distribution. Therefore, the distribution of sample sum of squares and crossproducts matrix, which has a Wishart distribution, plays an important role in almost all.
TAIWANESE JOURNAL OF MATHEMATICS Vol. 7, No. 3, pp. 477-491, September 2003 This paper is available online at http://www.math.nthu.edu.tw/tjm/
NON-CENTRAL MATRIX-VARIATE DIRICHLET DISTRIBUTION Luz Estela Sa´ nchez and Daya K. Nagar
Abstract. Let Xi » Wp (ni ; §; £) where £ = diag(µi2 ; 0; : : : ; 0), i = 1; : : : ; r + 1. In this article the authors have Pr+1derived the joint distribution of Ui = C ¡1 Xi C 0¡1 , i = 1; : : : ; r where i=1 Xi = CC 0 and C is a lower triangular matrix. The joint distribution of U1 ; : : : ; Ur is a non-central matrixvariate Dirichlet distribution. Several properties of this distribution such as marginal and conditional distributions, distribution of partial sums, moments and asymptotic results have also been studied.
1. INTRODUCTION The multivariate statistical analysis heavily depends upon multivariate normal distribution. Therefore, the distribution of sample sum of squares and crossproducts matrix, which has a Wishart distribution, plays an important role in almost all inferential procedures. A distribution closely connected to the Wishart, known as ‘matrix-variate beta’ was introduced by Prof. P. L. Hsu while studying distribution of roots of certain determinantal equation. The matrix-variate beta distribution arises in various problems in multivariate statistical analysis. Several test statistics in multivariate analysis of variance and covariance are functions of beta matrix. In Bayesian analysis, this distribution and some of its properties are utilized in preposterior analysis of parameters of normal multivariate regression models. An extension of the matrix-variate beta distribution is the “matrix-variate Dirichlet distribution”, which is useful in several testing problems in multivariate statistical analysis (Troskie [20]). For example, the likelihood ratio test statistic for testing homogeneity of several multivariate normal distributions is a function of Dirichlet matrices. Received Auguest 24, 2001; revised March 26, 2002. Communicated by H. M. Srivastava. 2000 Mathematics Subject Classification: Primary 62E15; Secondary 62H99. Key words and phrases: Asymptotic; confluent hypergeometric function; Dirichlet distribution; Kampe´ de Fe´ riet’s function; matrix variate; non-central; matrix-variate beta distribution; transformation.
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Luz Estela Sa´ nchez and Daya K. Nagar
In this article, we derive a non-central matrix-variate Dirichlet distribution. In Section 2, we give certain known definitions and results that are used to derive the main results. The non-central matrix-variate Dirichlet distribution has been derived in Section 3. Section 4 deals with certain properties and asymptotic expansion of this distribution. 2. SOME USEFUL RESULTS In this section we give definitions and results that will be used in the subsequent sections. The generalized hypergeometric functions of one and several variables will be used to derive the density function, marginal and conditional distributions and several moment expressions of random matrices which are jointly distributed as noncentral matrix-variate Dirichlet. Throughout this work we will use the Pochammer symbol (a)n defined by (a)n = a(a + 1) ¢ ¢ ¢ (a + n ¡ 1) = (a)n¡1 (a + n ¡ 1) for n = 1; 2; : : : ; and (a)0 = 1. The generalized hypergeometric function of scalar argument is defined by 1 X (a1 )k ¢ ¢ ¢ (ap )k z k ; p Fq (a1 ; : : : ; ap ; b1 ; : : : ; bq ; z) = (b1 )k ¢ ¢ ¢ (bq )k k!
(2.1)
k=0
where ai , i = 1; : : : ; p; bj ; j = 1; : : : ; q are complex numbers with suitable restrictions and z is a complex variable. Conditions for the convergence of the series in (2.1) are available in the literature, see Luke [13]. From (2.1) it is easy to see that P1 P1 (a)k xk xk 0 F1 (b; x) = k=0 (b)k k! and 1 F1 (a; b; x) = k=0 (b)k k! . Next, we will define confluent hypergeometric and generalized Kampe´ de Fe´ riet’s functions of several variables. For further results and properties of these functions the reader is referred to Srivastava and Kashyap [17, Section II.7] and Srivastava and Karlsson [16, Section 1.4]. The confluent hypergeometric function in m variables z1 ; : : : ; zm is defined by (2.2)
(m)
ª2 [a; c1 ; : : : ; cm ; z1 ; : : : ; zm ] =
1 X
j1 ;::: ;jm =0
jm (a)j1 +¢¢¢+jm z1j1 ¢ ¢ ¢ zm (c1 )j1 ¢ ¢ ¢ (cm )jm j1 ! ¢ ¢ ¢ jm !
where the series expansion is valid for all zi 2 R. Using the results Z 1 ¡(a + j) 1 (a)j = = exp(¡t)ta+j¡1 dt; Re(a) > 0; ¡(a) ¡(a) 0 for j = 0; 1; 2; : : : , and (2.3)
(m) ª2 [a; c1 ; : : :
P1
(tzi )ji ji =0 (ci )ji ji !
= 0 F1 (ci ; tzi ) in (2.2), one obtains
1 ; cm ; z1 ; : : : ; zm ] = ¡(a)
Z
1 0
exp(¡t)ta¡1
m Y i=1
0 F1 (ci ; tzi ) dt:
Non-Central Matrix-Variate Dirichlet Distribution
479
(m)
For m = 1, the function ª2 reduces to the confluent hypergeometric function (m) = ª2 is the Humbert’s confluent hypergeometric function 1 F1 . For m = 2, ª2 and (2.2) slides to ª2 [a; c1 ; c2 ; z1 ; z2 ] =
1 X (a)j1 z1j1 1 F1 (a + j1 ; c2 ; z2 ) (c1 )j1 j1 !
j1 =0
(2.4) =
1 X (a)j2 z2j2 1 F1 (a + j2 ; c1 ; z1 ): (c2 )j2 j2 !
j2 =0
The generalized Kampe´ de Fe´ riet’s function in m variables z1 ; : : : ; zm is defined as follows (Srivastava and Panda [18, p. 1127]): 1 ;¢¢¢ ;ºm F ¹:º ½:¾1 ;¢¢¢ ;¾m
(2.5) =
1 X
·
j1 ;::: ;jm =0
a1 ; : : : ; a¹ : b11 ; : : : ; b1º1 ; ¢ ¢ ¢ ; bm1 ; : : : ; bmºm ; z ; : : : ; zm c1 ; : : : ; c½ : d11 ; : : : ; d1¾1 ; ¢ ¢ ¢ ; dm1 ; : : : ; dm¾m ; 1
Qm Q1 Q¹ jm (b ) z j1 ¢ ¢ ¢ zm (b1i )j1 ¢ ¢ ¢ ºi=1 (ai )j1 +¢¢¢+jm ºi=1 i=1 Q¾m mi jm 1 Q¾ 1 Q½ i=1 (dmi )jm j1 ! ¢ ¢ ¢ jm ! i=1 (d1i )j1 ¢ ¢ ¢ i=1 (ci )j1 +¢¢¢+jm
¸
where the series is convergent if ¹ + ºk < ½ + ¾k + 1; k = 1; : : : ; m or ¹ + ºk = ½ + ¾k + 1; k = 1; : : : ; m with either ¹ > ½ and jz1 j1=(¹¡½) + ¢ ¢ ¢ + jzm j1=(¹¡½) < 1 or ¹ · ½ and maxfjz1 j; : : : ; jzm jg < 1. Further generalization of the multivariable generalized Kampe´ de Fe´ riet’s function, which is referred to in the literature as generalized Lauricella function, is due to Srivastava and Daoust [15, p. 454]. It may be recorded here that under certain conditions the generalized Kampe´ de Fe´ riet’s function reduces to Lauricella functions FA , FB , FC , FD and generalized hypergeometric function of one variable. For ½ = º1 = ¢ ¢ ¢ = ºm = 0 and ¹ = ¾1 = ¢ ¢ ¢ = ¾m = 1, the generalized Kampe´ de Fe´ riet’s function reduces to the confluent hypergeometric function of several variables. Substituting ¹ = ½ = º1 = ¢ ¢ ¢ = ºm = ¾1 = ¢ ¢ ¢ = ¾m = 1 in (2.5), the generalized Kampe´ de Fe´ riet’s function in m variables z1 ; : : : ; zm simplifies to ;1 F 1:1;¢¢¢ 1:1;¢¢¢ ;1
(2.6) =
·
1 X
a : b1 ; ¢ ¢ ¢ ; bm ; z ; : : : ; zm c : d1 ; ¢ ¢ ¢ ; dm ; 1
j1 ;::: ;jm =0
¸
jm (a)j1 +¢¢¢+jm (b1 )j1 ¢ ¢ ¢ (bm )jm z1j1 ¢ ¢ ¢ zm : (c)j1 +¢¢¢+jm (d1 )j1 ¢ ¢ ¢ (dm )jm j1 ! ¢ ¢ ¢ jm !
If b1 = d1 ; : : : ; bm¡1 = dm¡1 , then (2.6) reduces to
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Luz Estela Sa´ nchez and Daya K. Nagar
·
a : b1 ; ¢ ¢ ¢ ; bm¡1 ; bm ; z ; : : : ; zm c : b1 ; ¢ ¢ ¢ ; bm¡1 ; dm ; 1 ¸ · a : ¡; bm ; Pm¡1 1:0;1 = F 1:0;1 i=1 zi ; zm c : ¡; dm ;
1:1;¢¢¢ ;1 F 1:1;¢¢¢ ;1
(2.7)
=
1 j X (a)j (bm )j zm j=0
(c)j (dm )j j!
¸
m¡1 ³ X ´ zi : 1 F1 a + j; c + j; i=1
Next we will give a result which has been used in Section 4 to derive moment expression. Lemma 2.1. For Re(®i ) > 0; i = 1; : : : ; m and Re(¯) > 0; (2.8) Z
¢¢¢
Z
0
