D:\sbornik\...\Article.DVI


Mathematical Problems of Computer Science 40, 23{30, 2013.

On Application of Optimal M ultihypothesis T ests for

the B ounds Constr uction for the I ncr ease of Gr owth

Rate of Stock M ar ket

E vg u e n i A . H a r o u t u n ia n , A r t a k R . Ma r t ir o s ya n a n d A r a m O. Y e s s a ya n

Institute for Informatics and Automation Problems of NAS RA
Yerevan, Armenia

e-mail: eghishe@ipia.sci.am

Abstract

In the present paper new bounds for increase of the growth rate of stock market
(when an erroneous probability distribution is used) are obtained applying the matrix
of reliabilities of optimal tests.

Keywords: Hypothesis testing, Stock market, Growth rate, Portfolio.

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

Th e m o d e l o f s t o c k m a r ke t c o n s id e r e d in t h is p a p e r is p r e s e n t e d in [1 ], wh e r e t h e b o u n d
fo r t h e in c r e a s e o f t h e g r o wt h r a t e o f s t o c k m a r ke t in c a s e o f c o n t in u o u s d is t r ib u t io n s is
c o n s t r u c t e d . Th is m o d e l a n d t h e c o r r e s p o n d in g o p t im a l p o r t fo lio p r o b le m s we r e d e e p ly in -
ve s t ig a t e d in [1 ] { [8 ].

Th e a p p lic a t io n o f r e s u lt s o n t h e h yp o t h e s is 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 im a l
( L A O) t e s t in g ( [9 ] { [1 3 ]) in p o r t fo lio t h e o r y is in t r o d u c e d in [1 1 ], wh e r e u s in g e r r o r p r o b -
a b ilit ie s a n d r a lia b ilit ie s , n e w, m o r e e xa c t b o u n d s a r e fo u n d fo r t h e in c r e a s e o f t h e g r o wt h
r a t e o f s t o c k m a r ke t . U s in g t h e e xis t e n c e t h e o r e m o f L A O t e s t [1 0 ] b o u n d s o b t a in e d in [1 1 ]
a r e r e ¯ n e d in t h e p r e s e n t p a p e r .

W e s t u d y a s t o c k m a r ke t wit h K ( ¸ 1 ) s t o c ks d e ¯ n e d b y a r a n d o m ve c t o r X =
( X1; X2; :::; XK ) , Xk ¸ 0 ; k = 1 ; K; wh e r e t h e p r ic e r e la t ive s ( t h e r a t io o f t h e p r ic e a t
t h e e n d o f t h e d a y t o t h e p r ic e a t t h e b e g in n in g o f t h e d a y) Xk, k = 1 ; K; a r e 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 ) wit h u n kn o wn p r o b a b ilit y d is t r ib u t io n s ( P D s ) d e ¯ n e d o n t h e ¯ n it e
s e t X .

W e a s s u m e t h a t M d is t in c t P D s Gm; m = 1 ; M , a r e g ive n o n X wh ic h a r e p o s s i-
b le P D s fo r R V s Xk; k = 1 ; K. If t h e r ig h t P D o f t h e R V Xk is Gmk ; k = 1 ; K, t h e n

t h e jo in t P D o f t h e s t o c k m a r ke t ve c t o r X will b e Gm1;m2;:::;mK ( x) =
KQ

k=1
Gmk ( xk ) ; wh e r e

x = ( x1; x2; :::; xK ) 2 X K :
P o r t fo lio ve c t o r b = ( b1; b2; :::; bK ) , bk ¸ 0 , k = 1 ; K,

P
bk = 1 , is a n a llo c a t io n o f t h e

we a lt h a c r o s s t h e s t o c ks , wh e r e bk is t h e fr a c t io n o f o n e 's we a lt h in ve s t e d in s t o c k k [1 ].

2 3



2 4 On Application of Optimal Multihypothesis Tests for the Bounds Construction of Stock Market

W h e n a n in ve s t o r u s e s a p o r t fo lio b fo r s t o c k m a r ke t X, t h e n t h e we a lt h r e la t ive will b e
S = btX ( bt is t h e t r a n s p o s e d ve c t o r o f b) : A n d if t h e in ve s t o r in ve s t s in t h e s t o c k m a r ke t
fo r N c o n s e c u t ive d a ys a n d u s e s t h e s a m e b e t t in g s t r a t e g y b e a c h d a y, t h e n t h e we a lt h a t

t h e e n d o f N d a ys is g ive n b y SN =
NQ

n=1
btXn; wh e r e t h e s t o c k m a r ke t ve c t o r s X1; X2; ::: a r e

in d e p e n d e n t a n d id e n t ic a lly d is t r ib u t e d wit h P D Gm1;m2;:::;mK ( x ) :
Th e o p t im a l g r o wt h r a t e is W ¤ ( G ) = m a x

b
W ( b; G ) , wh e r e W ( b; G ) = E ( lo g btX ) is t h e

g r o wt h r a t e o f a s t o c k m a r ke t a t p o r t fo lio b = ( b1; b2; :::; bK ) wit h r e s p e c t t o a s t o c k P D
Gm1;m2;:::;mK ( if t h e lo g a r it h m is t o b a s e 2 , t h e g r o wt h r a t e is c a lle d a d o u b lin g r a t e [1 ]) .
Th e n a p o r t fo lio b¤ = A r g ( m a x

b
W ( b; G ) ) , is c a lle d a lo g { o p t im a l p o r t fo lio o r a g r o wt h

o p t im a l p o r t fo lio .
In t h is p a p e r we a s s u m e t h a t P D Gm1;m2;:::;mK is u n kn o wn a n d m u s t b e d e t e r m in e d o n t h e

b a s e o f s a m p le ( 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 ) xn = ( x1;n; x2;n; : : : ; xK;n ) , n = 1 ; N;
wh e r e xk;n is a r e s u lt o f t h e n { t h r e a liz a t io n o f t h e p r ic e r e la t ive Xk; k = 1 ; K: W it h t h e
s a m p le xN = ( x1; x2; :::; xN ) we c a n c o n s id e r t h e e m p ir ic a l P D o f t h e s t o c k m a r ke t a ft e r N
d a ys :

G(N)m1;m2;:::;mK ( x
N ) =

NY

n=1

Gm1;m2;:::;mK ( xn ) =
NY

n=1

KY

k=1

Gmk ( xk;n ) :

W e c o n s id e r t h e c a s e , wh e n e a c h P D fr o m t h e g ive n P D s Gm c a n b e a d o p t e d fo r s o m e
p r ic e r e la t ive s s im u lt a n e o u s ly.

Th e p r o c e d u r e o f d e c is io n m a kin g is a n o n -r a n d o m iz e d t e s t '(N); wh ic h c a n b e d e ¯ n e d
b y a d ivis io n o f t h e s a m p le s p a c e X KN in t o M K d is jo in t s u b s e t s

A(N)l1;l2;:::;lK = fx
N : '(N) ( xN ) = ( l1; l2; :::; lK ) g; lk = 1 ; M; k = 1 ; K:

Th 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 h yp o t h e s is Hl1;l2;:::;lK p r o vid e d t h a t
Hm1;m2;:::;mK is t r u e , fo r ( l1; l2; :::; lK ) 6= ( m1; m2; :::; mK ) ; is [9 ]

®
(N )
l1;l2;:::;lK jm1;m2;:::;mK = G

(N)
m1;m2;:::;mK

( A(N )l1;l2;:::;lK ) :
Fo r 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 Hm1;m2;:::;mK , wh e n it is t r u e t h e

fo llo win g d e ¯ n it io n is a d o p t e d :

®
(N )
m1;m2;:::;mK jm1;m2;:::;mK =

X

(l1;l2;:::;lK ) 6=(m1;m2;:::;mK )
®

(N )
l1;l2;:::;lK jm1;m2;:::;mK :

Fo r t h e s e qu e n c e ' o f t e s t s we will c o n s id e r e r r o r p r o b a b ilit y e xp o n e n t s wh ic h a r e c a lle d
r e lia b ilit ie s :

¡ lim
N !1

1

N
lo g ®

(N )
l1;l2;:::;lK jm1;m2;:::;mK = El1;l2;:::;lKjm1;m2;:::;mK ¸ 0 ; mk; lk = 1 ; M:

It is kn o wn , t h a t

Em1;m2;:::;mK jm1;m2;:::;mK = m in
(l1;l2;:::;lK )6=(m1;m2;:::;mK )

El1;l2;:::;lKjm1;m2;:::;mK :

Th e m a t r ix E = fEl1;l2;:::;lKjm1;m2;:::;mK g is a m a t r ix o f r e lia b ilit ie s o f t h e s e qu e n c e o f t e s t s
':

W e c a ll t h e t e s t s e qu e n c e '¤ L A O 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 m a t r ix o f r e lia b ilit ie s 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
t h e o t h e r e le m e n t s o f it .



E. Haroutunian, A. Martirosyan, A. Yessayan 2 5

2 . P r o b le m S t a t e m e n t

L e t t h e t r u e P D o f t h e s t o c k m a r ke t X = ( X1; X2; :::; XK ) b e Gm1;m2;:::;mK wit h
t h e c o r r e s p o n d in g lo g { o p t im a l p o r t fo lio bm1;m2;:::;mK a n d le t bl1;l2;:::;lK ; ( l1; l2; :::; lK ) 6=
( m1; m2; :::; mK ) b e a lo g { o p t im a l p o r t fo lio c o r r e s p o n d in g t o s o m e o t h e r , wr o n g P D
Gl1;l2;:::;lK :

W h e n t h e in c o r r e c t p o r t fo lio bl1;l2;:::;lK is u s e d in s t e a d o f bm1;m2;:::;mK , t h e n t h e in c r e a s e
o f t h e g r o wt h r a t e is t h e fo llo win g :

¢ W = W ( bm1;m2;:::;mK ; Gm1;m2;:::;mK ) ¡ W ( bl1;l2;:::;lK ; Gm1;m2;:::;mK ) ¸ 0 :

Th e in ve s t o r c a n u s e t h e s a m e in c o r r e c t p o r t fo lio fo r t h e n e xt p e r io d a n d h a ve s o m e
¯ n a n c ia l lo s s e s . Th e b o u n d s fo r ¢ W will s h o w t h e m a xim a l p o s s ib le a m o u n t o f lo s s e s .

Co n s id e r t h e fo llo win g qu a n t it ie s :

ql1;l2;:::;lKm1;m2;:::;mK ( x ) =
btm1;m2;:::;mK x

btl1;l2;:::;lK x
Gl1;l2;:::;lK ( x ) ; x 2 X K

gl1;l2;:::;lKm1;m2;:::;mK =
X

x2X K
Gm1;m2;:::;mK ( x) lo g

ql1;l2;:::;lKm1;m2;:::;mK ( x )

Gm1;m2;:::;mK ( x)
:

A s a d ir e c t c o n s e qu e n c e o f t h e K u h n { Tu c ke r c o n d it io n s , it wa s p r o ve d in [1 ] a n d [3 ]
t h a t E S

S¤ · 1 ; fo r a ll S; wh e r e S
¤ = b¤tX is t h e r a n d o m we a lt h r e s u lt in g fr o m t h e lo g {

o p t im a l p o r t fo lio b¤; a n d S = btX is t h e we a lt h r e s u lt in g fr o m a n y o t h e r p o r t fo lio b: Fr o m
in e qu a lit y E S

S¤ · 1 a n d t h e fa c t t h a t bl1;l2;:::;lK is a lo g { o p t im a l p o r t fo lio fo r Gl1;l2;:::;lK ; we
h a ve

ql1;l2;:::;lKm1;m2;:::;mK ( x ) ·
X

x2X K
ql1;l2;:::;lKm1;m2;:::;mK ( x) · 1 :

L e t D ( RkP ) = P
z2Z

R( z ) lo g
R(z)
P (z)

< 1 b y ( K u llb a c k - L e ib le r ) d ive r g e n c e fo r P D s
b y R a n d P; z 2 Z:
T heor em 1 [11]: If ( l1; l2; :::; lK ) 6= ( m1; m2; :::; mK ) ; then

4W = D ( Gm1;m2;:::;mK k Gl1;l2;:::;lK ) + gl1;l2;:::;lKm1;m2;:::;mK ;

where gl1;l2;:::;lKm1;m2;:::;mK · 0 with equality if and only if q
l1;l2;:::;lK
m1;m2;:::;mK

( x) = Gm1;m2;:::;mK ( x) ;
for all x 2 X K :
Cor ollar y 1: The analog of the bound exposed in [1] in the case of discrete P D s is

4W · D ( Gm1;m2;:::;mK k Gl1;l2;:::;lK ) :

Fr o m Th e o r e m 1 it fo llo ws t h a t fo r e a c h p a ir ( l1; l2; :::; lK ) 6= ( m1; m2; :::; mK )
t h e in ve s t o r ¯ r s t m u s t c h e c k t h e c o n d it io n ql1;l2;:::;lKm1;m2;:::;mK ( x) = Gm1;m2;:::;mK ( x) ; fo r a ll
x 2 X K : In s u c h c a s e s h is lo s s e s will b e m a xim a l.

A ll o t h e r c a s e s ( l1; l2; :::; lK ) 6= ( m1; m2; :::; mK ) ( a n d t h e c o r r e s p o n d in g p o r t fo lio s
bl1;l2;:::;lK 6= bm1;m2;:::;mK ) we will c a ll c a s e s ( p o r t fo lio s ) o f \ nonmaximal losses " a n d will
a s s u m e t h a t t h e fo llo win g c o n d it io n is fu l¯ lle d :

gl1;l2;:::;lKm1;m2;:::;mK ¸ ¡D ( Gl1;l2;:::;lK k Gm1;m2;:::;mK ) : ( 1 )



2 6 On Application of Optimal Multihypothesis Tests for the Bounds Construction of Stock Market

3 . N e w B o u n d fo r Gr o wt h R a t e

L e t N ( xjx ) b e t h e n u m b e r o f r e p e t it io n s o f t h e e le m e n t x 2 X in t h e ve c t o r x 2 X N ;
a n d le t

Q = fQ( x ) = N ( xjx ) =N; x 2 X g
b e t h e e m p ir ic a l d is t r ib u t io n ( t yp e ) o f t h e s a m p le x:

Fo r t h e g ive n p o s it ive d ia g o n a l e le m e n t s E1j1; E2j2; : : : ; EM¡1jM ¡1 o f t h e r e lia b ilit y m a t r ix
we c o n s id e r s e t s o f P D s

Rl = fQ : D ( QjjGl ) · Eljlg; l = 1 ; M ¡ 1 ; ( 2 )

RM = fQ : D ( QjjGl ) > Eljl; l = 1 ; M ¡ 1 g; ( 3 )
a n d d e ¯ n e t h e va lu e s fo r t h e e le m e n t s o f t h e fu t u r e r e lia b ilit y m a t r ix o f t h e L A O t e s t s
s e qu e n c e a s fo llo ws :

E¤ljl = E
¤
ljl ( Eljl ) = Eljl; l = M ¡ 1 ; ( 4 )

E¤ljm = E
¤
ljm ( Eljl ) = in f

Q2R l
D ( QjjGm ) ; m = 1 ; M ; m 6= l; l = 1 ; M ¡ 1 ; ( 5 )

E¤Mjm = E
¤
Mjm ( E1j1; : : : ; EM¡1jM ¡1 ) = in f

Q2R M
D ( QjjGm ) ; m = 1 ; M ¡ 1 ; ( 6 )

E¤MjM = E
¤
M jM ( E1j1; : : : ; EM¡1jM¡1 ) = m in

l=1;M¡1
E¤Mjl: ( 7 )

Th e L A O t e s t e xis t in g t h e o r e m c o n c e r n in g o n e o b je c t is t h e fo llo win g [1 0 ]:
T heor em 2 [10]: If the distributions Gm are di®erent, that is all divergences D ( GljjGm ) ,
l 6= m, l; m = 1 ; M , are strictly positive, then two statements hold:
a) when the given numbers E1j1; E2j2; : : : ; EM¡1jM ¡1 satisfy the conditions

0 < E1j1 < m in
l=2;M

D ( GljjG1 ) ; ( 8 )

0 < Emjm < m in [ m in
l=1;m¡1

E¤ljm ( Eljl ) ; m in
l=m+1;M

D ( GljjGm ) ]; m = 2 ; M ¡ 1 ; ( 9 )

then there exists a L AO sequence of tests '¤, the elements of the reliability matrix of which
E ( '¤ ) = fE¤ljmg are de¯ned in (4){(7) and all of them are strictly positive;
b) even if one of the conditions (8) or (9) is violated, then the reliability matrix of any such
test includes at least one element equal to zero.
T heor em 3: W hen the ¯rst M -1 diagonal elements of reliability matrix are ¯xed so that the
numbers E1j1; E2j2; : : : ; EM¡1jM ¡1 satisfy the conditions (8)-(9), then in case of condition (1),
for portfolios bl1;l2;:::;lK 6= bm1;m2;:::;mK ; ( l1; l2; :::; lK ) 6= ( m1; m2; :::; mK ) ; of non - maximal
losses the following bound is true:

¢ W · D ( Gm1;m2;:::;mK k Gl1;l2;:::;lK ) ¡E¤l1;l2;:::;lKjm1;m2;:::;mK ;

wh e r e

E¤l1;l2;:::;lKjm1;m2;:::;mK =
X

k=1;K: mk 6=lk
E¤lkjmk ( 'k ) :

P r oof: B y Co r o lla r y 1 fr o m [1 1 ] in c a s e o f c o n d it io n ( 1 ) , fo r p o r t fo lio s bl1;l2;:::;lK 6=
bm1;m2;:::;mK ; ( l1; l2; :::; lK ) 6= ( m1; m2; :::; mK ) ; o f n o n -m a xim a l lo s s e s we h a ve

¢ W · D ( Gm1;m2;:::;mK k Gl1;l2;:::;lK ) ¡El1;l2;:::;lKjm1;m2;:::;mK :



E. Haroutunian, A. Martirosyan, A. Yessayan 2 7

Th e r e fo r e t h e b e s t b o u n d fo r t h e in c r e a s e o f t h e g r o wt h r a t e will b e in c a s e o f m a xim a l
va lu e s o f r e lia b ilit ie s .

W h e n t h e ¯ r s t M-1 d ia g o n a l e le m e n t s o f o n e d im e n s io n a l r e lia b ilit y m a t r ix a r e ¯ xe d
c o r r e s p o n d in g t o t h e c o n d it io n s ( 8 ) -( 9 ) t h e n fr o m Th e o r e m 2 it fo llo ws t h a t t h e r e e xis t s a
L A O s e qu e n c e o f t e s t s '¤, t h e e le m e n t s E ( '¤ ) = fE¤ljmg o f o n e d im e n s io n a l r e lia b ilit y m a t r ix
o f wh ic h a r e d e ¯ n e d in ( 4 ) { ( 7 ) wit h m a xim a l va lu e s .

Th e c o m p o u n d t e s t '(N) fo r K s t o c ks c a n b e r e p r e s e n t e d a s a c o lle c t io n o f K in d ivid u a l

t e s t s '
(N)
1 , '

(N )
2 , ..., '

(N )
K fo r e a c h o f K s t o c ks [1 2 ], t h e n t h e in ¯ n it e c o m p o u n d t e s t ' is

a c o lle c t io n o f in ¯ n it e t e s t s '1, '2,..., 'K : B y t h e L e m m a fr o m [1 2 ] if e le m e n t s Elijmi ( '
i ) ,

mi; li = 1 ; M , i = 1 ; K; a r e s t r ic t ly p o s it ive , t h e n fo r ' = ( '1; '2; :::; 'K )

El1;l2;:::;lKjm1;m2;:::;mK ( ') =
X

i=1;K: mi 6=li
Elijmi ( 'i ) :

S in c e t h e e le m e n t s E¤lijmi ( 'i ) a r e m a xim a l, t h e r e fo r e t h e s u m E
¤
l1;l2;:::;lKjm1;m2;:::;mK ( ') will

a ls o b e m a xim a l.
T heor em is pr oved.

Th e fo llo win g e xa m p le illu s t r a t e s t h e r e s u lt o f t h e Th e o r e m 3 .
E xample: Co n s id e r t h e s e t X = f 1 ; 2 g a n d t wo P D s G1 = f 1 =2 ; 1 =2 g a n d
G2 = f 1 = 3 ; 2 = 3 g d e ¯ n e d o n X : Co r o lla r y 1 g ive s t h e fo llo win g b o u n d s :

¢ W = ¢ W ( G1 k G2 ) · D ( G1kG2 ) ¼ 0 :0 5 8 8 9

a n d

¢ W = ¢ W ( G2 k G1 ) · D ( G2kG1 ) ¼ 0 :0 5 6 6 3 :

A p p lyin g Th e o r e m 3 we s h a ll im p r o ve t h e b o u n d s . Fo r t wo h yp o t h e s is we c a n
t a ke E1j1 = 0 :0 5 : L e t Q = fq1; q2g b e t h e t yp e o f t h e ve c t o r x 2 X N ; wh e r e
q1 = N ( 1 jx ) =N a n d q2 = N ( 2 jx ) =N:

Fig. 1. q log 2q + (1 ¡ q) log 2(1 ¡ q)



2 8 On Application of Optimal Multihypothesis Tests for the Bounds Construction of Stock Market

Fig. 2. q log 3q + (1 ¡ q) log( 3
2
(1 ¡ q))

W e n e e d t o c a lc u la t e ( 5 ) a n d ( 6 )

E¤1j2 = in f
D(QjjG1)·E1j1

D ( QjjG2 ) = in f
q1 log 2q1+q2 log 2q2·E1j1

( q1 lo g 3 q1 + q2 lo g (
3

2
q2 ) )

a n d
E¤2j1 = in f

D(QjjG1)>E1j1
D ( QjjG1 ) = in f

q1 log 2q1+q2 log 2q2>E1j1
( q1 lo g 2 q1 + q2 lo g ( 2 q2 ) ) :

Ta kin g in t o a c c o u n t t h a t q1 + q2 = 1 ; fr o m t h e e xp r e s s io n q1 lo g 2 q1 + q2 lo g 2 q2 =
0 : 0 5 we o b t a in t h e e qu a t io n q lo g 2 q + ( 1 ¡ q ) lo g 2 ( 1 ¡ q ) = 0 : 0 5 : Th e a p p r o xim a t e
s o lu t io n s o f t h is e qu a t io n a r e q¡ ¼ 0 : 3 4 3 2 2 a n d q¡ ¼ 0 : 6 5 6 7 8 : In Fig . 1 a n d Fig . 2
t h e g r a p h s o f t h e fu n c t io n s q lo g 2 q +( 1 ¡q ) lo g 2 ( 1 ¡q ) a n d q lo g 3 q+( 1 ¡q ) lo g ( 3

2
( 1 ¡q ) )

a r e r e p r e s e n t e d . Th e s o lu t io n o f t h e in e qu a lit y q lo g 2 q + ( 1 ¡ q ) lo g 2 ( 1 ¡ q ) · E1j1 =
0 : 0 5 is q 2 [q¡; q¡]: Th e r e fo r e , n o w it is e a s y t o s e e t h a t

E¤1j2 = in f
[q¡;q¡]

( q lo g 3 q + ( 1 ¡ q ) lo g ( 3
2
( 1 ¡ q ) ) ) ¼ 0 :0 0 0 2 2 :

It is a ls o o b vio u s t h a t

E¤2j1 = in f
[0;q¡)[(q¡;1]

( q lo g 2 q + ( 1 ¡ q ) log ( 2 ( 1 ¡ q ) ) ) = 0 : 0 5 :

Th u s , n e w b o u n d s a r e
¢ W ( G1 k G2 ) · 0 : 0 0 8 8 9 ;
¢ W ( G2 k G1 ) · 0 : 0 5 6 4 1 :

4 . Co n c lu s io n

Th is p a p e r r e p r e s e n t s t h e a p p lic a t io n o f c o e ± c ie n t s o f t h e r e lia b ilit y m a t r ix o f L A O
t e s t s in S t o c k Ma r ke t p o r t fo lio t h e o r y. In ve s t o r s c a n u s e t h e o b t a in e d fo r m u la fo r b o u n d t o
e va lu a t e t h e fu t u r e r is ks , m in im a l in c o m e s a n d d o n e c e s s a r y c o r r e c t io n s in c u r r e n t p o r t fo lio .
Ch o o s in g s o m e P D fr o m t h e g ive n s e t o f P D s , t h e in ve s t o r will h a ve s o m e e r r o r s . Th e
e xis t in g e m p ir ic a l d a t a a llo w t o e s t im a t e s t a t is t ic a lly t h e p r o b a b ilit ie s o f t h o s e e r r o r s . A n d
t h e r e lia b ilit y c o e ± c ie n t s o f 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 r e u s e d fo r o b t a in in g o f
n e w b o u n d s . It is s h o wn t h a t t h e b o u n d s c a n b e m in im iz e d u s in g L A O t e s t s . Fixin g va lu e s



E. Haroutunian, A. Martirosyan, A. Yessayan 2 9

fo r d ia g o n a l e le m e n t s o f r e lia b ilit y m a t r ix t h e L A O t e s t a llo ws t o o b t a in 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 wh ic h e n a b le s o b t a in in g o f le s s e r b o u n d s fo r in c r e a s e o f g r o wt h r a t e o f t h e
s t o c k m a r ke t . Th e d e r ive d b o u n d c a n b e e ®e c t ive in m o d e lin g o f s t o c k m a r ke t . E va lu a t in g
wit h n e w b o u n d t h e in ve s t o r h a vin g h is t o r ic a l d a t a c a n m in im iz e h is lo s s e s m o r e t h a n t h e
in ve s t o r wit h o u t u s in g t h e h is t o r ic a l d a t a .

A c kn o wle d g e m e n t

Th is wo r k wa s s u p p o r t e d in p a r t b y S CS o f ME S o f R A u n d e r Th e m a t ic P r o g r a m N o S CS
1 3 { 1 A 2 9 5 .

Refer ences

[1 ] 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 , N e w
Je r s e y, 2 0 0 6 .

[2 ] T. M. Co ve r , \ A n a lg o r it h m fo r m a xim iz in g e xp e c t e d lo g in ve s t m e n t r e t u r n " , IE E E
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2 0 0 0 .

[7 ] P . H . A lg o e t a n d T. M. Co ve r , \ A s ym p t o t ic o p t im a lit y a n d a s ym p t o t ic e qu ip a r t it io n
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vo l. 3 4 , n o . 6 , p p . 7 2 4 { 7 3 3 , 1 9 8 8 .

[9 ] E . H a r o u t u n ia n , M. H a r o u t u n ia n a n d A . H a r u t yu n ya n , \ R e lia b ilit y c r it e r ia in in -
fo r m a t io n t h e o r y a n d in s t a t is t ic a l h yp o t h e s is t e s t in g " , F oundations and Trends in
Communications and Information Theory, vo l. 4 , n o . 2 , p p . 9 7 -2 6 3 , 2 0 0 8 .

[1 0 ] E . 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 ly 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 , n o . 5 -6 , p p . 4 1 3 -4 2 1 ,
1 9 9 0 .

[1 1 ] A . R . Ma r t ir o s ya n , \ N e w b o u n d s c o n s t r u c t io n b y m e a n s o f r e lia b ilit ie s fo r t h e in c r e a s e
o f g r o wt h r a t e o f s t o c k" , P roceedings of International Conference CSIT 2011, p p . 1 8 6
{ 1 8 9 , Y e r e va n , 2 0 1 1 .

[1 2 ] E . H a r o u t u n ia n a n d P . H a ko b ya n , \ Mu lt ip le h yp o t h e s is L A O t e s t in g fo r m a n y in d e -
p e n d e n t o b je c t s " , Scholarly R esearch E xchange, 2 0 0 9 .

[1 3 ] R . A h ls we d e a n d E . H a r o u t u n ia 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 t e s t in g
o f h yp o t h e s e s a n d id e n t ī c a t io n " , Information Transfer and Combinatorics, L ecture



3 0 On Application of Optimal Multihypothesis Tests for the Bounds Construction of Stock Market

Notes in Computer Science, vo l. 4 1 2 3 , p p . 5 5 3 { 5 7 1 , S p r in g e r , N e w Y o r k, N Y , U S A ,
2 0 0 6 .

Submitted 16.08.2013, accepted 01.10.2013.

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Ýáñ ·Ý³Ñ³ï³Ï³Ý:

Î ïðèìåíåíèè îïòèìàëüíûõ òåñòîâ îòíîñèòåëüíî ìíîãèõ
ãèïîòåç äëÿ ïîñòðîåíèÿ îöåíêè óâåëè÷åíèÿ ñêîðîñòè ðîñòà

ðûíêà öåííûõ áóìàã
Å. Àðóòþíÿí, À. Ìàðòèðîñÿí è À. Åñàÿí

Àííîòàöèÿ

Ïóòåì èñïîëüçîâàíèÿ ìàòðèöû íàäåæíîñòåé îïòèìàëüíûõ òåñòîâ ïîëó÷åíà
íîâàÿ îöåíêà óâåëè÷åíèÿ ñêîðîñòè ðîñòà ðûíêà öåííûõ áóìàã ïðè ïðèìåíåíèè
îøèáî÷íîãî ðàñïðåäåëåíèÿ âåðîÿòíîñòåé.