D:\sbornik\...\NOIC.DVI


Mathematical Problems of Computer Science 25, 2006, 92{100.

On LAO T esting of M ultiple H ypotheses

for P air of Objects

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-mail evhar@ipia.sci.am, par h@ipia.sci.am

Abstract

Many hypotheses testing for a model consisting of two independent by functioning
objects is considered. It is known that M(¸ 2) probability distributions are given
and objects independently of other follows to one of them. The matrix of asymptotic
interdependencies (reliability{reliability functions) of all possible pairs of the error
probability exponents (reliabilities) in optimal testing for this model is studied.

This problem was introduced (and solved for the case with two given probability
distributions) by Ahlswede and Haroutunian. The situation with three hypotheses was
examined by Haroutunian and Hakobyan.

Refer ences

[1 ] R . F. A h ls we d e a n d E . A . H a r o u t u n ia n , " Te s t in g o f h yp o t h e s e s a n d id e n t i¯ c a t io n " ,
E lectronic Notes on D iscrete M athematics, vol. 21, p p . 1 8 5 { 1 8 9 , 2 0 0 5 .

[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 S t a t is t ic a l H yp o t h e s e s Op t im a l Te s t in g
a n d Id e n t ī c a t io n " . M athematical P roblems of Computer Science 24, p p . 1 6 { 3 3 , 2 0 0 5 .

[3 ] E . A . H a r o u t u n ia n , " R e lia b ilit y in Mu lt ip le H yp o t h e s e s Te s t in g a n d Id e n t i¯ c a t io n
P r o b le m s " . 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 { vo l. 1 9 8 , p p . 1 8 9 { 2 0 1 . IOS P r e s s , 2 0 0 5 .

[4 ] R . E . B e c h h o fe r , J. K ie fe r , a n d M. S o b e l, S e qu e n t ia l id e n t ī c a t io n a n d r a n kin g p r o c e -
d u r e s . Th e U n ive r s it y o f Ch ic a g o P r e s s , Ch ic a g o , 1 9 6 8 .

[5 ] R . F. A h ls we d e a n d I. W e g e n e r , S e a r c h p r o b le m s . W ile y, N e w Y o r k, 1 9 8 7 .

[6 ] 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 ( 5 -6 ) , p p .
4 1 3 { 4 2 1 , 1 9 9 0 .

[7 ] 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 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 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 c e o b je c t s " , M athemat-
ical P roblems of Computer Science vol. 24, p p . 7 6 { 8 1 , 2 0 0 5 .

9 2



E. A. Haroutunian and P. M. Hakobyan 9 3

[8 ] I. Cs is z ¶a r a n d J. K Äo r n e r , Information Theory: Coding Theorems for D iscrete M emo-
ryless Systems, A c a d e m ic P r e s s , N e w Y o r k, 1 9 8 1 .

[9 ] E . A . H a r o u t u n ia n , " 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 a n y s t a t is t ic a l h yp o t h e s e s
c o n c e r n in g Ma r ko v c h a in " , 5th Intern. Vilnius Conference on P robability Theory and
M athem. Statistics, vo l. 1 , ( A -L ) , p p . 2 0 2 { 2 0 3 , 1 9 8 9 .

[1 0 ] E . Tu n c e l, " On e r r o r e xp o n e n t s in h yp o t h e s is t e s t in g " . IE E E Trans. on IT, vo l. 5 1 ,
n o . 8 , p p . 2 9 4 5 { 2 9 5 0 , 2 0 0 5 .

[1 1 ] L . B ir g ¶ e , " V it e s s e s m a xim a ls d e d ¶ e c r o is s a 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 .

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