A Determinant Representation for the Distribution of Quadratic Forms in Complex Normal Vectors

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1 Journal of ultivariate Analysis 73, 5565 (000 doi:0.006jmva , availale online at on A Determinant Representation for the Distriution of Quadratic Forms in Complex ormal Vectors Hongsheng Gao and Peter J. Smith Institute of Statistics and Operations Research, Victoria University of Wellington, P.O. Box 600, Wellington, ew Zealand Received Septemer 4, 996 Let the column vectors of X: _, <, e distriuted as independent complex normal vectors with the same covariance matrix 7. Then the usual quadratic form in the complex normal vectors is denoted y Z=XLX H where L: _ is a positive definite hermitian matrix. This paper deals with a representation for the density function of Z in terms of a ratio of determinants. This representation also yields a compact form for the distriution of the generalized variance Z. 000 Academic Press AS 99 suject classifications: 6H0, 33C70. Key words and phrases: quadratic form, complex normal vector, hypergeometric functions, distriutions.. ITRODUCTIO If the complex random matrix X: _, <, is distriuted as Gaussian whose density is given y? & 7 & B & exp(&tr 7 & XB & X H ( where 7 and B are hermitian positive definite, then the density function of Z=XLX H (L eing a hermitian positive definite matrix is given y Khatri [] as where f(z=( ( LB 7 & Z & _exp(&q & tr 7 & Z 0 F 0 (T*, q & 7 & Z ( q>0, T*=I &ql & B & L &, (=? &(& ` 55 (&i X Copyright 000 y Academic Press All rights of reproduction in any form reserved.

2 56 GAO AD SITH Although compact and simple to state, the density function given in ( is extremely difficult to compute due to the hypergeometric function in matrix argument. A straightforward power series expansion of 0 F 0 (. in terms of zonal polynomials is unlikely to produce a satisfactory numerical procedure due to the slow convergence of the series and the difficulty of working with partitions of large integers [, 3]. For example the algorithm due to claren [4] for computing the coefficients of zonal polynomials is restricted to partitions of the integers up to 3 and pulished tales go as high as [5]. Alternative representations for hypergeometric functions are availale in terms of partial differential equations [6], series of Laguerre polynomials [, 7, 8], series of chi-square distriutions [8] and Wishart type representations [7]. one of these approaches is satisfactory in general for numerical work and so in this paper we derive yet another representation ased on the work of Gross and Richards [9]. Our result expresses the density function of Z in terms of a ratio of determinants. For = this collapses to the simple scalar distriution discussed for example in [0]. For small the expression can e expanded to give a numerically practical formulation. The outline of the paper is as follows. In Section we state and prove the new representation for the density of Z. In Section 3 we discuss the use of this result in computational work, give some special cases and provide an integral for the distriution of the generalized variance Z.. A DETERIAT REPRESETATIO FOR THE DESITY OF Z Lemma. Let x, x,..., x e nonzero eigenvalues of 7 & Z, and #, #,..., # e eigenvalues of the matrix L BL, then the density function, f(z, of Z is given y where f(z=? (& 7 & 3 (3 =} # # # # && # && # && # && exp(&x # & # && exp&x # & # && exp(&x # & # && exp(&x # & # && exp(&x # & # && exp(&x # & } =} x x x x x x x & x & x & }=` (x i &x j i> j

3 QUADRATIC FORS I COPLEX ORAL VECTORS 57 3 =} # # # # # # # & # & # & }= ` (# i &# j i> j Proof. First we set up some notation which is necessary for the proof. 0<x <x <<x, #=diag(#, #,..., #, T=diag(t, t,..., t =I &q# & =diag(&q# &,&q# &,..., &q# & S=diag(s,..., s =diag(0, q & =, q & =,..., q & = &&, q & x, q & x,..., q & x and 0<q & = <q & = <..., q & = && <q & x <q & x <<q & x h i (x=exp[&# & i x], g i (x=x i&,,,..., h(x=(h (x, h (x,..., h (x H, g(x=(g (x, g (x,..., g (x H h (n (x= d n h(x g (n (x= d n g(x = \d n h (x = \d n g (x H dx + n, d n h (x,..., d n h (x H dx + n, d n g (x,..., d n g (x The algera ehind the proof is cumersome ut the steps are simple. First we note that the exact result in ( is simple to compute except for the hypergeometric function. Hence we appeal to a result due to Gross and Richards [9] who give a representation for the hypergeometric function in terms of a ratio of determinants. Unfortunately their representation requires all the eigenvalues of the matrix arguments to e unequal. This is not the case for the matrix argument q & 7 & Z since we must inflate this _ matrix to an _ matrix y adding a order of zero elements. This results in & eigenvalues equal to zero. Hence we pertur the zero diagonal elements y =, =,... in such a way that the pertured matrix S has q & 7 & Z as the non-zero principal minor in the limit as [= i ] 0. From the properties of the hypergeometric function we have 0F 0 (T*, q & 7 & Z= 0 F 0 (T, q & 7 & Z. Also we can choose q>0 and a set [= i ] such that &T& &S&<, thus 0 F 0 (T, S converges asolutely. Hence all the conditions required y Gross and Richard's [9] result are satisfied and we have det( 0F 0 (T, q & 7 & Z= lim F [= i ] (T, S= lim ; 0 F 0 (s i t j [= i ] 0 V(S V(T (4

4 58 GAO AD SITH where ; = ` det( 0 F 0 (s i t j ] j= V(T =(& (& ( j, V(S=(& (& ` }0F 0(s t 0F 0 (s t 0F 0 (s t ` i< j 0F 0 (s t 0F 0 (s t 0F 0 (s t (t j &t i i< j 0F 0 (s t 0F 0 (s t } 0F 0 (s t (s j &s i, and the functions 0 F 0 (. are the standard scalar hypergeometric functions. Using the fact that V(S and V(T are determinants of Vandermonde matrices the three determinants in (4 can e expressed as 0 = = && x x } V(S=(& (& q &(&}0 = = && x x 0 = & = & && x & x & =(& (& q &(& det[ g(0, g(=,..., g(= &&, g(x,..., g(x ] (&\ # # # V(T =(&q ` # i+&+ } } # # # # & # & # & =(&q (& LB &+ 3 det( 0 F 0 (s i t j =exp _ q& : x & _} =exp _ q& exp[&# & = ] exp[&# & = ] exp[&# & = ] : exp[&# & = && ] exp[&# & = && ] exp[&# & = &&] exp[&# & x ] exp[&# & x ] exp[&# & x ] x i& det[h(0, h(=,..., h(= &&, h(x,..., h(x ] exp[&# & x ] exp[&# & x ]} ] exp[&# & x ote that we have omitted the term exp[q & && = i ] in the aove as it disappears in the limit. The difficult part of (4 is the ratio of

5 QUADRATIC FORS I COPLEX ORAL VECTORS 59 det( 0 F 0 (s i t j to V(S since oth vanish as [= i ] 0. We evaluate this ratio using Cauchy's mean value theorem as elow: det[h(0, h(=,..., h(= &&, h(x,..., h(x ] det[ g(0, g(=,..., g(= &&, g(x,..., g(x ] = det[h(0, h( (!, h(=,..., h(= &&, h(x,..., h(x ] det[ g(0, g ( (!, g(=,..., g(= &&, g(x,..., g(x ] = det[h(0, h( (!, h ( (!,..., h ( (! &&, h(x,..., h(x ] det[ g(0, g ( (!, g ( (!,..., g ( (! &&, g(x,..., g(x ] = det[h(0, h( (!, h ( (!,..., h ( (! &&, h(x,..., h(x ] det[ g(0, g ( (!, g ( (!,..., g ( (! &&, g(x,..., g(x ] = det[h(0, h( (!, h ( (!,..., h ( (! &&, h(x,..., h(x ] det[ g(0, g ( (!, g ( (!,..., g ( (! &&, g(x,..., g(x ] = det[h(0, h( (!, h ( (!,..., h (&& (! &&, h(x,..., h(x ] det[ g(0, g ( (!, g ( (!,..., g (&& (! &&, g(x,..., g(x ] where 0! i = i,,,..., &&. Repeated application of the mean value theorem is required since the j th column requires j& differentiations efore it gives a column which in the limit does not cause the determinants to vanish. Since h ( j (x and g ( j (x are vectors which are continuous functions at x=0, we have lim [= i ] 0 = lim [= i ] 0 det[h(0, h(=,..., h(= &&, h(x,..., h(x ] det[ g(0, g(=,..., g(= &&, g(x,..., g(x ] det[h(0,h ( (!, h ( (!,..., h (&&,(! &&, h(x,..., h(x ] det[g(0, g ( (!, g ( (!,..., g (&& (! &&, g(x,..., g(x ] = det[h(0, h( (0, h ( (0,..., h (&& (0, h(x,..., h(x ] det[ g(0, g ( (0, g ( (0,..., g (&& (0, g(x,..., g(x ]

6 60 GAO AD SITH From (4 and the aove we otain 0F 0 (T, q & 7 & Z det( 0 F 0 (s i t j = lim ; [= i ] 0 V(S V(T = ; V(T lim [= i ] 0 \ exp[q& x i] _det[h(0, h(=,..., h(= &&, h(x,..., h(x ]+ \ (&(& q &(& _det[ g(0, g(=,..., g(= &&, g(x,..., g(x ]+ exp[q&( x i] = \; _det[h(0, h ( (0,..., h (&& (0, h(x,..., h(x ]+ LB &+ 3 q &(& _det[ g(0, g ( (0,..., g (&& (0, g(x,..., g(x ]+ \ q(& That is 0F 0 (T, q & 7 & Z exp[q & tr(z] = \; _det[h(0, h ( (0,..., h (&& (0, h(x,..., h(x ]+ \ LB &+ 3 _det[ g(0, g ( (0,..., g (&& (0, g(x,..., g(x ]+ (5 ow it is necessary to evaluate the matrices in (5 which now contain derivatives of h(. and g(.. Since h ( j (0=(# &j, # &j,..., # &j, we have in the numerator det[h(0, h ( (0, h ( (0,..., h (&& (0, h(x,..., h(x ] =} (&# & (&# & (&# & (&# & && (&# & && (&# & && exp(&x # & exp(&x # & exp(&x # & exp(&x # & exp(&x # & exp(&x # & } = ` # &++ i } # # # # &+ # && # && # && exp(&x # & # && exp(&x # & # && exp(&x # & # && exp(&x # & # && exp(&x # & # && exp(&x # & }

7 QUADRATIC FORS I COPLEX ORAL VECTORS 6 Hence det[h(0, h ( (0, h ( (0,..., h (&& (0, h(x,..., h(x ] = LB &++ (6 Also in the denominator we have det[ g(0, g ( (0,..., g (&& (0, g(x,..., g(x ] Hence ! 0 0 x x 0 0! 0 x x = (&&! x && x && x & x & x & x & & = ` j= ( j&! ` x & i } x x x } x x x x & x & x & det[ g(0, g ( (0,..., g (&& (0, g(x,..., g(x ] & = 7 & Z & ` j= Sustituting (6 and (7 into (5 gives ( j (7 0F 0 (T, q & 7 & Z= ; exp[q & tr(7 & Z] LB &++ LB &+ 3 7 & Z & > & j= ( j (8 Hence 0F 0 (T, q & 7 & Z= ; exp[q & tr(7 & Z] LB 3 7 & Z & > & j= ( j (9

8 6 GAO AD SITH From Eqs. ( and (9 we have f(z=( ( LB 7 & Z & _exp(&q & tr7 & Z 0 F 0 (T*, q & 7 & Z =( ( LB 7 & Z & _exp(&q & tr7 & Z 0 F 0 (T, q & 7 & Z =( ( LB 7 & Z & exp(&q & tr7 & Z _ ; exp[q & tr(7 & Z] LB 3 7 & Z & > & j= ( j ; = ( 3 7 > & j= ( j > j= = ( j? &(& > (&i+ 3 7 > & j= ( j =? (& 7 & 3 3. COCLUSIO The quadratic form Z has een studied for many years and several representations are already availale for the density of Z. However these representations are usually complex series solutions, often involving summations over partitions, which are difficult to use in numerical work. In Section we have derived a new expression for density of Z in terms of a ratio of determinants ased on the work of Gross and Richard [9]. This expression gives an alternative form of solution which is an appealing formulation in its own right and may e useful in numerical work for small values of. For example when =, B=I, L=diag(#, #,..., # and 7=I we have the well known scalar quadratic form Z= : # i / i

9 QUADRATIC FORS I COPLEX ORAL VECTORS 63 where / i are iid chi-square random variales with d.o.f. In this case Eq. (3 gives } # # & # & exp(&z# & # # & # & exp(&z# & # # & # & exp(&z# & f (z= (0 } # # # # # # # & # & # & } } Using Laplace's expansion theorem to expand the numerator in (0 gives the density as a sum of exponentials and y mathematical induction we can show that f(z=(& & : # & i exp(&z# & i > j{i (# j &# i ( This is the form given in [0] and [] for example. For small a similar approach leads to reasonaly simple expressions for the density. By repeated use of Laplace's expansion theorem a recursion can e developed to compute ( 3 which leads to finite doule sums in exponentials (=, triple sums in exponentials (=3, etc. The general recursion can e written in terms of the sudeterminants (r,..., r j (& j of, where (r,..., r j (& j is the determinant of the & j_& j sumatrix of gained y removing the last j columns of and the set of rows [r,..., r j ]. We also use (& j and 3 (& j to refer to the determinants of the leading principal minors, with sizes & j and & j, of the matrices with determinants and 3 respectively. The first step is to write (& = 3 > & (# &# i > & (x &x i : k= _exp(&x # & (k (& k (& 3 (& (& k # && k

10 64 GAO AD SITH Susequent recursions are given y (r,..., r j& (& j+ (& j+ 3 (& j+ = & j+ (& > & j (# & j+ &# i > & j (x & j+ &x i & j+ _ : k= (& k # && s k exp(&x & j+ # & s k _ (r,..., r j&, s k (& j (& j 3 (& j where [r,..., r j& ] _ [s,..., s & j+ ]=[,,..., ]. The recursion stops after steps since (... (&, (0 and 3 (& are all known Vandermonde determinants. As an example of this approach for = we can write? f(x, x = (x &x > i> j (# i &# j _ : k={h<k (& k+h+ (# k # h &3 _exp[&(x # &+x k # & h + h>k (& k+h (# k # h &3 _exp[&(x # &+x k # & h ] > i, i {k, h i >i (# i &# i i, i{k, h ] > i >i (# i &# i = Finally we can also use the new representation to give a new and compact expression for the distriution of the generalized variance Z F Z (z=p( Z z=p( 7 & Z 7 & z =? (& 7 & x x 7 & z where i,,, 3 are shown in (3. 3 dx dx REFERECES. C. G. Khatri, On certain distriution prolems ased on positive definite quadratic functions in normal vectors, Ann. ath. Statist. 37 (966, R. J. uirhead, Expressions for some hypergeometric functions of matrix argument with applications, J. ultivariate Anal. 5 (975, T. Sugiyama, Percentile points of the largest latent root of a matrix and power calculations for testing hypothesis 7=I, J. Japan Statist. Soc. 3 (97, L. claren, Coefficients of the zonal polynomials, Appl. Statist. 5 (976, 887.

11 QUADRATIC FORS I COPLEX ORAL VECTORS A.. Parkhurst and A. T. James, Zonal polynomials of order through, in ``Selected Tales in athematical Statistics'' (H. L. Harter and D. B. Owen, Eds., pp , American athematical Society, Providence, RI, R. J. uirhead, Partial differential equations for hypergeometric functions of matrix argument, Ann. ath. Statist. 4 (970, C. G. Khatri, Series representations of distriutions of quadratic form in the normal vectors and generalised variance, J. ultivariate Anal. (97, W. J. Conradie and A. K. Gupta, Quadratic forms in complex normal variates: asic results, Statistica XLVII ( (987, K. I. Gross, and D. ST. P. Richards, Total positivity, spherical series, and ypergeometric functions of matrix argument, J. Approx. Theory 59 (989, A.. athai and Serge B. Provost, ``Quadratic Forms in Random Variales: Theory and Applications,'' Dekker, ew York, 99.. C. Bingham, Approximating the matrix Fisher and Bingham distriutions: Applications to spherical regression and Procrustes analysis, J. ultivariate Anal. 4 (99,