D:\sbornik\...\13_YesHar\4.DVI


Mathematical Problems of Computer Science 29, 2007, 97{103.

On Logar ithmically Asymptotically Optimal

H ypothesis T esting of Distr ibutions for P air of

Statistically Dependent Objects

E vg u e n i H a r o u t u n ia n a n d A r a m Y e s s a ya n

Institue for Informatics and Automation Problems of NAS of RA

e-mail: evhar@ipia.sci.am

Abstract

The problem of hypotheses testing for a model consisting of two statistically depen-
dent objects is considered. It is supposed that two probability distributions are known
for the ¯rst object and the second object dependent on the ¯rst can be distributed
according to one of two given conditional distributions. The matrix of asymptotical
optimal interdependencies (reliability{reliability functions) of all possible pairs of the
error probability exponents (reliabilities) is studied. The case with two objects which
can't have the same probability distribution from two given was discussed by Ahlswede
and Haroutunian and for three hypotheses by Haroutunian and Yessayan.

Refer ences

[1 ] E . A . H a r o u t u n ia n , \ L o g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f m u lt ip le s t a t is t i-
c a l h yp o t h e s e s " , P roblems of Control and Information Theory, vo l. 1 9 ( 5 -6 ) , p p . 4 1 3 { 4 2 1 ,
1 9 9 0 .

[2 ] R . F. A h ls we d e a n d E . A . H a r o u t u n ia n , \ On lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l
t e s t in g o f h yp o t h e s e s a n d id e n t ī c a t io n " . L e c t u r e N o t e s in Co m p u t e r S c ie n c e , vo l. 4 1 2 3 ,\
Ge n e r a l Th e o r y o f In fo r m a t io n Tr a n s fe r a n d Co m b in a t o r ic s " S p r in g e r , p p . 4 6 2 { 4 7 8 ,
2 0 0 6 .

[3 ] E . A . H a r o u t u n ia n , " R e lia b ilit y in m u lt ip le h yp o t h e s e s t e s t in g a n d id e n t ī c a t io n " . P r o -
c e e d in g s o f t h e N A TO A S I, Y e r e va n 2 0 0 3 , N A TO S c ie n c e S e r ie s , III: Co m p u t e r a n d
S ys t e m s S c ie n c e s -V o l 1 9 8 , IOS P r e s s , p p . 1 8 9 -2 0 1 , 2 0 0 5 .

[4 ] E . A . H a r o u t u n ia n a n d P . M. H a ko b ya n , \ On lo g a r it h m ic a lly o p t im a l h yp o t h e s is t e s t -
in g o f t h r e e d is t r ib u t io n s fo r p a ir o f in d e p e n d e n t o b je c t s " , M athematical P roblems of
Computer Sciences, vo l. X X IV , Y e r e va n , p p . 7 6 { 8 1 , 2 0 0 5 .

[5 ] E . A . H a r o u t u n ia n a n d P . M. H a ko b ya n , \ On L A O t e s t in g o f m u lt ip le h yp o t h e s e s fo r
p a ir o f o b je c t " , M athematical P roblems of computer Science, vo l. X X V , Y e r e va n , p p .
9 3 { 1 0 1 , 2 0 0 6 .

9 7



9 8 On LAO Hypothesis Testing of Distributions for Pair of Statistically Dependent Objects

[6 ] E . A . H a r o u t u n ia n a n d A . O. Y e s s a ya n , \ On h yp o t h e s e s t e s t in g fo r t wo d i®e r e n t ly
d is t r ib u t e d o b je c t s " . M athematical P roblems of computer Science, vo l. X X V I, Y e r e va n ,
p p . 9 1 { 9 6 , 2 0 0 6 .

[7 ] T. M. Co ve r a n d J. A . Th o m a s , E lements of information theory. W ile y, N e w Y o r k, 1 9 9 1 .

[8 ] I. Cs is z ¶a r a n d P . C. S h ie ld s , \ In fo r m a t io n t h e o r y a n d s t a t is t ic s : a t u t o r ia l" . F oundations
and Trends in Communications and Information Theory, vo l.1 , n o .4 , 2 0 0 4 .

[9 ] E . A . H a r o u t u n ia n , M. A . H a r o u t u n ia n , a n d A . N . H a r u t yu n ya n , \ R e lia b ilit y c r it e r ia
in in fo r m a t io n t h e o r y a n d in s t a t is t ic a l h yp o t h e s e s t e s t in g " , A c c e p t e d fo r p u b lic a t io n
in F oundation and Trends in Communications and Information Theory.

[1 0 ] R . E . B e c h h o fe r , J. K ie fe r , a n d M. S o b e l, Sequential identi¯cation and ranking proce-
dures. Th e U n ive r s it y o f Ch ic a g o , 1 9 6 8 .

ìÇ׳ϳ·ñáñ»Ý ϳËÛ³É »ñÏáõ ûµÛ»ÏïÝ»ñÇ Ýϳïٳٵ
í³ñϳÍÝ»ñÇ ëïáõ·Ù³Ý Ù³ëÇÝ

º. ². гñáõÃÛáõÝÛ³Ý ¨ ². ú. ºë³Û³Ý

²Ù÷á÷áõÙ

¸Çï³ñÏí³Í ¿ íÇ׳ϳ·ñáñ»Ý ϳËÛ³É »ñÏáõ ûµÛ»ÏïÝ»ñÇ Ýϳïٳٵ í³ñϳÍÝ»ñÇ
Éá·³ñÇÃÙáñ»Ý ³ëÇÙåïáïáñ»Ý ûåïÇÙ³É ëïáõ·Ù³Ý ËݹÇñÁ: ²é³çÇÝ ûµÛ»ÏïÁ ϳñáÕ
¿ µ³ßËí³Í ÉÇÝ»É ïñí³Í »ñÏáõ ѳí³Ý³Ï³Ý³ÛÇÝ µ³ßËáõÙÝ»ñÇó Ù»Ïáí, ÇëÏ
»ñÏñáñ¹Áª ϳËí³Í ³é³çÇÝÇ µ³ßËáõÙÇó, ïñí³Í »ñÏáõ å³ÛÙ³Ý³Ï³Ý Ñ³í³Ý³Ï³Ý³ÛÇÝ
µ³ßËáõÙÝ»ñÇó Ù»Ïáí: àõëáõÙݳëÇñí»É ¿ í³ñϳÍÝ»ñÇ ûåïÇÙ³É ï»ëï³íáñÙ³Ý
¹»åùáõÙ »ñÏáõ ûµÛ»ÏïÝ»ñÇ Ýϳïٳٵ í³ñϳÍÝ»ñÇ ë˳ÉÝ»ñÇ Ñ³í³Ý³Ï³ÝáõÃÛáõÝÝ»ñÇ
óáõóÇãÝ»ñÇ (Ñáõë³ÉÇáõÃÛáõÝÝ»ñÇ) ÷áËϳËí³ÍáõÃÛáõÝÁ: