# Properties of Matrix Variate Confluent Hypergeometric Function Distribution

Authors: Arjun K. Gupta; Daya K. Nagar; Luz Estela Sánchez
Journal: Journal of Probability and Statistics
 (2016)

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## Abstract

We study matrix variate confluent hypergeometric function kind 1 distribution which is a generalization of the matrix variate gamma distribution. We give several properties of this distribution. We also derive density functions ofX2-1/2X1X2-1/2, (X1+X2)-1/2X1(X1+X2)-1/2, and X1+X2, where m×m independent random matrices X1 and X2 follow confluent hypergeometric function kind 1 and gamma distributions, respectively.


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## Body

## 1. Introduction

The matrix variate gamma distribution has many applications in multivariate statistical analysis. The Wishart distribution, which is the distribution of the sample variance covariance matrix when sampling from a multivariate normal distribution, is a special case of the matrix variate gamma distribution.The purpose of this paper is to give a generalization of the matrix variate gamma distribution and study its properties.We begin with a brief review of some definitions and notations. We adhere to standard notations (cf. Gupta and Nagar [1]). Let A=(aij) be an m×m matrix. Then, A′ denotes the transpose of A; tr(A)=a11+⋯+amm; etr(A)=exp⁡(tr(A)); det⁡(A) = determinant of A; norm of A=A = maximum of absolute values of eigenvalues of the matrix A; A>0 means that A is symmetric positive definite; and A1/2 denotes the unique symmetric positive definite square root of A>0. The multivariate gamma function Γm(a) is defined by(1)Γma=∫X>0etr⁡-Xdet⁡Xa-m+1/2dX=πmm-1/4∏i=1mΓa-i-12,Rea>m-12.Them×m symmetric positive definite random matrix X is said to have a matrix variate gamma distribution, denoted by X~Ga(m,ν,θ,Ω), if its probability density function (p.d.f.) is given by (2)det⁡Xν-m+1/2etr⁡-Ω-1X/θΓmνθmνdet⁡Ων,X>0,where Ω is a symmetric positive definite matrix of order m, θ>0, and ν>(m-1)/2. For Ω=Im, the above density reduces to a standard matrix variate gamma density and in this case we write X~Ga(m,ν,θ). Further, if X1~Ga(m,ν1,θ) and X2~Ga(m,ν2,θ) are independent gamma matrices, then the random matrix (X1+X2)-1/2X1(X1+X2)-1/2 follows a matrix variate beta type 1 distribution with parameters ν1 and ν2.By replacingetr(-Ω-1X/θ) by the confluent hypergeometric function of matrix argument F11α;β;-Ω-1X/θ, a generalization of the matrix variate gamma distribution can be defined by the p.d.f.:(3)Cν,α,β,θ,Ωdet⁡Xν-m+1/2F11α;β;-1θΩ-1X,where X>0 and C(ν,α,β,θ,Ω) is the normalizing constant. In Section 2, it has been shown that, for β-ν>(m-1)/2, α-ν>(m-1)/2, ν>(m-1)/2, θ>0, and Ω>0, the normalizing constant can be evaluated as (4)Cν,α,β,θ,Ω=ΓmαΓmβ-νdet⁡θΩ-νΓmνΓmβΓmα-ν.Therefore, the p.d.f. in (3) can be written explicitly as(5)ΓmαΓmβ-νdet⁡θΩ-νΓmνΓmβΓmα-νdet⁡Xν-m+1/2F11α;β;-1θΩ-1X,X>0,where β-ν>(m-1)/2, α-ν>(m-1)/2, ν>(m-1)/2, θ>0, Ω>0, and F11 is the confluent hypergeometric function of the first kind of matrix argument (Gupta and Nagar [1]). Since the density given above involves the confluent hypergeometric function, we will call the corresponding distribution aconfluent hypergeometric function distribution. We will write X~CHm(ν,α,β,θ,Ω,kind1) to say that the random matrix X has a confluent hypergeometric function distribution defined by the density (5). It has been shown by van der Merwe and Roux [2] that the above density can be obtained as a limiting case of a density involving the Gauss hypergeometric function of matrix argument. For α=β, the density (5) reduces to a matrix variate gamma density and for Ω=Im it slides to(6)ΓmαΓmβ-νθmνΓmνΓmβΓmα-νdet⁡Xν-m+1/2F11α;β;-1θX,X>0,where β-ν>(m-1)/2, α-ν>(m-1)/2, ν>(m-1)/2, and θ>0. In this case we will write X~CHm(ν,α,β,θ,kind1). The matrix variate confluent hypergeometric function kind 1 distribution occurs as the distribution of the matrix ratio of independent gamma and beta matrices. For m=1, (6) reduces to a univariate confluent hypergeometric function kind 1 density given by (Orozco-Castañeda et al. [3])(7)ΓαΓβ-νθνΓνΓβΓα-νxν-1F11α;β;-xθ,x>0,where β-ν>0, α-ν>0, ν>0, θ>0, and F11 is the confluent hypergeometric function of the first kind (Luke [4]). The random variable x having the above density will be designated by x~CH(ν,α,β,θ,kind1). Since the matrix variate confluent hypergeometric function kind 1 distribution is a generalization of the matrix variate gamma distribution, it is reasonable to say that the matrix variate confluent hypergeometric function kind 1 distribution can be used as an alternative to the gamma distribution quite effectively.Although ample information about matrix variate gamma distribution is available, little appears to have been done in the literature to study the matrix variate confluent hypergeometric function kind 1 distribution.In this paper, we study several properties including stochastic representations of the matrix variate confluent hypergeometric function kind 1 distribution. We also derive the density function of the matrix quotient of two independent random matrices having confluent hypergeometric function kind 1 and gamma distributions. Further, densities of several other matrix quotients and matrix products involving confluent hypergeometric function kind 1, beta type 1, beta type 2, and gamma matrices are derived.


## 2. Some Definitions and Preliminary Results

In this section we give some definitions and preliminary results which are used in subsequent sections.A more general integral representation of the multivariate gamma function can be obtained as(8)Γma=det⁡Ya∫R>0etr⁡-YRdet⁡Ra-m+1/2dR,where Re(a)>(m-1)/2 and Re(Y)>(m-1)/2. The above result can be established for real Y>0 by substituting X=Y1/2RY1/2 with the Jacobian J(X→R)=det⁡(Y)(m+1)/2 in (1) and it follows for complex Y by analytic continuation.The multivariate generalization of the beta function is given by(9)Bma,b=∫0Imdet⁡Ra-m+1/2det⁡Im-Rb-m+1/2dR=ΓmaΓmbΓma+b=Bmb,a,where Re(a)>(m-1)/2 and Re(b)>(m-1)/2.The generalized hypergeometric function of one matrix, defined in Constantine [5], is given by(10)​p⁡Fqa1,…,ap;b1,…,bq;X=∑k=0∞∑κ⊢ka1κ⋯apκb1κ⋯bqκCκXk!,where ai, i=1,…,p, bj, j=1,…,q are arbitrary complex numbers, X is an m×m complex symmetric matrix, Cκ(X) is the zonal polynomial of m×m complex symmetric matrix X corresponding to the ordered partition κ=(k1,…,km), k1≥⋯≥km≥0, k1+⋯+km=k, and ∑κ⊢k‍ denotes the summation over all partitions κ. The generalized hypergeometric coefficient (a)κ used above is defined by (11)aκ=∏i=1ma-i-12ki,where (a)r=a(a+1)⋯(a+r-1), r=1,2,…, with (a)0=1. Conditions for convergence of the series in (10) are available in the literature. From (10), it follows that(12)F00X=∑k=0∞∑κ⊢kCκXk!=etr⁡X,F01a;X=∑k=0∞∑κ⊢kaκCκXk!=det⁡Im-X-a,X<1,(13)F11a;c;X=∑k=0∞∑κ⊢kaκCκXcκk!,(14)F12a,b;c;X=∑k=0∞∑κ⊢kaκbκcκCκXk!,X<1.By taking ap=bq=c in (10), it can be observed that(15)
​
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p
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1
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p
-
1
;
b
1
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…
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b
q
-
1
;
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.Substituting p=2, q=1 in (15) and using (12), the Gauss hypergeometric function F12(a,c;c;X) is reduced as(16)F12a,c;c;X=F01a;X=det⁡Im-X-a,X<1.The integral representations of the confluent hypergeometric functionF11 and the Gauss hypergeometric function F12 are given by(17)F11a;c;X=1Bma,c-a∫0Imdet⁡Ra-m+1/2det⁡Im-Rc-a-m+1/2etr⁡XRdR,(18)F12a,b;c;X=1Bma,c-a∫0Imdet⁡Ra-m+1/2det⁡Im-Rc-a-m+1/2det⁡Im-XRbdR,where Re(a)>(m-1)/2 and Re(c-a)>(m-1)/2.Further generalizations of (8) and (9) in terms of zonal polynomials, due to Constantine [5], are given as(19)∫R>0etr⁡-YRdet⁡Ra-m+1/2CκXRdR=Γmaaκdet⁡Y-aCκY-1X,(20)∫0Imdet⁡Ra-m+1/2det⁡Im-Rb-m+1/2CκXRdR=Bma,baκa+bκCκX,respectively.ForRe(α)>(m-1)/2 and Re(β)>(m-1)/2, we have(21)
∫
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(22)
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m
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m
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m
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p
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β
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b
q
;
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.We can establish (21) and (22) by expanding ​p⁡Fq in series form by using (10) and integrating term by term by applying (19) and (20) and finally summing the resulting series.Note that the series expansions forF11 and F12 given in (13) and (14) can be obtained by expanding etr(XR) and det⁡(Im-XR)-b, XR<1, in (17) and (18) and integrating R using (20). Substituting X=Im in (18) and integrating, we obtain(23)F12a,b;c;Im=ΓmcΓmc-a-bΓmc-aΓmc-b,where Re(c-a-b)>(m-1)/2, Re(c-a)>(m-1)/2, Re(c-b)>(m-1)/2, and Re(c)>(m-1)/2. The hypergeometric function F11(a;c;X) satisfies Kummer’s relation(24)F11a;c;-X=etr⁡-XF11c-a;c;X.For properties and further results on these functions the reader is referred to Constantine [5], James [6], Muirhead [7], and Gupta and Nagar [1]. The numerical computation of a hypergeometric function of matrix arguments is very difficult. However, some numerical methods are proposed in recent years; see, Hashiguchi et al. [8] and Koev and Edelman [9].The generalized hypergeometric function withm×m complex symmetric matrices X and Y is defined by (25)​p⁡Fqma1,…,ap;b1,…,bq;X,Y=∑k=0∞∑κ⊢ka1κ⋯apκb1κ⋯bqκCκXCκYCκImk!.It is clear from the above definition that the order of X and Y is unimportant; that is, (26)
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a
p
;
b
1
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b
q
;
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,
X
.Also, if one of the argument matrices is the identity, this function reduces to the one argument function. Further, the two-matrix argument function ​p⁡Fqm can be obtained from the one-matrix function ​p⁡Fq by averaging over the orthogonal group O(m) using a result given in James [6, Equation 23]; namely, (27)∫OmCκXHYH′dH=CκXCκYCκIm,where (dH) denotes the normalized invariant measure on O(m). That is,(28)​p⁡Fqma1,…,ap;b1,…,bq;X,Y=∫Om​p⁡Fqma1,…,ap;b1,…,bq;XHYH′dH,given in James [6, Equation 30].Finally, we define the inverted matrix variate gamma, matrix variate beta type 1, and matrix variate beta type 2 distributions. These definitions can be found in Gupta and Nagar [1] and Iranmanesh et al. [10].Definition 1.
Anm×m random symmetric positive definite matrix X is said to have an inverted matrix variate gamma distribution with parameters μ, θ, and Ψ, denoted by X~InvGa(m,μ,θ,Ψ), if its p.d.f. is given by (29)det⁡X-μ-m+1/2etr⁡-Ψ-1X-1/θdet⁡θΨμΓmμ,X>0,where μ>(m-1)/2, θ>0, and Ψ is a symmetric positive definite matrix of order m.Definition 2.
Anm×m random symmetric positive definite matrix U is said to have a matrix variate beta type 1 distribution with parameters a(>(m-1)/2) and b(>(m-1)/2), denoted as U~B1(m,a,b), if its p.d.f. is given by(30)det⁡Ua-m+1/2det⁡Im-Ub-m+1/2Bma,b,0<U<Im.Definition 3.
Anm×m random symmetric positive definite matrix V is said to have a matrix variate beta type 2 distribution with parameters a(>(m-1)/2) and b(>(m-1)/2), denoted as V~B2(m,a,b), if its p.d.f. is given by(31)det⁡Va-m+1/2det⁡Im+V-a+bBma,b,V>0.Note that ifU~B1(m,a,b), then (Im-U)-1U~B2(m,a,b). Further, if X1 and X2 are independent, X1~Ga(m,ν1,θ) and X2~Ga(m,ν2,θ), then (X1+X2)-1/2X1(X1+X2)-1/2~B1(m,ν1,ν2) and X2-1/2X1X2-1/2~B2(m,ν1,ν2).We conclude this section by evaluating the normalizing constantC(ν,α,β,θ,Ω) in (3). Since the density over its support set integrates to one, we have (32)Cν,α,β,θ,Ω-1=∫X>0det⁡Xν-m+1/2F11α;β;-1θΩ-1XdX.By rewriting F11 using Kummer’s relation (24) and integrating X by applying (21), we get (33)Cν,α,β,θ,Ω-1=Γmνdet⁡θΩνF12ν,β-α;β;Im,where Re(ν)>(m-1)/2. Finally, writing F12ν,β-α;β;Im in terms of multivariate gamma functions by using (23), we obtain(34)Cν,α,β,θ,Ω-1=∫X>0det⁡Xν-m+1/2F11α;β;-1θΩ-1XdX=det⁡θΩνΓmνΓmβΓmα-νΓmαΓmβ-ν,where Re(β-ν)>(m-1)/2, Re(α-ν)>(m-1)/2, θ>0, Re(ν)>(m-1)/2, and Ω>0.


## 3. Properties

In this section we study several properties of the confluent hypergeometric function kind 1 distribution defined in Section1. For the sake of completeness, we first state the following results established in Gupta and Nagar [1].(1)
LetX~CHm(ν,α,β,θ,Ω,kind1) and let A be an m×m constant nonsingular matrix. Then, AXA′~CHm(ν,α,β,θ,AΩA′,kind1).(2)
LetX~CHm(ν,α,β,θ,kind1) and let H be an m×m orthogonal matrix whose elements are either constants or random variables distributed independent of X. Then, the distribution of X is invariant under the transformation X→HXH′ if H is a matrix of constants. Further, if H is a random matrix, then H and HXH′ are independent.(3)
LetX~CHm(ν,α,β,θ,Ω,kind1). Then, the cumulative distribution function (cdf) of X is derived as (35)PX≤Λ=ΓmαΓmβ-νΓmm+1/2Γmν+m+1/2ΓmβΓmα-νdet⁡θΩ-νdet⁡ΛνF22α,ν;β,ν+m+12;-1θΩ-1Λ,whereΛ>0.(4)
LetX=X11X12X21X22, where X11 is a q×q matrix. Define X11·2=X11-X12X22-1X21 and X22·1=X22-X21X11-1X12. If X~CHm(ν,α,β,θ,kind1), then (i) X11 and X22·1 are independent, X11~CHq(ν,α-(m-q)/2,β-(m-q)/2,θ,kind1) and X22·1~CHm-q(ν-q/2,α-q/2,β-q/2,θ, kind1), and (ii) X22 and X11·2 are independent, X22~CHm-q(ν,α-q/2,β-q/2,θ,kind1) and X11·2~CHq(ν-(m-q)/2,α-(m-q)/2,β-(m-q)/2,θ,kind1).(5)
LetA be a q×m constant matrix of rank q(≤m). If X~CHm(ν,α,β,θ,kind1), then AXA′~CHq(ν,α-(m-q)/2,β-(m-q)/2,θ,AA′,kind1) and (AX-1A′)-1~CHq(ν-(m-q)/2,α − (m-q)/2,β-(m-q)/2,θ,(AA′)-1,kind1).(6)
LetX~CHm(ν,α,β,θ,kind1) and let a be a nonzero m-dimensional column vector of constants, then (a′a)-1(a′Xa)~CH(ν,α-(m-1)/2,β-(m-1)/2,θ,kind1) and a′a(a′X-1a)-1~CH(ν-(m-1)/2,α − (m-1)/2,β-(m-1)/2,θ,kind1). Further, if y is an m-dimensional random vector, independent of X, and P(y≠0)=1, then it follows that (y′y)-1(y′Xy)~CH(ν,α-(m-1)/2,β-(m-1)/2,θ,kind1) and y′y(y′X-1y)-1~CH(ν-(m-1)/2,α-(m-1)/2,β-(m-1)/2,θ,kind1).It may also be mentioned here that properties (1)–(6) given above are modified forms of results given in Section 8.10 of Gupta and Nagar [1].If them×m random matrices X1 and X2 are independent, X1~CHm(ν,a1+ν,2ν,θ,kind1) and X2~CHm(ν,a2+ν,2ν,θ,kind1), ν=γ/2+(m+1)/4, then Roux and van der Merwe [11] have shown that X2-1/2X1X2-1/2 has matrix variate beta type 2 distributions with parameters a2 and a1.The matrix variate confluent hypergeometric function kind 1 distribution can be derived as the distribution of the matrix ratio of independent gamma and beta matrices. It has been shown in Gupta and Nagar [1] that if Y~Ga(m,ν,θ) and U~B1(m,a,b), then U-1/2YU-1/2~CHm(ν,a+ν,a+b+ν,θ,kind1).The expected values ofX and X-1, for X~CHm(ν,α,β,θ,kind1), can easily be obtained from the above results. For any fixed a∈Rm, a≠0,(36)Ea′Xaa′a=Ev1,where v1~CH(ν,α-(m-1)/2,β-(m-1)/2,θ,kind1), and(37)Ea′X-1aa′a=E1v2,where v2~CH(ν-(m-1)/2,α-(m-1)/2,β-(m-1)/2,θ,kind1). Hence, for all a∈Rm,(38)a′EXa=a′aEv1=νθβ-ν-m+1/2α-ν-m+1/2a′a,α-ν>m+12,β-ν>m+12,a′EX-1a=a′aE1v2=α-νθν-m+1/2β-νa′a,ν>m+12,which implies that (39)EX=νθβ-ν-m+1/2α-ν-m+1/2Im,α-ν>m+12,β-ν>m+12,EX-1=α-νθν-m+1/2β-νIm,ν>m+12.The Laplace transform of the density ofX, where X~CHm(ν,α,β,θ,kind1), is given by(40)θ-mνΓmαΓmβ-νΓmνΓmβΓmα-ν∫X>0etr⁡-ZX-Xθdet⁡Xν-m+1/2F11β-α;β;XθdX=ΓmαΓmβ-νΓmβΓmα-νdet⁡Im+θZ-νF12ν,β-α;β;Im+θZ-1,ReZ>0,where we have used (24) and (21). From the above expression, the Laplace transform of the density of X, where X~CHm(ν,α,β,θ,Ω,kind1), is derived as (41)ΓmαΓmβ-νΓmβΓmα-νdet⁡Im+θΩZ-νF12ν,β-α;β;Im+θΩZ-1,ReZ>0.Theorem 4.
LetX~CHm(ν,α,β,θ,Ω,kind1); then(42)Edet⁡Xh=det⁡θΩhΓmβ-νΓmν+hΓmα-ν-hΓmνΓmα-νΓmβ-ν-h,where Re(h+ν)>(m-1)/2, Re(h)<α-ν-(m-1)/2, and Re(h)<β-ν-(m-1)/2.Proof.
From the density ofX, we have (43)Edet⁡Xh=det⁡θΩ-νΓmαΓmβ-νΓmνΓmβΓmα-ν∫X>0det⁡Xν+h-m+1/2F11α;β;-1θΩ-1XdX.Now, evaluating the above integral by using (34), we get (44)Edet⁡Xh=det⁡θΩ-νΓmαΓmβ-νΓmνΓmβΓmα-νdet⁡θΩν+hΓmν+hΓmβΓmα-ν-hΓmαΓmβ-ν-h,where Re(h+ν)>(m-1)/2, Re(h)<α-ν-(m-1)/2, and Re(h)<β-ν-(m-1)/2. Finally, simplifying the above expression, we get the desired result.Corollary 5.
Letx~CH(ν,α,β,θ,kind1); then(45)Exh=θhΓβ-νΓν+hΓα-ν-hΓνΓα-νΓβ-ν-h,where Re(h+ν)>0, Re(h)<α-ν, and Re(h)<β-ν.Using (42) the mean and the variance of det⁡(X) are derived as(46)Edet⁡X=θmdet⁡Ω∏j=1mν-j-1/2β-ν-j+1/2α-ν-j+1/2,where β>ν+(m+1)/2, α>ν+(m+1)/2, and(47)Var⁡det⁡X=θ2mdet⁡Ω2∏j=1mν-j-1/2β-ν-j+1/2α-ν-j+1/2ν-j-3/2β-ν-j+3/2α-ν-j+3/2-ν-j-1/2β-ν-j+1/2α-ν-j+1/2,where β>ν+(m+3)/2 and α>ν+(m+3)/2. For m×m symmetric matrix A, E[Cκ(AX)] is derived as (48)ECκAX=det⁡θΩ-νΓmαΓmβ-νΓmνΓmβΓmα-ν∫X>0CκAXdet⁡Xν-m+1/2F11α;β;-1θΩ-1XdX.Replacing F11α;β;-Ω-1X/θ by its integral representation, namely, (49)F11α;β;-1θΩ-1X=ΓmβΓmαΓmβ-α∫0Imetr⁡-1θΩ-1/2XΩ-1/2Ydet⁡Yα-m+1/2det⁡Im-Yβ-α-m+1/2dY,where Re(β-α)>(m-1)/2 and Re(α)>(m-1)/2, one obtains(50)ECκAX=det⁡θΩ-νΓmβ-νΓmνΓmβ-αΓmα-ν∫0Imdet⁡Yα-m+1/2det⁡Im-Yβ-α-m+1/2∫X>0CκAXdet⁡Xν-m+1/2etr⁡-1θΩ-1/2XΩ-1/2YdXdY.Now, evaluating the above integral by using (19), we obtain(51)ECκAX=θkΓmβ-ννκΓmβ-αΓmα-ν∫0Imdet⁡Yα-ν-m+1/2det⁡Im-Yβ-α-m+1/2CκΩ1/2AΩ1/2Y-1dY.Finally, evaluating the integral involving Y by using (Khatrzi [12])(52)∫0Imdet⁡Ra-m+1/2det⁡Im-Rb-m+1/2CκXR-1dR=-a-b+m+1/2κΓmaΓmb-a+m+1/2κΓma+bCκX,Rea>m-12+k1,Reb>m-12,we get(53)ECκAX=θkν-β+m+1/2κνκν-α+m+1/2κCκΩA,where Re(α-ν)>(m-1)/2+k1 and Re(β-ν)>(m-1)/2+k1.Proceeding similarly and using the result (Khatrzi [12]) (54)∫R>0etr⁡-XRdet⁡Ra-m+1/2CκR-1SdR=-1kΓma-a+m+1/2κdet⁡X-aCκXS,Rea>m-12+k1,the expected value of Cκ(AX-1) is derived as(55)ECκAX-1=det⁡θΩ-νΓmβ-νΓmνΓmβ-αΓmα-ν∫0Imdet⁡Yα-m+1/2det⁡Im-Yβ-α-m+1/2∫X>0CκAX-1det⁡Xν-m+1/2etr⁡-1θΩ-1/2XΩ-1/2YdXdY=-1kθ-kΓmβ-νΓmβ-αΓmα-ν-ν+m+1/2κ∫0Imdet⁡Yα-ν-m+1/2det⁡Im-Yβ-α-m+1/2CκΩ-1/2AΩ-1/2YdY,where Re(ν)>(m-1)/2+k1. Finally, evaluating the above integral using (20) we obtain (56)ECκAX-1=-1kθ-kα-νκβ-νκ-ν+m+1/2κCκΩ-1A,Reν>m-12+k1.In the next theorem, we derive the confluent hypergeometric function kind 1 distribution using independent beta and gamma matrices.Theorem 6.
LetX1 and X2 be independent, X1~Ga(m,ν,θ) and X2~B1(m,a,b). Then, X2-1/2X1X2-1/2~CHm(ν,a+ν,a+b+ν,θ,kind1).Proof.
See Gupta and Nagar [1].Theorem 7.
LetX1 and X2 be independent, X1~Ga(m,ν,θ) and X2~B1(m,a,b). Then, X11/2X2-1X11/2~CHm(ν,a+ν,a+b+ν,θ,kind1).Proof.
The result follows from Theorem6 and the fact that X2-1/2X1X2-1/2 and X11/2X2-1X11/2 have same eigenvalues, and the matrix variate confluent hypergeometric function kind 1 distribution is orthogonally invariant.Theorem 8.
LetX1 and X2 be independent, X1~Ga(m,ν,θ) and X2~B1(m,a,b). Then, (Im-X2)-1/2X1(Im-X2)-1/2~CHm(ν,b+ν,a+b+ν,θ,kind1).Proof.
Noting thatIm-X2~B1(m,b,a) and using Theorem 6 we get the result.Theorem 9.
LetX1 and X2 be independent, X1~Ga(m,ν,θ) and X2~B2(m,a,b). Then, (Im+X2)1/2X1(Im+X2)1/2~CHm(ν,b+ν,a+b+ν,θ,kind1).Proof.
The desired result is obtained by observing that(Im+X2)-1~B1(m,b,a) and using Theorem 6.Theorem 10.
LetX1 and X2 be independent, X1~Ga(m,ν,θ) and X2~B2(m,a,b). Then, (Im+X2-1)1/2X1(Im+X2-1)1/2~CHm(ν,a+ν,a+b+ν,θ,kind1).Proof.
Noting that(Im+X2-1)-1~B1(m,a,b) and using Theorem 6 we get the result.Theorem 11.
LetX1 and X2 be independent, X1~Ga(m,ν1,θ) and X2~Ga(m,ν2,θ). Then, (X1+X2)X1-1(X1+X2)~CHm(ν1+ν2,2ν1+ν2,2ν1+2ν2,θ,kind1).Proof.
It is well known thatX1+X2 and (X1+X2)-1/2X1(X1+X2)-1/2 are independent, X1+X2~Ga(m,ν1+ν2,θ) and (X1+X2)-1/2X1(X1+X2)-1/2~B1(m,ν1,ν2). Therefore, using Theorem 7, (X1+X2)X1-1(X1+X2)~CHm(ν1+ν2,2ν1+ν2,2ν1+2ν2,θ,kind1).Theorem 12.
LetX1 and X2 be independent, X1~Ga(m,ν1,θ) and X2~Ga(m,ν2,θ). Then, (X1+X2)X2-1(X1+X2)~CHm(ν1+ν2,ν1+2ν2,2ν1+2ν2,θ,kind1).Proof.
The proof is similar to the proof of Theorem11.


## 4. Distributions of Sum and Quotients

In statistical distribution theory it is well known that ifX1 and X2 are independent, X1~Ga(m,ν1,θ) and X2~Ga(m,ν2,θ), then X2-1/2X1X2-1/2~B2(m,ν1,ν2), (X1+X2)-1/2X1(X1+X2)-1/2~B1(m,ν1,ν2), and X1+X2~Ga(m,ν1+ν2). In this section we derive similar results when X1 and X2 are independent confluent hypergeometric function kind 1 and gamma matrices, respectively.Theorem 13.
LetX1 and X2 be independent, X1~CHm(ν1,α1,β1,θ,kind1) and X2~Ga(m,ν2,θ). Then, the p.d.f. of Z=X2-1/2X1X2-1/2 is given by (57)Γmα1Γmβ1-ν1Γmν1+ν2Γmβ1Γmα1-ν1Γmν1Γmν2det⁡Zν1-m+1/2det⁡Im+Zν1+ν2F12β1-α1,ν1+ν2;β1;Im+Z-1Z,Z>0.Proof.
Using the independence, the joint p.d.f. ofX1 and X2 is given by(58)Kdet⁡X1ν1-m+1/2det⁡X2ν2-m+1/2etr⁡-1θX1+X2F11β1-α1;β1;1θX1,X1>0,X2>0,where(59)K=θ-mν1+ν2Γmα1Γmβ1-ν1Γmν1Γmν2Γmβ1Γmα1-ν1.Making the transformation Z=X2-1/2X1X2-1/2, X2=X2, with the Jacobian J(X1,X2→Z,X2)=det(X2)(m+1)/2 we obtain the joint p.d.f. of Z and X2 as(60)Kdet⁡Zν1-m+1/2det⁡X2ν1+ν2-m+1/2exp⁡-1θIm+ZX2F11β1-α1;β1;1θZX2,Z>0,X2>0.Now, integrating X2 in (60) by applying (21) and substituting for K, we obtain the desired result.Corollary 14.
LetX1 and X2 be independent, X1~CHm(ν1,α1,β1,θ,kind1) and X2~Ga(m,ν2,θ). Then, the p.d.f. of Z1=X21/2X1-1X21/2 is given by (61)Γmα1Γmβ1-ν1Γmν1+ν2Γmβ1Γmα1-ν1Γmν1Γmν2det⁡Z1ν2-m+1/2det⁡Im+Z1ν1+ν2F12β1-α1,ν1+ν2;β1;Im+Z1-1,Z1>0.Corollary 15.
LetX1 and X2 be independent, X1~Ga(m,ν1,θ) and X2~CHm(ν2,α2,β2,θ,kind1). Then, the p.d.f. of Z3=X2-1/2X1X2-1/2 is given by (62)Γmα2Γmβ2-ν2Γmν1+ν2Γmβ2Γmα2-ν2Γmν1Γmν2det⁡Z3ν1-m+1/2det⁡Im+Z3ν1+ν2F12β2-α2,ν1+ν2;β2;Im+Z3-1,Z3>0.Proof.
Interchanging subscripts1 and 2 in Corollary 14, the p.d.f. of Z2=X11/2X2-1X11/2 is given by (63)Γmα2Γmβ2-ν2Γmν1+ν2Γmβ2Γmα2-ν2Γmν1Γmν2det⁡Z2ν1-m+1/2det⁡Im+Z2ν1+ν2F12β2-α2,ν1+ν2;β2;Im+Z2-1,Z2>0,where X1~Ga(m,ν1,θ) and X2~CHm(ν2,α2,β2,θ,kind1). Now, the result follows from the fact that Z2=X11/2X2-1X11/2 and Z3=X2-1/2X1X2-1/2 have the same eigenvalues, and the distribution Z2 is orthogonally invariant.Corollary 16.
LetX1 and X2 be independent, X1~CHm(ν1,α1,ν1+ν2,θ,kind1) and X2~Ga(m,ν2,θ). Then, X2-1/2X1X2-1/2~B2(m,ν1,α1-ν1).Corollary 17.
Let the random matricesX1 and X2 be independent, X1~Ga(m,ν1,θ) and X2~CHm(ν2,α2,ν2+ν1,θ,kind1). Then, X2-1/2X1X2-1/2~B2(m,α2-ν2,ν2).Corollary 18.
LetX1 and X2 be independent, X1~Ga(m,ν1,θ) and X2~Ga(m,ν2,θ). Then, X2-1/2X1X2-1/2~B2(m,ν1,ν2).Theorem 19.
LetX1, X2, and X3 be independent, X1~Ga(m,μ,θ), X2~B1(m,a,b), and X3~Ga(m,ν,θ). Then, the p.d.f. of Z=X3-1/2X2-1/2X1X2-1/2X3-1/2 is given by (64)Γma+μΓma+bΓmμ+νΓma+b+μΓmaΓmμΓmνdet⁡Zμ-m+1/2det⁡Im+Zμ+νF12b,μ+ν;a+b+μ;Im+Z-1Z,Z>0.Proof.
Using the independence ofX1 and X2 and Theorem 6, X2-1/2X1X2-1/2~CHm(μ,a+μ,a+b+μ,kind1). Further, using independence of X2-1/2X1X2-1/2 and X3 and Theorem 13, we obtain the desired result.Corollary 20.
LetX1, X2, and X3 be independent, X1~Ga(m,μ,θ), X2~B1(m,a,b), and X3~Ga(m,a+b,θ). Then, X3-1/2X2-1/2X1X2-1/2X3-1/2~B2(m,μ,a).Proof.
Forν=a+b, the p.d.f. of Z=X3-1/2X2-1/2X1X2-1/2·X3-1/2 given in the above theorem slides to (65)Γma+μΓmaΓmμdet⁡Zμ-m+1/2det⁡Im+Zμ+a+bF12b,μ+a+b;a+b+μ;Im+Z-1Z,Z>0.Now, simplifying the Gauss hypergeometric function as (66)F12b,μ+a+b;a+b+μ;Im+Z-1Z=F01b;Im+Z-1Z=det⁡Im-Im+Z-1Z-b=det⁡Im+Zb,where we have used (16), the desired result is obtained.Corollary 21.
LetV and U be independent, V~B2(m,μ,ν) and U~B1(m,a,b). Then, the p.d.f. of Z=U-1/2VU-1/2 is given by (67)Γma+μΓma+bΓmμ+νΓma+b+μΓmaΓmμΓmνdet⁡Zμ-m+1/2det⁡Im+Zμ+νF12b,μ+ν;a+b+μ;Im+Z-1Z,Z>0.Further, if ν=a+b, then U-1/2VU-1/2~B2(m,μ,a).Proof.
Observe thatZ=X3-1/2X2-1/2X1X2-1/2X3-1/2 and X2-1/2X3-1/2X1X3-1/2X2-1/2 have same eigenvalues and the distribution of Z is orthogonally invariant. Therefore, the random matrices X3-1/2X2-1/2X1X2-1/2X3-1/2 and X2-1/2X3-1/2X1X3-1/2X2-1/2 have identical distribution. Now, setting V=X3-1/2X1X3-1/2 and U=X2, where X3-1/2X1X3-1/2~B2(m,μ,ν) and X2~B1(m,a,b), we observe that X3-1/2X2-1/2X1X2-1/2X3-1/2 and U-1/2VU-1/2 have identical distribution.Theorem 22.
LetX1 and X2 be independent, X1~CHm(ν1,α1,β1,θ,kind1) and X2~Ga(m,ν2,θ). Then, the p.d.f. of R=(X1+X2)-1/2X1(X1+X2)-1/2 is given by (68)Γmα1Γmβ1-ν1Γmν1+ν2Γmβ1Γmα1-ν1Γmν1Γmν2det⁡Rν1-m+1/2det⁡Im-Rν2-m+1/2F12β1-α1,ν1+ν2;β1;R,0<R<Imand the p.d.f. of S=X1+X2 is given by (69)θ-mν1+ν2Γmα1Γmβ1-ν1Γmβ1Γmα1-ν1Γmν1+ν2det⁡Sν1+ν2-m+1/2etr⁡-1θSF22ν1,β1-α1;ν1+ν2,β1;1θS,S>0.Proof.
SubstitutingR=(X1+X2)-1/2X1(X1+X2)-1/2 and S=X1+X2 with the Jacobian J(X1,X2→R,S)=det(S)(m+1)/2 in (58) we obtain the joint p.d.f. of R and S as (70)Kdet⁡Rν1-m+1/2det⁡Im-Rν2-m+1/2det⁡Sν1+ν2-m+1/2etr⁡-1θSF11β1-α1;β1;1θRS,0<R<Im,S>0.Now, integration of S by using (21) yields the density of R. The marginal density of S is obtained by integrating R by using (22).It may be remarked here that the density ofR given in the above theorem can also be obtained from the density of Z=X2-1/2X1X2-1/2 derived in Theorem 13 by making the transformation R=(Im+Z)-1Z.Corollary 23.
LetX1 and X2 be independent random matrices, X1~Ga(m,ν1,θ) and X2~CHm(ν2,α2,β2,θ,kind1). Then, the p.d.f. of R1=(X1+X2)-1/2X1(X1+X2)-1/2 is given by (71)Γmα2Γmβ2-ν2Γmν1+ν2Γmβ2Γmα2-ν2Γmν1Γmν2det⁡R1ν1-m+1/2det⁡Im-R1ν2-m+1/2F12β2-α2,ν1+ν2;β2;Im-R1,0<R1<Imand the p.d.f. of S1=X1+X2 is given by (72)θ-mν1+ν2Γmα2Γmβ2-ν2Γmβ2Γmα2-ν2Γmν1+ν2det⁡S1ν1+ν2-m+1/2etr⁡-1θS1F22ν2,β2-α2;ν1+ν2,β2;1θS1,S1>0.Proof.
Interchanging subscripts1 and 2 in Theorem 22, the p.d.f. of R2=(X1+X2)-1/2X2(X1+X2)-1/2 is given by (73)Γmα2Γmβ2-ν2Γmν1+ν2Γmβ2Γmα2-ν2Γmν1Γmν2det⁡R2ν2-m+1/2det⁡Im-R2ν2-m+1/2F12β2-α2,ν1+ν2;β2;R2,0<R2<Im,where now X1~Ga(m,ν1,θ) and X2~CHm(ν2,α2,β2,θ,kind1). The desired result is now obtained by observing that R2=Im-R1. Similarly, the p.d.f. of S1 is obtained by interchanging subscripts 1 and 2 in the p.d.f. of S.Corollary 24.
Let the random matricesX1 and X2 be independent, X1~CHm(ν1,α1,ν1+ν2,θ,kind1) and X2~Ga(m,ν2,θ). Then, (X1+X2)-1/2X1(X1+X2)-1/2~B1(m,ν1,α1-ν1).Proof.
The desired result is obtained by substitutingβ1=ν1+ν2 in the p.d.f. of R and simplifying the resulting expression by using (16).Corollary 25.
Let the random matricesX1 and X2 be independent, X1~Ga(m,ν1,θ) and X2~CH(m,ν2,α2,ν1+ν2,θ,kind1). Then, (X1+X2)-1/2X1(X1+X2)-1/2~B1(m,α2-ν2,ν2).Proof.
The desired result is obtained by substitutingβ2=ν1+ν2 in the p.d.f. of R1 and simplifying the resulting expression by using (16).Corollary 26.
Let the random matricesX1 and X2 be independent, X1~CHm(ν1,α1,α1+ν1+ν2,θ,kind1) and X2~Ga(m,ν2,θ). Then X1+X2~CHm(ν1+ν2,α1+ν2,α1+ν1+ν2,θ,kind1).Proof.
The result is obtained by substitutingβ1=α1+ν1+ν2 in the p.d.f. of S and simplifying the resulting expression by using (15).Corollary 27.
Let the random matricesX1 and X2 be independent, X1~Ga(m,ν1,θ) and X2~CHm(ν2,α2,α2+ν1+ν2,θ,kind1). Then X1+X2~CHm(ν1+ν2,α2+ν1,α2+ν1+ν2,θ,kind1).Proof.
The result is obtained by substitutingβ2=α2+ν1+ν2 in the p.d.f. of S and simplifying the resulting expression by using (15).Corollary 28.
Let the random matricesX1 and X2 be independent, X1~Ga(m,ν1,θ) and X2~Ga(m,ν2,θ). Then, (X1+X2)-1/2X1(X1+X2)-1/2~B1(m,ν1,ν2) and X1+X2~Ga(m,ν1+ν2,θ).Proof.
Substituteβ1=α1 in the p.d.f. of R or β2=α2 in the p.d.f. of R1 and simplify the resulting expression to get the desired result.


## 5. Related Distributions

This section gives distributional results for the determinant of the random matrix distributed as confluent hypergeometric function kind 1.In an unpublished report, Coelho et al. [13] have shown that if z is a positive random variable and E(zh) is defined for h in some neighborhood of zero, then the moments E(zh) uniquely identify the distribution of z. In the next theorem, we will use this result to derive the distribution of the product of two independent confluent hypergeometric function kind 1 variables.Theorem 29.
Ifx1~CH(ν,α,β,θ,kind1) and x2~CH(ν+1/2,α+1,β+1,θ,kind1) are independent, then 2x1x2~CH(2ν,2α,2β,θ,kind1).Proof.
Thehth moment of 2x1x2 is derived as(74)E2x1x2h=2θhΓβ-νΓν+h/2Γα-ν-h/2ΓνΓα-νΓβ-ν-h/2Γβ-ν+1/2Γν+1/2+h/2Γα-ν+1/2-h/2Γν+1/2Γα-ν+1/2Γβ-ν+1/2-h/2.
Now, using the duplication formula for gamma function, namely,(75)Γ2z=ΓzΓz+1/221-2zπ,the hth moment of 2x1x2 is rewritten as(76)E2x1x2h=θhΓ2β-2νΓ2ν+hΓ2α-2ν-hΓ2νΓ2α-2νΓ2β-2ν-h,where ν>0, α-ν>0, β-ν>0, 2ν>h, 2(α-ν)>h, and 2(β-ν)>h.
Finally, comparison of the above expression with the one given in (45) yields the desired result.Theorem 30.
IfX~CHm(ν,α,β,Ω,θ,kind1), then det(Ω-1/2XΩ-1/2) is distributed as ∏i=1m‍zi, where z1,…,zm are independent, zi~CH(ν-(i-1)/2,α-i+1,β-i+1,θ,kind1), i=1,…,m.Proof.
Writing multivariate gamma functions in terms of ordinary gamma function, (42) is rewritten as (77)Edet⁡Ω-1/2XΩ-1/2h=∏i=1mθhΓβ-ν-i-1/2Γν-i-1/2Γα-ν-i-1/2Γν-i-1/2+hΓα-ν-i-1/2-hΓβ-ν-i-1/2-h.Now, comparing the above expression with (45), we get E[det(Ω-1/2XΩ-1/2)h]=∏i=1m‍E(zih).Corollary 31.
IfX~CH2(ν,α,β,θ,kind1), then(78)2detX1/2~CH2ν-1,2α-2,2β-2,θ,kind1.Proof.
Form=2, det(X) is distributed as z1z2, where z1 and z2 are independent, z1~CH(ν,α,β,θ,kind1) and z2~CH(ν-1/2,α-1,β-1,θ,kind1). From Theorem 29, we have 2z1z2~CH(2ν-1,2α-2,2β-2,θ,kind1).Corollary 32.
IfX~Ga(m,ν,θ,Ω), then det(Ω-1/2XΩ-1/2) is distributed as ∏i=1m‍zi, where z1,…,zm are independent, zi~Ga(ν-(i-1)/2,θ), i=1,…,m.


## 6. Distribution of Eigenvalues

In this section, we derive density of eigenvalues of random matrix distributed as confluent hypergeometric function kind 1.Theorem 33.
LetA be a positive definite random matrix of order m with the p.d.f. f(A). Then, the joint p.d.f. of the eigenvalues l1,l2,…,lm of A is given by(79)πm2/2Γmm/2∏i<jmli-lj∫OmfHLH′dH,where l1>l2>⋯>lm>0, L=diag⁡(l1,l2,…,lm), and (dH) is the unit invariant Haar measure on the group of orthogonal matrices O(m).Proof of Theorem33 and several other results can be found in Muirhead [7].Theorem 34.
IfX~CHm(ν,α,β,θ,Ω,kind1), then the joint p.d.f. of the eigenvalues x1,x2,…,xm of X is given by (80)πm2/2ΓmαΓmβ-νdet⁡θΩ-νΓmm/2ΓmνΓmβΓmα-ν∏i<jmxi-xj∏i=1mxiν-m+1/2F1m1α;β;-1θΩ-1,L,where 0<xm<⋯<x1<∞, L=diag⁡(x1,…,xm), and F1(m)1 is the two-matrix argument confluent hypergeometric function.Proof.
The p.d.f. ofX is given by (5). Applying Theorem 33, we obtain the joint p.d.f. of the eigenvalues x1,…,xm of X as (81)πm2/2ΓmαΓmβ-νdet⁡θΩ-νΓmm/2ΓmνΓmβΓmα-ν∏i<jmxi-xj∏i=1mxiν-m+1/2∫OmF1m1α;β;-1θΩ-1HLH′dH.Now, using (28), we obtain the desired result.


## 7. A Generalized Form

In this section, we give a more general form of the matrix variate confluent hypergeometric function kind 1 distribution by introducing an additional factoretr(-Ψ-1X/θ) in the p.d.f. (5). The p.d.f. of X, in this case, is given by(82)θ-mνdet⁡Ψ-1+Ω-1νΓmνF21ν,β-α;β;Ψ-1+Ω-1-1Ω-1etr⁡-1θdet⁡Xν-m+1/2F11α;β;-1θΩ-1X,where X>0. We will write X~CHm(ν,α,β,θ,Ω,Ψ,kind1) if the density of X is given by (82). For Ψ=Im and θ=1, the above p.d.f. reduces to (83)det⁡Im+Ω-1νΓmνF12ν,β-α;β;Im+Ω-1etr⁡-Xdet⁡Xν-m+1/2F11α;β;-Ω-1X,X>0,which is a special case of the generalized hypergeometric function density defined by Roux [14].Theorem 35.
LetZ∣Σ~InvGa(m,μ,θ,Σ-1). Further, let the prior distribution of Σ be a generalized matrix variate confluent hypergeometric kind 1 distribution with parameters ν, α, β, θ, Ω, and Ψ, Σ~CHm(ν,α,β,θ,Ω,Ψ,kind1). Then, the marginal distribution of Z is a generalized inverted matrix variate beta with the density (84)mZ=Γmμ+νdet⁡Ψ-1+Ω-1νΓmνΓmμF12ν,β-α;β;Ψ-1+Ω-1-1Ω-1det⁡Z-μ-m+1/2det⁡Ψ-1+Ω-1+Z-1μ+νF12μ+ν,β-α;β;Ψ-1+Ω-1+Z-1-1Ω-1,where Z>0.Proof.
By definition, the marginal density ofZ, denoted by m(Z), is obtained as(85)mZ=∫Σ>0fZ∣ΣπΣdΣ.Now, substituting for f(Z∣Σ) and π(Σ), we get(86)mZ=θ-mν+μdet⁡Ψ-1+Ω-1νΓmνΓmμF12ν,β-α;β;Ψ-1+Ω-1-1Ω-1det⁡Z-μ-m+1/2∫Σ>0det⁡Σμ+ν-m+1/2etr⁡-1θΨ-1+Ω-1+Z-1ΣF11β-α;β;1θΩ-1ΣdΣ.
Finally, evaluating the above expression by using (21) and simplifying, we get the desired result.Theorem 36.
LetZ∣Σ~InvGa(m,μ,Σ-1). Further, let the prior distribution of Σ be a generalized matrix variate confluent hypergeometric function kind 1 distribution parameters ν, α, β, θ, Ω, and Ψ, Σ~CHm(ν,α,β,θ,Ω,Ψ,kind1). Then, the posterior distribution of Σ is a generalized matrix variate confluent hypergeometric kind 1 distribution with parameters μ+ν, α, β, θ, Ω, and (Ψ-1+Z-1)-1.Proof.
By definition and Theorem35, we have (87)πΣ∣Z=fZ∣ΣπΣmZ.Now, substituting appropriately, we get(88)πΣ∣Z=θ-mν+μdet⁡Ψ-1+Ω-1+Z-1ν+μΓmν+μF12ν+μ,β-α;β;Ψ-1+Ω-1+Z-1-1Ω-1etr⁡-1θΨ-1+Z-1Σdet⁡Σν+μ-m+1/2F11α;β;-1θΩ-1Σ,Σ>0,which is the desired result.From the above results it is quite clear that the generalized matrix variate confluent hypergeometric function kind 1 distribution as the prior distribution is conjugate. Thus, this distribution may be used as an alternative to matrix variate gamma distribution.



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