The distribution of symmetric matrix quotients

ARTICLE IN PRESS

Journal of Multivariate Analysis 87 (2003) 413–417

The distribution of symmetric matrix quotients A.K. Guptaa,
and D.G. Kabeb a

Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403-0221, USA b Department of Math and Computer Science, St. Mary’s University, Halifax, NS, Canada Received 8 October 2002

Abstract Phillips (J. Multivariate Anal. 16 (1985) 157) generalizes Cramer’s (Mathematical Methods of Statistics, Princeton University Press, Princeton, NJ, 1946) inversion formula for the distribution of a quotient of two scalar random variables to the matrix quotient case. However, he gives the result for the asymmetric matrix quotient case. This note extends Phillips’ (1985) result to the symmetric matrix quotient case. r 2003 Elsevier Science (USA). All rights reserved. AMS 2000 subject classifications: primary 62H10; secondary 62E15 Keywords: Matrix variate; Positive definite; Density; Transformation; Moment generating function; Inversion formula

1. Introduction Let x and y be two random variables with the joint characteristic function fðy1 ; y2 Þ: Then Cramer [1], and Geary [2] state the following formula for the density gðzÞ of the quotient z ¼ x=y: gðzÞ ¼ ð2piÞ1

Z

N N

where i ¼


@fðy1 ; y2 Þ @y2

 dy1 ; y2 ¼zy1

pffiffiffiffiffiffiffi 1:

Corresponding author. E-mail address: [email protected] (A.K. Gupta).

0047-259X/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0047-259X(03)00046-0

ð1Þ

ARTICLE IN PRESS A.K. Gupta, D.G. Kabe / Journal of Multivariate Analysis 87 (2003) 413–417

414

Now Phillips [4] records the following matrix generalization of (1). Given a positive definite symmetric matrix A : ðp þ qÞ  ðp þ qÞ " # A11 A12 A¼ ð2Þ ; A11 ðp  pÞ; A22 ðq  qÞ; A21 A22 he finds the density of the quotient X ¼ A1 22 A21 : If fðF21 ; F22 Þ denotes the joint characteristic function of A21 and A22 ; then the density of X is  Z N   @ p pq   fðF21 ; F22 Þ gðX Þ ¼ ð2piÞ dF21 ; ð3Þ @F  1 22 N F22 ¼ ðXF 0 þF21 X 0 Þ 2

j@F@22 j

21

detð@f@rs Þ; frs

¼ being the ðr; sÞth element of F22 ; and we assume all integrals where to be properly evaluated. Using (3), and assuming A to have the central Wishart density with n degrees of freedom and population covariance matrix identity for convenience, Phillips [4] illustrates (3) by showing the density of X to be gðX Þ ¼ KjI þ XX 0 jn ;

ð4Þ

where K denotes the normalizing constants of density functions in this paper. However, in multivariate statistical analysis the quotients of two symmetric matrices are often used than the quotients X of the above type. Thus, e.g., if A has the Wishart density  1 1 gðAÞ ¼ K exp  tr A jAj2ðnqp1Þ ; ð5Þ 2 then the canonical correlation matrix R defined by 

1



1

2 R ¼ A112 A12 A1 22 A21 A11

ð6Þ

is a symmetric quotient whose density is desired. We now proceed to extend (3) to study the density of the random matrix R of type (6) in the next section.

2. Symmetric matrix quotient density We first record the following known formulas. If Y is a p  p positive definite symmetric matrix and T is another p  p positive definite symmetric matrix, then we have that  t  @    expftrTY g ¼ ð1Þpt expftrTY gjTjt ; ð7Þ @Y  and hence obviously  t  @  1 n   jT þ Y j2 n pjT þ Y jð2þtÞ : @Y 

ð8Þ

ARTICLE IN PRESS A.K. Gupta, D.G. Kabe / Journal of Multivariate Analysis 87 (2003) 413–417

415

Now assuming two p  p random positive definite symmetric matrices A and B to have the joint density gðA; BÞ; we write the density of the matrix G defined by 1

1

B ¼ A2 GA2 to be Z 1 1 gðGÞ ¼ K gðA; A2 GA2 ÞjAjp dA;

ð9Þ

where jAjp is the Jacobian of transformation from B to G (see [3]). We now observe that hðA; BÞ ¼ K½EjAjp 1 jAjp gðA; BÞ;

ð10Þ

where EðjAjp Þ denotes the expected value of jAjp ; defines a new density whose joint characteristic function fðF11 ; F22 Þ is Z p 1 ½EjAj  expftrðiF11 A þ iF22 BÞgjAjp gðA; BÞ dA dB  Z   @ p 1 1 1 1   expftrðiF11 A þ iF22 A2 GA2 ÞggðA; A2 GA2 Þ dA dG ¼K  @F11     @ p   fðF11 ; F22 Þ; ¼ K ð11Þ @F11  where F11 ðp  pÞ and F22 are defined in the usual way for evaluation of the characteristic functions of symmetric matrices. We now assume that the density of G is invariant under the transformation 1

1

G-HGH 0 for any p  p orthogonal matrix H: Thus we may write HA2 GA2 H 0 ¼ 1

1

G 2 AG 2 ; in which case (3) modifies to the formula   Z   @ p   gðGÞ ¼ K 1 1 dF22 : @F  fðF11 ; F22 Þ 11 F11 ¼G2 F22 G2

ð12Þ

The differentiation in (12) may become involved for the noncentral density of G; however, it can be avoided by integrating (11) with respect to A only, and writing (12) as Z gðGÞ ¼ KcðGÞ ½fðF11 ; F22 ; GÞ ð13Þ 1 1 dF22 : F11 ¼G2 F22 G2

We illustrate (13) by using (6). The joint characteristic function of A11 and A12 A1 22 A21 is  Z 1 1 trA trA A Þg exp   fðF11 ; F22 Þ ¼ K expftrðiF11 A þ iF22 A12 A1 11 22 22 21 2 2 1

1

2ðnqp1Þ jA22 j2ðnqp1Þ dA11 dA12 dA22 :  jA11  A12 A1 22 A21 j

ð14Þ

ARTICLE IN PRESS A.K. Gupta, D.G. Kabe / Journal of Multivariate Analysis 87 (2003) 413–417

416

Setting A12 A1 and integrating out A12 ; A22 ; we reduce (14) to the integral 22 A21 ¼ G;  Z 1 fðF11 ; F22 ; GÞ ¼ K expftrðiF11 A11 þ iF22 GÞg exp  trA11 2 1

1

 jA11  Gj2ðnqp1Þ jGj2ðqp1Þ dA11 : 1

1

1

ð15Þ

1

Further setting G ¼ HA211 RA211 H 0 ¼ R2 A11 R2 ; (15) yields the result  Z 1 1 1 2 2 fðF11 ; F22 ; GÞ ¼ K expftrðiF11 A11 þ iR F22 R A11 Þg exp  trA11 2 1

 jA11 j2ðnp1Þ cðRÞ dA11 1

1

1

¼ KjI  2F11  2R2 F22 R2 j2 n cðRÞ; and hence the density of R is Z 1 1 1 n gðRÞ ¼ KcðRÞ jI  2F11  2R2 F22 R2 j 2

1 1 F11 ¼R2 F22 R2

ð16Þ

ð17Þ

dF22 ;

where 1

1

cðRÞ ¼ KjI  Rj2ðnqp1Þ jRj2ðqp1Þ :

ð18Þ

However, note that 1

1

jI  2F11  2R2 F22 R2 jn

1 1 F11 ¼R2 F22 R2

¼ I;

ð19Þ

and hence the integration in (17) with respect to F22 must be dropped or else assume its value to be some constant. Thus the density of R is given by (18). The essence of the above example may be stated as follows. If A and B two p  p positive definite symmetric matrices have  the joint density 1 1 1 gðA; BÞ ¼ K exp  trðA þ BÞ jAj2ðnp1Þ jBj2ðqp1Þ ; ð20Þ 2 1

1

then the density of G ¼ A2 BA2 is obtained by the evaluation of the integral  Z 1 1 1 fðF11 ; F22 ; GÞ ¼ K expftrðiF11 A þ iG 2 F22 G 2 AÞg exp  trðI þ GÞA 2 1

1

 jAj2ðnþqp1Þ jGj2ðqp1Þ dA Z 1 1 1 ¼ KjGj2ðqp1Þ ½jI þ G  2F11  2G2 j2ðnþqÞ 

1 1 F11 ¼G2 F22 G2

dF22 ; ð21Þ

and dropping the integration with respect to F22 if the function under the integral sign is a constant. Thus the density of G is 1

1

gðGÞ ¼ KjGj2ðqp1Þ jI þ Gj2ðnþqÞ :

ð22Þ

ARTICLE IN PRESS A.K. Gupta, D.G. Kabe / Journal of Multivariate Analysis 87 (2003) 413–417

417

To evaluate (12) directly, we need multivariate analog of the integrals of the type Z N ðq  iyÞa ð1 þ iyzÞb dy ¼ ½Bða; bÞ1 ð1 þ zÞðaþbÞ ; ð23Þ N

which appears to be unknown for the matrix case. However, now from (22) and (23) we conclude that Z 1 1 jI  iF11 ja jI þ iG 2 F11 G2 jb dF11 ¼ ½Bp ða; bÞ1 jI þ GjðaþbÞ : In fact, Phillips’ [4, p. 160, Eq. (12)] integral

 Z N 1 1 1 1 jA þ GG 0 jn dG ¼ jAjðn2 qÞ p2 pq p4 pð1pÞ GP ðnÞ=GP q ; 2 N where G is p  q; qXp; and A is p  p; is a generalized version of the known integral

 Z N 1 1 1 1 ðax2 þ bx þ cÞn dx ¼ an2 22n1 ð4ac  b2 Þ2n B ; n  : 2 2 N

References [1] [2] [3] [4]

H. Cramer, Mathematical Methods of Statistics, Princeton University Press, Princeton, NJ, 1946. R.C. Geary, Extension of a theorem by Harold Cramer, J. Roy. Statist. Soc. 17 (1944) 56–57. A.K. Gupta, D.K. Nagar, Matrix Variate Distributions, Chapman & Hall/CRC, Boca Raton, 2000. P.C.B. Phillips, Distribution of matrix quotients, J. Multivariate Anal. 16 (1985) 157–161.