = (xN1 +1 , xN1 +2 , . . . , xN ), x2 ∈ X N2 ,
x = (x1 , x2 ), x ∈ X N , X N = X N1 × X N2 .
An achievable region of the problem of many hypotheses two-stage testing concerning an object at two stages of
test is investigated. At first stage of test, one family of PDs is denoted, then at the next stage, object’s distribution
between selected family of PDs is detected.
The statistician must accept one of the families of PDs in the first stage on the base of a sample x1 and then he
accept one of the PDs in the second stage, on the base of other sample x2 .
The procedure of making decision on the base of N observations called the test and denote it by φN for the
N2
1
one-stage test and by ΦN for the two-stage test. The test ΦN may be composed of the pair of tests ϕN
1 and ϕ2
N2
N1
N
for two consecutive stages and it is written by Φ = (ϕ1 , ϕ2 ). The first stage of making decision for selecting
1
a family of PDs is a non-randomized test ϕN
1 (x1 ). The next stage of making decision for PD acceptance in the
N1
2
determined family of PDs is a non-randomized test ϕN
2 (x2 |ϕ1 (x1 ) = i), i = 1, 2, on the base of the sample x2 and
N1
on the result 1 or 2 of the test ϕ1 (x1 ).
We need some formulations of method of types. Let N (x|x) be the number of repetitions of the element x ∈ X
in the vector x ∈ X N . The distribution
4 N (x|x)
Qx (x) =
, x ∈ X,
N
is called in information theory, “the type” of x [1, 3, 4].
Let P N (X ) be the set of all possible types of vectors from X N and TQN be type class of PD Q, the set of all
vectors x of the type Q ∈ P N (X ) [3]. The divergence (Kullback-Leibler distance) for PDs P and Q, is defined as
follows [3, 4, 6]:
Q(x)
4 X
D (Q k P ) =
Q(x) log
.
P (x)
x∈X
The number of distinct types for vectors of length N grows at most polynomially with N [2, 3]. More specifically
quantity of elements of P N (X ) can be estimated as follows
|P N (X )| ≤ (N + 1)|X | .
For any type Q, population of the type class TQN is estimated as follows [2, 3]
©
ª
©
ª
(N + 1)−|X | exp N H(Q) ≤ |TQN | ≤ exp N H(Q) ,
where H(Q) denotes the entropy of distribution Q [2, 3]
X
4
H(Q) = −
Q(x) log Q(x).
x∈X
Probability of members x of a type class
TQN
provided x is i.i.d. by PD P is given by the formula [2, 3]
©
ª
PN (x) = exp − N [H(Q) + D(QkP )] .
Bounds on the total probability of a type class TQN when x is i.i.d. by PD P are [2, 3]
©
ª
©
ª
(N + 1)−|X | exp − N D(QkP ) ≤ PN (TQN ) ≤ exp − N D(QkP ) .
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International Journal of Applied Mathematical Research
3.
Achievable region at the one-stage test
In multiple hypotheses testing, decision making on the base of N -sample x for accepting a realized PD between
S PDs Ps , s = 1, S, the test φN can be defined by partitioning the sample space X N into S disjoint subsets
(N )
(N )
Gs , s = 1, S. The set Gs consists of all vectors x for which s-th PD is adopted.
ª
4 ©
Gs(N ) = x : φN (x) = s , s = 1, S,
S
[
Gs(N ) = X N .
s=1
Let αl|k be the probability of the erroneous acceptance of PD Pl provided Pk is true
³
´
4
(N )
αl|k (φN ) = PkN Gl
, l, k = 1, S, l 6= k.
The probability to reject Pk , when it is true, is
³ (N ) ´ X
4
αk|k (φN ) = PkN G k
=
αl|k (φN ), l, k = 1, S.
l6=k
The infinite sequences of tests is denoted by φ, corresponding the reliabilities of φ are defined as follows
½
¾
¡ ¢
1
4
El|k (φ) = lim inf − log αl|k φN
, l, k = 1, S.
N →∞
N
We now formally define the concept of achievability of a set of error probability exponents in S-ary hypotheses
testing. Consider the S(S − 1)-dimensional Euclidean space, and index the dimensions as {l, k}, where l, k =
1, S. We denote by E = {El|k , l 6= k} the vector, element of which correspond to the set of error exponents
©
(N ) ª
− N1 log PkN (Gl ) .
Definition 3.1 [11] The set of error exponents indicated by the vector E is called achievable if for all ε > 0
(N )
and large enough N there exists a decision scheme Gs , s = 1, S, satisfying for l, k = 1, S, l 6= k the following
conditions:
¡ ¢
1
− log αl|k φN > El|k (φ) − ε.
N
The set of all achievable vectors is defined by R.
Let us define a region E in the error-exponent space as follows:
4
E = {E : ∀Q, ∃l, D(QkPk ) > El|k (φ), k 6= l, l, k = 1, S}.
Theorem 3.2 [11] The following inclusion take place E ⊂ R. Conversely if E ∈ R, then for any δ > 0, Eδ ∈ E,
where Eδ = {El|k (φ) − δ}.
4.
Achievable region at the first stage of two-stage test
Suppose D1 = {1, R} and D2 = {R + 1, S} and the pair of disjoint families of PDs P1 and P2 :
P1 = {Ps , s ∈ D1 },
P2 = {Ps , s ∈ D2 }.
1
The first stage of decision making for selection of a family of PDs is a test ϕN
1 (x1 ), which can be defined by
partitioning the sample space X N1 into the pair of disjoint subsets
4
1
1
AN
= {x1 : ϕN
1 (x1 ) = i}, i = 1, 2,
i
N1
N1
1
AN
.
1 ∪ A2 = X
1
The set AN
consists of all vectors x1 for which i-th family of PDs is adopted.
i
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International Journal of Applied Mathematical Research
N1
0
1
The test ϕN
1 (x1 ) have two kinds of errors for the pair of hypotheses P ∈ Pi , i = 1, 2. Let α2|1 (ϕ1 ) be the
probability of the erroneous acceptance of the second family of PDs provided the first family of PDs is true. And
0
1
α1|2
(ϕN
1 ) be the probability of the erroneous acceptance of the first family of PDs provided the second family of
PDs is true:
4
N1
0
N1
1
α2|1
(ϕN
1 ) = max Ps (A2 ),
s:s∈D1
4
N1
0
N1
1
α1|2
(ϕN
1 ) = max Ps (A1 ).
s:s∈D2
And corresponding reliabilities of the infinite sequence of tests ϕ1 , are:
½
¾
1
4
0
0
1
(ϕ1 ) = lim inf −
log αi|j
(ϕN
Ei|j
)
, i, j = 1, 2.
1
N1 →∞
N1
0
We denote by E0 = {Ei|j
, i 6= j, i, j = 1, 2} the vector, element of which correspond to the set of error exponents
©
ª
N1
1
0
− N1 log αi|j (ϕ1 ) .
0
N1
1
For given E2|1
, the test ϕN
into
1 is defined by partitioning X
(N1 )
A1
[
=
TQNx1
0
Qx1 : min D(Qx1 ||Ps )≤E2|1
1
s:s∈D1
(N1 )
and A2
(N1 )
= X N1 \ A 1
disjoint subsets [8].
Definition 4.1 [11] The set of error exponents indicated by vector E0 is called achievable if for all ε > 0 and large
(N )
enough N1 there exists a decision schemes Ai 1 , i = 1, 2 satisfying for i, j = 1, 2, i 6= j the following conditions:
−
³
´
1
0
0
1
log αi|j
ϕN
> Ei|j
(ϕ1 ) − ε.
1
N1
The set of all achievable vectors is defined by R0 .
Let us define a region E 0 in the error-exponent space as follows:
ª
4 ©
0
E 0 = E0 : ∀Q, ∃i, min D(QkPs ) > Ei|j
(ϕ1 ), j 6= i, i, j = 1, 2 .
s:s∈Dj
So we have
©
ª
0
0
E 0 = E0 : ∀Q, min D(QkPs ) > E2|1
(ϕ1 ), or min D(QkPs ) > E1|2
(ϕ1 ) .
s:s∈D1
s:s∈D2
Theorem 4.2 The following inclusion take place E 0 ⊂ R0 . Conversely if E0 ∈ R0 , then for any δ > 0, E0δ ∈ E 0 ,
0
(ϕ1 ) − δ}.
where E0δ = {Ei|j
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International Journal of Applied Mathematical Research
Proof. If E0 ∈ E 0 , then
X
N1
0
N1
1
α2|1
(ϕN
1 ) = max Ps (A2 ) = max
s:s∈D1
s:s∈D1
X
= max
s:s∈D1
PsN1 (x1 ∈ TQNx1 )
N
1
N
Qx1 :TQx1 ⊂A2 1
1
X
= max
s:s∈D1
|TQNx1 | exp
N
1
X
s:s∈D1
N
N
1
X
s:s∈D1
N
N
n
exp
o
− N1 D(Qx1 kPs )
N
Qx1 :TQx1 ⊂A2 1
1
X
=
o
− N1 {H(Qx1 ) + D(Qx1 kPs )}
n
o
n
o
exp N1 H(Qx1 ) exp − N1 {H(Qx1 ) + D(Qx1 kPr )}
Qx1 :TQx1 ⊂A2 1
= max
n
1
N
Qx1 :TQx1 ⊂A2 1
≤ max
PsN1 (x1 )
N
x1 ∈A2 1
n
exp
o
− N1 min D(Qx1 kPs )
s:s∈D1
N
Qx1 :TQx1 ⊂A2 1
1
X
≤
N
0
exp{−N1 E2|1
}
N
Qx1 :TQx1 ⊂A2 1
1
0
< (N1 + 1)|X | exp{−N1 E2|1
}
0
= exp{−N1 E2|1
+ |X | log(N1 + 1)}
0
= exp{−N1 {E2|1
− ε}}.
So we have
−
1
0
0
1
log α2|1
(ϕN
1 ) > E2|1 − ε,
N1
and similarly we get
1
0
0
1
log α1|2
(ϕN
1 ) > E1|2 − ε.
N1
0
0
Therefore E0 = {E2|1
, E1|2
} is achievable and E0 ∈ R0 , so the first part of Theorem accomplished.
Conversely in the second part of Theorem if E0 ∈ R0 then for any N1 > N1 (ε) we have:
−
−
1
1
0
0
0
0
1
1
log α2|1
(ϕN
log α1|2
(ϕN
1 ) > E2|1 − ε and −
1 ) > E1|2 − ε.
N1
N1
If E0δ ∈
/ E 0 then there exist a distribution Q such that:
0
min D(QkPs ) ≤ E2|1
− δ, and
s:s∈D1
0
min D(QkPs ) ≤ E1|2
− δ.
s:s∈D2
Using the continuity properties of D(QkPs ) for a fixed s and P N1 (X ) is dense in all PDs, we can find for large
enough N1 and Qx1 ∈ P N1 (X ),
D(Qx1 kPs ) ≤ D(QkPs ) + δ/2.
Then it is clear that for all s : s ∈ D1 , we receive to
min D(Qx1 kPs ) ≤ min D(QkPs ) + δ/2
s:s∈D1
≤
s:s∈D1
0
E2|1
−
δ + δ/2,
0
0
and we receive to min D(Qx1 kPs ) ≤ E2|1
− δ/2 < E2|1
, and consequently we get
s:s∈D1
(N1 )
A2
[
=
0
Qx1 : min D(Qx1 ||Ps )>E2|1
TQNx1 = ∅,
1
s:s∈D1
that it contradict to Definition 4.1 and there is E0δ ∈ E 0 and proof of Theorem is completed.
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5.
International Journal of Applied Mathematical Research
Achievable region at the second stage of two-stage test
N1
2
At the second stage of decision making the test is ϕN
2 (x2 |ϕ1 (x1 ) = i), i = 1, 2, if the first (or the second) family
of PDs is accepted, then it can be defined by partitioning the sample space X N2 to R (or S − R) disjoint subsets
(N )
(N )
(N )
Bs 2 , s ∈ D1 (or Bs 2 , s ∈ D2 ). The set Bs 2 consists of all vectors x2 for which s-th PD is adopted. So if the
i-th family of PDs is accepted, then
³
´
ª
4 ©
2
1
Bs(N2 ) = x2 : ϕN
x2 |ϕN
2
1 (x1 ) = i = s , s ∈ Di , i = 1, 2,
[
(N2 )
Bs(N2 ) = X N2 ,
Bs(N2 ) ∩ Bl
= ∅,
l 6= s ∈ Di , i = 1, 2.
s∈Di
³
´
00
2
Let αl|k
ϕN
be the probability of the erroneous acceptance of PD Pl at the second stage of test provided Pk is
2
true:
³
´4
³
´
(N )
00
2
αl|k
ϕN
= PkN2 Bl 2 , l ∈ Di , l 6= k, i = 1, 2.
2
The probability to reject Pk , when it is true, is
³
³
´4
³ (N ) ´ X
´
2
00
00
2
2
αk|k
=
αl|k
ϕN
= PkN2 Bk
ϕN
, l ∈ Di , i = 1, 2.
2
2
(1)
l6=k
For the infinite sequences of tests ϕ2 , corresponding reliabilities are defined as
½
³
´¾
1
4
00
00
2
El|k
(ϕ2 ) = lim inf −
log αl|k
ϕN
, l ∈ Di , i = 1, 2.
2
N2 →∞
N2
It follows from (1) and (2) that
(2)
00
00
Ek|k
(ϕ2 ) = min El|k
(ϕ2 ).
l6=k
00
00
{El|k
,
We denote by E =
©
(N ) ª
− N12 log PkN2 (Bl 2 ) .
l 6= k, l, k = 1, S} the vector, element of which correspond to the set of error exponents
Similar to Definition 3.1. the set of error exponents indicated by vector E00 is called achievable if for all ε > 0
(N )
and large enough N2 there exists a decision schemes Bl 2 , l ∈ Di , i = 1, 2, satisfying for l, k = 1, S, l 6= k the
following conditions:
´
³
1
00
00
2
−
> El|k
(ϕ2 ) − ε.
log αl|k
ϕN
2
N2
The set of all achievable vectors is defined by R00 .
Let us define a region E 00 in the error-exponent space as follows:
4
00
E 00 = {E00 : ∀Q, ∃l, D(QkPk ) > El|k
(ϕ2 ), k 6= l, l, k = 1, S}.
Theorem 5.1 [11] The following inclusion take place E 00 ⊂ R00 . Conversely if E00 ∈ R00 , then for any δ > 0,
00
E00δ ∈ E 00 , where E00δ = {El|k
(ϕ2 ) − δ}.
6.
Achievable region at the two-stage test
N2
N1
N2
1
The test ΦN = (ϕN
1 , ϕ2 ) is composed by a pair of tests ϕ1 and ϕ2 . In the two-stage decision making, if at the
first stage of test, the i-th family of PDs is accepted then the test ΦN , can be defined by partitioning the sample
space X N to S disjoint subsets as follows:
4
(N1 )
Cs(N ) = Ai
× Bs(N2 ) , s ∈ Di , i = 1, 2,
(N1 )
x = (x1 , x2 ) ∈ Cs(N ) : x1 ∈ Ai
, x2 ∈ Bs(N2 ) , s ∈ Di , i = 1, 2.
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International Journal of Applied Mathematical Research
(N )
and the set Cs
consists of all vectors x for which in the two-stage test s-th PD is adopted and
SS
s=1
(N )
Cs
= XN.
000
Let αl|k
be the probability of the erroneous acceptance of PD Pl at the two-stage test, provided Pk is true:
4
(N )
000
αl|k
(ΦN ) = PkN (Cl
), l, k = 1, S, l 6= k.
And the probability to reject Pk at the two-stage test, when it is true, is
¡ N ¢ 4 N ³ (N ) ´ X 000 ¡ N ¢
000
Φ = Pk C k
αk|k
=
αl|k Φ , l, k = 1, S.
l6=k
The reliabilities of the infinite sequences of tests Φ = (ϕ1 , ϕ2 ) are defined as
½
¾
¡ N¢
1
4
000
000
, l, k = 1, S.
El|k (Φ) = lim inf − log αl|k Φ
N →∞
N
000
We denote by E000 = {El|k
, l 6= k, l, k = 1, S} the vector, element of which correspond to the set of error exponents
©
ª
(N )
1
N
− N log Pk (Cl ) .
Definition 6.1 [11] The set of error exponents indicated by vector E000 is called achievable if for all ε > 0 and large
(N )
enough N there exists a decision schemes Cl , l = 1, S satisfying for l, k = 1, S, l 6= k the following conditions:
−
¡ N¢
1
000
000
log αl|k
Φ > El|k
(Φ) − ε.
N
The set of all achievable vectors is defined by R000 .
Let us define a region E 000 in the error-exponent space as follows:
4
000
E 000 = {E000 : ∀Q, ∃l, D(QkPk ) > El|k
(Φ), k 6= l, l, k = 1, S}.
Theorem 6.2 The following inclusion take place E 000 ⊂ R000 . Conversely if E000 ∈ R000 , then for any δ > 0,
000
000
000
E000
δ ∈ E , where Eδ = {El|k (Φ) − δ}.
000
Proof. If E000 ∈ E 000 then for any type Q there exist at least one l such that D(QkPk ) > El|k
(Φ) for all k 6= l. If
(N )
there ara multiple such l , select one arbitrarily and assign the all type class TQN to Cl
of error corresponding satisfies the following chain of inequalities:
(N )
PkN (Cl
)=
X
. The result of probability
PkN (x)
(N )
x∈Cl
X
=
©
ª
|TQN | exp − N {H(Q) + D(QkPk )}
(N )
N ⊂C
Q:TQ
l
X
≤
exp{−N D(QkPk )}
N ⊂C (N )
Q:TQ
l
< (N + 1)|X | exp{−N D(QkPk )}
000
≤ exp{−N {El|k
− ε}}
So we have
© 000 ¡ N ¢ ª
©
1
1
(N ) ª
000
log αl|k
(Φ) − ε.
Φ
= − log PkN (Cl ) > El|k
N
N
Consequently the first part of Theorem is proved.
Conversely in the second part of Theorem if E000 ∈ R000 then for every ε > 0, there exists some sequence of decision
(N )
schemes Cl , l = 1, S and some number N > N (ε) such that for all l 6= k
−
−
©
1
(N ) ª
000
log PkN (Cl ) > El|k
(Φ) − ε.
N
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International Journal of Applied Mathematical Research
If E000
/ E 000 , then there exists a distribution Q and δ > 0 such that
δ ∈
000
∃l : D(QkPk ) ≤ El|k
(Φ) − δ, k 6= l, l, k = 1, S.
Using the continuity properties of D(QkPk ) for a fixed k and P N (X ) is dense in all PDs, we can find for large
enough N and Qx ∈ P N (X ),
D(Qx kPk ) ≤ D(QkPk ) + δ/2
000
≤ El|k
(Φ) − δ/2.
The corresponding result of error probabilities satisfy the following chain of inequalities:
X
(N )
PkN (Cl ) =
PkN (x)
(N )
x∈Cl
=
X
©
ª
|TQNx | exp − N {H(Qx ) + D(Qx kPk )}
(N )
N ⊂C
Qx :TQ
l
x
©
ª
≥ (N + 1)−|X | exp − N D(Qx kPk )
ª
©
000
≥ (N + 1)−|X | exp − N {El|k
(Φ) − δ/2}
©
ª
000
= exp − N {El|k
(Φ) − δ ∗ }
So we have
© 000 ¡ N ¢ ª
©
1
1
(N ) ª
000
log αl|k
Φ
= − log PkN (Cl ) ≤ El|k
(Φ) − δ ∗ .
N
N
But this last inequality contradicts with Definition 6.1. and therefore E000 ∈
/ R000 . Consequently if E000 ∈ R000 , then
000
000
Eδ ∈ E and the converse part of Theorem is accomplished.
−
7.
Conclusion
The achievable region of reliabilities in multiple hypothesis testing problem for an object that follows a PD among
one of the pair of disjoint families of PDs is exposed . In the two-stage test, at the first stage of test ϕ1 , an achievable
region of reliabilities R0 and at the second stage of test ϕ2 an achievable region of reliabilities R00 are investigated
and consequently at the two-stage test Φ = (ϕ1 , ϕ2 ) is shown by Theorem 6.2. that there exists an achievable region
of reliabilities R000 and the defined region for the vectors of reliabilities of the two-stage test characterizes the set of
all achievable vectors. Also advantages of procedures of the two-stage LAO testing was revealed in [8].
Acknowledgments
This work was supported by Ahvaz branch, Islamic Azad University, grant number 51063910508006.
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