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Funciones Beta y Gama generalizadas extendidas y sus aplicaciones.

Generalized Extended Matrix Variate Beta and Gamma Functions and Their Applications

1 Introduction

The gamma function was first introduced by Leonard Euler in 1729, as the limit of a discrete expression and later as an absolutely convergent improper integral, namely,

[GAMMA](a) = [[integral].sup.[infinity].sub.0] [t.sup.a-1]exp(-t)dt, Re(a) > 0. (1)

The gamma function has many beautiful properties and has been used in almost all the branches of science and engineering.

One year later, Euler introduced the beta function defined for a pair of complex numbers a and b with positive real parts, through the integral

B(a, b) = [[integral].sup.1.sub.0][t.sup.a-1][(1 - t).sup.b-1]dt, Re(a) > 0, Re(b) > 0. (2)

The beta function has many properties, including symmetry, B(a, b) = B(b, a), and its relationship to the gamma function,

B(a, b) = [GAMMA](a)[GAMMA](b)/[GAMMA](a + b).

In statistical distribution theory, gamma and beta functions have been used extensively. Using integrands of gamma and beta functions, the gamma and beta density functions are usually defined.

Recently, the domains of gamma and beta functions have been extended to the whole complex plane by introducing in the integrands of (1) and (2), the factors exp(-[sigma]/t) and exp[-[sigma]/i(1 - t)], respectively, where Re([sigma]) > 0. The functions so defined have been named extended gamma and extended beta functions.

In 1994, Chaudhry and Zubair [7] defined the extended gamma function, [GAMMA]f(a; [alpha]), as

[GAMMA](a; [sigma]) = [[integral].sup.[infinity].sub.0][t.sup.a-1]exp(-t - [sigma]/t)dt, (3)

where Re([sigma]) > 0 and a is any complex number. For Re(a) > 0 and [sigma] = 0, it is clear that the above extension of the gamma function reduces to the classical gamma function, [GAMMA](a, 0) = [GAMMA](a). The extended gamma function is a special case of Kratzel function defined in 1975 by Kratzel [16]. The generalized gamma function (extended) has been proved very useful in various problems in engineering and physics, see for example, Chaudhry and Zubair [8].

In 1997, Chaudhry et al. [6] defined the extended beta function

B(a, b; [sigma]) = [[integral].sup..sup.10][t.sup.a-1][(1 - t).sup.b]exp[-[sigma]/t(1 - t)]dt, (4)

where Re([sigma]) > 0 and parameters a and b are arbitrary complex numbers. When [sigma] = 0, it is clear that for Re(a) > 0 and Re(b) > 0, the extended beta function reduces to the classical beta function B(a, b).

Recently, Ozergin, Ozarslan and Altin [24] have further generalized the extended gamma and extended beta functions as

[[GAMMA].sup.([alpha],[beta])](a; [sigma]) = [[integral].sup.[infinity].sub.0][t.sup.a-1][PHI]([alpha]; [beta]; -t - [sigma]/t)dt, (5)

[[beta].sup.([alpha],[beta])]([alpha]; [beta]; *) = [[integral].sup.1.sub.0][t.sup.a-1][(1 - t).sup.b-1][PHI]([alpha]; [beta]; -[sigma]/t(1 - t))dt, (6)

where [PHI]([alpha]; [beta]; *) is the type 1 confluent hypergeometric function. The gamma function, the extended gamma function, the beta function, the extended beta function, the gamma distribution, the beta distribution and the extended beta distribution have been generalized to the matrix case in various ways. These generalizations and some of their properties can be found in Olkin [23], Gupta and Nagar [10], Muirhead [18], Nagar, Gupta, and Sanchez [19], Nagar, Roldan-Correa and Gupta [20], Nagar and RoldanCorrea [21], and Nagar, Moran-Vasquez and Gupta [22]. For some recent advances the reader is refereed to Hassairi and Regaig [12], Farah and Hassairi [4], Gupta and Nagar [11], and Zine [25]. However, generalizations of the extended gamma and extended beta functions defined by (5) and (6), respectively, to the matrix case have not been defined and studied. It is, therefore, of great interest to define generalizations of the extended gamma and beta functions to the matrix case, study their properties, obtain different integral representations, and establish the connection of these generalizations with other known special functions of matrix argument.

This paper is divided into seven sections. Section 2 deals with some well known definitions and results on matrix algebra, zonal polynomials and special functions of matrix argument. In Section 3, the extended matrix variate gamma function has been defined and its properties have been studied. Definition and different integral representations of the extended matrix variate beta function are given in Section 4. Some integrals involving zonal polynomials and generalized extended matrix variate beta function are evaluated in Section 5. In Section 6, the distribution of the sum of dependent generalized inverted Wishart matrices has been derived in terms of generalized extended matrix variate beta function. We introduce the generalized extended matrix variate beta distribution in Section 7.

2 Some known definitions and results

In this section we give several known definitions and results. We first state the following notations and results that will be utilized in this and subsequent sections. Let A = ([a.sup.ij]) be an m x m matrix of real or complex numbers. Then, A denotes the transpose of A; tr(A) = [a.sub.11] + ... + [a.sub.mm]; etr(A) = exp(tr(A)); det(A) = determinant of A; [parallel]A[parallel] = spectral norm of A; A = A' > 0 means that A is symmetric positive definite, 0 < A < [I.sub.m] means that both A and [I.sub.m] - A are symmetric positive definite, and [A.sup.1/2] denotes the unique positive definite square root of A > 0.

Several generalizations of the Euler's gamma function are available in the scientific literature. The multivariate gamma function, which is frequently used in multivariate statistical analysis, is defined by

[[GAMMA].sub.m](a) = [[integral].sub.X>0]etr(-X)det[(X).sup.a-(m+1)/2]dX, (7)

where the integration is carried out over m x m symmetric positive definite matrices. By evaluating the above integral it is easy to see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

The multivariate generalization of the beta function is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

The generalized hypergeometric function of one matrix argument as defined by Constantine [9] and James [15] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

where [C.sub.[kappa]](X) is the zonal polynomial of mxm complex symmetric matrix X corresponding to the ordered partition [kappa] = ([k.sub.1], ..., [k.sub.m]), [k.sub.1] [greater than or equal to] ... [greater than or equal to] [k.sub.m] [greater than or equal to] 0, [k.sub.1] + ... + [k.sub.m] = k and [[summation].sub.[kappa][??]k] denotes summation over all partitions [kappa]. The generalized hypergeometric coefficient [(a).sub.[kappa]] used above is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

where [(a).sub.r] = a(a + 1) ... (a + r - l), r = 1, 2, ... with [(a).sub.0] = 1. The parameters [a.sub.i], i = 1, ..., p, [b.sub.j], j = 1, ..., q are arbitrary complex numbers. No denominator parameter bj is allowed to be zero or an integer or half-integer [less than or equal to] (m - 1)/2. If any numerator parameter [a.sub.i] is a negative integer, [a.sub.1] = -r, then the function is a polynomial of degree mr. The series converges for all X if p [less than or equal to] q, it converges for [parallel]X[parallel] < 1 if p = q + 1, and, unless it terminates, it diverges for all X [not equal to] 0 if p > q.

If X is an m x m symmetric matrix, and R is an m x m symmetric positive definite matrix, then the eigenvalues of RX are same as those of [R.sup.1/2][XR.sup.1/2], where [R.sup.1/2] is the unique symmetric positive definite square root of R. In this case [C.sub.[kappa]](RX) = [C.sub.[kappa]]([R.sup.1/2][XR.sup.1/2]) and

[sub.q][F.sub.q] ([a.sub.1], ..., [a.sub.p]; [b.sub.1], ..., [b.sub.q]; RX)

= [sub.q][F.sub.q]([a.sub.1], ..., [a.sub.p]; [b.sub.1], ..., [b.sub.q]; [R.sup.1/2][XR.sup.1/2]). (12)

Two special cases of (10) are the confluent hypergeometric function and the Gauss hypergeometric function denoted by [PHI] and F, respectively. They are given by

[PHI](a; c; X) = [[infinity].summation over (k=0)][summation over ([kappa][??]k)][[(a).sub.[kappa]]/[(c).sub.[kappa]]][C.sub.[kappa]](X)/k!

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

The integral representations of the confluent hypergeometric function [PHI] and the Gauss hypergeometric function F are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

and for X < [I.sub.m],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (15)

where Re(a) > (m - 1)/2 and Re(c - a) > (m - 1)/2.

For properties and further results on these functions the reader is referred to Herz [13], Constantine [9], James [15], and Gupta and Nagar [10].

The confluent hypergeometric function [PHI] satisfies the Kummer's relation

[PHI](a; c; X) = etr(X)[PHI](c - a; c; -X). (16)

From (15), it is easy to see that

G(a, b; c; [I.sub.m]) = [[GAMMA].sub.m](c)[[GAMMA].sub.m](c - a - b)/[[GAMMA].sub.m](c - a)[[GAMMA].sub.m](c - b).

Lemma 2.1. Let Z be an mxm complex symmetric matrix with Re(Z) > 0 and let Y be an m x m complex symmetric matrix. Then, for Re(t) > (m - 1)/2, we have

[[integral].sub.S>0]etr(-ZS)det[(S).sup.t-(m+1)/2][C.sub.[kappa]](YS)dS

= [[GAMMA].sub.m](t, [kappa]) det[(Z).sup.-t][C.sub.[kappa]]([Z.sup.-1]Y). (17)

Lemma 2.2. Let Z be an mxm complex symmetric matrix with Re(Z) > 0 and let Y be an m x m complex symmetric matrix. Then, for Re(t) > [k.sub.1] + (m - 1)/2, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

Lemma 2.3. Let Y be an m x m complex symmetric matrix, then for Re(a) > (m - 1)/2 and Re(b) > (m - 1)/2, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)

Lemma 2.4. Let Y be an m x m complex symmetric matrix. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (20)

where Re(a) > [k.sub.1] + (m - 1)/2 and Re(b) > (m - 1)/2.

Results given in Lemma 2.1 and Lemma 2.3 were given by Constantine [9] while Lemma 2.2 and Lemma 2.4 were derived in Khatri [17]. In the expressions (17) and (18), [[GAMMA].sub.m](a, [rho]) and [[GAMMA].sub.m](a, -[rho]), for an ordered partition [rho] of r, [rho] = ([r.sub.1], ..., [r.sub.m]), are defined by

[[GAMMA].sub.m](a, [rho]) = [(a).sub.[rho]][[GAMMA].sub.m](a), [[GAMMA].sub.m](a, 0) = [[GAMMA].sub.m](a)

and

[[GAMMA].sub.m](a, -[rho]) = [(-1).sup.r][[GAMMA].sub.m](a)/[(-a + (m + 1)/2).sub.[rho]], Re(a) > [r.sub.1] + m - 1/2,

respectively.

Definition 2.1. The extended matrix variate gamma function, denoted by [[GAMMA].sub.m](a; [summation]), is defined by

[[GAMMA].sub.m](a; [summation]) = [[integral].sub.Z>0]det[(Z).sup.a-(m+1)/2]etr(-Z - [summation][Z.sup.-1])dZ, (21)

where Re([summation]) > 0 and a is an arbitrary complex number.

From (21), one can easily see that for Re([summation]) > 0 and H [member of] O(m), [[GAMMA].sub.m](a; H[summation]H') = [[GAMMA].sub.m](a; [summation]) thereby [[GAMMA].sub.m](a; [summation]) depends on the matrix [summation] only through its eigenvalues if [summation] is a real matrix.

From the definition, it is clear that if [summation] = 0, then for Re(a) > (m-1)/2, the extended matrix variate gamma function reduces to the multivariate gamma function [[GAMMA].sub.m](a).

Definition 2.2. The extended matrix variate beta function, denoted by [B.sub.m](a, b; [summation]), is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (22)

where where Re([summation]) > 0 and a and b are arbitrary complex numbers. If [summation] = 0, then Re(a) > (m - 1)/2 and Re(b) > (m - 1)/2.

3 Generalized extended matrix variate Gamma function

A matrix variate generalization of the generalized extended gamma function can be defined in the following way:

Definition 3.1. The generalized extended matrix variate gamma function, denoted by [[GAMMA].sup.([alpha],[beta]).sub.m]([alpha]; [summation]), is defined by

[[GAMMA].sup.([alpha],[beta]).sub.m](a; [summation]) = [[integral].sub.Z>0]det[(Z).sup.a-(m+1)/2] , (23)

where [summation] > 0 and a is an arbitrary complex number.

From the definition it is clear that for [alpha] = [beta] the generalized extended matrix variate gamma function reduces to an extended matrix variate gamma function, i.e., [[GAMMA].sup.([alpha],[alpha]).sub.m](a; [summation]) = [[GAMMA].sub.m](a; [summation]). Further, if [alpha] = [beta] and [summation] = 0, then for Re(a) > (m - 1)/2, the generalized extended matrix variate gamma function reduces to the multivariate gamma function [[GAMMA].sub.m](a).

Replacing [PHI]([alpha]; [beta]; -Z - [Z.sup.-1/2][summation][Z.sup.-1/2]) by its integral representation, namely,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (24)

where Re([alpha]) > (m - 1)/2 and Re([beta] - [alpha]) > (m - 1)/2, in (23), an alternative integral representation of the generalized extended matrix variate gamma function can be given as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (25)

Two special cases of (25) are worth mentioning. For m = l, this expression simplifies to

[[GAMMA].sup.([alpha],[beta])](a; [sigma]) = [1/B([alpha], [beta] - [alpha])][[integral].sup.1.sub.0][y.sup.[alpha]-a-1][(1 - y).sup.[beta]-[alpha]-1][GAMMA](a; [sigma][y.sup.2])dy.

Further, for [summation] = [I.sub.m], substituting X = [Y.sup.1/2][ZY.sup.1/2] with the Jacobian J(Z [right arrow] X) = det[(Y).sup.-(m+1)/2] in (25) and applying (21), the expression for [[GAMMA].sup.([alpha],[beta]).sub.m](a; [I.sub.m]) is derived as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (26)

In the following theorem we establish a relationship between generalized extended gamma function of matrix argument and multivariate gamma function through an integral involving the generalized extended gamma function of matrix argument and zonal polynomials.

Theorem 3.1. For Re(s) > (m - 1)/2, Re(s + a) > (m - 1)/2, Re([alpha] - a - 2s) > (m - 1)/2 and Re([beta] - a - 2s) > (m - 1)/2, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (27)

Proof. Replacing [[GAMMA].sup.([alpha],[beta]).sub.m](a; [summation]) by its integral representation given in (25) and changing the order of integration, the left hand side integral in (27) is re-written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (28)

where Re([alpha]) > (m - 1)/2 and Re([beta] - [alpha]) > (m - 1)/2.

Further, using Lemma 2.1, the integral involving S is evaluated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (29)

Replacing (29) in (28), to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (30)

Now, integration of Z using Lemma 2.1 yields the desired result. ?

Theorem 3.2. For a symmetric positive definite matrix T of order m, Re(s) > (m - 1)/2, Re(s + a) > (m - 1)/2, Re([alpha] - a - 2s) > (m - 1)/2 and Re([beta] - a - 2s) > (m - 1)/2, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (31)

Corollary 3.2.1. For Re(s) > (m - 1)/2, Re(s + a) > (m - 1)/2, Re([alpha] - a - 2s) > (m - 1)/2 and Re([beta] - a - 2s) > (m - 1)/2, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (32)

Note that the above corollary gives an interesting relationship between the generalized extended gamma function of matrix argument and multivariate gamma function. Substituting s = (m + 1)/2, in (32), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 3.3. For Re(s) > [k.sub.1] + (m - 1)/2 and Re(s + a) > [k.sub.1] + (m -1)/2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (33)

where [kappa] = ([k.sub.1], ..., [k.sub.m]), [k.sub.1] [greater than or equal to] ... [greater than or equal to] [k.sub.m] [greater than or equal to] 0 and [k.sub.1] + .... + [k.sub.m] = k.

Proof. Replacing [[GAMMA].sup.([alpha],[beta]).sub.m](a; [summation]) by its integral representation given in (25), the left hand side integral in (33) is re-written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (34)

Now, integrating first with respect to [summation] and then with respect to Z by using Lemma 2.2, we obtain the desired result.

Theorem 3.4. For [summation] > 0 and a > (m - 1)/2, [alpha] - a > (m - 1)/2 and [beta] - a > (m - 1)/2, we have

[[GAMMA].sup.([alpha],[beta]).sub.m](a; [summation]) [[GAMMA].sub.m](a)[[GAMMA].sub.m]([beta])[[GAMMA].sub.m]([alpha] - a)/[[GAMMA].sub.m]([alpha])[[GAMMA].sub.m]([beta] - a).

Proof. Let Z, Y and [summation] be symmetric positive definite matrices of order m. Further, let [[lambda].sub.1], ..., [[lambda].sub.m] be the characteristic roots of the matrix [Y.sup.1/2][Z.sup.-1/2][summation][Z.sup.-1/2][Y.sup.1/2]. Then

etr(-[Z.sup.-1/2][summation][Z.sup.-1/2]Y) = exp[-([[lambda].sub.1] + ... + [[lambda].sub.m])].

Since Z > 0, Y > 0 and [summation] > 0, we have [Y.sup.1/2][Z.sup.-1/2][summation][Z.sup.-1/2][Y.sup.1/2] > 0, and therefore [[lambda].sub.1] + ... + [[lambda].sub.m] > 0. Further, as exp(-t) < 1, for all t > 0, we have

etr(-[Z.sup.-1/2][summation][Z.sup.-1/2]Y) = exp[-([[lambda].sub.1] + ... + [[lambda].sub.m])] < 1. (35)

Now, using the above inequality in (25), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (36)

Finally, integrating Z and Y by using multivariate gamma and multivariate beta integrals and simplifying the resulting expression, we obtain the desired result.

Theorem 3.5. Suppose that [[sigma].sub.1] and [[sigma].sub.n] are the smallest and largest eigenvalues of the matrix [summation] > 0. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Note that

[[sigma].sub.1]tr([Z.sup.-1/2]Y[Z.sup.-1/2]) [less than or equal to] tr([summation][Z.sup.-1/2]Y[Z.sup.-1/2]) [less than or equal to] [[sigma].sub.n]tr([Z.sup.-1/2]Y[Z.sup.-1/2])

and therefore

exp[-[[sigma].sub.n]([Z.sup.-1/2]Y[Z.sup.-1/2])] [less than or equal to] etr(-[Z.sup.-1/2][summation][Z.sup.-1/2]Y)

[less than or equal to] exp[-[[sigma].sub.1]([Z.sup.-1/2]Y[Z.sup.-1/2])].

Now, applying the above inequality in (25), and using the integral representation of the generalized extended gamma function given in (25), we obtain the desired result.

By Holder's inequality, it is possible to obtain an interesting inequality that follows.

Theorem 3.6. Let l < p < [infinity] and (1/p) + (1/q) = l. Then, for [summation] [greater than or equal to] 0, x > (m - 1)/2 and y > (m - 1)/2, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (37)

Proof. Substituting a = x/p + y/q in (23) and using Holder's inequality, we obtain the desired result.

Substituting p = q = 2 in (37), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where x > (m - 1)/2), y > (m - 1)/2, and [summation] [greater than or equal to] 0.

4 Generalized extended matrix variate Beta function

In this section, a matrix variate generalization of (6) is defined and several of its properties are studied.

Definition 4.1. The generalized extended matrix variate beta function, denoted by [B.sup.([alpha],[beta]).sub.m](a, b; [summation]), is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (38)

where a and b are arbitrary complex numbers and [summation] > 0. If [summation] = 0, then Re(a) > (m - 1)/2, Re(b) > (m - 1)/2.

Using Kummer's relation (16), the above expression can also be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (39)

From (38), it is apparent that [B.sup.([alpha],[beta]).sub.m](a, b; [summation]) = [B.sup.([alpha],[beta]).sub.m](b, a; [summation]). Further, [[beta].sup.([alpha],[beta]).sub.m](a, b; [summation]) = [[beta].sup.([alpha],[beta]).sub.m](a, b; H[summation]H'), H [member of] O(m), thereby [B.sup.([alpha],[beta]).sub.m](a, b; [summation]) is a function of the eigenvalues of the matrix [summation] > 0.

Replacing the confluent hypergeometric function by its integral representation in (38), changing the order of integration, and integrating Z by using (22), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (40)

Theorem 4.1. For Re(s) > (m -1)/2, Re(s + a) > (m -1)/2, Re(s + b) > (m - 1)/2, Re([alpha] - s) > (m - 1)/2 and Re([beta] - s) > (m - 1)/2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (41)

Proof. Replacing [B.sup.([alpha],[beta]).sub.m](a, b; [summation]) by its equivalent integral representation given in (39) and changing the order of integration, the integral in (41) is rewritten as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (42)

where we have used the result

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Finally, evaluating (42) by using the definition of multivariate beta function, we obtain the desired result

By letting s = (m + 1)/2, in (41), we obtain an interesting relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

between the multivariate beta function and the generalized extended beta function of matrix argument.

Theorem 4.2. For a and b arbitrary complex numbers and Re([summation]) > 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (43)

Proof. Substituting Z = [([I.sub.m] + Y).sup.-1] Y with the Jacobian J(Z [right arrow] Y) = det[([I.sub.m] + Y).sup.-(m+1)] in (38), we obtain the desired result.

Theorem 4.3. For a and b arbitrary complex numbers and Re([summation]) > 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Noting that

[B.sup.([alpha],[beta]).sub.m](a, b; [summation]) = 1/2[[B.sup.([alpha],[beta]).sub.m](a, b; [summation]) + [B.sup.([alpha],[beta]).sub.m](b, [alpha]; [summation])]

and substituting for [B.sup.([alpha],[beta]).sub.m](a, b; [summation]) and [B.sup.([alpha],[beta]).sub.m](b, a; [summation]) from (43), we obtain the desired result.

In the following theorem, we present an important inequality that shows how the extended matrix variate beta function decreases exponentially compared to the multivariate beta function.

Theorem 4.4. For [summation] > 0, Re([alpha]) > (m + 1)/2, Re([beta] - -[alpha]a) > (m - 1)/2, a > (m - 1)/2 and b > (m - 1)/2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. For [summation] > 0, a > (m - 1)/2 and b > (m - 1)/2, Nagar, Roldan and Gupta [20] have shown that

[absolute value of [B.sub.m](a, b; [summation])] [less than or equal to] etr(-4[summation])[B.sub.m](a, b) [less than or equal to] [exp(-m)/det(4[summation])][B.sub.m](a, b).

Now, using (40) and the above inequality, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The desired result is now obtained by evaluating integrals using (14) and (9) and simplifying the resulting expression.

Theorem 4.5. Suppose that [[sigma].sub.1] and [[sigma].sub.n] are the smallest and largest eigenvalues of the matrix S. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Similar to the proof of Theorem 3.5.

The next result is obtained by applying the Minkowski inequality for determinants. The famous Minkowski inequality states that if A and B are symmetric positive definite matrices of order m, then

det[(A + B).sup.1/m] [greater than or equal to] det[(A).sup.1/m] + det[(B).sup.1/m].

Theorem 4.6. For the generalized extended beta function of matrix argument, we have

[B.sup.([alpha],[beta]).sub.m](a + 1/m, b; [summation]) + [B.sup.([alpha],[beta]).sub.m](a, b + 1/m; [summation]) [less than or equal to] [B.sup.([alpha],[beta]).sub.m](a, b; [summation]).

Proof. Replacing [B.sup.([alpha],[beta]).sub.m] (a + l/m, b; [summation]) and [B.sup.([alpha],[beta]).sub.m](a, b + 1/m; [summation]) by their respective integral representation, one obtains

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now, by noting that det[(Z).sup.1/m] + det[([I.sub.m] - Z).sup.1/m] - 1 we obtain the desired result.

5 Results involving zonal polynomials

In this section, we will compute the integral

[[integral].sub.Z>0] det[([summation]).sup.s-(m+1)/2][C.sub.[kappa]]([summation])[B.sup.([alpha],[beta]).sub.m](a, b; [summation])d[summation], (44)

where Re(s) > (m - 1)/2, Re(s + a) > (m - 1)/2 and Re(s + b) > (m - 1)/2.

The calculation of this integral requires evaluation of the integrals of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (45)

where Re(a) > (m - 1)/2 and Re(b) > (m - 1)/2.

Recently, Nagar, Roldan-Correa and Gupta [20] have given computable representations of (45) for k = l and k = 2 which we state in the following three lemmas.

Lemma 5.1. For Re(a) > (m - 1)/2 and Re(b) > (m - 1)/2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (46)

where

[K.sub.(1)](a, b) = a/a + b[1 - (a + 1)(m + 2)/3(a + b + 1) + (a - 1/2)(m - 1)/3(a + b - 1/2)].

Proof. See Nagar, Roldan-Correa and Gupta [20].

Lemma 5.2. For Re(a) > (m - 1)/2 and Re(b) > (m - 1)/2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (47)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. See Nagar, Roldan-Correa and Gupta [20].

Lemma 5.3. For Re(a) > (m - 1)/2 and Re(b) > (m - 1)/2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (48)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. See Nagar, Roldan-Correa and Gupta [20].

In the following theorem and three corollaries we give closed form representations of the integral (44) for k = l and k = 2.

Theorem 5.1. For Re(s) > (m -1)/2, Re(s + a) > (m -1)/2, Re(s + b) > (m - 1)/2, Re(a - s) > [k.sub.1] + (m - 1)/2 and Re([beta] - s) > [k.sub.1] + (m - 1)/2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (49)

Proof. Replacing (a, b; S) by its equivalent integral representation given in (40) and changing the order of integration, the integral in (49) is rewritten as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (50)

Now, evaluating the integral containing [summation] using Lemma 2.1, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (51)

Further, substituting (51) in (50) and integrating with respect to X by using Lemma 2.4, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (52)

where Re([alpha] - s) > [k.sub.1+] (m - 1)/2 and Re([beta] - s) > [k.sub.1+] (m - 1)/2. Now, substituting appropriately, we obtain the result.

Corollary 5.1.1. For Re(s) > (m - 1)/2, Re(s + a) > (m - 1)/2 and Re(s + b) > (m - 1)/2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (53)

where Re([alpha] - s) > (m + 1)/2 and Re([beta] - s) > (m + 1)/2.

Proof. Substituting [kappa] = (l) in (49) and using Lemma 5.1, we obtain the desired result.

Corollary 5.1.2. For Re(s) > (m - 1)/2, Re(s + a) > (m - 1)/2 and Re(s + b) > (m - 1)/2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (54)

where Re([alpha] - s) > (m + 3)/2 and Re([beta] - s) > (m + 3)/2.

Proof. Substituting [kappa] = (2) in (49) and using Lemma 5.2, we obtain the desired result.

Corollary 5.1.3. For Re(s) > (m - 1)/2, Re(s + a) > (m - 1)/2 and Re(s + b) > (m - 1)/2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (55)

where Re([alpha] - s) > (m + 1)/2 and Re([beta] - s) > (m + 1)/2.

Proof. Substituting k = (l, l) in (49) and using Lemma 5.3, we obtain the desired result.

Corollary 5.1.4. For Re(s) > (m - 1)/2, Re(s + a) > (m - 1)/2 and Re(s + b) > (m - l)/2, Re([alpha] - s) > (m + 3)/2 and Re([beta] - s) > (m + 3)/2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. We obtain the desired result by summing (54) and (55) and using the result [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Corollary 5.1.5. For Re(s) > (m - 1)/2, Re(s + a) > (m - 1)/2 and Re(s+b) > (m-1)/2, Re([alpha] - s) > (m + 3)/2 and Re([beta] - s) > [k.sub.1] + (m + 3)/2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. We obtain the desired result by using [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

6 Application to multivariate statistics

The Wishart distribution, which is the distribution of the sample variance covariance matrix when sampling from a multivariate normal distribution, is an important distribution in multivariate statistival analysis. Recently, Bekker et al. [2, 3] and Bekker, Roux and Arashi [1] have used Wishart distribution in deriving a number of matrix variate distributions. Further, Bodnar, Mazur and Okhrin [5] have considered exact and approximate distribution of the product of a Wishart matrix and a Gaussian vector. The inverted Wishart distribution is widely used as a conjugate prior in Bayesian statistics (Iranmanesh et al. [14]). Knowledge of densities of functions of inverted Wishart matrices is useful for the implementation of several statistical procedures and in this regard we show that the distribution of the sum of dependent generalized inverted Wishart matrices can be written in terms generalized extended beta function of matrix argument. If W ~ [IW.sub.m](v, [PSI]), v > m - 1, [PSI] > 0, then its p.d.f. is given by

det[([PSI]).sup.(v-m-1)/2]etr(-[PSI][W.sup.-1]/2)det[(W).sup.-v/2]/[2.sup.m(v-m-1)/2][[GAMMA].sub.m][(v - m - 1/2), W > 0.

By replacing etr (-[PSI][W.sup.-1]/2) by the confluent hypergeometric function of matrix argument [Phi]([alpha]; [beta]; -[PSI][W.sup.-1]/2) and evaluating the normalizing constant, a generalization of the inverted Wishart distribution can be defined by the density

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Further, a bi-matrix variate generalization of the above density can be defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (56)

where [W.sub.1] > 0 and [W.sub.2] > 0. Note that if we take [alpha] = [beta] in the above density, then [W.sub.1] and [W.sub.2] are independent, [W.sub.1] ~ [IW.sub.m]([v.sub.1], [PSI]) and [W.sub.2] ~ [IW.sub.m]([v.sub.2, [PSI]). Further, the marginal density of [W.sub.1] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (57)

where [W.sub.1] > 0. Likewise, the marginal density of [W.sub.2] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (58)

where [W.sub.2] > 0.

Theorem 6.1. Suppose that the joint density of the random matrices [W.sub.1] and [W.sub.2] is given by (56). Then, the p.d.f. of the sum W = [W.sub.1] + [W.sub.2] is derived as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Proof. Transforming W = [W.sub.1] + [W.sub.2], R = [W.sup.-1/2][W.sub.1][W.sup.-1/2] with the Jacobian J([W.sub.1], [W.sub.2] [right arrow] R, W) = det[(W).sup.(m+1)/2] in (56), the joint p.d.f. of R and W is obtained as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where W > 0 and 0 < R < [I.sub.m]. Now, integrating R by using (38), the marginal p.d.f. of W is obtained.

Corollary 6.1.1. Suppose that the joint density of the random matrices [W.sub.1] and [W.sub.2] is given by (56). Then, the p.d.f. of S = [([W.sub.1] + [W.sub.2]).sup.-1] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Next, we will derive results like [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 6.2. Suppose that the joint density of the random matrices [W.sub.1] and [W.sub.2] is given by (56) with [PSI] = [I.sub.m] and S = [([W.sub.1] + [W.sub.2]).sup.-1]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are given in Lemma 5.1, Lemma 5.2 and Lemma 5.3, respectively.

Proof. The expected value of CK(S) is derived as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now, setting s = ([v.sub.1] + [v.sub.2] - 2m - 2)/2, a = -([v.sub.1] - m - 1)/2, b = -([v.sub.2] - m - 1)/2 and using Corollary 5.1.1 for [kappa] = (1), Corollary 5.1.2 for [kappa] = (2), Corollary 5.1.3 for [kappa] = (1[bar]2), we obtain the desired result.

Theorem 6.3. Suppose that the joint density of the random matrices Wi and W2 is given by (56) with [PSI] = [I.sub.m] and S = [([W.sub.1] + [W.sub.2]).sup.-1]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are given in Lemma 5.1, Lemma 5.2 and Lemma 5.3, respectively.

Proof. We obtain the desired result by using [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and Theorem 6.2.

Theorem 6.4. Suppose that the joint density of the random matrices W' and W2 is given by (56) with [PSI] = [I.sub.m] and S = [([W.sub.1] + [W.sub.2]).sup.-1]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are given in Lemma 5.1, Lemma 5.2 and Lemma 5.3, respectively.

Proof. Since, for any m x m orthogonal matrix H, the random matrices S and HSH' have the same distribution, we have E(S) = [c.sub.1][I.sub.m], E([S.sup.2]) = [c.sub.2][I.sub.m] and E[(tr S)S] = [c.sub.3][I.sub.m] and hence E(tr S) = [c.sub.1]m, E(tr [S.sup.2]) = [c.sub.2]m and E[[(tr S).sup.2]] = [c.sub.3]m. Thus, the coefficient of m in the expressions for E(tr S), E(tr [S.sup.2]) and E[[(tr S).sup.2]] are [c.sub.1], [c.sub.2] and [c.sub.3], respectively. Finally, using Theorem 6.3, we obtain the desired result.

7 Generalized extended matrix variate Beta distribution

Recently, Nagar, Roldan-Correa and Gupta [20] and Nagar and Roldan-Correa [21], by using the integrand of the extended matrix variate beta function, generalized the conventional matrix variate beta distribution and studied several of its properties. We define the generalized extended matrix variate beta density as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where -[infinity] < p < [infinity], -[infinity] < q < [infinity] and [summation] > 0. If r and s are real numbers, then

E[det[(X).sup.r]det[([I.sub.m] - X).sup.s]] = [B.sup.([alpha],[beta]).sub.m](p + r, q + s; [summation])/[B.sup.([alpha],[beta]).sub.m](p, q; [summation]).

Specializing r and s in the above expression, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],.

doi: 10.17230/ingciencia.12.24.3

Received: 06-02-2016 | Accepted: 12-09-2016 | Online: 15-11-2016

MSC: 33E99, 62H99

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Daya K. Nagar (1), Sergio Alexander Gomez-Noguera (2) and Arjun K. Gupta (3)

(1) Universidad de Antioquia, dayaknagar@yahoo.com, ORCID:http//orcid.org/0000. 0003-4337-6334, Medellin, Colombia.

(2) Universidad de Antioquia, sgomez1987@gmail.com, Medellin, Colombia.

(3) Bowling Green State University, gupta@bgsu.edu, Bowling Green, USA
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