D:\sbornik\...\Article.DVI


Mathematical Problems of Computer Science 37, 25{34, 2012.

On T wo-stage Logar ithmically Asymptotically Optimal

T esting of M ultiple H ypotheses Concer ning

Distr ibutions fr om the P air of Families

E vg u e n i H a r o u t u n ia n 1, P a r a n d z e m H a ko b ya n 1 a n d Fa r s h in H o r m o z i n e ja d 2

Institute for Informatics and Automation Problems of NAS of RA.1

Islamic Azad University, Ahvaz Branch, Iran.2

evhar@ipia.sci.am, par h@ipia.sci.am, hormozi-nejad@iauahvaz.ac.ir

Abstract

Two-stage testing of multiple hypotheses for a model with two given families of
hypothetical probability distributions is considered. The matrix of reliabilities of log-
arithmically asymptotically optimal hypotheses testing by a pair of stages is studied
and compared with the case of similar one-stage testing.

Keywords: Logarithmically asymptotically optimal (LAO) test, multiple hypothe-
ses testing, multistage tests, reliabilities matrix, error probability exponent.

Refer ences

[1 ] R . A h ls we d e a n d E .A . H a r o u t u n ia n . \ On s t a t is t ic a l h yp o t h e s e s o p t im a l t e s t in g a n d id e n -
t ī c a t io n " , L ecture Notes in Computer Science, vol. 4123, General Theory of Information
Transfer and Combinatorics, S p r in g e r , p p . 4 6 2 -4 7 8 , 2 0 0 6 .

[2 ] T.M Co ve r a n d J.A . To m a s . E lements of Information Theory, S e c o n d e d it io n , W ile y,
N e wY o r k, 2 0 0 6 .

[3 ] I. Cs is z ¶a r a n d J. K Äo r n e r . Information Theory: Coding Theorems for D iscrete M emoryless
Systems, A c a d e m ic p r e s s , N e wY o r k, 1 9 8 1 .

[4 ] I. Cs is z ¶a r a n d G. L o n g o . \ On t h e e r r o r e xp o n e n t fo r s o u r c e c o d in g a n d fo r t e s t in g s im p le
s t a t is t ic a l h yp o t h e s e s " , Studia Sc. M ath. Hungarica, vo l. 6 , p p . 1 8 1 { 1 9 1 , 1 9 7 1 .

[5 ] 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 ic a l
h yp o t h e s e s " , P roblems of Control and Information Theory, vo l 1 9 , n o . 5 -6 , p p . 4 1 3 -4 2 1 ,
1 9 9 0 .

[6 ] 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 ro-
ceedings of the NATO-ASI Conference, vol. 198 of NATO Science Series III: Computer
and Systems Sciences, Y e r e va n , A r m e n ia , p p . 1 8 9 -2 0 1 , IOS P r e s s , 2 0 0 8 .

[7 ] E .A . H a r o u t u n ia n , M.E . H a r o u t u n ia n a n d A .N . H a r o u t u n ia 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 is t e s t in g , F oundations and Trends in Com-
munications and Information Theory, vo l. 4 , n o . 2 -3 , 2 0 0 8 .

2 5



2 6 On Two-stage Logarithmically Asymptotically Optimal Testing of Multiple Hypotheses

[8 ] E .A . H a r o u t u n ia n a n d P .M. H a ko b ya n , \ Mu lt ip le h yp o t h e s e s L A O t e s t in g fo r m a n y
in d e p e n d e n t o b je c t s " , International J ournal \Scholarly R esearch E xchange", p p . 1 { 6 ,
2 0 0 9 .

[9 ] W . H o e ®d in g . \ A s ym p t o t ic a lly o p t im a l t e s t s fo r m u lt in o m ia l d is t r ib u t io n s " , Annals of
M athematical Statistics, vo l. 3 6 , p p . 3 6 9 -4 0 1 , 1 9 6 5 .

[1 0 ] G. Tu s n a d y. \ On a s ym p t o t ic a lly o p t im a l t e s t s " , Annals of Statistics, vo l. 5 , n o . 2 , p p .
3 8 5 -3 9 3 , 1 9 7 7 .

ºñÏáõ ÁÝï³ÝÇùÝ»ñ ϳ½ÙáÕ µ³ßËáõÙÝ»ñÇ í»ñ³µ»ñÛ³É µ³½Ù³ÏÇ í³ñϳÍÝ»ñÇ
»ñÏ÷áõÉ Éá·³ñÇÃÙáñ»Ý ³ëÇÙåïáïáñ»Ý ûåïÇÙ³É ï»ëï³íáñáõÙ

º. гñáõÃÛáõÝÛ³Ý, ö. гÏáµÛ³Ý ¨ ü. ÐáñÙá½Ç ݻų¹

²Ù÷á÷áõÙ

¸Çï³ñÏí»É ¿ µ³ßËáõÙÝ»ñÇ »ñÏáõ ÁÝï³ÝÇùÝ»ñÇó ϳ½Ùí³Í Ùá¹»ÉÇ í»ñ³µ»ñÛ³É
µ³½Ù³ÏÇ íÇ׳ϳ·ñ³Ï³Ý í³ñϳÍÝ»ñÇ ëïáõ·Ù³Ý ·áñÍÁÝóóÁ: лﳽáïí»É »Ý
»ñÏáõ ÷áõÉ»ñÇó Ûáõñ³ù³ÝãÛáõñÇ ë˳ÉÝ»ñÇ ½áõÛ·»ñÇ óáõóÇãÝ»ñÇ (Ñáõë³ÉÇáõÃÛáõÝÝ»ñÇ)
÷áËϳåí³ÍáõÃÛáõÝÝ»ñÇ Ù³ïñÇóÝ»ñÁ: гٻٳïí»É »Ý »ñÏ÷áõÉ ï»ëïÇ ¨ Ùdz÷áõÉ
ï»ëïÇ Ñáõë³ÉÇáõÃÛáõÝÝ»ñÁ ¨ ·áñÍÁÝóóÝ»ñÇ »ñϳñáõÃÛáõÝÝ»ñÁ:

Î äâóõ-ýòàïíîì ëîãàðèôìè÷åñêè àñèìïòîòè÷åñêè
îïòèìàëüíîì òåñòèðîâàíèè ìíîãèõ ãèïîòåç

îòíîñèòåëüíî ðàñïðåäåëåíèé èç äâóõ ñåìåéñòâ
Å. Àðóòþíÿí, Ï. Àêîïÿí è Ô. Õîðìîçè íåæàä

Àííîòàöèÿ

Ðàññìîòðåíî òåñòèðîâàíèå ìîäåëè ñîñòîÿøåé èç äâóõ ñåìåéñòâ âîçìîæíûõ
ðàñïðåäåëåíèé âåðîÿòíîñòåé. Ìàòðèöà íàäåæíîñòåé ëîãîðèôìè÷åñêè àñèìïòî-
òè÷åñêè îïòèìàëüíîãî òåñòèðîâàíèÿ â äâà ýòàïà èçó÷åíà è ñðàâíåíà ñî ñëó÷àåì
àíàëîãè÷íîãî îäíîýòàïíîãî òåñòèðîâàíèÿ.