D:\sbornik\...\OnMultiple.DVI


Mathematical Problems of Computer Science 23, 2004, 36{46.

On M ultiple H ypotheses T esting by I nfor med

Statistician for Ar bitr ar ily Var ying Object and

Application to Sour ce Coding¤

E vg u e n i A . H a r o u t u n ia n a n d P a r a n d z e m M. H a ko b ya n

Institue for Informatics and Automation Problems of NAS of RA

e-mails evhar@ipia.sci.am, par h@ipia.sci.am

Abstract

The matrix of asymptotic interdependencies (reliability{reliability functions) of all
possible pairs of the error probability exponents (reliabilities) in testing of multiple
statistical hypotheses is studied for arbitrarily varying object with the current states
sequence known to the statistician. The case of two hypotheses when state sequences
are not known to the decision maker was studied by Fu and Shen, and when decision is
founded on the known states sequence was considered by Ahlswede, Haroutunian and
Aloyan.

In the same way as Fu and Shen we obtain from the main result rate-reliability and
reliability-rate functions for arbitrarily varying source coding with side information.
An illustrative example is presented.

Refer ences

[1 ] R . F. A h ls we d e , " Co lo r in g h yp e r g r a p h s : a n e w a p p r o a c h t o m u lt i-u s e r s o u r c e c o d in g "
I, II, J . Combin. Inform. and Syst. Sci., vo l. 4 , n o . 1 , p p . 7 6 -1 1 5 , 1 9 7 9 , vo l. 5 , n o . 3 ,
p p . 2 2 0 { 2 6 8 , 1 9 8 0 .

[2 ] E . A . H a r o u t u n ia n , " Ma n y s t a t is t ic a l h yp o t h e s e s : in t e r d e p e n d e n c e o f o p t im a l t e s t 's
e r r o r p r o b a b ilit ie s e xp o n e n t s " , ( In R u s s ia n ) , A b s t r a c t o f t h e r e p o r t o n t h e 3 r d A ll-
U n io n s c h o o l-s e m in a r , "P rogram-algorithmical software for applied multi-variate statis-
tical analysis", Ts a kh ka d z o r , P a r t 2 , p p . 1 7 7 { 1 7 8 , 1 9 8 8 .

[3 ] 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 .

[4 ] R . L . D o b r u s h in , P e r s o n a l c o m m u n ic a t io n , 1 9 8 7 .

[5 ] R . F. A h ls we d e , E . A . H a r o u t u n ia n a n d E . V . A lo ya 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 h yp o t h e s is t e s t in g fo r a r b it r a r ily va r yin g s o u r c e wit h s id e in fo r m a t io n " ,
p r e s e n t e d fo r p u b lic a t io n 2 0 0 4 .

¤The work was partially supported by INTAS, project 00{738.

3 6



E. A. Haroutunian and P. M. Hakobyan 3 7

[6 ] L . B ir g ¶ e , " V it e s s e s m a xim a ls d e d ¶ c r o is s e n c e d e s e r r e u r s e t t e s t s o p t im a u x a s s o c ie ¶ s " . Z.
W a h r s c h . ve r w. Ge b ie t e , vo l. 5 5 , p p . 2 6 1 { 2 7 3 , 1 9 8 1 .

[7 ] I. Cs is z ¶a r a n d J. K Äo r n e r , " In fo r m a t io n Th e o r y: Co d in g Th e o r e m s fo r D is c r e t e Me m o -
r yle s s S ys t e m s " , Academic press., N e w Y o r k, 1 9 8 1 .

[8 ] I. Cs is z ¶a r , " Th e m e t h o d o f t yp e s " , IE E E Trans. Inform. Theory, vo l. 4 4 , n o . 6 ,
p p . 2 5 0 5 { 2 5 2 3 , 1 9 9 8 .

[9 ] 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 .

[1 0 ] V . A n a n t a r a m , " A la r g e d e r ia t io n s a p p r o a c h t o e r r o r e xp o n e n t in s o u r c e c o d in g a n d
h yp o t h e s e s t e s t in g " , IE E E Trans. Inform. Theory, vo l. 3 6 , n o . 4 , p p . 9 3 8 { 9 4 3 , 1 9 9 0 .

[1 1 ] S . N a t a r a ja n , " L a r g e d e r iva t io n s h yp o t h e s is t e s t in g a n d s o u r c e c o d in g fo r ¯ n it e Ma r ko v
c h a in s " , IE E E Trans. Inform. Theory, vo l. 3 1 , n o . 3 , p p . 3 6 0 { 3 6 5 , 1 9 8 5 .

[1 2 ] F.-W . Fu a n d S .-Y . S h e n , " H yp o t h e s is t e s t in g fo r a r b it r a r ily va r yin g s o u r c e wit h
e xp o n e n t ia l-t yp e c o n s t r a in t " , IE E E Trans. Inform. Theory, vo l. 4 4 , n o . 2 , p p . 8 9 2 {
8 9 5 , 1 9 9 8 .

î»Õ»Ï³Ï³óí³Í íÇ׳ϳ·ñÇ ÏáÕÙÇó ϳٳ۳ϳÝáñ»Ý ÷á÷áËíáÕ ûµÛ»ÏïÇ
Ýϳïٳٵ µ³½Ù³ÏÇ í³ñϳÍÝ»ñÇ ëïáõ·áõÙ³Ý ¨ ³ÕµÛáõñÇ Ïá¹³íáñÙ³Ý

ѳٳñ ÏÇñ³éáõÃÛ³Ý Ù³ëÇÝ

º. ². гñáõÃÛáõÝÛ³Ý ¨ ö. Ø. гÏáµÛ³Ý

²Ù÷á÷áõÙ

àõëáõÙݳëÇñí»É ¿ µ³½Ù³ÏÇ í³ñϳÍÝ»ñÇ ï»ëï³íáñÙ³Ý ÁÝóóùáõÙ µáÉáñ Ñݳñ³íáñ
½áõÛ·»ñÇ ë˳ÉÝ»ñÇ Ñ³í³Ý³Ï³ÝáõÃáõÝÝ»ñÇ Ñáõë³ÉÇáõÃÛ³Ý óáõóÇãÝ»ñÇ ÷áËϳËí³Íáõ-
ÃÛáõÝÝ»ñÇ ÷á÷áËíáÕ ûµÛ»ÏïÇ Ñ³Ù³ñ, áñÇ íÇ׳ÏÝ»ñÁ ѳÛïÝÇ »Ý íÇ׳ϳ·ñÇÝ: ºñÏáõ
í³ñϳÍÝ»ñÇ ¹»åùÁ, »ñµ áñáßáõÙ ÁݹáõÝáÕÇÝ ³ÝѳÛï ¿ íÇ׳ÏÝ»ñÇ Ñ³çáñ¹³Ï³ÝáõÃÛáõÝÁ,
ùÝݳñÏí»É ¿ üáõÇ ¨ Þ»ÝÇ ÏáÕÙÇó, ÇëÏ Ñ³ÛïÝÇ íÇ׳ÏÝ»ñáí ï³ñµ»ñ³ÏÁ ¹Çï³ñÏí»É
¿ ²Éëí»¹»Ç, гñáõÃÛáõÝÛ³ÝÇ ¨ ²ÉáÛ³ÝÇ ÏáÕÙÇó: ÆÝãå»ë üáõÝ ¨ Þ»ÝÁ, Ù»Ýù ÝáõÛÝå»ë
ëï³ó»É »Ýù ÏáÕÙݳÏÇ ÇÝýáñÙ³ódzÛáí ϳٳ۳ϳÝáñ»Ý ÷á÷áËíáÕ ³ÕµÛáõñÇ Ñ³Ù³ñ
³ñ³·áõÃÛáõÝ-Ñáõë³ÉÇáõÃÛáõÝ ¨ Ñáõë³ÉÇáõÃÛáõÝ-³ñ³·áõÃÛáõÝ ýáõÝÏódzݻñÁ: Ü»ñϳ-
Û³óí³Í ¿ å³ñ½³µ³ÝáÕ ûñÇݳÏ: