Haroutunian_37_43.DVI


Mathematical Problems of Computer Science 46, 37{43, 2016.

On Optimality of T ests of M -ar y H ypotheses for Fixed

N umber of I ndependent Obser vations

E vg u e n i A . H a r o u t u n ia n

Institute for Informatics and Automation Problems of NAS RA

e-mail: eghishe@sci.am

Abstract

The paper is devoted to the design of optimal approaches of testing of multiple
simple hypotheses with samples of a ¯xed number of independent observations.

Keywords: Statistical hypothesis testing, Error probability, Compairing of tests,
Optimal test, Neyman - Pearson approach.

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

W e a d d r e s s t h e c la s s ic a l d e t e c t io n p r o b le m . L e t M ¸ 2 s im p le h yp o t h e s e s , wh ic h we n o t e
H1, ..., HM , c o r r e s p o n d t o d is t r ib u t io n s G1; :::; GM , g ive n o n t h e ¯ n it e s p a c e X . It is n o t
kn o wn wh ic h h yp o t h e s is is t r u e . Th e o b s e r ve d s a m p le is a ve c t o r x = ( x1; x2; ; :::; xN ) in
N-d im e n t io n a l s p a c e X N o f N in d e p e n d e n t o b s e r va t io n s id e n t ic a lly d is t r ib u t e d a c c o r d in g t o
o n e o f h yp o t h e s e s . Th e s t a t is t ic ia n u s in g t h e o b t a in e d s a m p le m u s t g u e s s wh ic h h yp o t h e s is
is t r u e . It is n e c e s s a r y t o c h o o s e a n o p t im a l c r it e r io n fo r m a kin g s u c h a d e c is io n . Th e
p r o c e d u r e o f d e c is io n is c a lle d a t e s t .

D e ¯ n in g a t e s t c a n b e fo r m u la t e d a s p a r t it io n in g o f t h e s p a c e X N in t o M d is jo in t s e t s
X N1 ,..., X NM o n e a c h o f wh ic h o n e o f t h e h yp o t h e s e s H1, ..., HM will b e , r e s p e c t ive ly, a c c e p t e d .

Th e qu a lit y o f a t e s t ' m a y b e c h a r a c t e r iz e d b y t h e m a t r ix o f e r r o r p r o b a b ilit ie s

®ljm = ®ljm ( ') = Gm ( x 2 X Nl ) ; l; m = 1 ; M ; l 6= m;

®m = ®mjm ( ') = Gm ( x =2 X Nm ) ; m = 1 ; M;
wh e r e ®ljm is t h e p r o b a b ilit y t o a c c e p t Hl, wh e n Hm is t r u e , a n d ®mjm ( ') is t h e p r o b a b ilit y
o f r e je c t io n o f h yp o t h e s is Hm wh e n it is t r u e . E vid e n t ly

®m =
X

l 6=m
®ljm; m = 1 ; M:

R ic h lit e r a t u r e is d e d ic a t e d t o t h e p r o b le m o f h yp o t h e s is t e s t in g , we qu o t e [1 -3 ]. Th e c a s e
o f m u lt ip le h yp o t h e s e s t e s t in g is c o n s id e r e d in d e t a il in [4 ] a n d [5 ].

To c o m p a r e t e s t s t h e r e e xis t d i®e r e n t a p p r o a c h e s . A B a ye s a p p r o a c h is b a s e d o n a s s u m p -
t io n t h a t t h e o b s e r va t io n s a r e g o ve r n e d b y g ive n p r o b a b ilit ie s c a lle d t h e a p r io r i p r o b a b ilit ie s .
In t h is c a s e t h e s e t o f t e s t s is o r d e r e d b y t h e va lu e s o f t h e a ve r a g e p r o b a b ilit ie s o f e r r o r s o f
t e s t s [4 ].

3 7



3 8 On Optimality of Tests of M -ary Hypotheses for Fixed Number of Independent Observations

N e ym a n - P e a r s o n a p p r o a c h [6 ] p r im a r ily wa s fo r m u la t e d fo r t wo h yp o t h e s e s . In c a s e
o f M h yp o t h e s e s t h e e r r o r p r o b a b ilit ie s a r e ¯ xe d fo r m = 1 ; M ¡ 1 . Th e t e s t is fo u n d b y
m a xim iz in g t h e p r o b a b ilit y 1 ¡ ®M o f c o r r e c t d e t e c t io n o f t h e h yp o t h e s is HM .

A n o t h e r d ir e c t io n o f in ve s t ig a t io n s o f qu a lit y a n d o p t im a lit y o f t e s t s is t h e a s ym p t o t ic
a p p r o a c h wh e n t h e le n g t h o f s a m p le is n o t lim it e d a n d wh e n t h e vo lu m e n o f t h e s a m p le s
g o e s t o t h e in ¯ n it y, e r r o r p r o b a b ilit ie s ®ljm d e c r e a s e e xp o n e n t ia lly a s 2

¡nEljm a n d t h e in t e r -
d e p e n d e n c e o f t h e s e e xp o n e n t ia l c o e ± c ie n t s Eljm, c a lle d r e lia b ilit ie s , c h a r a c t e r iz e t h e t e s t .
Th is d ir e c t io n is fo u n d e d b y H o e ®d in g [7 ] a n d t h e c o r r e s p o n d in g s e qu e n c e o f t e s t s is c a lle d
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 ( L A O) , o r e xp o n e n t io n a l r a t e o p t im a l ( E R O) [7 ,8 ].
Th e p r o b le m s o f L A O t e s t s fo r m u lt ip le h yp o t h e s e s fo r d ive r s e m o d e ls we r e s t u d ie d in [9 -1 4 ].

In t h e m in im a x a p p r o a c h t h e c o m p a r is io n o f m a xim a l va lu e s o f e r r o r p r o b a b ilit ie s o f a
t e s t a ls o a llo ws t o o r d e r t h e s e t o f a ll t e s t s .

Th e in t e r c h a n g e o f id e a s a n d m e t h o d s o f S t a t is t ic s a n d In fo r m a t io n t h e o r y is p r e s e n t e d
in [1 5 ,1 6 ].

Ou r p u r p o s e in t h is p a p e r is c o m p a r in g d i®e r e n t t e s t s a n d in t r o d u c in g a n o r d e r o n t h e
s e t o f a ll t e s t s fo r M ¸ 2 h yp o t h e s e s wh e n b o t h , lis t o f h yp o t h e t ic a l d is t r ib u t io n s a n d t h e
vo lu m e N o f s a m p le s , a r e ¯ xe d . W e p r o c e e d a n a lo g ic a lly a s in [4 ], wh e r e t h e B a ye s ia n t e s t s
a r e s t u d ie d .

2 . Fo r m u la t io n o f R e s u lt s

W e b e g in b y c o n s id e in g t h e c a s e o f M h yp o t h e t ic a l d is t r ib u t io n s wh e n a p r io r i p r o b a b ilit ie s
a r e a b s e n t . It is c le a r t h a t t h e t e s t is b e t t e r if it s e r r o r p r o b a b ilit e s a r e s m a lle r .
De¯nition 1: An average error probability ®( ') for test ' is

®( ') =
1

2 M

X

l;m

®ljm =
1

M

MX

1

®m:

De¯nition 2: The test '0 we call optimal if ®( '0 ) = m in
'

®( ') :

N o w it is p o s s ib le t o a r r a n g e t h e s e t o f t e s t s ' a c c o r d in g t o t h e va lu e s o f t h e a va r a g e
e r r o r p r o b a b ilit ie s ®( ') . In t h e fo llo win g t h e o r e m we s t a t e t h e c o n s t r u c t io n o f o p t im a l t e s t .

L e t p r o b a b ilit y d is t r ib u t io n s Gm, m = 1 ; M , h a ve d e n s it ie s fm ( x ) wit h r e s p e c t t o s o m e

m e a s u r e ¹, t h e n t h e like lih o o d fu n c t io n is fm ( x) =
NQ

n=1
fm ( xn ) ; m = 1 ; M:

T heor em 1:
1. The average probability of error ®( ') of any test ' satis¯es the inequality

®( ') ¸ 1 ¡ 1
M

Z
m a x

m
fm ( x) ¹

n ( dx ) : ( 1 )

2. In order the test '0 be optimal it is necessary and su± cient that for almost all with

respect to measure G ( G( x ) = 1
M

MP
m=1

Gm ( x ) ) values of x the test satis¯es the relations

'0 ( x ) = Hm; if fm ( x) = m a x
l

fl ( x ) : ( 2 )

F or such '0 relation (1) becomes an equality.



E. Haroutunian 3 9

It is n e c e s s a r y t o n o t e t h a t wh e n t wo o r m o r e va lu e s fm ( x ) a r e m a xim a l it d o e s n o t
m a t t e r wh ic h o f t h e s e h yp o t h e s e s t o c h o o s e a n d we c a n m a ke a r a n d o m d e c is io n , t h is will
n o t c h a n g e t h e va lu e o f ®( '0 ) .

It is n o t d i± c u lt t o a p p ly t h is c r it e r io n t o t wo p a r t ic u la r e xt r e m a l c a s e s . If a ll d e n s it ie s
a r e e qu a l f1 ( x) = :::: = fM ( x) it is n o t p o s s ib le t o c o n s t r u c t a t e s t wit h ® ( '

0 ) le s s t h a n

M¡1
M

. In t h e o t h e r c a s e , wh e n X N =
MS

m=1
X Nm a n d Gm ( X Nm ) = 1 , Gm ( X Nl ) = 0 , m = 1 ; M,

l 6= m, t h e b e s t t e s t h a s ® ( '0 ) = 0 .
P r oof: 1 . B y D e ¯ n it io n 1 fo r e a c h t e s t ' fo r M h yp o t h e s e s t h e a va r a g e e r r o r p r o b a b ilit y o f
t h e t e s t ' is

® ( ') =
1

M

MX

m=1

®m;

wh e r e
®m = Gm ( '( x ) 6= m) =

Z

x:'(x) 6=m

fm ( x ) ¹
n ( dx ) :

Th e n

® ( ') =
1

M

MX

m=1

Z

x:'(x)6=m

fm ( x) ¹
n ( dx ) =

= 1 ¡ 1
M

MX

m=1

Z

x:'(x)=m

fm ( x ) ¹
n ( dx ) ¸

¸ 1 ¡ 1
M

Z
m a x

m
fm ( x ) ¹

n ( dx ) :

2 . Op t im a l t e s t '0 d e ¯ n e d in ( 2 ) r e a c h e s t h e lo we r b o u n d in ( 1 ) , t h a t is c o n d it io n ( 2 ) is
s u ± c ie n t . To p r o ve t h e n e c e s s it y o f ( 2 ) s u p p o s e t h a t t h e o p t im a l t e s t ' is s u c h t h a t '( x) =
Hm wit h fm ( x ) < m a xl fl ( x ) fo r x 2 A wit h a s e t A o f p o s it ive p r o b a b ilit y G ( A) > 0 . S u c h
t e s t ' c a n b e im p r o ve d o n A b y g ivin g fo r x 2 A, '0 ( x) = Hm1 , wit h fm1 ( x ) = m a xl fl ( x ) .

R e a lly ® ( '0 ) < ®( ') b e c a u s e

®( ') ¡ ®( '0 ) = 1
M

Z

A
[fm1 ( x ) ¡ fm ( x) ]¹n ( dx) > 0 :

N o w we s h a ll c o n s id e r t h e m in im a x a p p r o a c h . W e will c o m p a r e t e s t s wit h m a xim a l va lu e s
o f e r r o r p r o b a b ilit ie s .
De¯nition 3: D enote maximal error probability in the matrix for each test ' by

®( ') = m a x
m

®mjm ( ') :

It is c le a r t h a t it is p o s s ib le t o p u t in o r d e r t e s t s b y va lu e s o f ® ( ') . W e n a m e m in im a x
t h e t e s t ' wh ic h h a s t h e m in im a l va lu e o f ®( ')

®( ') = m in
'

® ( ') :

T heor em 2: The optimal test ' such that

®1 ( ') = ®2 ( ') = ::: = ®M ( ') ( 3 )



4 0 On Optimality of Tests of M -ary Hypotheses for Fixed Number of Independent Observations

will be a minimax test.
P r oof: Fo r e ve r y t e s t ' u s in g ( 3 ) we h a ve

®( ') ¸
MX

m=1

1

M
®m ( ') ¸

MX

m=1

1

M
®m ( ') = ®( ') :

N o w we t a ke in t o c o n s id e r a t io n t h e n o t io n o f t h e m o s t p o we r fu l t e s t in a c la s s o f t e s t s .
In o r d e r t o c o m p a r e t e s t s it is p o s s ib le t o in t r o d u c e t h e c la s s e s h a vin g g ive n va lu e s o f e r r o r
p r o b a b ilit ie s ®1, ..., ®M ¡1

K®1;:::;®M¡1 = f' : ®m ( ') = ®m; m = 1 ; M ¡ 1 g

a n d t h e n o r d e r t e s t s b y t h e va lu e s o f ®M ( ') , n a t u r a lly t h e s m a lle r ®m ( ') c o r r e s p o n d s t o t h e
b e t t e r t e s t .
De¯nition 4: A test e' 2 K®1;:::;®M¡1 is the most powerful test in the class K®1;:::;®M¡1 if for
any ' from the same class

®m ( e') · ®m ( ') :
If M = 2 t h e o p t im a l t e s t e' is g ive n b y t h e Fu n d a m e n t a l L e m m a o f N e ym a n -P e a r s o n

[2 ,3 ,6 ,1 6 ]. Fo r m o r e t h a n t wo h yp o t h e s e s M ¸ 2 t h e s o lu t io n c a n b e fo u n d b y g e n e r a liz a t io n
o f t h e N e ym a n -P e a r s o n L e m m a [1 7 ]. Th is im p o r t a n t r e s u lt d e s e r ve s t o b e r e c u r e d .

A s it wa s n o t e d in [1 5 ] t h e c a s e N = 1 c o n t a in s t h e g e n e r a l o n e a n d t h e r e is n o n e e d t o
r e s t r ic t a t t e n t io n t o m u lt ip le in d e p e n d e n t d r a win g s .

Fo r g ive n p r e a s s in g n e d va lu e s 0 < ®¤1j1; ®
¤
2j2; :::; ®

¤
M¡1jM ¡1 < 1 we c h o o s e n u m b e r s

T1; T2; :::; TM ¡1 a n d s e t s A¤m, m = 1 ; M, s u c h t h a t

A¤1 =
(
x : m in

Ã
G1 ( x)

G2 ( x)
; :::;

G1 ( x)

GM ( x)

!
> T1

)
; 1 ¡ G1 ( A¤1 ) = ®¤1j1;

A¤2 = A¤1 \
(
x : m in

Ã
G2 ( x )

G3 ( x )
; :::;

G2 ( x )

GM ( x )

!
> T2

)
; 1 ¡ G2 ( A¤2 ) = ®¤2j2;

...........

A¤M¡1 = A¤1 \A 2 ¤ \ :::: \ A¤M ¡2 \
(
x :

GM¡1 ( x )

GM ( x)
) > TM¡1

)
; 1 ¡GM¡1 ( A¤M ¡1 ) = ®¤M ¡1jM¡1;

a n d

A¤M = X N ¡ ( A¤1 [ A¤2 [ ::: [ A¤M ¡1 ) = A¤1 \ A 2 ¤ \ :::: \ A¤M¡2 \ A¤M¡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 d e n o t e d b y

®¤ljm ( 'N ) ; l; m = 1 ; M :

T heor em 3: ( Ge n e r a liz a t io n o f N e ym a n -P e a r s o n L e m m a )
The test determined by sets A¤1, A¤2, ..., A¤M is optimal in the sense that, for each other test

de¯ned by the sets B1, B2, ..., BM with the corresponding error probabilities ¯ljm, l; m = 1 ; M ,
if ¯mjm · ®¤mjm, for some m 2 [1 ; M ¡ 1 ], then there exists at least one index j, j 2 [m+1 ; M]
such that ¯mjj ¸ ®¤mjj:



E. Haroutunian 4 1

P r oof: L e t ©A ¤m a n d © B m b e t h e in d ic a t o r fu n c t io n s o f t h e d e c is io n r e g io n s A¤m a n d Bm; m =
1 ; M . Fo r a ll x = ( x1; x2; :::; xN ) 2 X N , t h e fo llo win g in e qu a lit y is c o r r e c t

( ©A ¤m ( x) ¡ ©B m ( x) ) ( Gm ( x) ¡ m a x( TmGm+1 ( x) ; :::; TmGm ( x ) ) ) ¸ 0 :

Mu lt ip lyin g a n d t h e n s u m m in g o ve r X N we o b t a in
X

x:x2X N

h
© A ¤m ( x ) Gm ( x ) ¡ ©A ¤m ( x) m a x( TmGm+1 ( x) ; :::; TmGm ( x ) ) ¡

¡©B m ( x) Gm ( x) + ©B m ( x ) m a x( TmGm+1 ( x ) ; :::; TmGm ( x ) ) ¸ 0 ] ;
X

x:x2A m¤
[Gm ( x ) ¡ Tm m a x( Gm+1 ( x ) ) ; :::; GM ( x ) ] ¡

¡
X

x:x2B m
[Gm ( x) ¡ Tm m a x( Gm+1 ( x ) ) ; :::; GM ( x) ] ¸ 0 :

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 y we o b t a in t h e fo llo win g in e qu a lit y

1 ¡ ®¤mjm ¡ Tm m a x( ®¤mjm+1; :::; ®¤mjM ) ¡ ( 1 ¡ ¯mjm ) + Tm m a x( ¯mjm+1; :::; ¯mjM ) ¸ 0 ;

¡¯mjm + ®¤mjm · Tm
h
¡ m a x( ®¤mjm+1; :::; ®¤mjM ) + m a x( ¯mjm+1; :::; ¯mjM )

i
:

W e s e e n o w t h a t fr o m ¯mjm · ®¤mjm it fo llo ws t h a t

m a x( ¯mjm+1; :::; ¯mjM ) ¸ m a x( ®¤mjm+1; :::; ®¤mjM ) :

Fr o m t h is it fo llo ws t h a t if t h e m a xim a l is ¯mjj , j 2 [m + 1 ; M], t h e n ¯mjj ¸ ®¤mjj.

Disscussion: Th is p a p e r d e a ls wit h s o m e c e n t r a l b a s ic r e s u lt s o f t h e Th e o r y o f t e s t in g
s t a t is t ic a l h yp o t h e s e s o u g h t t o b e in c lu d e d in t e xt b o o ks . D i®e r e n t a p p r o a c h e s o f c o n s t r u c t -
in g o p t im a l t e s t s fo r ¯ n it e ly m a n y s im p le h yp o t h e s e s b y a p p lic a t io n o f s a m p le s o f ¯ xe d le n g t h
a r e p r e s e n t e d .

A c kn o wle d g m e n t

Th e a u t h o r t h a n ks t h e r e vie we r fo r s u g g e s t io n s o n im p r o vin g t h e e xp o s it io n o f t h e m a t e r ia l.

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4 2 On Optimality of Tests of M -ary Hypotheses for Fixed Number of Independent Observations

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Submitted 12.08.2016, accepted 10.11.2016.

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E. Haroutunian 4 3

Îá îïòèìàëüíîñòè òåñòîâ M-àðíûõ ãèïîòåç ïðè
ôèêñèðîâàííîì ÷èñëå íåçàâèñèìûõ íàáëþäåíèé

Å. Àðóòþíÿí

Àííîòàöèÿ

Ñòàòüÿ ïîñâÿùåíà ïîñòðîåíèþ îïòèìàëüíûõ ïîäõîäîâ ê òåñòèðîâàíèþ
ìíîãèõ ïðîñòûõ ãèïîòåç ïðè ôèêñèðîâàííîì ÷èñëå íåçàâèñèìûõ íàáëþäåíèé.