D:\User\...\main.DVI


Mathematical Problems of Computer Science 49, 110{114, 2018.

On M ultiple H ypotheses LAO T esting With Rejection

of Decision for T wo Dependent Objects

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

Institute for Informatics and Automation Problems of NAS RA

e-mail: eghishe@sci.am, armfrance@yahoo.fr, narineharutyunyan57@gmail.com

Abstract

Multiple statistical hypotheses testing with possibility of rejecting of decision is
considered for model consisting of two dependent objects characterized by joint discrete
probability distribution. The matrix of error probabilities exponents (reliabilities) of
asymptotically optimal tests is studied.

Keywords: Multiple hypotheses testing, Optimal tests, Rejection option, Two
object.

1 . In t r o d u c t io n

Th is p a p e r is d e vo t e d t o t h e s t u d y o f c h a r a c t e r is t ic s o f lo g a r it h m ic a lly a s ym p t o t ic a lly o p t i-
m a l ( L A O) h yp o t h e s e s t e s t in g wit h p o s s ib ilit y o f r e je c t io n o f d e c is io n fo r t h e m o d e l o f t wo
d e p e n d e n t o b je c t s wit h jo in t p t o b a b ilit y d is t r ib u t io n s ( P D s ) . Th e c o r r e s p o n d e n c e p r e s e n t s
a c o m p le m e n t t o p r o b le m s s t u d ie d in [1 , 2 ], wh e r e t h e c o m p o n e n t s o f r a n d o m ve c t o r c h a r a c -
t e r iz in g t wo o b je c t s we r e in d e p e n d e n t , s o it wa s p o s s ib le t o c o n s id e r t h e t e s t p r o c e d u r e wit h
s e p a r a t e t e s t s fo r t wo o b je c t s . In [2 ] t wo d i®e r e n t m o d e ls a r e s t u d ie d : ¯ r s t , wh e n r e je c t io n
wa s a llo we d o n ly fo r o n e o f t h e o b je c t s a n d s e c o n d , wh e n r e je c t io n wa s a llo we d fo r b o t h
o b je c t s . Th e p r o b le m wit h a n a lo g o u s s t a t e m e n t fo r o n e a r b it r a r ily va r yin g o b je c t wit h s id e
in fo r m a t io n wa s e xa m in e d in [3 ].

It is wo r t h t o r e c a ll t h e p r e vio u s r e s u lt s o n L A O t e s t in g o f m a n y h yp o t h e s e s p u b lis h e d
in [4 -6 ].
Me t h o d s a n d b a s ic r e s u lt s o f L A O t e s t in g a r e a ls o p r e s e n t e d in b o o ks [7 -9 ]

2 . P r o b le m Fo r m u la t io n a n d R e s u lt

L e t P ( X ) b e a s p a c e o f a ll p r o b a b ilit y d is t r ib u t io n s G( x ) o n ¯ n it e s e t X . L e t ( X1; X2 ) b e
r a n d o m ve c t o r t a kin g va lu e s in t h e s e t X £ X wit h o n e o f M 2; M ¸ 2 jo in t P D s Gm1;m2 2
P ( X £ X ) , m1; m2 = 1 ; M. L e t ( x1; x2 )

4
= ( ( x11; x

2
1 ) ; :::; ( x

1
n; x

2
n ) ; :::; ( x

1
N ; x

2
N ) ) , x

1
n; x

2
n 2 X ;

n = 1 ; N, b e a ve c t o r o f r e s u lt s o f N in d e p e n d e n t o b s e r va t io n s o f t h e ve c t o r ( X1; X2 ) , it is
c a lle d a s a m p le .

Th e s t a t is t ic ia n h a s t o d e t e r m in e u n kn o wn P D s fr o m t h e s e t o f h yp o t h e s e s : Hm1;m2 :
G = Gm1;m2, m1; m2 = 1 ; M o r wit h d r a w t o d o a n y ju d g e m e n t u s in g o b t a in e d s a m p le .

1 1 0



E. Haroutunian, A. Yesayan and N. Harutyunyan 1 1 1

W e c a ll t h is p r o c e d u r e a c o m p o u n d t e s t a n d d e n o t e it b y ©N .
Th e t e s t ©N c a n b e d e ¯ n e d b y t h e d ivis io n o f t h e s p a c e X N £ X N in t o M 2 + 1 d is jo in t

s u b s e t s , wh e r e Am1;m2, m1; m2 = 1 ; M , c o n t a in s a ll ve c t o r s ( x1; x2 ) fo r wh ic h t h e h yp o t h e s is
Hm1;m2 is a d o p t e d , a n d AM +1 c o n t a in s a ll ve c t o r s fo r wh ic h we r e fu s e t o t a ke a c e r t a in
a n s we r .

L e t ®l1;l2jm1;m2 ( ©N ) b e t h e p r o b a b ilit y o f t h e e r r o n e o u s a c c e p t a n c e o f t h e h yp o t h e s is
Hl1;l2 b y t h e t e s t ©N p r o vid e d t h a t t h e h yp o t h e s is Hm1;m2 is t r u e , wh e r e ( m1; m2 ) 6= ( l1; l2 ) ,
m1; m2; l1; l2 = 1 ; M,

®l1;l2jm1;m2 ( ©N ) = G
N
m1;m2

( Al1;l2 ) :

W h e n t h e h yp o t h e s is Hm1;m2 is t r u e , b u t we d e c lin e t h e d e c is io n c o n c e r n in g t o t h e
h yp o t h e s e s , t h e c o r r e s p o n d in g p r o b a b ilit y o f e r r o r is :

®M +1;M+1jm1;m2 ( ©N ) = G
N
m1;m2

( AM +1 ) :

Th e p r o b a b ilit y n o t t o a c c e p t a t r u e h yp o t h e s e s Hm1;m2, m1; m2 = 1 ; M is t h e fo llo win g :

®m1;m2jm1;m2 ( ©N ) =
X

(l1;l2) 6=(m1;m2); l1;l2=1;M ; (l1;l2)=(M+1;M+1)
®l1;l2jm1;m2 ( ©N ) : ( 1 )

W e s t u d y t h e c o r r e s p o n d in g r e lia b ilit ie s El1;l2jm1;m2 ( ©) o f t h e s e qu e n c e o f t e s t s ©,

El1;l2jm1;m2 ( ©)
4
= lim

N!1
¡ 1

N
lo g ®l1;l2jm1;m2 ( ©N ) ;

m1; m2; l1; l2 = 1 ; M; ( l1; l2 ) = ( M + 1 ; M + 1 ) : ( 2 )

D e ¯ n it io n s ( 1 ) a n d ( 2 ) im p ly t h a t

Em1;m2jm1;m2 ( ©) = m in
(l1;l2) 6=(m1;m2)

El1;l2jm1;m2 ( ©) ;

m1; m2; l1; l2 = 1 ; M; ( l1; l2 ) = ( M + 1 ; M + 1 ) : ( 3 )

W e c a ll t h e t e s t s e qu e n c e ©¤ L A O fo r t h e m o d e l wit h t wo o b je c t s if fo r t h e g ive n p o s it ive
va lu e s o f c e r t a in p a r t o f e le m e n t s o f t h e r e lia b ilit y m a t r ix E ( ©¤ ) t h e p r o c e d u r e ©¤ p r o vid e s
m a xim a l va lu e s fo r a ll o t h e r e le m e n t s o f it .
Fo r M = 2 t h e m a t r ix will b e a s fo llo ws :

E ( ©) =

0
BBB@

E1;1j1;1 E1;2j1;1E2;1j1;1 E2;2j1;1 E3;3j1;1
E1;1j1;2 E1;2j1;2E2;1j1;2 E2;2j1;2 E3;3j1;2
E1;1j2;1 E1;2j2;1E2;1j2;1 E2;2j2;1 E3;3j2;1
E1;1j2;2 E1;2j2;2E2;1j2;2 E2;2j2;2 E3;3j2;2

1
CCCA :

W it h t h e g ive n e le m e n t s E1;1j1;1; E1;2j1;2; E2;1j2;1; E2;2j2;2 we d e ¯ n e t h e r e g io n s o f a c c e p t a n c e
o f t h e t e s t .

In t h e g e n e r a l c a s e o f M h yp o t h e s e s fo r g ive n r e lia b ilit ie s E1;1j1;1; E1;2j1;2; E2;1j2;1; :::; EM;MjM;M
we d e ¯ n e t h e fo llo win g r e g io n s :

Rm1;m2
4
= fQ : D ( QjjGm1;m2 ) · Em1;m2jm1;m2g; m1; m2 = 1 ; M; ( 4 )



1 1 2 On Multiple Hypotheses LAO Testing with Rejection of Decision for Two Dependent Objects

RM+1;M +1
4
= fQ : D ( QjjGm1;m2 ) > Em1;m2jm1;m2 ; m1; m2 = 1 ; Mg; ( 5 )

E¤m1;m2jm1;m2 = E
¤
m1;m2jm1;m2 ( Em1;m2jm1;m2 )

4
=

4
= Em1;m2jm1;m2; m1; m2 = 1 ; M; ( 6 )

E¤l1;l2jm1;m2 = E
¤
l1;l2jm1;m2 ( El1;l2jl1;l2 )

4
= in f

Q2R l1;l2
D ( QjjGm1;m2 ) ; l1; l2; m1; m2 = 1 ; M; ( m1; m2 ) 6= ( l1; l2 ) ( 7 )

E¤M +1;M+1jm1;m2 = E
¤
M+1;M+1jm1;m2 ( E1;1j1;1; E1;2j1;2; :::; EM;MjM;M )

4
= in f

Q2R M +1;M+1
D ( QjjGm1;m2 ) ; m1; m2 = 1 ; M: ( 8 )

L e t u s d e n o t e b y ( m1; m2 )
¡ t h e s e t o f a ll p a ir in d ic e s in r o w o f ( m1; m2 ) va r yin g fr o m ( 1 ; 1 )

t ill p r e vio u s o f ( m1; m2 ) a n d b y ( m1; m2 )
+ t h e s e t o f a ll p a ir in d ic e s in r o w o f ( m1; m2 )

va r yin g fr o m n e xt o f ( m1; m2 ) t ill ( M; M ) .

T heor em: If all distributions Gm1;m2 = fGm1;m2 ( x1; x2 ) ; x1; x2 2 X g, m1; m2 =
1 ; M , are di®erent in the sense that D ( Gl1;l2jjGm1;m2 ) > 0 , and the positive numbers
E1;1j1;1; E2;2j2;2; :::; EM;MjM;M are such that the following inequalities hold

E1;1j1;1 < m in
l1;l2=1;M ; (l1;l2) 6=(1;1)

D ( Gl1;l2jjG1;1 ) ; ( 9 )

Em1;m2jm1;m2 < m in [ m in
(l1;l2)2(m1;m2)¡

E¤l1;l2jm1;m2 ( El1;l2jl1;l2 ) ;

m in
(l1;l2)2(m1;m2)+

D ( Gl1;l2jjGm1;m2 ) ];

m1; m2 = 1 ; M; ( m1; m2 ) 6= ( 1 ; 1 ) ; ( m1; m2 ) 6= ( M; M ) ; ( 1 0 )

EM;MjM;M < m in
l1;l2=1;M ; (l1;l2) 6=(M;M )

E¤l1;l2jM;M ( El1;l2jl1;l2 ) ; ( 1 1 )

then there exists a LAO sequence of tests, all elements of the reliability matrix of which
E ¤ = fE¤l1;l2jm1;m2g are positive and are de¯ned in ( 6 ) ¡ ( 8 ) .

W hen one of the inequalities ( 9 ) ¡ ( 1 1 ) is violated, then at least one element of the matrix
( 6 ) ¡ ( 8 ) is equal to 0 .

Th e p r o o f o f t h e t h e o r e m c o n s is t s in p r e s e n t a t io n o f t h e p r o b le m fo r t wo o b je c t s a s a p r o b -
le m fo r o n e c a p a c io u s o b je c t . If we r e n u m e r a t e a s fo llo ws ( 1 ; 1 ) = 1 ; ( 1 ; 2 ) = 2 ; :::; ( 1 ; M ) =
M; ( 2 ; 1 ) = M + 1 ; :::; ( 2 ; M ) = 2 M; :::; ( M; M ) = M 2 a n d d e n o t e ( X1; X2 ) = Y , X £ X = Y
we will h a ve p r o b le m o f M 2 h yp o t h e s e s t e s t in g fo r o n e o b je c t wit h p o s s ib ilit y o f d e c is io n
r e je c t io n . S o u s in g t h is n u m e r a t io n we will h a ve t h e c o r r e s p o n d in g e r r o r p r o b a b ilit ie s a n d
r e lia b ilit ie s fo r l = 1 ; M 2 + 1 ; ; m = 1 ; M 2, wh e n we a p p ly Th e o r e m 2 o f [3 ].

Ge n e r a liz a t io n o f t h e r e s u lt is p o s s ib le in m a n y d ir e c t io n s .



E. Haroutunian, A. Yesayan and N. Harutyunyan 1 1 3

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Submitted 10.10.2017, accepted 15.02.2018.

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ëïáõ·Ù³Ý Ù³ëÇÝ »ñÏáõ ϳËÛ³É ûµÛ»ÏïÝ»ñÇ ¹»åùáõÙ

º. гñáõÃÛáõÝÛ³Ý, ². ºë³Û³Ý ¨ Ü. гñáõÃÛáõÝÛ³Ý

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гٳï»Õ Áݹѳï ѳí³Ý³Ï³Ý³ÛÇÝ µ³ßËٳٵ µÝáõó·ñíáÕ »ñÏáõ ϳËÛ³É
ûµÛ»ÏïÝ»ñÇ Ýϳïٳٵ ¹Çï³ñÏíáõÙ ¿ µ³½Ù³ÏÇ í³ñϳÍÝ»ñÇ áñáßáõÙÇó Ññ³Å³ñÙ³Ý
Ñݳñ³íáñáõÃÛ³Ùµ ëïáõ·áõÙÁ: àõëáõÙݳëÇñí»É ¿ ë˳ÉÇ Ñ³í³Ý³Ï³ÝáõÃÛáõÝÝ»ñÇ
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1 1 4 On Multiple Hypotheses LAO Testing with Rejection of Decision for Two Dependent Objects

Î LAO òåñòèðîâàíèè ìíîãèx ãèïîòåç ñ îòêàçîì îò ðåøåíèÿ
äëÿ äâóõ çàâèñèìûõ îáúåêòîâ

Å. Àðóòþíÿí, À. Åñàÿí è Í. Àðóòþíÿí

Àííîòàöèÿ

Äëÿ ìîäåëè ñîñòîÿùeé èç äâóõ çàâèñèìûõ îáúåêòîâ, xàðàêòåðèçóåìûx ñîâ-
ìåñòíûì äèñêðåòíûì ðàñïðåäåëåíèåì âåðîÿòíîñòåé ðàññìàòðèâàåòñÿ òåñòèðî-
âàíèå ìíîãèx ñòàòèñòè÷åñêèõ ãèïîòåç ñ âîçìîæíîñòüþ îòêàçà îò ðåøåíèÿ.
Èçó÷åíà ìàòðèöà àñèìïòîòè÷åñêè îïòèìàëüíûõ ýêñïîíåíò âåðîÿòíîñòåé îøèáîê
(íàäåæíîñòåé).