# Properties of Matrix Variate Beta Type 3 Distribution

Authors: Arjun K. Gupta; Daya K. Nagar
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

We study several properties of matrix variate beta type 3 distribution.
We also derive probability density functions of the product of two independent random
matrices when one of them is beta type 3. These densities are expressed in terms of Appell's
first hypergeometric functionF1 and Humbert's confluent hypergeometric function Φ1 of matrix arguments. Further, a bimatrix variate generalization of the beta type 3 distribution
is also defined and studied.


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## Body

## 1. Introduction

The beta  families of distributions are defined by the density functions(1.1)uα-1(1-u)β-1B(α,β),0<u<1,(1.2)vα-1(1+v)-(α+β)B(α,β),v>0,
respectively, where α>0, β>0, and(1.3)B(α,β)=Γ(α)Γ(β)Γ(α+β).
The beta type 1 and beta type 2 are very flexible distributions for positive random variables and have wide applications in statistical analysis, for example, see Johnson et al. [1].  Recently, Cardeño et al. [2] have defined and studied family of beta type 3 distributions. A random variable w is said to follow a beta type 3 distribution if its density function is given by(1.4)2αwα-1(1-w)β-1B(α,β)(1+w)α+β,0<w<1.If a random variableu has the p.d.f (1.1), then we will write u~B1(α,β), and if the p.d.f. of a random variable v is given by (1.2), then v~B2(α,β). The density (1.4) will be designated by w~B3(α,β). The  matrix variate generalizations of (1.1) and (1.2) have been studied extensively in the literature, for example, see Gupta and Nagar [3]. The matrix variate beta type 3 distribution has been defined, and some of its properties have been studied by Gupta and Nagar [4].In this paper, we study several properties of matrix variate beta type 3 distribution. We also derive probability density functions of the product of two independent random matrices when one of them is beta type 3. We also define bimatrix beta type 3 distribution and study some of its properties.


## 2. Some Known Results and Definitions

We begin with a brief review of some definitions and notations. We adhere to standard notations (cf. Gupta and Nagar [3]). 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; ∥A∥= norm of 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 which is frequently used in multivariate statistical analysis is defined by(2.1)Γm(a)=∫X>0etr(-X)det(X)a-(m+1)/2dX=πm(m-1)/4∏i=1mΓ(a-i-12),Re(a)>m-12.
The multivariate generalization of the beta function is given by(2.2)Bm(a,b)=∫0Imdet(X)a-(m+1)/2det(Im-X)b-(m+1)/2dX=Γm(a)Γm(b)Γm(a+b)=Bm(b,a),
where Re(a)>(m-1)/2 and Re(b)>(m-1)/2.The generalized hypergeometric coefficient(a)ρ is defined by(2.3)(a)ρ=∏i=1m(a-i-12)ri,
where ρ=(r1,…,rm), r1≥⋯≥rm≥0, r1+⋯+rm=r, and (a)k=a(a+1)⋯(a+k-1), k=1,2,… with (a)0=1. The generalized hypergeometric function of one matrix is defined by(2.4)pFq(a1,…,ap;b1,…,bq;X)=∑k=0∞∑κ⊢k(a1)κ⋯(ap)κ(b1)κ⋯(bq)κCκ(X)k!,
where ai, i=1,…,p, bj, j=1,…,q are arbitrary complex numbers, X(m×m) is a complex symmetric matrx, and ∑κ⊢k denotes summation over all partitions κ. Conditions for convergence of the series in (2.4) are available in the literature. From (2.4) it follows that(2.5)0F0(X)=∑k=0∞∑κ⊢kCκ(X)k!=∑k=0∞(trX)kk!=etr(X),(2.6)1F0(a;X)=∑k=0∞∑κ⊢k(a)κCκ(X)k!=det(Im-X)-a,∥X∥<1,(2.7)1F1(a;c;X)=∑k=0∞∑κ⊢k(a)κ(c)κCκ(X)k!,(2.8)2F1(a,b;c;X)=∑k=0∞∑κ⊢k(a)κ(b)κ(c)κCκ(X)k!,∥X∥<1.
The integral representations of the confluent hypergeometric function 1F1 and the Gauss hypergeometric function 2F1 are given by(2.9)1F1(a;c;X)=Γm(c)Γm(a)Γm(c-a)∫0Imetr(RX)det(R)a-(m+1)/2det(Im-R)c-a-(m+1)/2dR,(2.10)2F1(a,b;c;X)=Γm(c)Γm(a)Γm(c-a)∫0Imdet(R)a-(m+1)/2det(Im-R)c-a-(m+1)/2det(Im-XR)-bdR,
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 Constantine [5] and Gupta and Nagar [3].Davis [6, 7] introduced a class of polynomials Cϕκ,λ(X,Y) of m×m symmetric matrix arguments X and Y, which are invariant under the transformation (X,Y)→(HXH′,HYH′), H∈O(m). For properties and applications of invariant polynomials we refer to Davis [6, 7], Chikuse [8], and Nagar and Gupta [9]. Let κ, λ, ϕ, and ρ be ordered partitions of the nonnegative integers k, ℓ, f=k+ℓ and r, respectively, into not more than m parts. Then(2.11)Cϕκ,λ(X,X)=θϕκ,λCϕ(X),θϕκ,λ=Cϕκ,λ(Im,Im)Cϕ(Im),(2.12)Cϕκ,λ(X,Im)=θϕκ,λCϕ(Im)Cκ(X)Cκ(Im),(2.13)Cκκ,0(X,Y)≡Cκ(X),Cλ0,λ(X,Y)≡Cλ(Y),(2.14)Cκ(X)Cλ(Y)=∑ϕ∈κ·λθϕκ,λCϕκ,λ(X,Y),
where ϕ∈κ·λ signifies that irreducible representation of Gl(m,R) indexed by 2ϕ occurs in the decomposition of the Kronecker product 2κ⊗2λ of the irreducible representations indexed by 2κ and 2λ. Further(2.15)∫0Imdet(R)t-(m+1)/2det(Im-R)u-(m+1)/2Cϕκ,λ(R,Im-R)dR=Γm(t,κ)Γm(u,λ)Γm(t+u,ϕ)θϕκ,λCϕ(Im),(2.16)∫0Imdet(R)t-(m+1)/2det(Im-R)u-(m+1)/2Cϕκ,λ(AR,BR)dR=Γm(t,ϕ)Γm(u)Γm(t+u,ϕ)Cϕκ,λ(A,B).
In expressions (2.15) and (2.16), Γm(a,ρ) is defined by(2.17)Γm(a,ρ)=(a)ρΓm(a).
Note that Γm(a,0)=Γm(a), which is the multivariate gamma function.The matrix variate generalizations of (1.1), (1.2), and (1.4) are given as follows (Gupta and Nagar [3, 4]).Definition 2.1.
Anm×m random symmetric positive definite matrix U is said to have a matrix variate beta type 1 distribution with parameters (α,β), denoted as U~B1(m,α,β), if its p.d.f. is given by
(2.18)det(U)α-(m+1)/2det(Im-U)β-(m+1)/2Bm(α,β),0<U<Im,
where α>(m-1)/2 and β>(m-1)/2.IfU~B1(m,α,β), then the cumulative distribution function F(Λ)=P(U<Λ) is given by
(2.19)F(Λ)=Γm(α+β)Γm[(m+1)/2]Γm(β)Γm[α+(m+1)/2]det(Λ)α×2F1(α,-β+m+12;α+m+12;Λ),0<Λ<Im,(2.20)E[det(U)r1det(Im-U)r2]=Γm(α+r1)Γm(β+r2)Γm(α+β)Γm(α)Γm(β)Γm(α+β+r1+r2).Definition 2.2.
Anm×m random symmetric positive definite matrix V is said to have a matrix variate beta type 2 distribution with parameters (α,β), denoted as V~B2(m,α,β), if its p.d.f. is given by
(2.21)det(V)α-(m+1)/2det(Im+V)-(α+β)Bm(α,β),V>0,
where α>(m-1)/2 and β>(m-1)/2.Definition 2.3.
Anm×m random symmetric positive definite matrix W is said to have a matrix variate beta type 3 distribution with parameters (α,β), denoted as W~B3(m,α,β), if its p.d.f. is given by
(2.22)2mαdet(W)α-(m+1)/2det(Im-W)β-(m+1)/2Bm(α,β)det(Im+W)α+β,0<W<Im,
where α>(m-1)/2 and β>(m-1)/2.


## 3. Hypergeometric Functions of Two Matrices

In this section we define Appell’s first hypergeometric functionF1 and Humbert’s confluent hypergeometric function Φ1 of m×m symmetric matrices Z1 and Z2 and give their series expansions involving invariant polynomials. Following Prudnikov et al. [10, equations 7.2.4(43), (48)], F1 and Φ1 are defined as
(3.1)F1(a,b1,b2;c;Z1,Z2)=Γm(c)Γm(a)Γm(c-a)∫0Imdet(V)a-(m+1)/2det(Im-V)c-a-(m+1)/2dVdet(Im-VZ1)b1det(Im-VZ2)b2,(3.2)Φ1[a,b1;c;Z1,Z2]=Γm(c)Γm(a)Γm(c-a)∫0Imdet(V)a-(m+1)/2det(Im-V)c-a-(m+1)/2dVdet(Im-VZ1)b1etr(-VZ2),
respectively, where Re(a)>(m-1)/2 and Re(c-a)>(m-1)/2. Note that for b1=0, F1 and Φ1 reduce to 2F1 and 1F1 functions, respectively. Expanding det(Im-VZ1)-b1, ∥VZ1∥<1, det(Im-VZ2)-b2, ∥VZ2∥<1 and etr(VZ2) using (2.6) and (2.5), and applying (2.14), one can write
(3.3)det(Im-VZ1)-b1det(Im-VZ2)-b2=∑k=0∞∑ℓ=0∞∑κ⊢k∑λ⊢ℓ∑ϕ∈κ·λ(b1)κ(b2)λk!ℓ!Cϕκ,λ(VZ1,VZ2),∥Z1∥<1,∥Z2∥<1,(3.4)det(Im-VZ1)-b1etr(VZ2)=∑k=0∞∑ℓ=0∞∑κ⊢k∑λ⊢ℓ∑ϕ∈κ·λ(b1)κk!ℓ!Cϕκ,λ(VZ1,VZ2),∥Z1∥<1.
Now, substituting (3.3) and (3.4) in (3.1) and (3.2), respectively, and integrating V using (2.16), the series expansions for F1 and Φ1 are derived as
(3.5)F1(a,b1,b2;c;Z1,Z2)=∑k=0∞∑ℓ=0∞∑κ⊢k∑λ⊢ℓ∑ϕ∈κ·λ(b1)κ(b2)λk!ℓ!(a)ϕ(c)ϕCϕκ,λ(Z1,Z2),Φ1[a,b1;c;Z1,Z2]=∑k=0∞∑ℓ=0∞∑κ⊢k∑λ⊢ℓ∑ϕ∈κ·λ(b1)κk!ℓ!(a)ϕ(c)ϕCϕκ,λ(Z1,Z2).


## 4. Properties

In this section we derive several properties of the matrix variate beta type 3 distribution. For the sake of completeness we first state the following results established in Gupta and Nagar [4].(1)
LetW~B3(m,α,β) and A(m×m) be a constant nonsingular matrix. Then, the density of X=AWA′ is
(4.1)2mαdet(X)α-(m+1)/2det(AA′-X)β-(m+1)/2det(AA′)-(m+1)/2Bm(α,β)det(AA′+X)α+β,0<X<AA′.(2)
LetW~B3(m,α,β) and H(m×m) be an orthogonal matrix, whose elements are either constants or random variables distributed independent of W. Then, the distribution of W is invariant under the transformation W→HWH′, and is independent of H in the latter case.(3)
LetW~B3(m,α,β). Then, the density of Y=W-1 is
(4.2)2mαdet(Y-Im)β-(m+1)/2Bm(α,β)det(Im+Y)α+β,Y>Im.(4)
IfU~B1(m,α,β), then (Im+U)-1(Im-U)~B3(m,β,α) and (2Im-U)-1U~B3(m,α,β).(5)
IfV~B2(m,α,β), then (2Im+V)-1V~B3(m,α,β) and (Im+2V)-1~B3(m,β,α).(6)
IfW~B3(m,α,β), then 2(Im+W)-1W~B1(m,α,β), (Im+W)-1(Im-W)~B1(m,β,α), 2(Im-W)-1W~B2(m,α,β), and (1/2)(Im-W)W-1~B2(m,β,α).(7)
LetW=(W11W12W21W22), W11(q×q). Define W11·2=W11-W12W22-1W21 and W22·1=W22-W21W11-1W12. If W~B3(m,α,β), then W22·1~B3(m-q,α-q/2,β) and W11·2~B3(q,α-(m-q)/2,β).(8)
LetA(q×m) be a constant matrix of rank q (≤m). If W~B3(m,α,β), then [(AA′)-1/2AW-1A′(AA′)-1/2]-1~B3(q,α-(m-q)/2,β).(9)
LetW~B3(m,α,β) and a∈ℝm, a≠0, then a′a(a′W-1a)-1~B3(α-(m-1)/2,β). Further, if y(m×1) is a random vector, independent of W, and P(y≠0)=1, then it follows that y′y(y′W-1y)-1~B3(α-(m-1)/2,β).From the above results it is straightforward to show that, ifc (m×1) is a nonzero constant vector or a random vector independent of W with P(c≠0)=1, then
(4.3)c′(W-1-Im)cc′(W-1+Im)c~B1(β,α-m-12),2c′cc′(W-1+Im)c~B1(α-m-12,β),2c′cc′(W-1-Im)c~B2(α-m-12,β),c′(W-1-Im)c2c′c~B2(β,α-m-12).
The expectation of W-1, E(W-1), can easily be obtained from the above results. For any fixed c∈ℝm, c≠0,
(4.4)E[c′(W-1-Im)c2c′c]=E(v),
where v~B2(β,α-(m-1)/2). Hence, for all c∈ℝm,
(4.5)c′E(W-1-Im)c=2c′cE(v)=2βα-(m+1)/2c′c,α>m+12,
which implies that
(4.6)E(W-1)=2β+α-(m+1)/2α-(m+1)/2Im,α>m+12.
The matrix variate beta type 3 distribution can be derived by using independent gamma matrices. An m×m random symmetric positive definite matrix Y is said to have a matrix variate gamma distribution with parameters Ψ (>0), and κ (>(m-1)/2), denoted by Y~Ga(m,κ,Ψ), if its p.d.f. is given by
(4.7)etr(-Ψ-1Y)det(Y)κ-(m+1)/2Γm(κ)det(Ψ)κ,Y>0.
It is well known that if Y1 and Y2 are independent, Yi~Ga(m,κi,Im), i=1,2, then (i) (Y1+Y2)-1/2Y1(Y1+Y2)-1/2 and Y1+Y2 are independent and (ii) Y2-1/2Y1Y2-1/2 and Y1+Y2 are independent. Further, (Y1+Y2)-1/2Y1(Y1+Y2)-1/2~B1(m,κ1,κ2), Y2-1/2Y1Y2-1/2~B2(m,κ1,κ2) and Y1+Y2~Ga(m,κ1+κ2,Im). In the following theorem we derive similar result for matrix variate beta type 3 distribution.Theorem 4.1.
Let  them×m random matrices  Y1 and Y2 be independent, Yi~Ga(m,κi,Im), i=1,2. Then, (Y1+2Y2)-1/2Y1(Y1+2Y2)-1/2~B3(m,κ1,κ2).Proof.
The joint density function ofY1 and Y2 is given by
(4.8)etr[-(Y1+Y2)]det(Y1)κ1-(m+1)/2det(Y2)κ2-(m+1)/2Γm(κ1)Γm(κ2),Y1>0,Y2>0.
Making the transformation W=Y-1/2Y1Y-1/2 and Y=Y1+2Y2 with the Jacobian J(Y1,Y2→W,Y)=2-m(m+1)/2det(Y)(m+1)/2 in the joint density of Y1 and Y2, we obtain the joint density of W and Y as
(4.9)det(W)κ1-(m+1)/2det(Im-W)κ2-(m+1)/22mκ2Γm(κ1)Γm(κ2)×etr[-12(Im+W)Y]det(Y)κ1+κ2-(m+1)/2,0<W<Im,Y>0.
Now, the desired result is obtained by integrating Y using (2.1).Next, we derive the cumulative distribution function (cdf) and several expected values of functions of beta type 3 matrix.IfW~B3(m,α,β), then the cdf of W, denoted by G(Ω), is given by
(4.10)G(Ω)=P(W<Ω)=P(U<(Im+Ω)-1(Im-Ω)),
where U~B1(m,β,α). Now, using (2.19), the cdf G(Ω) is obtained as
(4.11)G(Ω)=Γm(α+β)Γm[(m+1)/2]Γm(α)Γm[β+(m+1)/2]det((Im+Ω)-1(Im-Ω))β×2F1(β,-α+m+12;β+m+12;(Im+Ω)-1(Im-Ω)),
where 0<Ω<Im.Theorem 4.2.
LetW~B3(m,α,β), then
(4.12)E[det(W)rdet(Im-W)sdet(Im+W)t]=2-m(β+t)Γm(α+r)Γm(β+s)Γm(α+β)Γm(α)Γm(β)Γm(α+β+r+s)×2F1(β+s,α+β+t;α+β+r+s;Im2),
where Re(α+r)>(m-1)/2 and Re(β+s)>(m-1)/2.Proof.
By definition(4.13)E[det(W)rdet(Im-W)sdet(Im+W)t]=2mαBm(α,β)∫0Imdet(W)α+r-(m+1)/2det(Im-W)β+s-(m+1)/2dWdet(Im+W)α+β+t.
Writing
(4.14)det(Im+W)-(α+β+t)=2-m(α+β+t)det(Im-12(Im-W))-(α+β+t)
and substituting Z=Im-W, we have
(4.15)E[det(W)rdet(Im-W)sdet(Im+W)t]=12m(β+t)Bm(α,β)∫0Imdet(Z)β+s-(m+1)/2det(Im-Z)α+r-(m+1)/2dZdet(Im-Z/2)α+β+t=Bm(α+r,β+s)2m(β+t)Bm(α,β)2F1(β+s,α+β+t;α+β+r+s;Im2),
where the integral has been evaluated using integral representation of the Gauss hypergeometric function given in (2.10).Corollary 4.3.
LetW~B3(m,α,β), then for Re(h)>-α+(m-1)/2, one has
(4.16)E[det(W)hdet(Im+W)h]=Γm(α+β)Γm(α+h)2mhΓm(α)Γm(α+β+h),E[det(W)h]=Γm(α+β)Γm(α+h)2mβΓm(α)Γm(α+β+h)2F1(β,α+β;α+β+h;Im2).
Further, for Re(h)>-β+(m-1)/2,
(4.17)E[det(Im+W)h]=Γm(α+β)Γm(β+h)2mβΓm(β)Γm(α+β+h)×2F1(β+h,α+β;α+β+h,Im2).From the density ofW, we have
(4.18)E[Cκ(W)]=2mαBm(α,β)×∫0ImCκ(W)det(W)α-(m+1)/2det(Im-W)β-(m+1)/2dW(Im+W)α+β.
Now, expanding (Im+W)-(α+β) in series involving zonal polynomials using (2.6), the above expression is rewritten as
(4.19)E[Cκ(W)]=12mβBm(α,β)∑ℓ=0∞∑λ⊢ℓ(α+β)λ2ℓℓ!×∫0ImCκ(W)det(W)α-(m+1)/2det(Im-W)β-(m+1)/2Cλ(Im-W)dW.
Further, writing
(4.20)Cκ(W)Cλ(Im-W)=∑ϕ∈κ·λθϕκ,λCϕκ,λ(W,Im-W)
and integrating W using (2.15), we get
(4.21)E[Cκ(W)]=12mβBm(α,β)∑ℓ=0∞∑λ⊢ℓ(α+β)λ2ℓℓ!∑ϕ∈κ·λθϕκ,λ×∫0Imdet(W)α-(m+1)/2det(Im-W)β-(m+1)/2Cϕκ,λ(W,Im-W)dW=12mβ∑ℓ=0∞∑λ⊢ℓ(α+β)λ2ℓℓ!∑ϕ∈κ·λ(θϕκ,λ)2(α)κ(β)λ(α+β)ϕCϕ(Im).


## 5. Distributions of Random Quadratic Forms

In this section we obtain distributional results for the product of two independent random matrices involving beta type 3 distribution.Theorem 5.1.
LetX1~B1(m,α1,β1) and X2~B3(m,α2,β2) be independent. Then, the p.d.f. of Z=X21/2X1X21/2 is
(5.1)2-mβ2Γm(α1+β1)Γm(α2+β2)Γm(α1)Γm(α2)Γm(β1+β2)det(Z)α1-(m+1)/2det(Im-Z)β1+β2-(m+1)/2×F1(β2,α1+β1-α2,α2+β2,β1+β2;Im-Z,Im-Z2),0<Z<Im.Proof.
Using the independence, the joint p.d.f. ofX1 and X2 is given by
(5.2)K1det(X1)α1-(m+1)/2det(Im-X1)β1-(m+1)/2×det(X2)α2-(m+1)/2det(Im-X2)β2-(m+1)/2det(Im+X2)α2+β2,
where 0<Xi<Im, i=1,2, and
(5.3)K1=2α2m{Bm(α1,β1)Bm(α2,β2)}-1.
Transforming Z=X21/2X1X21/2, 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
(5.4)K1det(Z)α1-(m+1)/2det(X2-Z)β1-(m+1)/2det(Im-X2)β2-(m+1)/2det(X2)α1+β1-α2det(Im+X2)α2+β2,
where 0<Z<X2<Im. To find the marginal p.d.f. of Z, we integrate (5.4) with respect to X2 to get
(5.5)K1det(Z)α1-(m+1)/2×∫ZImdet(X2-Z)β1-(m+1)/2det(Im-X2)β2-(m+1)/2dX2det(X2)α1+β1-α2det(Im+X2)α2+β2.
In (5.5) change of variable V=(Im-Z)-1/2(Im-X2)(Im-Z)-1/2 with the Jacobian J(X2→V)=det(Im-Z)(m+1)/2 yields
(5.6)K12-m(α2+β2)det(Z)α1-(m+1)/2det(Im-Z)β1+β2-(m+1)/2×∫0Imdet(V)β2-(m+1)/2det(Im-V)β1-(m+1)/2dVdet(Im-(Im-Z)V)α1+β1-α2det(Im-(Im-Z)V/2)α2+β2=K12-m(α2+β2)det(Z)α1-(m+1)/2det(Im-Z)β1+β2-(m+1)/2×Γm(β1)Γm(β2)Γm(β1+β2)F1(β2,α1+β1-α2,α2+β2,β1+β2;Im-Z,Im-Z2),
where the last step has been obtained by using the definition of F1. Finally, substituting for K1 we obtain the desired result.Corollary 5.2.
LetX1 and X2 be independent random matrices, X1~B1(m,α1,β1) and X2~B3(m,α2,β2). If α2=α1+β1, then the p.d.f. of Z=X21/2X1X21/2 is given by
(5.7)2-mβ2Γm(α1+β1+β2)Γm(α1)Γm(β1+β2)det(Z)α1-(m+1)/2det(Im-Z)β1+β2-(m+1)/2×2F1(β2,α1+β1+β2;β1+β2;Im-Z2),0<Z<Im.Theorem 5.3.
LetX1 and X2 be independent random matrices, X1~B3(m,α1,β1) and X2~B2(m,α2,β2). Then, the p.d.f. of Z=X11/2X2X11/2 is given by
(5.8)2-mβ1Bm(β1,α1+β2)Bm(α1,β1)Bm(α2,β2)det(Z)α2-(m+1)/2det(Im+Z)α2+β2×F1(β1,α1+β1,α2+β2;α1+β1+β2;Im2,(Im+Z)-1),Z>0.Proof.
SinceX1 and X2 are independent, their joint p.d.f. is given by
(5.9)K2det(X1)α1-(m+1)/2det(Im-X1)β1-(m+1)/2det(X2)α2-(m+1)/2det(Im+X1)α1+β1det(Im+X2)α2+β2,
where 0<X1<Im, X2>0, and
(5.10)K2=2mα1{Bm(α1,β1)Bm(α2,β2)}-1.
Now consider the transformation Z=X11/2X2X11/2 and V=Im-X1 whose Jacobian is J(X1,X2→V,Z)=det(Im-V)-(m+1)/2. Thus, we obtain the joint p.d.f. of V and Z as
(5.11)K2det(Z)α2-(m+1)/22m(α1+β1)det(Im+Z)α2+β2det(V)β1-(m+1)/2det(Im-V)α1+β2-(m+1)/2det(Im-V/2)α1+β1det(Im-(Im+Z)-1V)α2+β2,
where Z>0 and 0<V<Im. Finally, integrating V using (3.1) and substituting for K2, we obtain the desired result.In the next theorem we derive the density ofZ1=X-1/2YX-1/2, where the random matrices X and Y are independent, X~B3(m,α,β), and the distribution of Y is matrix variate gamma.Theorem 5.4.
Let them×m random matrices X and Y be independent, X~B3(m,α,β) and Y~Ga(m,κ,Im). Then, the p.d.f. of Z1=X-1/2YX-1/2 is given by
(5.12)Γm(α+κ)Γm(α+β)det(Z1)κ-(m+1)/2etr(-Z1)2mβΓm(κ)Γm(α)Γm(α+β+κ)Φ1(β,α+β;α+β+κ;Im2,Z1),
where Z1>0.Proof.
The joint p.d.f. ofX and Y is given by
(5.13)det(X)α-(m+1)/2det(Im-X)β-(m+1)/2det(Y)κ-(m+1)/22-mαΓ(κ)B(α,β)det(Im+X)α+βetr(Y),
where 0<X<Im and Y>0. Now, transforming Z1=X-1/2YX-1/2 and W=Im-X, with the Jacobian J(X,Y→W,Z1)=det(Im-W)(m+1)/2, we obtain the joint p.d.f. of Z1 and W as
(5.14)etr(-Z1)det(Z1)κ-(m+1)/22mβΓ(κ)B(α,β)det(W)β-(m+1)/2det(Im-W)α+κ-(m+1)/2det(Im-W/2)α+βetr(-WZ1),
where 0<W<Im and Z1>0. Now, integrating W using (3.2), we get the marginal density of Z1.


## 6. Bimatrix Beta Type 3 Distribution

The bimatrix generalization of the beta type 1 density is defined by(6.1)det(U1)α1-(m+1)/2det(U2)α2-(m+1)/2det(Im-U1-U2)β-(m+1)/2Bm(α1,α2,β),U1>0,U2>0,U1+U2<Im,
where α1>(m-1)/2, α2>(m-1)/2, β>(m-1)/2, and
(6.2)Bm(α1,α2,β)=Γm(α1)Γm(α2)Γm(β)Γm(α1+α2+β).
This distribution, denoted by (U1,U2)~D1(m,α1,α2;β), is a special case of the matrix variate Dirichlet type 1 distribution. The m×m random symmetric positive definite matrices V1 and V2 are said to have a bimatrix variate generalization of the beta type 2 distribution, denoted as (V1,V2)~D2(m,α1,α2;β), if their joint p.d.f. is given by
(6.3)det(V1)α1-(m+1)/2det(V2)α2-(m+1)/2Bm(α1,α2,β)det(Im+V1+V2)α1+α2+β,V1>0,V2>0,
where α1>(m-1)/2, α2>(m-1)/2, and β>(m-1)/2.A natural bimatrix generalization of the beta type 3 distribution can be given as follows.Definition 6.1.
Them×m symmetric positive definite random matrices W1 and W2 are said to have a bimatrix beta type 3 distribution, denoted as (W1,W2)~D3(m,α1,α2;β), if their joint p.d.f. is given by
(6.4)det(W1)α1-(m+1)/2det(W2)α2-(m+1)/2det(Im-W1-W2)β-(m+1)/22-m(α1+α2)Bm(α1,α2,β)det(Im+W1+W2)α1+α2+β,W1>0,W2>0,W1+W2<Im,
where α1>(m-1)/2, α2>(m-1)/2, and β>(m-1)/2.The bimatrix beta type 3 distribution belongs to the Liouville family of distributions and can be obtained using independent gamma matrices as shown in the following theorem.Theorem 6.2.
LetY1, Y2, and Y3 be independent, Yi~Ga(m,κi,Im), i=1,2,3. Define Wi=(Y1+Y2+2Y3)-1/2Yi(Y1+Y2+2Y3)-1/2, i=1,2. Then, (W1,W2)~D3(m,κ1,κ2;κ3).Proof.
Similar to the proof of Theorem4.1.The next two theorems derive the bimatrix beta type 3 distribution from the bimatrix beta type 1 and type 2 distributions.Theorem 6.3.
Let(U1,U2)~D1(m,α1,α2;β) and define
(6.5)Wi=(2Im-U1-U2)-1/2Ui(2Im-U1-U2)-1/2,i=1,2.
Then, (W1,W2)~D3(m,α1,α2;β).Proof.
LetZ=2Im-U1-U2 and W1=Z-1/2U1Z-1/2. Then, W2=2Z-1-(Im+W1). The Jacobian of the transformation (6.5) is given by
(6.6)J(U1,U2→W1,W2)=J(U1,U2→W1,Z)J(W1,Z→W1,W2)=det(Z)(m+1)/22-m(m+1)/2det(Z)m+1=2m(m+1)det(Im+W1+W2)-3(m+1)/2.
Now, substituting Ui=2(Im+W1+W2)-1/2Wi(Im+W1+W2)-1/2, i=1,2 and the Jacobian in the joint density of U1 and U2 given in (6.1), we get the desired result.Theorem 6.4.
Let(V1,V2)~D2(m,α1,α2;β) and define
(6.7)Wi=(2Im+V1+V2)-1/2Vi(2Im+V1+V2)-1/2,i=1,2.
Then, (W1,W2)~D3(m,α1,α2;β).Proof.
LetZ=2Im+V1+V2 and W1=Z-1/2V1Z-1/2. Then, W2=Im-W1-2Z-1. The Jacobian of the transformation (6.7) is given by
(6.8)J(V1,V2→W1,W2)=J(V1,V2→W1,Z)J(W1,Z→W1,W2)=det(Z)(m+1)/22-m(m+1)/2det(Z)m+1=2m(m+1)det(Im-W1-W2)-3(m+1)/2.
Now, substitution of Vi=2(Im-W1-W2)-1/2Wi(Im-W1-W2)-1/2, i=1,2, along with the Jacobian in the joint density of V1 and V2 given in (6.3) yields the desired result.The marginal distribution ofW1, when the random matrices W1 and W2 follow a bimatrix beta type 3 distribution, is given next.Theorem 6.5.
Let(W1,W2)~D3(m,α1,α2;β). Then, the marginal p.d.f. of W1 is given by
(6.9)det(W1)α1-(m+1)/2det(Im-W1)α2+β-(m+1)/22-m(α1+α2)Bm(α1,α2+β)det(Im+W1)α1+α2+β×2F1(α2,α1+α2+β;α2+β;-(Im+W1)-1(Im-W1)),
where 0<W1<Im. Further, (Im-W1)-1/2W2(Im-W1)-1/2~B3(m,α2,β).Proof.
SubstitutingX2=(Im-W1)-1/2W2(Im-W1)-1/2 with the Jacobian J(W2→X2)=det(Im-W1)(m+1)/2 in (6.4), the joint density of W1 and X2 is derived as
(6.10)2m(α1+α2)det(W1)α1-(m+1)/2det(Im-W1)α2+β-(m+1)/2Bm(α1,α2,β)det(Im+W1)α1+α2+β×det(X2)α2-(m+1)/2det(Im-X2)β-(m+1)/2det(Im+(Im+W1)-1(Im-W1)X2)α1+α2+β,0<W1<Im,0<X2<Im.
Now, integration of the above expression with respect to X2 yields the marginal density of W1. Further, by integrating (6.10) with respect to W1 we find the marginal density of X2 as
(6.11)2m(α1+α2)det(X2)α2-(m+1)/2det(Im-X2)β-(m+1)/2Bm(α1,α2,β)det(Im+X2)α1+α2+β×∫0Imdet(W1)α1-(m+1)/2det(Im-W1)α2+β-(m+1)/2dW1det(Im+(Im+X2)-1(Im-X2)W1)α1+α2+β,0<X2<Im.
Now, by evaluating the above integral using results on Gauss hypergeometric function, we obtain
(6.12)∫0Imdet(W1)α1-(m+1)/2det(Im-W1)α2+β-(m+1)/2dW1det(Im+(Im+X2)-1(Im-X2)W1)α1+α2+β=Γm(α1)Γm(α2+β)Γm(α1+α2+β)2F1(α1,α1+α2+β;α1+α2+β;-(Im+X2)-1(Im-X1))=Γm(α1)Γm(α2+β)Γm(α1+α2+β)1F0(α1;-(Im+X2)-1(Im-X1))=Γm(α1)Γm(α2+β)Γm(α1+α2+β)2-mα1det(Im+X2)α1.
Finally, substituting (6.12) in (6.11) and simplifying the resulting expression we obtain the desired result.Using the result(6.13)2F1(a,b;c;X)=det(Im-X)-b2F1(c-a,b;c;-X(Im-X)-1),
the Gauss hypergeometric function given in (6.9) can be rewritten as
(6.14)2F1(α2,α1+α2+β;α2+β;-(Im+W1)-1(Im-W1))=det(Im+W1)α1+α2+β2m(α1+α2+β)2F1(β,α1+α2+β;α2+β;Im-W12).
Hence, the density of W1 can also be written as
(6.15)det(W1)α1-(m+1)/2det(Im-W1)α2+β-(m+1)/22mβBm(α1,α2+β)×2F1(β,α1+α2+β;α2+β;Im-W12),0<W1<Im.
It can clearly be observed that the p.d.f. in (6.9) is not a beta type 3 density and differs by a factor involving 2F1. In the next theorem we give distribution of sum of random matrices distributed jointly as bimatrix beta type 3.Theorem 6.6.
Let(W1,W2)~D3(m,α1,α2;β). Define U=W-1/2W1W-1/2 and W=W1+W2. Then, (i) U and W are independently distributed, (ii) U~B1(m,α1,α2), and (iii) W~B3(m,α1+α2,β).Proof.
Making the transformationU=W-1/2W1W-1/2 and W=W1+W2 with the Jacobian J(W1,W2→U,W)=det(W)(m+1)/2 in the joint density of (W1,W2) given by (6.4), we get the joint density of U and W as
(6.16)det(U)α1-(m+1)/2det(Im-U)α2-(m+1)/2Bm(α1,α2)×det(W)α1+α2-(m+1)/2det(Im-W)β-(m+1)/22-m(α1+α2)Bm(α1+α2,β)det(Im+W)α1+α2+β,
where 0<U<Im and 0<W<Im. From the above factorization, it is easy to see that U and W are independently distributed. Further, U~B1(m,α1,α2) and W~B3(m,α1+α2,β).Using Theorem6.6, the joint moments of det(W1) and det(W2) are given by
(6.17)E[det(W1)r1det(W2)r2]=E[det(U)r1det(Im-U)r2]E[det(W)r1+r2],
where U~B1(m,α1,α2) and W~B3(m,α1+α2,β). Now, computing E[det(W)r1+r2] and E[det(U)r1det(Im-U)r2] using Corollary 4.3 and (2.20) and simplifying the resulting expression, we obtain
(6.18)E[det(W1)r1det(W2)r2]=Γm(α1+r1)Γm(α2+r2)Γm(α1+α2+β)2mβΓm(α1)Γm(α2)Γm(α1+α2+β+r1+r2)×2F1(β,α1+α2+β;α1+α2+β+r1+r2;Im2).



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*Source: 308518-2009-07-28.xml*