Bivariate distributions with pareto conditionals

Statistics & Probability Letters 5 (1987) 263-266 North-Holland

June 1987

BIVARIATE D I S T R I B U T I O N S W I T H P A R E T O C O N D I T I O N A L S

Barry C. A R N O L D University of California, Riverside, CA, USA Received May 1986 Revised November 1986

Abstract: For a fixed a > 0, the totality of bivariate densities with all conditionals being of the Pareto (a) form is identified. The resulting family is of the form

f ( x , y ) O~[1 + ~klx + A2y + OXiA2xy ] - ( ' + 1 ) for suitable choices of X1, ~2 and 4-

AMS (1980) Subject Classification: 62H05. Keywords: Pareto distribution, conditional densities, logistic distribution.

1. Introduction Castillo and Galambos (1985) identified the class of all analytic bivariate densities f(x, y) defined on R z for which f ( x l Y ) is a normal density for every y and f ( y Ix) is a normal density for every x, i.e. all conditional distributions are normal. The family they derived includes and provides an interesting extension of the classical bivariate normal family. In m a n y economic contexts, Pareto distributions rather than normal distributions play a central role. The present note derives the class of all bivariate densities on R + 2 for which f ( x [ y ) for every y and f ( y [ x ) for every x are Pareto (a) densities. The class of densities derived includes as special cases the case of independent marginals and the popular family of bivariate Pareto densities introduced by Mardia (1962).

2. Densities with Pareto (a) conditionals

This density can be identified with the Pareto (II) (0, o, a) density in the hierarchy of Pareto distributions described in Arnold (1983). We wish to identify the class of all bivariate densities f(x, y) with the property that all of their conditional densities are of the form (1). Thus f(x, y) must be such that

f ( x l Y) = ao~(Y)/(°(Y) + x

a+l

(2)

and

f ( y l x ) = ar~(x)/(r(x) + y)~+l

(3)

for some functions o(y) and ~(x). Here and henceforth all equations involving x and y are to be understood as holding over the region x > 0, y>0. Let h(x) and g(y) be the marginal densities of f(x, y). From (2) and (3), recalling that a joint density can be written as the product of a marginal density and a conditional density in two ways, we may write

For a > 0, a random variable X has a Pareto (a) distribution if its density if of the form

ag(y)o°(y) ah(x)~°(x) = . ( o ( y ) + x ) °+1 ( , ( z ) + y ) ° + l

fx(x)=

We wish to solve (4) for g, h, • and o. To this end

1+

,

x>0.

(1)

0167-7152/87/$3.50 ~ 1987, Elsevier Science Publishers B.V. (North-Holland)

(4)

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Proof. By (4) and (9), the inequality (10) is equivalent to the following

ffuB~(o)h(u, v)du dv~
ffuB (o)h(u,

v) d u d v = - ( 1 - r ) M ( r ,

h)+(r/(l-r))l/2frl

foluh(u , v) d u d v

-%<- (1 - r )J~( h ) - rJ~( h ) = - J,( h ), since M(r, h) >1J~(h) and N(1, h) = 0 implies (r(1-

flJ r f'uh(u, v)du d v ~ ,lO

-J~(h).

Acknowledgment

The author would like to thank Dr. T. Ledwina for her advise and her interest during the preparation o this paper and the referee for many helpful comments.

References Bahadur, R.R. (1967), Rates of convergence of estimates and test statistics, Ann. Math. Statist. 38, 303-324. Farlie, D.J.G. (1960), The performance of some correlation coefficients for a general bivariate distribution, Biometrika 47, 307-323. Kremer, E. (1981), Local Bahadur efficiency of rank tests for the independence problem, J. Multivar. Anal. 11, 532-543. Ledwina, T. (1986a), Large deviation and Bahadur slopes of some rank tests of independence, Sankhy~ Series A 48, 188-207. Ledwina, T. (1986b), On the limiting Pitman efficiency of some

262

rank tests of independence, Z Multivar. Anal 20, 265-271 Ledwina, T. (1987), An expansion of the index of large devia tions for linear rank statistics, Statistics & Probability Letter

5 (5). Mardia, K.V. (1970), Families of Bivariate Distributions (Gilt fin, London). Nikitin, J.J. (1984), Local Bahadur optimality and characteri zation problems, Theor. Probability Appl. 29, 79-92 (iJ Russian). Woodworth, G.G. (1970), Large deviations and Bahadur et ficiency of linear rank statistics, Ann. Math. Statist. 44 251 283.

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If 0 < q, ~< ½, an alternative hypergeometric function representation is appropriate.

that

l=-l~£°C(l +)~.2y)-'~(Xl +)`12Y)-l dy.

(22)

In order for this to be integrable we must have )'2 > 0. An analogous argument integrating first with respect to y indicates that Xl > 0 is also necessary. If )t12 = 0 then in order for expression (22) to be finite we must have a > 1. If )t12 > 0 then (22) exists for any a > 0. In summary the general bivariate Pareto distribution is of the form (20) where )t 1, )'2 > 0, and )k12 • 0 if a > 1, o r )~12> 0 if 0 < a ~< 1. It represents all bivariate densities for which both families of conditional densities are of the Pareto (a) type (i.e. of the form (1)).

/)a -- 1

[0<~ ½] I

--F(a,

a; a+ l" 1-q~)

0¢21kl)`2 ~a 1 [ -

O/2~kl)`2

0~2 1 +

0¢2(0~ q- 1) + 2(a+2)

(1 - q~)

] (1-(/)12+ ""]" (24b)

In particular we have for every ~ > O, (25)

[a = 1] and [a=2]

3. T h e normalizing constant

The only detail lacking for a complete specification of the general bivariate Pareto distribution is the determination of the constant of proportionality in (20). We must evaluate the integral (21). Making the change of variable u = XlX, v = ~kzy and defining

, f0V0°r 1 + U+ V+~bUV] -('~+ 11 d u d v

I = X-7~2 1

fo~(1 + u)-"(1

+ flu)-'

du.

Making the change of variable t = we find

eou/(1 + eou)

Provided q, > ½ so that 1 - 1/q~ > - 1 this can be represented as a hyper-geometric function, i.e. [q~>½]

(26/ The complicated nature of the dependence of I on a and (p portends difficulties in fitting the model to data.

(23)

we may argue as follows:

a)`11`2

21kl)`2 (1 - ~)2

4. Relation to earlier bivariate Pareto densities

= )k12/~kl)` 2

_

I

(

I=I~F

a, 1 ; a + l ; 1 - ~

~21kl ~k2~ 1

~

1)

If ~ = 1, i.e. if ~k12-~")`l)k2 in (20), then the joint density factors and X and Y are independent Pareto (a) variables with corresponding scale factors )~11 and 2t21. If ~ = 0 , i.e. )`12=0 in (201, then we reduce to the bivariate Pareto distribution introduced by Mardia (1962). In this case we must have a > 1 in order to have a bona fide density. If desired, location parameters # and v for X and Y can be introduced in the family (20). The support of the density would then be (/~, m ) × (v, ~ ) . 5. E s t i m a t i o n when a = 1

When a = 1, the bivariate density (20) assumes the form

f(x, y ) -

J

-- O/2)`I)`2-------Gj_E0(OC----~---jj)(1--~)

)`1)`2(1 - ~) - log

X [1 + )`ix + )`2Y + ~ikl)`aXY] 2 (24a)

x>O,

y>O,

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If we substitute (11) and (12) in (13) we obtain

define two new functions

(5)

8(x) =

co(y)T(x)

+ d r ( x ) + c o ( y ) y + dy

= c ' r ( x ) o ( y ) + e o ( y ) + c ' r ( x ) x + ex.

(141

and

p(y) =

[ a g ( y ) a " ( y ) ] 1/~+1.

(6)

With these definitions, evidently

p(y) o(y) + x

8(x) +y

(7)

• ( x ) = (3' - e x ) / ( c x - d)

First we must consider the trivial case where r ( x ) -= r, a constant. If this is the case then from (4) we conclude that f ( x , y) factors into a product of a function of x and a f u n c t i o n of y. We have independence and consequently o ( y ) is also a constant, say o, and f ( x , y) can be written in the form

f(x, y)=

(O+ X)a+l(,r + y ) a+l

8(xi)

(161

Substituting these in (11) and (12), with c = c ' yields

6 ( x ) = (c3" - d e ) / ( c x - d)

(17;

(18)

Finally using (15)-(18) we have, in the non-independent case, a+l

(9)

f(x, y)=

which, since r ( x l ) 4=~'(x2), yield

,r(x.,)

o(y)

x2 (x2) ]

(10)

(11)

for some constants c and d. Analogous arguments lead to the conclusion that, either independence obtains (i.e. (8) holds), or a ( x ) is non-constant and

6(x) = c'r(x) + e

=

r(x)+y (19)

In order that (19) should represent a genuine density (i.e. non-negative and integrating to 1) it must be true that c 3 ' - d e ~ O , 3"~0 and 3"(c3" - de) > 0. F r o m (19), and (8), it follows that any joint density with the desired Pareto ( a ) conditional densities will be of the form

f ( x , y ) oc [1 + X,x + X2y + X,2xy] -("+1)

Thus

O(y)=co(y)+d

o(y)+x

3"- ex - dy + cxy

8(x2)] (xl) -

o( y ) = (3' - dy ) / ( cy - e ).

oCY) = (c3" - a e ) / ( c y - e).

r(xi) + y'

[ xl

and

(8)

If r ( x ) is not a constant then there exist x 1 :# x 2 with r(xl) 4: r(x2). F r o m (7) we have, for i = 1,2,

= L,.,.(xl)

(151

and

0~20a,i-a

p(y) o(y) + x i

F o r (14) to hold uniformly in x and y, when r(xl and o ( y ) are non-constant functions, it must b~ the case that c = c' and both sides of (14) arc equal to some constant, say 3'. It then follows that

(20)

It remains to identify necessary constraints on the X's and perhaps on a to ensure that f ( x , y) is non-negative and integrates to 1. To ensure that f ( x , y ) > 0 we must have X1 >~ 0, X2 >~ 0 and )k12 > 0. N o w we wish to determine for which X~ >/0, ~k2 ~ 0, )k12 ) 0 and a > 0 does the integral

(12)

I = f o ~ £ ~ [ l + 2tlx + X2Y+ X12xy]-~+') d x d y for some constants c' a n d e. F r o m (7) we have

0(y),(x)+0(),)y=8(x)o(y)+8(x)x. 264

(21)

(13)

converge. Integrating first with respect to x we see