Parandzem.DVI


Mathematical Problems of Computer Science 46, 103{106, 2016.

On N eyman-P ear son T esting for P air of I ndependent

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

Institute for Informatics and Automation Problems of NAS RA

e-mail: eghishe@ipia.sci.am, par h@ipia.sci.am

Abstract

The Neyman-Pearson principle for a model consisting of two independent objects
is considered. It is supposed that two probability distributions are known and each
object follows one of them independently. Two approaches of Neyman-Pearson test-
ing are considered for this model. The aim is to compare error probabilities to the
corresponding two cases.

Keywords: Independent objects, Statistical hypotheses, Neyman-Pearson testing,
Error probability.

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

In t h is p a p e r we d is c u s s N e ym a n -P e a r s o n h yp o t h e s e s t e s t in g p r o b le m fo r a m o d e l c o n s is t in g
o f t wo in d e p e n d e n t o b je c t s . Th is m o d e l wa s p r o p o s e d b y A h ls we d e a n d H a r o u t u n ia n [1 ].
Th e c h a r a c t e r is t ic s o f t h e o b je c t s a r e X1 a n d X2 in d e p e n d e n t r a n d o m va r ia b le s ( R V s ) t a kin g
va lu e s in t h e s a m e ¯ n it e s e t X . S o , t h a t c o n s id e r e d m o d e l is d e s c r ib e d b y t h e r a n d o m ve c t o r
( X1; X2 ) , wh ic h a s s u m e s va lu e s ( x

1; x2 ) 2 X £ X .
Th e N e ym a n -P e a r s o n le m m a is t h e fo u n d a t io n o f t h e m a t h e m a t ic a l t h e o r y o f s t a t is t ic a l

h yp o t h e s is t e s t in g . Th is p r in c ip le p la ys a c e n t r a l r o le in t h e t h e o r y a n d p r a c t ic e o f s t a t is t ic s .
Th e r e e xis t m a n y wo r ks wh e r e t h e N e ym a n -P e a r s o n le m m a is e xp o u n d e d fo r t h e c a s e o f t wo
h yp o t h e s e s [2 ]{ [1 0 ]. Th is p r in c ip le t o t h e c a s e o f m u lt ip le h yp o t h e s e s is s o lve d in [1 1 , 1 2 ].

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

L e t X1 a n d X2 b e in d e p e n d e n t r a n d o m va r ia b le s t a kin g va lu e s in t h e s a m e ¯ n it e s e t X ,
e a c h o f t h e m wit h o n e o f t wo h yp o t h e t ic a l p r o b a b ilit y d is t r ib u t io n s Gm = fGm ( x ) ; x 2 X ,
m = 1 ; 2 .

L e t ( x1; x2 ) = ( ( x
1
1; x

2
1 ) ; :::; ( x

1
n; x

2
n ) ; :::; ( x

1
N ; x

2
N ) ) , x

i
n 2 X , i = 1 ; 2 , n = 1 ; N, b e a

s e qu e n c e 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 n e c e s s a r y t o
d e ¯ n e u n kn o wn P D s o f t h e o b je c t s o n t h e b a s e o f t h e o b s e r ve d d a t a .

Th e d e c is io n fo r e a c h o b je c t m u s t b e m a d e fr o m t h e s a m e s e t o f h yp o t h e s e s : Hm : G =
Gm, m = 1 ; 2 . Th e r e a r e fo u r h yp o t h e t ic a l p r o b a b ilit y d is t r ib u t io n s fo r r a n d o m ve c t o r
( X1; X2 ) :

1 0 3



1 0 4 On Neyman-Pearson Testing for Pair of Independent Objects

G1 ± G1 ( x1; x2 ) = fG1 ( x1 ) _G1 ( x2 ) ; ( x1; x2 ) 2 X £ X g,
G1 ± G2 ( x1; x2 ) = fG1 ( x1 ) _G2 ( x2 ) ; ( x1; x2 ) 2 X £ X g,
G2 ± G1 ( x1; x2 ) = fG2 ( x1 ) _G1 ( x2 ) ; ( x1; x2 ) 2 X £ X g
a n d
G2 ± G2 ( x1; x2 ) = fG2 ( x1 ) _G1 ( x2 ) ; ( x1; x2 ) 2 X £ X g.
W e c a ll t h e p r o c e d u r e o f m a kin g d e c is io n o n t h e b a s e o f N o b s e r va t io n s t h e t e s t . It c a n

b e d e ¯ n e d b y d ivis io n o f t h e s a m p le s p a c e X N £ X N o n 4 d is jo in t s u b s e t s BNi;j, i; j = 1 ; 2 .
Th e s e t BNi;j c o n s is t s o f 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 Gi ± Gj is a d o p t e d .

L e t ®l1;l2jm1;m2 = Gm1 ±GNm2 ( B
N
l1;l2

) 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 Gl1 ±Gl2
b y t h e t e s t p r o vid e d t h a t Gm1 ± Gm2 is t r u e , wh e r e ( l1; l2 ) 6= ( m1; m2 ) , li; mi = 1 ; 2 , i = 1 ; 2 .
Th e p r o b a b ilit y t o r e je c t a t r u e d is t r ib u t io n Gm1 ± Gm2 is a s fo llo ws :

®m1;m2jm1;m2 =
X

(l1;l2)6=(m1;m2)
®l1;l2jm1;m2 :

Th e m a t r ix o f e r r o r p r o b a b ilit ie s is a s fo llo ws :

0
BBB@

®1;1j1;1 ®1;2j1;1 ®2;1j1;1 ®2;2j1;1
®1;1j1;2 ®1;2j1;2 ®2;1j1;2 ®2;2j1;2
®1;1j2;1 ®1;2j2;1 ®2;1j2;1 ®2;2j2;1
®1;1j2;2 ®1;2j2;2 ®2;1j2;2 ®2;2j2;2

1
CCCA

N o w we will c o n s id e r N e ym a n -P e a r s o n t e s t in g fo r t h is m o d e l wit h t wo a p p r o a c h e s : a )
d ir e c t m e t h o d a n d b ) r e n u m b e r in g m e t h o d o f h yp o t h e s e s p a ir s .

Ou r a im is t o c o m p a r e 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 o f t h o s e t wo a p p r o a c h e s .
L e t u s d e s c r ib e t h o s e c a s e s .
a ) D ir e c t a p p r o a c h
L e t ®I1j1 a n d ®

I
1j1 b e e r r o r p r o b a b ilit ie s o f t h e ¯ r s t a n d t h e s e c o n d o b je c t s , r e s p e c t ive ly.

A c c o r d in g t o fu n d a m e n t a l N e ym a n -P e a r s o n le m m a , we c a n c h o o s e T I a n d T II p o s it ive n u m -
b e r s a n d we c a n d ivid e t h e s a m p le s p a c e s o f t h e ¯ r s t a n d t h e s e c o n d o b je c t s a s fo llo ws :

1 ) Fo r t h e ¯ r s t o b je c t :

AI1 =
(
x1 :

GN1 ( x1 )

GN2 ( x1 )
> T I

)
; 1 ¡ GN1 ( AI1 ) = ®I1j1;

a n d

AI2 = X N =AI1:
2 ) Fo r t h e s e c o n d o b je c t :

AII1 =
(
x2 :

GN1 ( x2 )

GN2 ( x2 )
> T II

)
; 1 ¡ GN1 ( AII1 ) = ®II1j1;

a n d

AII2 = X N =AII1 :
Fin a lly, t h e N e ym a n -P e a r s o n d ivis io n o f t h e s a m p le s p a c e X N £ X N is t h e fo llo win g :

A1;1 = AI1 £ AII1 ; A1;2 = AI1 £ AII2 ; A2;1 = AI2 £ AII1 ; A2;2 = AI2 £ AII2 :



E. Haroutunian and P. Hakobyan 1 0 5

A c c o r d in g t o t h e d e ¯ n it io n o f e r r o r p r o b a b ilit ie s a n d in d e p e n d e n t s o f o b je c t s we c a n s e e
t h a t t h e e r r o r p r o b a b ilit ie s o f t h is t e s t a r e fo r m u la t e d a s fo llo ws :

®l1;l2jm1;m2 = G
N
m1

± GNm2 ( A
N
l1;l2

) =
X

x 1 2A Nl1

Gm1 ( x1 ) £
X

x2 2A Nl2

Gm2 ( x2 ) :

b ) R e n u m b e r in g a p p r o a c h
H e r e we h a ve t o r e n u m b e r t h e e a c h p a ir o f h yp o t h e s e s o n e a t a t im e a n d we h a ve t o

a p p ly N e ym a n -P e a r s o n le m m a fo r m u lt ip le h yp o t h e s e s , wh ic h is in ve s t ig a t e d in [1 2 ].
In t h a t c a s e fo r t h e g ive n p o s it ive va lu e s ®¤1;1j1;1, ®

¤
1;2j1;2 a n d ®

¤
2;1j2;1 a n d c h o s e n n u m b e r s

T1, T2 a n d T3 s e t s Ai;j, i; j = 1 ; 2 , a r e t h e fo llo win g :

A1;1 =
(

( x1; x2 ) : m in

Ã
GN1 ( x1 )

GN2 ( x2 )
;
GN1 ( x2 )

GN2 ( x2 )
;
GN1 ( x1 ) ¢ GN1 ( x2 )
GN2 ( x1 ) ¢ GN2 ( x2 )

!
> T1

)
; 1 ¡GN1 ±GN1 ( A1;1 ) = ®¤1;1j1;1;

A1;2 = A1;1
(

( x1; x2 ) : m in

Ã
GN1 ( x1 )

GN2 ( x2 )
;
GN1 ( x1 ) ¢ GN2 ( x2 )
GN2 ( x1 ) ¢ GN1 ( x2 )

!
> T2

)
; 1 ¡GN1 ±GN2 ( A1;2 ) = ®¤1;2j1;2

A2;1 = A1;1 \ A1;2
(

( x1; x2 ) :
GN1 ( x2 )

GN2 ( x2 )
> T3

)
; 1 ¡ GN2 ± GN1 ( A2;1 ) = ®¤2;1j2;1

a n d

A¤2;2 = A1;1 \ A1;2 \ A2;1:
Th 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 r e fo r m u la t e d a s a b o ve .
L e t u s a s s u m e t h a t we h a ve fo u n d t h e e r r o r p r o b a b ilit ie s b y d ir e c t a p p r o a c h . B y c o n s id e r -

in g t h e fo llo win g ®1;1j1;1, ®1;2j1;2 a n d ®2;1j2;1 e r r o r p r o b a b ilit ie s a n d a p p lyin g t h e r e n u m b e r in g
a p p r o a c h we will ¯ n d a ll t h e o t h e r e r r o r p r o b a b ilit ie s . B e c a u s e in [1 2 ] it is p r o ve d t h a t t h e s e
e r r o r p r o b a b ilit ie s a r e t h e s m a lle s t . H e n c e , we c a n in s is t t h a t t h e r e n u m b e r in g a p p r o a c h o f
N e ym a n -P e a r s o n t e s t in g is t h e b e s t m e t h o d .

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 ] J. N e ym a n a n d E . S . P e a r s o n , \ On t h e p r o b le m o f t h e m o s t e ± c ie n t t e s t s o f s t a t is t ic a l
h yp o t h e s e s " , P h il. Tr a n s . R o y. S o c . L o n d o n , S e r . A , 2 3 1 , p p . 2 8 9 -3 3 7 , 1 9 3 3 .

[3 ] J. N e ym a n , F irst Course in P robability and Statistics, H o lt , R in e h a r t a n d W in s t o n ,
N e w Y o r k, 1 9 5 0 .

[4 ] E . L . L e h m a n a n d J.P . R o m a n o , Testing statistical hypotheses, Th ir d E d it io n . S p r in g e r ,
N e w Y o r k, 2 0 0 5 .

[5 ] A . A . B o r o vko v, M athematical Statistics, in R u s s ia n , N a u ka , Mo s c o w, 1 9 9 7 .
[6 ] H . L . V a n Tr e e s , D etection, E stimation and M odulation Theory, P a r t 1 . N e w Y o r k:

W ile y, 1 9 6 8 .
[7 ] M. H . D e Gr o o t , P robability and Statistics, 2 n d e d ., R e a d in g , MA , A d d is o n -W e s le y,

1 9 8 6 .
[8 ] M. G. K e n d a ll a n d A . S t u a r t , The Advanced Theory of Statistics, 2 , In fe r e n c e a n d

r e la t io n s h ip , Th ir d e d it io n . H a fn e r p u b lis h in g c o m p a n y, L o n d o n , 1 9 6 1 .



1 0 6 On Neyman-Pearson Testing for Pair of Independent Objects

[9 ] A . K . B e r a , \ H yp o t h e s is t e s t in g in t h e 2 0 -t h c e n t u r y wit h a s p e c ia l r e fe r e n c e t o t e s t in g
wit h m is s p e c i¯ e d m o d e ls " , In : \ S t a t is t ic s fo r t h e 2 1 -s t c e n t u r y. Me t h o d o lo g ie s fo r
a p p lic a t io n s o f t h e Fit ir e " , Ma r c e l D e kke r , In c ., N e w Y o r k, B a s e l, p p . 3 3 -9 2 , 2 0 0 0 .

[1 0 ] T. M. Co ve r a n d J. A . Th o m a s , E lements of Information Theory, S e c o n d E d it io n .
W ile y, N e w Y o r k, 2 0 0 6 .

[1 1 ] E . A . H a r o u t u n ia n , \ N e ym a n -P e a r s o n p r in c ip le fo r m o r e t h a n t wo h yp o t h e s e s " , Ab-
stracts of Armenian M athematical Union Annual Session D edicated to 90 Anniversary
of R afael Alexandrian, Y e r e va n , p p .4 9 { 5 0 , 2 0 1 3 .

[1 2 ] E . A . H a r o u t u n ia n a n d P . H a ko b ya n , \ N e ym a n -P e a r s o n p r in c ip le fo r m o r e t h a n t wo
h yp o t h e s e s " , M athematical P roblems of Computer Science vo l. 4 0 , p p . 3 4 { 3 8 , 2 0 1 3 .

Submitted 24.08.2016, accepted 07.11.2016.

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ï»ëï³íáñÙ³Ý Ù³ëÇÝ

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äÇñëáÝÇ ëϽµáõÝùÁ: ºÝó¹ñíáõÙ ¿, áñ ѳÛïÝÇ »Ý »ñÏáõ ѳí³Ý³Ï³Ý³ÛÇÝ µ³ßËáõÙÝ»ñ
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ѳí³Ý³Ï³ÝáõÃÛáõÝÝ»ñÁ:

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íåçàâèñèìûõ îáúåêòîâ

Å. Àðóòþíÿí è Ï. Àêîïÿí

Àííîòàöèÿ

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