A conditional characterization of the multivariate normal distribution

Statistics & Probability North-Holland

Letters

15 March

19 (1994) 313-315

1994

A conditional characterization of the multivariate normal distribution Barry C. Arnold University of California, Riverside, USA

Enrique Castillo and Jose Maria Sarabia University of Cantabria, Santander, Spain Received December Revised April 1993

1992

Abstract: If X is a k-dimensional random vector, we denote by Xcl, j) the vector X with coordinates i and j deleted. If for each i, j the conditional distribution of X,, X, given Xcl, j) = xci, j) is classical bivariate normal for each xci, j) GZIWke2 then it is shown that X has a classical k-variate normal distribution. Keywords: Bivariate

conditionals;

classical

normal

distribution;

1. Introduction

conditional

specification

The classic example is provided by the following joint distribution.

If X has a classical k-dimensional normal distribution with mean vector p and variance covariante matrix ,Z then it is well known that all marginals of the distribution of X are again classical multivariate normal, It is also true that all conditional distributions of X given X (where X and X are non-overlapping subvectors of X) are multivariate normal. Efforts to characterize k-dimensional normality assuming only marginal or conditional normal specifications have been, in general, unsuccessful. If we assume only that X i, . . . , X, are marginally univariate normal then an enormous panoply of k dimensional distributions can be constructed with such a marginal normal specification. If we require that higher dimensional marginals are all multivariate normal we still fail to guarantee k-dimensional normality.

fx(

x)

=

(ZT)

-k/Q ,-cL~5/*

H fixi 1

x 1+

i=l

Z()xJ
1 .

This clearly has all marginals of order
fAx>

Correspondence too:Dr. B.C. Arnold, University of California, Department of Statistics, Riverside, CA 92521-0138, USA. 0167-7152/94/$07.00 0 1994 - Elsevier SSDI 0167-7152(93)E0120-I

Science

B.V. All rights resewed

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STATISTICS

& PROBABILITY

When we turn to bivariate conditionals a surprise is in store. It turns out that if for each i, j, (X,, Xj> given Xci,j), is classical bivariate normal, then X is multivariate normal.

2. The characterization

LETTERS

15 March

1994

This forces certain of the 6,, i, , ik’s in (2.1) to be zero. For example the coefficients of xix,? and xixj2xkxI, etc. By varying the choice of i, j, more of the coefficients can be shown to be zero. In fact, if all the conditionals of (Xi, Xj> given Xci,j) are to be classical bivariate normal, the joint density must assume the form (using new notation for the reduced number of parameters)

Let X denote a k-dimensional random vector. For each i, j let Xci,j) denote the vector X with coordinates i and j deleted. In similar fashion x represents a generic point in R“ and xci,j) denotes x with its ith and jth coordinates deleted.

fx(x)=q-

Theorem. Suppose that for each i, j and for each x(i,j) E W-2, the conditional distribution of (Xi, X,) given XCi,j) = A+,j) is classical bivariate normal with mean vector (pi(xCi, j,), ~j(X~i,j,)) and variance covariance matrix

where S is the set of all vectors of O’s and l’s of dimension k. However the number of non-zero parameters can be reduced even further. Consider the conditional distribution of (Xi, X2) given Xc12) which is to be classical bivariate normal. From (2.2) it will be of the form

(

a11( x(i,

j))

(T12(

x(i.

j))

g21( x(i,

j) >

v22(

x(i,

j))

iPj’J+ [

Es,

j=l

SES

f x,.x*~x(&l~

x*l-%*J

I ’ a=?-

i

P~“~+P*~,2+~000...0+~100...0~c1 +6

OlO...OX2 + ~11,...0x,x*

II

Proof. Since given Xci,j) = xci,j) is classi-

cal bivariate normal, it has univariate normal conditionals. It then follows that Xi given Xi and Xci,j), i.e. given Xci, is univariate normal. But then from Chapter 8 of Arnold, Castillo and Sarabia (1992) we can conclude (as noted in the introduction) that the joint density of X is of the form fx(x)

[

5 i,=o

i,=o

II

(2.2)

It follows that X has a classical k-variate normal distribution.

= exp -;;

,filx?

i

” ’ iI i,=O

‘i,,i,,...,ik

*

IfIx2

j=l

I

+

c 6, ,&P SE-S’ i

(2.3)

where S’ is the set of all vectors of O’s and l’s with a 1 in the first two coordinates and at least one other coordinate being a 1. For (2.3) to represent a classical bivariate normal density the parameters pi and p2 must be positive and the coefficient matrix of the quadratic form in (xi, x2> must be non-negative definite. This means that we must have (coeff of (xix,))’

G 4 (coeff of xf)(coeff

of x2”)

(2.1)

Now for any i and j, the conditional density of (Xi, Xj> given Xci,j) = x(~,~) is postulated to be classical bivariate normal and so it will be of the form f X,X,1+,,,,(xi’ xil xU,j,) a

exp [ a quadratic form in xi, xi with coefficients which may depend on x(~,~,].

314

i.e. 2 6 11000+

i

c %]$3x;1 SES’

G

4P1P2.

(2.4)

i

Note that (2.4) must hold for every choice of . . , xk. The only way that this can hold is if 6,=OforeverysES’. By considering, in an analogous fashion, the conditional density of given Xcl,j, for

x3, x4,.

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STATISTICS

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(i, j) # (1, 2) we eventually conclude that, in (2.21, any 6, for which s contains more than 2 ones must be zero. It follows that the joint density is expressible in the form (again with new simplified parameters)

fx(x)

= exp -

I

ff + i:

j=l

hjxj + t j=l

i I==1

y,/xjxr

1

LETTERS

15 March

1994

Rk-‘, the conditional distribution of Xi given Xci, = xci) is classical univariate normal and in addition assume that the regression of each X, on Xcj, is linear (or assume the conditional variance of X, given Xci, = xci) does not depend on x& Then X has a classical k-variate normal distribution. This extends the result for k = 2, reported in Castillo and Galambos (1989).

(2.5)

where the matrix (~~~),k:‘,,~=1 is non-negative definite. This of course indicates that X has a classical k-variate normal distribution. q Remark. Arguments similar to those used in this proof can be used to justify the following statement. Suppose that for each i and for each xci) E

References Arnold, B.C., E. Castillo and J.M. Sarabia (19921, Conditionally specified distributions, Lecture Notes in Statist. No. 73 (Springer, Berlin). Castillo, E. and J. Galambos (19891, Conditional distributions and the bivariate normal distribution, Metrika 36, 209-214.

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