A characterization of a family of distributions

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXI (1979) ROCZNIK1 POLSK.IEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXI (1979)

H

P

(Lôdz)

A characterization of a family of distributions

Abstract. Tamhankar [4] obtained a characterization of normality and of distributions with densities axb exp (cx2).

Flusser [1] gave a generalization of the above results. Similarly, Kotlarski [2], [3]

obtained another generalization of Tamhankar’s results. In the present paper a generalization of Flusser’s results will be obtained on the basis of a more general system of statistics than that employed in [1].

1. Introduction. Let us consider a random vector ( A ^ , X„) for n ^ 2 with the joint density p continuous and positive in an n-dimensional region A containing the origin of a coordinate system, a random vector (Y j,..., Yn) with the joint density q defined and positive in an n-dimensional region В and

^ *( = ^{I й} (Yi)]/3‘ ^{Vt ( y 2}, • • •, ^{y n -} 1) ^Щ(y„) (/ = 1 1 ) ,

= l u ( y i ) f nun (>’„),

where ft # 0, х.еД- for i = 1 ,.. ., n , y x e B x, (y2, y n_ x) e B 2, yne B 3; the sets Ai, B l , B 2, B 3 are such that A = A x x ... x A„, В = B x x B2 x B3. The functions in (1) satisfy the conditions

(2) V = °.

be В i

(3) Y Д Ui(c) = 0 л un(c) ф 0;

c e B3 i e { l , . . . , n — 1}

moreover, there exist functions <p, continuous in A t for i = 1 , .. ., n , and a function (p continuous and monotonie in B x such that

n

(4 ) < p { y i ) = z <Pi(x i)>

i = 1

(5) A 9 ,(0 ) = 0

and the functions

(6) ^ (x) = C, exp [(txpi (x)] (i = 1,..., n) are densities for suitably chosen constants C,, a .

? — Roczniki PTM — Prace Matematyczne XXI

3 4 8 H. P i s a r e w s k a

2. Characterization. We now make the following assumption:

(I) Differentiable and one-to-one transformation (1) with the Jacobian J ( y i , - -,yn) = D ( x i , . . . , x n)/D(y1, . . . , y n) is such that

(7) P (*i, ...,*„) =

and the right-hand side is well-defined in В and continuous in y x.

Theorem.

Under assumption (I), for independence of the pair of vectors ( X x, . . . , X s), (X s + X„) f or some s < n and of the pair of vectors ( T j , >9, ( l ^ + i , Y„) for some l < n, if Y1, ..., Yn are related to X x, ..., X„

by (1), it suffices that X 1, . . . , X n be mutually independent with densities of the form (6).

P ro o f. The Jacobian of the transformation (1) can be expressed in the form J ( y i , - - , y n) = h(y1) H ( y 2,...,y„),

X ft-i

where h(yJ = uf(yx) [w(yi)] ~ and H is a determinant obtained from the Jacobian J when h(yx) has been removed from it. In view of the assumed independence and by (7) we have

(8) p 1(x1, . . . , x s)p2(xs+l, . . . , x n)

= < h { y i , - - - , y i ) / \ h CKi)l • 2 O'/ +1 > • • ■, y n ) / \ H ( y 2 , . . . , У J ,

where Pi, p2, qi, q2 are densities of the suitable random vectors. Passing to the limit at y x^>b on either side of equation (8) in view of (1), (2), gives:

(9) where

p 1 ( 0 , . . . , 0 ) p 2 ( 0 , . . . , 0 ) = g i ( y 2 , . . . , y i )

q2 (У1+и---,Уп)

\ H ( y 2 ,

9 i ( y 2 , - - - , >’/) = lim

У1~>Ь

Dividing equation (8) by (9) and introducing notations

w1(x1,..,.,x s) = ■9 Xs)/Pt (0, ...,0 ), (10) W2 (xs+1, .. - ,x n) = P2(xs+i, •••9

(0 ,..., 0),

we get

У(Уи •■ •, y t) = я Л уи - -- ,yi)/\h(yi)\gi{y2, •••

(11) Wi(Xi, • • • ? xs) w2 (xs+1,..••9*„) = y(yi,...,T/)

Letting in the latter equation y„ = c, it can easily be seen, thanks to (3), that the left-hand side of (11) is a function of y x and so we can write y ( y i,...,y 9 = yi(vi); hence equation (11) will take the form:

( 1 2 ) W j ( * ! , . . . , x s) w 2 ( x s + 1 , . . . , X „ ) = 7 i ( T l ) .

Family of distributions 3 4 9

Determining from relation (4) and substituting the obtained results in (12) , we get

n

(13) Wj ,x a)w2(xs + X„) = w[ £ <M*i)],

i = 1

where w(z) = yi [<p_1 (z)], q>~1 is an inverse function of cp. Letting in (13) x l = ... = x s = 0, and then xs+1 = ... = x„ = 0, in view of assumption (5) and considering that w^O, . . . , 0) = w2(0 , . . . , 0) = 1, we obtain

(14) w2(xs+1, x „ ) = w[ X *M *i, •••,*») = w[ £ <?,•(*,•)]•

i = s + 1 , i = 1

Hence equation (13) may be expressed in the form

(15) w(z1)w(z2) = w (z j+ z 2),

where

s n

Z l = Z <Pi(X i)> Z 2 = S ( P i M -

i = 1 i = s + 1

A continuous solution of equation (15) are the functions w(f) = eat, a # 0.

Multiplying equation (14) by sides, we get

П

W ifo , ...,x s)w2(xs+1, ...,x„) = exp [a <р4(х,)].

i = 1

Going back to notations (10), it can easily be seen that joint density of the random vector ( X 1^ . . . , X n) can be presented in the form

П

p (Xj , ..., X„) = П C i eXP t a(Pi (X*)] »

i= 1

with the constants Cf > 0 ; this completes the proof.

It follows from the above theorem that by suitably selecting the system of statistics (1) we can obtain a characterization of various distributions.

In particular, if <р,-(х,-) = x?‘, а,- Ф 0, we obtain a characterization of distri­

butions with densities:

f (xf) = C, exp (шф), x, e A, (i = 1 ,..., n).

As the above family contains, among others, a normal distribution, for At = ( —00, 00), a, = 2, and the system of statistics considered in [1] is a particular case of (1), the above theorem is a generalization of Flusser’s results in [ 1].

Example.

Let us consider transformation (16) and the inverse transfor­

mation (17)

350 H. P i s a r e w s k a

(16)

(17)

*1 = (1 У2 Уз)У4 In)/’!,

*3 = Уз У4 ln y lf

y t = exp (XJ + X2 + X3 + X4),

Уз =

X i+ x f + x3 ’

*2 = J y i У4 V ln У1,

X 4 = ( 1 — y 4 ) l n y x \

У 2 = --- — 2 --- ,

xi

* l + * 2 + * 3 _

X!+X2 + X3

Xj + x2 -Ь x3 x4 ’

(xl5 x2, x 3, х4) е Л , A = A x x A 2

A 3 x A4, A ( = <0, ~ ) (i = 1 , 4 ) ; ^ ef i i , = <1, ~) , (у2,Уз,У4) е ^1 x B 3 xB 4, Bt = <0, 1> (i = 2, 3,4).

Transformation (16) satisfies conditions (2}-(6) for b = 1, c = 0, (p(yx)

= 1пу15 (pi(x) = x (i = 1 ,3 ,4 ), (p2{x) = x2; and density p ( x ! , ..., x„)

n n

= П Cj ^{exp (я X} (Pi (x,)) satisfies assumption (I), when X t are related to

i= 1 i=l

1Î by (16).

References

[1] P. F lu s s e r , A generalization o f a theorem by M. V. Tamhankar, J. Multivariate Anal.

1.3 (1971).

[2] 1.1. K o t la r s k i, Una caratteruzzazione della distribuzione gamma per mezzo di statistiche indipendenti, Rend. Mat. Appl. 2 (1969), p. 1-5.

[3] — On characterization o f probability distributions by means of independent statistics, Ann.

Mat. Рига Appl. 83 (1969), p. 253-260.

[4] M. V. T a m h a n k a r, A characterization of normality, Ann. Math. Statist. 38 (1967), p. 1924-1927.

INSTITUTE O F M ATHEMATICS

TECHNICAL UNIVERSITY, LÔ D É, POLAN D