Properties of Matrix Variate Beta Type 3 Distribution

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 function 𝐹₁ 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.

1. Introduction

The beta families of distributions are defined by the density functions

𝑢^𝛼−1(1−𝑢)^𝛽−1𝑣𝐵(𝛼,𝛽),0<𝑢<1,_𝛼−1(1+𝑣)^{−(𝛼+𝛽)}𝐵(𝛼,𝛽),𝑣>0, (1.1)(1.2) The integral representations of the confluent hypergeometric function ₁𝐹₁ and the Gauss hypergeometric function ₂𝐹₁ are given by

₁𝐹₁Γ(𝑎;𝑐;𝑋)=^𝑚(𝑐)Γ_𝑚(𝑎)Γ_𝑚(𝑐−𝑎)^𝐼𝑚₀etr(𝑅𝑋)det(𝑅)^{𝑎−(𝑚+1)/2}det(𝐼_𝑚−𝑅)^{𝑐−𝑎−(𝑚+1)/2}𝑑𝑅,₂𝐹₁Γ(𝑎,𝑏;𝑐;𝑋)=^𝑚(𝑐)Γ_𝑚(𝑎)Γ_𝑚(𝑐−𝑎)^𝐼𝑚₀det(𝑅)^{𝑎−(𝑚+1)/2}det(𝐼_𝑚−𝑅)^{𝑐−𝑎−(𝑚+1)/2}det(𝐼_𝑚−𝑋𝑅)^−𝑏𝑑𝑅, (2.9)(2.10) 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

^𝐼𝑚₀det(𝑅)^{𝑡−(𝑚+1)/2}det(𝐼_𝑚−𝑅)^{𝑢−(𝑚+1)/2}𝐶_𝜙^𝜅,𝜆(𝑅,𝐼_𝑚Γ−𝑅)d𝑅=^𝑚(𝑡,𝜅)Γ_𝑚(𝑢,𝜆)Γ_𝑚𝜃(𝑡+𝑢,𝜙)^{𝜙𝜅,𝜆}𝐶_𝜙(𝐼_𝑚),_𝐼𝑚0det(𝑅)^{𝑡−(𝑚+1)/2}det(𝐼_𝑚−𝑅)^{𝑢−(𝑚+1)/2}𝐶_𝜙^𝜅,𝜆Γ(𝐴𝑅,𝐵𝑅)d𝑅=^𝑚(𝑡,𝜙)Γ_𝑚(𝑢)Γ_𝑚𝐶(𝑡+𝑢,𝜙)^{𝜙𝜅,𝜆}(𝐴,𝐵). (2.15)(2.16) In expressions (2.15) and (2.16), Γ_𝑚(𝑎,𝜌) is defined by

Γ_𝑚(𝑎,𝜌)=(𝑎)_𝜌Γ_𝑚(𝑎). (2.17) Note that Γ_𝑚(𝑎,0)=Γ_𝑚(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. An 𝑚×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 det(𝑈)^{𝛼−(𝑚+1)/2}det(𝐼_𝑚−𝑈)^{𝛽−(𝑚+1)/2}𝐵_𝑚(𝛼,𝛽),0<𝑈<𝐼_𝑚, (2.18) where 𝛼>(𝑚−1)/2 and 𝛽>(𝑚−1)/2.

If 𝑈∼𝐵1(𝑚,𝛼,𝛽), then the cumulative distribution function 𝐹(Λ)=𝑃(𝑈<Λ) is given by Γ𝐹(Λ)=^𝑚(𝛼+𝛽)Γ_𝑚[(𝑚+1)/2]Γ_𝑚(𝛽)Γ_𝑚[𝛼+(𝑚+1)/2]det(Λ)^𝛼×₂𝐹₁(𝛼,−𝛽+𝑚+12;𝛼+𝑚+12;Λ),0<Λ<𝐼_𝑚,𝐸[det(𝑈)^𝑟1det(𝐼_𝑚−𝑈)^𝑟2Γ]=^𝑚(𝛼+𝑟₁)Γ_𝑚(𝛽+𝑟₂)Γ_𝑚(𝛼+𝛽)Γ_𝑚(𝛼)Γ_𝑚(𝛽)Γ_𝑚(𝛼+𝛽+𝑟₁+𝑟₂). (2.19)(2.20)

Definition 2.2. An 𝑚×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 det(𝑉)^{𝛼−(𝑚+1)/2}det(𝐼_𝑚+𝑉)^{−(𝛼+𝛽)}𝐵_𝑚(𝛼,𝛽),𝑉>0, (2.21) where 𝛼>(𝑚−1)/2 and 𝛽>(𝑚−1)/2.

Definition 2.3. An 𝑚×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^𝑚𝛼det(𝑊)^{𝛼−(𝑚+1)/2}det(𝐼_𝑚−𝑊)^{𝛽−(𝑚+1)/2}𝐵_𝑚(𝛼,𝛽)det(𝐼_𝑚+𝑊)^𝛼+𝛽,0<𝑊<𝐼_𝑚, (2.22) 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 function 𝐹₁ 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 𝐹₁(𝑎,𝑏₁,𝑏₂;𝑐;𝑍₁,𝑍₂Γ)=^𝑚(𝑐)Γ_𝑚(𝑎)Γ_𝑚(𝑐−𝑎)^𝐼𝑚₀det(𝑉)^{𝑎−(𝑚+1)/2}det(𝐼_𝑚−𝑉)^{𝑐−𝑎−(𝑚+1)/2}𝑑𝑉det(𝐼_𝑚−𝑉𝑍₁)^𝑏1det(𝐼_𝑚−𝑉𝑍₂)^𝑏2Φ₁,[𝑎,𝑏₁;𝑐;𝑍₁,𝑍₂Γ]=^𝑚(𝑐)Γ_𝑚(𝑎)Γ_𝑚(𝑐−𝑎)^𝐼𝑚₀det(𝑉)^{𝑎−(𝑚+1)/2}det(𝐼_𝑚−𝑉)^{𝑐−𝑎−(𝑚+1)/2}𝑑𝑉det(𝐼_𝑚−𝑉𝑍₁)^𝑏1etr(−𝑉𝑍₂), (3.1)(3.2) 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(𝐼_𝑚−𝑉𝑍₁)^−𝑏1, ‖𝑉𝑍₁‖<1, det(𝐼_𝑚−𝑉𝑍₂)^−𝑏2, ‖𝑉𝑍₂‖<1 and etr(𝑉𝑍₂) using (2.6) and (2.5), and applying (2.14), one can write det(𝐼_𝑚−𝑉𝑍₁)^−𝑏1det(𝐼_𝑚−𝑉𝑍₂)^−𝑏2=^∞∞_{𝑘=0ℓ=0𝜅⊢𝑘𝜆⊢ℓ𝜙∈𝜅⋅𝜆}(𝑏₁)_𝜅(𝑏₂)_𝜆𝐶𝑘!ℓ!^{𝜙𝜅,𝜆}(𝑉𝑍₁,𝑉𝑍₂),‖𝑍₁‖<1,‖𝑍₂‖<1,det(𝐼_𝑚−𝑉𝑍₁)^−𝑏1etr(𝑉𝑍₂)=^∞∞_{𝑘=0ℓ=0𝜅⊢𝑘𝜆⊢ℓ𝜙∈𝜅⋅𝜆}(𝑏₁)_𝜅𝐶𝑘!ℓ!^{𝜙𝜅,𝜆}(𝑉𝑍₁,𝑉𝑍₂),‖𝑍₁‖<1. (3.3)(3.4) 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 𝐹₁(𝑎,𝑏₁,𝑏₂;𝑐;𝑍₁,𝑍₂)=^∞∞_{𝑘=0ℓ=0𝜅⊢𝑘𝜆⊢ℓ𝜙∈𝜅⋅𝜆}(𝑏₁)_𝜅(𝑏₂)_𝜆𝑘!ℓ!(𝑎)_𝜙(𝑐)_𝜙𝐶_𝜙^𝜅,𝜆(𝑍₁,𝑍₂Φ),₁[𝑎,𝑏₁;𝑐;𝑍₁,𝑍₂]=^∞∞_{𝑘=0ℓ=0𝜅⊢𝑘𝜆⊢ℓ𝜙∈𝜅⋅𝜆}(𝑏₁)_𝜅𝑘!ℓ!(𝑎)_𝜙(𝑐)_𝜙𝐶_𝜙^𝜅,𝜆(𝑍₁,𝑍₂). (3.5)

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)Let 𝑊∼𝐵3(𝑚,𝛼,𝛽) and 𝐴(𝑚×𝑚) be a constant nonsingular matrix. Then, the density of 𝑋=𝐴𝑊𝐴^′ is 2^𝑚𝛼det(𝑋)^{𝛼−(𝑚+1)/2}det(𝐴𝐴^′−𝑋)^{𝛽−(𝑚+1)/2}det(𝐴𝐴^′)^{−(𝑚+1)/2}𝐵_𝑚(𝛼,𝛽)det(𝐴𝐴^′+𝑋)^𝛼+𝛽,0<𝑋<𝐴𝐴^′. (4.1)(2)Let 𝑊∼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)Let 𝑊∼𝐵3(𝑚,𝛼,𝛽). Then, the density of 𝑌=𝑊⁻¹ is 2^𝑚𝛼det(𝑌−𝐼_𝑚)^{𝛽−(𝑚+1)/2}𝐵_𝑚(𝛼,𝛽)det(𝐼_𝑚+𝑌)^𝛼+𝛽,𝑌>𝐼_𝑚. (4.2)(4)If 𝑈∼𝐵1(𝑚,𝛼,𝛽), then (𝐼_𝑚+𝑈)⁻¹(𝐼_𝑚−𝑈)∼𝐵3(𝑚,𝛽,𝛼) and (2𝐼_𝑚−𝑈)⁻¹𝑈∼𝐵3(𝑚,𝛼,𝛽). (5)If 𝑉∼𝐵2(𝑚,𝛼,𝛽), then (2𝐼_𝑚+𝑉)⁻¹𝑉∼𝐵3(𝑚,𝛼,𝛽) and (𝐼_𝑚+2𝑉)⁻¹∼𝐵3(𝑚,𝛽,𝛼).(6)If 𝑊∼𝐵3(𝑚,𝛼,𝛽), then 2(𝐼_𝑚+𝑊)⁻¹𝑊∼𝐵1(𝑚,𝛼,𝛽), (𝐼_𝑚+𝑊)⁻¹(𝐼_𝑚−𝑊)∼𝐵1(𝑚,𝛽,𝛼), 2(𝐼_𝑚−𝑊)⁻¹𝑊∼𝐵2(𝑚,𝛼,𝛽), and (1/2)(𝐼_𝑚−𝑊)𝑊⁻¹∼𝐵2(𝑚,𝛽,𝛼).(7)Let 𝑊=(^𝑊11𝑊12_𝑊21𝑊22), 𝑊₁₁(𝑞×𝑞). Define 𝑊_11⋅2=𝑊₁₁−𝑊₁₂𝑊⁻¹₂₂𝑊₂₁ and 𝑊_22⋅1=𝑊₂₂−𝑊₂₁𝑊⁻¹₁₁𝑊₁₂. If 𝑊∼𝐵3(𝑚,𝛼,𝛽), then 𝑊_22⋅1∼𝐵3(𝑚−𝑞,𝛼−𝑞/2,𝛽) and 𝑊_11⋅2∼𝐵3(𝑞,𝛼−(𝑚−𝑞)/2,𝛽).(8)Let 𝐴(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)Let 𝑊∼𝐵3(𝑚,𝛼,𝛽) and 𝐚∈ℝ^𝑚, 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, if 𝐜 (𝑚×1) is a nonzero constant vector or a random vector independent of 𝑊 with 𝑃(𝐜≠0)=1, then 𝐜^′(𝑊⁻¹−𝐼_𝑚)𝐜𝐜^′(𝑊⁻¹+𝐼_𝑚)𝐜∼𝐵1(𝛽,𝛼−𝑚−12),2𝐜^′𝐜𝐜^′(𝑊⁻¹+𝐼_𝑚)𝐜∼𝐵1(𝛼−𝑚−12,𝛽),2𝐜^′𝐜𝐜^′(𝑊⁻¹−𝐼_𝑚)𝐜∼𝐵2(𝛼−𝑚−12𝐜,𝛽),_′(𝑊⁻¹−𝐼_𝑚)𝐜2𝐜^′𝐜∼𝐵2(𝛽,𝛼−𝑚−12). (4.3) The expectation of 𝑊⁻¹, 𝐸(𝑊⁻¹), can easily be obtained from the above results. For any fixed 𝐜∈ℝ^𝑚, 𝐜≠0, 𝐜𝐸[^′(𝑊⁻¹−𝐼_𝑚)𝐜2𝐜^′𝐜]=𝐸(𝑣), (4.4) where 𝑣∼𝐵2(𝛽,𝛼−(𝑚−1)/2). Hence, for all 𝐜∈ℝ^𝑚, 𝐜^′𝐸(𝑊⁻¹−𝐼_𝑚)𝐜=2𝐜^′𝐜𝐸(𝑣)=2𝛽𝐜𝛼−(𝑚+1)/2^′𝐜,𝛼>𝑚+12, (4.5) which implies that 𝐸(𝑊⁻¹)=2𝛽+𝛼−(𝑚+1)/2𝐼𝛼−(𝑚+1)/2^𝑚,𝛼>𝑚+12. (4.6) 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 etr(−Ψ⁻¹𝑌)det(𝑌)^{𝜅−(𝑚+1)/2}Γ_𝑚(𝜅)det(Ψ)^𝜅,𝑌>0. (4.7) It is well known that if 𝑌₁ and 𝑌₂ are independent, 𝑌_𝑖∼𝐺𝑎(𝑚,𝜅_𝑖,𝐼_𝑚), 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,𝐼_𝑚). In the following theorem we derive similar result for matrix variate beta type 3 distribution.

Theorem 4.1. Let the 𝑚×𝑚 random matrices 𝑌₁ and 𝑌₂ be independent, 𝑌_𝑖∼𝐺𝑎(𝑚,𝜅_𝑖,𝐼_𝑚), 𝑖=1,2. Then, (𝑌₁+2𝑌₂)^−1/2𝑌₁(𝑌₁+2𝑌₂)^−1/2∼𝐵3(𝑚,𝜅₁,𝜅₂).

Proof. The joint density function of 𝑌₁ and 𝑌₂ is given by etr[−(𝑌₁+𝑌₂)]det(𝑌₁)^{𝜅1−(𝑚+1)/2}det(𝑌₂)^{𝜅2−(𝑚+1)/2}Γ_𝑚(𝜅₁)Γ_𝑚(𝜅₂),𝑌₁>0,𝑌₂>0. (4.8) 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)/2}det(𝑌)^(𝑚+1)/2 in the joint density of 𝑌₁ and 𝑌₂, we obtain the joint density of 𝑊 and 𝑌 as det(𝑊)^{𝜅1−(𝑚+1)/2}det(𝐼_𝑚−𝑊)^{𝜅2−(𝑚+1)/2}2^𝑚𝜅2Γ_𝑚(𝜅₁)Γ_𝑚(𝜅₂)1×etr[−2(𝐼_𝑚+𝑊)𝑌]det(𝑌)^{𝜅1+𝜅2−(𝑚+1)/2},0<𝑊<𝐼_𝑚,𝑌>0. (4.9) 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.

If 𝑊∼𝐵3(𝑚,𝛼,𝛽), then the cdf of 𝑊, denoted by 𝐺(Ω), is given by 𝐺(Ω)=𝑃(𝑊<Ω)=𝑃(𝑈<(𝐼_𝑚+Ω)⁻¹(𝐼_𝑚−Ω)), (4.10) where 𝑈∼𝐵1(𝑚,𝛽,𝛼). Now, using (2.19), the cdf 𝐺(Ω) is obtained as Γ𝐺(Ω)=^𝑚(𝛼+𝛽)Γ_𝑚[(𝑚+1)/2]Γ_𝑚(𝛼)Γ_𝑚[𝛽+(𝑚+1)/2]det((𝐼_𝑚+Ω)⁻¹(𝐼_𝑚−Ω))^𝛽×₂𝐹₁(𝛽,−𝛼+𝑚+12;𝛽+𝑚+12;(𝐼_𝑚+Ω)⁻¹(𝐼_𝑚−Ω)), (4.11) where 0<Ω<𝐼_𝑚.

Theorem 4.2. Let 𝑊∼𝐵3(𝑚,𝛼,𝛽), then 𝐸[𝑑𝑒𝑡(𝑊)^𝑟𝑑𝑒𝑡(𝐼_𝑚−𝑊)^𝑠𝑑𝑒𝑡(𝐼_𝑚+𝑊)^𝑡]=2^{−𝑚(𝛽+𝑡)}Γ_𝑚(𝛼+𝑟)Γ_𝑚(𝛽+𝑠)Γ_𝑚(𝛼+𝛽)Γ_𝑚(𝛼)Γ_𝑚(𝛽)Γ_𝑚×(𝛼+𝛽+𝑟+𝑠)₂𝐹₁𝐼(𝛽+𝑠,𝛼+𝛽+𝑡;𝛼+𝛽+𝑟+𝑠;^𝑚2), (4.12) where Re(𝛼+𝑟)>(𝑚−1)/2 and Re(𝛽+𝑠)>(𝑚−1)/2.

Proof. By definition 𝐸[det(𝑊)^𝑟det(𝐼_𝑚−𝑊)^𝑠det(𝐼_𝑚+𝑊)^𝑡]=2^𝑚𝛼𝐵_𝑚(𝛼,𝛽)^𝐼𝑚₀det(𝑊)^{𝛼+𝑟−(𝑚+1)/2}det(𝐼_𝑚−𝑊)^{𝛽+𝑠−(𝑚+1)/2}𝑑𝑊det(𝐼_𝑚+𝑊)^{𝛼+𝛽+𝑡}. (4.13) Writing det(𝐼_𝑚+𝑊)^{−(𝛼+𝛽+𝑡)}=2^{−𝑚(𝛼+𝛽+𝑡)}det(𝐼_𝑚−12(𝐼_𝑚−𝑊))^{−(𝛼+𝛽+𝑡)} (4.14) and substituting 𝑍=𝐼_𝑚−𝑊, we have 𝐸[det(𝑊)^𝑟det(𝐼_𝑚−𝑊)^𝑠det(𝐼_𝑚+𝑊)^𝑡]=12^{𝑚(𝛽+𝑡)}𝐵_𝑚(𝛼,𝛽)^𝐼𝑚₀det(𝑍)^{𝛽+𝑠−(𝑚+1)/2}det(𝐼_𝑚−𝑍)^{𝛼+𝑟−(𝑚+1)/2}𝑑𝑍det(𝐼_𝑚−𝑍/2)^{𝛼+𝛽+𝑡}=𝐵_𝑚(𝛼+𝑟,𝛽+𝑠)2^{𝑚(𝛽+𝑡)}𝐵_𝑚(𝛼,𝛽)²𝐹₁𝐼(𝛽+𝑠,𝛼+𝛽+𝑡;𝛼+𝛽+𝑟+𝑠;^𝑚2), (4.15) where the integral has been evaluated using integral representation of the Gauss hypergeometric function given in (2.10).

Corollary 4.3. Let 𝑊∼𝐵3(𝑚,𝛼,𝛽), then for Re(ℎ)>−𝛼+(𝑚−1)/2, one has 𝐸[𝑑𝑒𝑡(𝑊)^ℎ𝑑𝑒𝑡(𝐼_𝑚+𝑊)^ℎΓ]=^𝑚(𝛼+𝛽)Γ_𝑚(𝛼+ℎ)2^𝑚ℎΓ_𝑚(𝛼)Γ_𝑚,(𝛼+𝛽+ℎ)𝐸[𝑑𝑒𝑡(𝑊)^ℎΓ]=^𝑚(𝛼+𝛽)Γ_𝑚(𝛼+ℎ)2^𝑚𝛽Γ_𝑚(𝛼)Γ_𝑚(𝛼+𝛽+ℎ)²𝐹₁𝐼(𝛽,𝛼+𝛽;𝛼+𝛽+ℎ;^𝑚2). (4.16) Further, for Re(ℎ)>−𝛽+(𝑚−1)/2, 𝐸[𝑑𝑒𝑡(𝐼_𝑚+𝑊)^ℎΓ]=^𝑚(𝛼+𝛽)Γ_𝑚(𝛽+ℎ)2^𝑚𝛽Γ_𝑚(𝛽)Γ_𝑚×(𝛼+𝛽+ℎ)₂𝐹₁𝐼(𝛽+ℎ,𝛼+𝛽;𝛼+𝛽+ℎ,^𝑚2). (4.17)

From the density of 𝑊, we have 𝐸[𝐶_𝜅2(𝑊)]=^𝑚𝛼𝐵_𝑚×(𝛼,𝛽)_𝐼𝑚0𝐶_𝜅(𝑊)det(𝑊)^{𝛼−(𝑚+1)/2}det(𝐼_𝑚−𝑊)^{𝛽−(𝑚+1)/2}𝑑𝑊(𝐼_𝑚+𝑊)^𝛼+𝛽. (4.18) Now, expanding (𝐼_𝑚+𝑊)^{−(𝛼+𝛽)} in series involving zonal polynomials using (2.6), the above expression is rewritten as 𝐸[𝐶_𝜅1(𝑊)]=2^𝑚𝛽𝐵_𝑚(𝛼,𝛽)^∞_{ℓ=0𝜆⊢ℓ}(𝛼+𝛽)_𝜆2^ℓ×ℓ!_𝐼𝑚0𝐶_𝜅(𝑊)det(𝑊)^{𝛼−(𝑚+1)/2}det(𝐼_𝑚−𝑊)^{𝛽−(𝑚+1)/2}𝐶_𝜆(𝐼_𝑚−𝑊)𝑑𝑊. (4.19) Further, writing 𝐶_𝜅(𝑊)𝐶_𝜆(𝐼_𝑚−𝑊)=_{𝜙∈𝜅⋅𝜆}𝜃_𝜙^𝜅,𝜆𝐶_𝜙^𝜅,𝜆(𝑊,𝐼_𝑚−𝑊) (4.20) and integrating 𝑊 using (2.15), we get 𝐸[𝐶_𝜅1(𝑊)]=2^𝑚𝛽𝐵_𝑚(𝛼,𝛽)^∞_{ℓ=0𝜆⊢ℓ}(𝛼+𝛽)_𝜆2^ℓℓ!_{𝜙∈𝜅⋅𝜆}𝜃_𝜙^𝜅,𝜆×^𝐼𝑚₀det(𝑊)^{𝛼−(𝑚+1)/2}det(𝐼_𝑚−𝑊)^{𝛽−(𝑚+1)/2}𝐶_𝜙^𝜅,𝜆(𝑊,𝐼_𝑚=1−𝑊)𝑑𝑊2^∞𝑚𝛽_{ℓ=0𝜆⊢ℓ}(𝛼+𝛽)_𝜆2^ℓℓ!_{𝜙∈𝜅⋅𝜆}(𝜃_𝜙^𝜅,𝜆)²(𝛼)_𝜅(𝛽)_𝜆(𝛼+𝛽)_𝜙𝐶_𝜙(𝐼_𝑚). (4.21)

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. Let 𝑋₁∼𝐵1(𝑚,𝛼₁,𝛽₁) and 𝑋₂∼𝐵3(𝑚,𝛼₂,𝛽₂) be independent. Then, the p.d.f. of 𝑍=𝑋₂^1/2𝑋₁𝑋₂^1/2 is 2^−𝑚𝛽2Γ_𝑚(𝛼₁+𝛽₁)Γ_𝑚(𝛼₂+𝛽₂)Γ_𝑚(𝛼₁)Γ_𝑚(𝛼₂)Γ_𝑚(𝛽₁+𝛽₂)𝑑𝑒𝑡(𝑍)^{𝛼1−(𝑚+1)/2}𝑑𝑒𝑡(𝐼_𝑚−𝑍)^{𝛽1+𝛽2−(𝑚+1)/2}×𝐹₁(𝛽₂,𝛼₁+𝛽₁−𝛼₂,𝛼₂+𝛽₂,𝛽₁+𝛽₂;𝐼_𝑚𝐼−𝑍,^𝑚−𝑍2),0<𝑍<𝐼_𝑚. (5.1)

Proof. Using the independence, the joint p.d.f. of 𝑋₁ and 𝑋₂ is given by 𝐾₁det(𝑋₁)^{𝛼1−(𝑚+1)/2}det(𝐼_𝑚−𝑋₁)^{𝛽1−(𝑚+1)/2}×det(𝑋₂)^{𝛼2−(𝑚+1)/2}det(𝐼_𝑚−𝑋₂)^{𝛽2−(𝑚+1)/2}det(𝐼_𝑚+𝑋₂)^𝛼2+𝛽2, (5.2) where 0<𝑋_𝑖<𝐼_𝑚, 𝑖=1,2, and 𝐾₁=2^𝛼2𝑚{𝐵_𝑚(𝛼₁,𝛽₁)𝐵_𝑚(𝛼₂,𝛽₂)}⁻¹. (5.3) Transforming 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 𝐾₁det(𝑍)^{𝛼1−(𝑚+1)/2}det(𝑋₂−𝑍)^{𝛽1−(𝑚+1)/2}det(𝐼_𝑚−𝑋₂)^{𝛽2−(𝑚+1)/2}det(𝑋₂)^{𝛼1+𝛽1−𝛼2}det(𝐼_𝑚+𝑋₂)^𝛼2+𝛽2, (5.4) where 0<𝑍<𝑋₂<𝐼_𝑚. To find the marginal p.d.f. of 𝑍, we integrate (5.4) with respect to 𝑋₂ to get 𝐾₁det(𝑍)^{𝛼1−(𝑚+1)/2}×^𝐼𝑚_𝑍det(𝑋₂−𝑍)^{𝛽1−(𝑚+1)/2}det(𝐼_𝑚−𝑋₂)^{𝛽2−(𝑚+1)/2}𝑑𝑋₂det(𝑋₂)^{𝛼1+𝛽1−𝛼2}det(𝐼_𝑚+𝑋₂)^𝛼2+𝛽2. (5.5) In (5.5) change of variable 𝑉=(𝐼_𝑚−𝑍)^−1/2(𝐼_𝑚−𝑋₂)(𝐼_𝑚−𝑍)^−1/2 with the Jacobian 𝐽(𝑋₂→𝑉)=det(𝐼_𝑚−𝑍)^(𝑚+1)/2 yields 𝐾₁2^{−𝑚(𝛼2+𝛽2)}det(𝑍)^{𝛼1−(𝑚+1)/2}det(𝐼_𝑚−𝑍)^{𝛽1+𝛽2−(𝑚+1)/2}×^𝐼𝑚₀det(𝑉)^{𝛽2−(𝑚+1)/2}det(𝐼_𝑚−𝑉)^{𝛽1−(𝑚+1)/2}𝑑𝑉det(𝐼_𝑚−(𝐼_𝑚−𝑍)𝑉)^{𝛼1+𝛽1−𝛼2}det(𝐼_𝑚−(𝐼_𝑚−𝑍)𝑉/2)^𝛼2+𝛽2=𝐾₁2^{−𝑚(𝛼2+𝛽2)}det(𝑍)^{𝛼1−(𝑚+1)/2}det(𝐼_𝑚−𝑍)^{𝛽1+𝛽2−(𝑚+1)/2}×Γ_𝑚(𝛽₁)Γ_𝑚(𝛽₂)Γ_𝑚(𝛽₁+𝛽₂)𝐹₁(𝛽₂,𝛼₁+𝛽₁−𝛼₂,𝛼₂+𝛽₂,𝛽₁+𝛽₂;𝐼_𝑚𝐼−𝑍,^𝑚−𝑍2), (5.6) 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. Let 𝑋₁ 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=𝑋₂^1/2𝑋₁𝑋₂^1/2 is given by 2^−𝑚𝛽2Γ_𝑚(𝛼₁+𝛽₁+𝛽₂)Γ_𝑚(𝛼₁)Γ_𝑚(𝛽₁+𝛽₂)𝑑𝑒𝑡(𝑍)^{𝛼1−(𝑚+1)/2}𝑑𝑒𝑡(𝐼_𝑚−𝑍)^{𝛽1+𝛽2−(𝑚+1)/2}×₂𝐹₁(𝛽₂,𝛼₁+𝛽₁+𝛽₂;𝛽₁+𝛽₂;𝐼_𝑚−𝑍2),0<𝑍<𝐼_𝑚. (5.7)

Theorem 5.3. Let 𝑋₁ and 𝑋₂ be independent random matrices, 𝑋₁∼𝐵3(𝑚,𝛼₁,𝛽₁) and 𝑋₂∼𝐵2(𝑚,𝛼₂,𝛽₂). Then, the p.d.f. of 𝑍=𝑋₁^1/2𝑋₂𝑋₁^1/2 is given by 2^−𝑚𝛽1𝐵_𝑚(𝛽₁,𝛼₁+𝛽₂)𝐵_𝑚(𝛼₁,𝛽₁)𝐵_𝑚(𝛼₂,𝛽₂)𝑑𝑒𝑡(𝑍)^{𝛼2−(𝑚+1)/2}𝑑𝑒𝑡(𝐼_𝑚+𝑍)^𝛼2+𝛽2×𝐹₁(𝛽₁,𝛼₁+𝛽₁,𝛼₂+𝛽₂;𝛼₁+𝛽₁+𝛽₂;𝐼_𝑚2,(𝐼_𝑚+𝑍)⁻¹),𝑍>0. (5.8)

Proof. Since 𝑋₁ and 𝑋₂ are independent, their joint p.d.f. is given by 𝐾₂det(𝑋₁)^{𝛼1−(𝑚+1)/2}det(𝐼_𝑚−𝑋₁)^{𝛽1−(𝑚+1)/2}det(𝑋₂)^{𝛼2−(𝑚+1)/2}det(𝐼_𝑚+𝑋₁)^𝛼1+𝛽1det(𝐼_𝑚+𝑋₂)^𝛼2+𝛽2, (5.9) where 0<𝑋₁<𝐼_𝑚, 𝑋₂>0, and 𝐾₂=2^𝑚𝛼1{𝐵_𝑚(𝛼₁,𝛽₁)𝐵_𝑚(𝛼₂,𝛽₂)}⁻¹. (5.10) Now consider the transformation 𝑍=𝑋₁^1/2𝑋₂𝑋₁^1/2 and 𝑉=𝐼_𝑚−𝑋₁ whose Jacobian is 𝐽(𝑋₁,𝑋₂→𝑉,𝑍)=det(𝐼_𝑚−𝑉)^{−(𝑚+1)/2}. Thus, we obtain the joint p.d.f. of 𝑉 and 𝑍 as 𝐾₂det(𝑍)^{𝛼2−(𝑚+1)/2}2^{𝑚(𝛼1+𝛽1)}det(𝐼_𝑚+𝑍)^𝛼2+𝛽2det(𝑉)^{𝛽1−(𝑚+1)/2}det(𝐼_𝑚−𝑉)^{𝛼1+𝛽2−(𝑚+1)/2}det(𝐼_𝑚−𝑉/2)^𝛼1+𝛽1det(𝐼_𝑚−(𝐼_𝑚+𝑍)⁻¹𝑉)^𝛼2+𝛽2, (5.11) where 𝑍>0 and 0<𝑉<𝐼_𝑚. Finally, integrating 𝑉 using (3.1) and substituting for 𝐾₂, we obtain the desired result.

In the next theorem we derive the density of 𝑍₁=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 the 𝑚×𝑚 random matrices 𝑋 and 𝑌 be independent, 𝑋∼𝐵3(𝑚,𝛼,𝛽) and 𝑌∼𝐺𝑎(𝑚,𝜅,𝐼_𝑚). Then, the p.d.f. of 𝑍₁=𝑋^−1/2𝑌𝑋^−1/2 is given by Γ_𝑚(𝛼+𝜅)Γ_𝑚(𝛼+𝛽)𝑑𝑒𝑡(𝑍₁)^{𝜅−(𝑚+1)/2}etr(−𝑍₁)2^𝑚𝛽Γ_𝑚(𝜅)Γ_𝑚(𝛼)Γ_𝑚Φ(𝛼+𝛽+𝜅)¹𝐼(𝛽,𝛼+𝛽;𝛼+𝛽+𝜅;^𝑚2,𝑍₁), (5.12) where 𝑍₁>0.

Proof. The joint p.d.f. of 𝑋 and 𝑌 is given by det(𝑋)^{𝛼−(𝑚+1)/2}det(𝐼_𝑚−𝑋)^{𝛽−(𝑚+1)/2}det(𝑌)^{𝜅−(𝑚+1)/2}2^−𝑚𝛼Γ(𝜅)𝐵(𝛼,𝛽)det(𝐼_𝑚+𝑋)^𝛼+𝛽etr(𝑌), (5.13) where 0<𝑋<𝐼_𝑚 and 𝑌>0. Now, transforming 𝑍₁=𝑋^−1/2𝑌𝑋^−1/2 and 𝑊=𝐼_𝑚−𝑋, with the Jacobian 𝐽(𝑋,𝑌→𝑊,𝑍₁)=det(𝐼_𝑚−𝑊)^(𝑚+1)/2, we obtain the joint p.d.f. of 𝑍₁ and 𝑊 as etr(−𝑍₁)det(𝑍₁)^{𝜅−(𝑚+1)/2}2^𝑚𝛽Γ(𝜅)𝐵(𝛼,𝛽)det(𝑊)^{𝛽−(𝑚+1)/2}det(𝐼_𝑚−𝑊)^{𝛼+𝜅−(𝑚+1)/2}det(𝐼_𝑚−𝑊/2)^𝛼+𝛽etr(−𝑊𝑍₁), (5.14) where 0<𝑊<𝐼_𝑚 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 det(𝑈₁)^{𝛼1−(𝑚+1)/2}det(𝑈₂)^{𝛼2−(𝑚+1)/2}det(𝐼_𝑚−𝑈₁−𝑈₂)^{𝛽−(𝑚+1)/2}𝐵_𝑚(𝛼₁,𝛼₂,𝑈,𝛽)₁>0,𝑈₂>0,𝑈₁+𝑈₂<𝐼_𝑚, (6.1) where 𝛼₁>(𝑚−1)/2, 𝛼₂>(𝑚−1)/2, 𝛽>(𝑚−1)/2, and 𝐵_𝑚(𝛼₁,𝛼₂Γ,𝛽)=^𝑚(𝛼₁)Γ_𝑚(𝛼₂)Γ_𝑚(𝛽)Γ_𝑚(𝛼₁+𝛼₂+𝛽). (6.2) 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 det(𝑉₁)^{𝛼1−(𝑚+1)/2}det(𝑉₂)^{𝛼2−(𝑚+1)/2}𝐵_𝑚(𝛼₁,𝛼₂,𝛽)det(𝐼_𝑚+𝑉₁+𝑉₂)^{𝛼1+𝛼2+𝛽},𝑉₁>0,𝑉₂>0, (6.3) 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. The 𝑚×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 det(𝑊₁)^{𝛼1−(𝑚+1)/2}det(𝑊₂)^{𝛼2−(𝑚+1)/2}det(𝐼_𝑚−𝑊₁−𝑊₂)^{𝛽−(𝑚+1)/2}2^{−𝑚(𝛼1+𝛼2)}𝐵_𝑚(𝛼₁,𝛼₂,𝛽)det(𝐼_𝑚+𝑊₁+𝑊₂)^{𝛼1+𝛼2+𝛽},𝑊₁>0,𝑊₂>0,𝑊₁+𝑊₂<𝐼_𝑚, (6.4) 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. Let 𝑌₁, 𝑌₂, and 𝑌₃ be independent, 𝑌_𝑖∼𝐺𝑎(𝑚,𝜅_𝑖,𝐼_𝑚), 𝑖=1,2,3. Define 𝑊_𝑖=(𝑌₁+𝑌₂+2𝑌₃)^−1/2𝑌_𝑖(𝑌₁+𝑌₂+2𝑌₃)^−1/2, 𝑖=1,2. Then, (𝑊₁,𝑊₂)∼𝐷3(𝑚,𝜅₁,𝜅₂;𝜅₃).

Proof. Similar to the proof of Theorem 4.1.

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 𝑊_𝑖=(2𝐼_𝑚−𝑈₁−𝑈₂)^−1/2𝑈_𝑖(2𝐼_𝑚−𝑈₁−𝑈₂)^−1/2,𝑖=1,2. (6.5) Then, (𝑊₁,𝑊₂)∼𝐷3(𝑚,𝛼₁,𝛼₂;𝛽).

Proof. Let 𝑍=2𝐼_𝑚−𝑈₁−𝑈₂ and 𝑊₁=𝑍^−1/2𝑈₁𝑍^−1/2. Then, 𝑊₂=2𝑍⁻¹−(𝐼_𝑚+𝑊₁). The Jacobian of the transformation (6.5) is given by 𝐽(𝑈₁,𝑈₂→𝑊₁,𝑊₂)=𝐽(𝑈₁,𝑈₂→𝑊₁,𝑍)𝐽(𝑊₁,𝑍→𝑊₁,𝑊₂)=det(𝑍)^(𝑚+1)/22^{−𝑚(𝑚+1)/2}det(𝑍)^𝑚+1=2^𝑚(𝑚+1)det(𝐼_𝑚+𝑊₁+𝑊₂)^{−3(𝑚+1)/2}. (6.6) Now, substituting 𝑈_𝑖=2(𝐼_𝑚+𝑊₁+𝑊₂)^−1/2𝑊_𝑖(𝐼_𝑚+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 𝑊_𝑖=(2𝐼_𝑚+𝑉₁+𝑉₂)^−1/2𝑉_𝑖(2𝐼_𝑚+𝑉₁+𝑉₂)^−1/2,𝑖=1,2. (6.7) Then, (𝑊₁,𝑊₂)∼𝐷3(𝑚,𝛼₁,𝛼₂;𝛽).

Proof. Let 𝑍=2𝐼_𝑚+𝑉₁+𝑉₂ and 𝑊₁=𝑍^−1/2𝑉₁𝑍^−1/2. Then, 𝑊₂=𝐼_𝑚−𝑊₁−2𝑍⁻¹. The Jacobian of the transformation (6.7) is given by 𝐽(𝑉₁,𝑉₂→𝑊₁,𝑊₂)=𝐽(𝑉₁,𝑉₂→𝑊₁,𝑍)𝐽(𝑊₁,𝑍→𝑊₁,𝑊₂)=det(𝑍)^(𝑚+1)/22^{−𝑚(𝑚+1)/2}det(𝑍)^𝑚+1=2^𝑚(𝑚+1)det(𝐼_𝑚−𝑊₁−𝑊₂)^{−3(𝑚+1)/2}. (6.8) Now, substitution of 𝑉_𝑖=2(𝐼_𝑚−𝑊₁−𝑊₂)^−1/2𝑊_𝑖(𝐼_𝑚−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 of 𝑊₁, 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 𝑑𝑒𝑡(𝑊₁)^{𝛼1−(𝑚+1)/2}𝑑𝑒𝑡(𝐼_𝑚−𝑊₁)^{𝛼2+𝛽−(𝑚+1)/2}2^{−𝑚(𝛼1+𝛼2)}𝐵_𝑚(𝛼₁,𝛼₂+𝛽)𝑑𝑒𝑡(𝐼_𝑚+𝑊₁)^{𝛼1+𝛼2+𝛽}×₂𝐹₁(𝛼₂,𝛼₁+𝛼₂+𝛽;𝛼₂+𝛽;−(𝐼_𝑚+𝑊₁)⁻¹(𝐼_𝑚−𝑊₁)), (6.9) where 0<𝑊₁<𝐼_𝑚. Further, (𝐼_𝑚−𝑊₁)^−1/2𝑊₂(𝐼_𝑚−𝑊₁)^−1/2∼𝐵3(𝑚,𝛼₂,𝛽).

Proof. Substituting 𝑋₂=(𝐼_𝑚−𝑊₁)^−1/2𝑊₂(𝐼_𝑚−𝑊₁)^−1/2 with the Jacobian 𝐽(𝑊₂→𝑋₂)=det(𝐼_𝑚−𝑊₁)^(𝑚+1)/2 in (6.4), the joint density of 𝑊₁ and 𝑋₂ is derived as 2^{𝑚(𝛼1+𝛼2)}det(𝑊₁)^{𝛼1−(𝑚+1)/2}det(𝐼_𝑚−𝑊₁)^{𝛼2+𝛽−(𝑚+1)/2}𝐵_𝑚(𝛼₁,𝛼₂,𝛽)det(𝐼_𝑚+𝑊₁)^{𝛼1+𝛼2+𝛽}×det(𝑋₂)^{𝛼2−(𝑚+1)/2}det(𝐼_𝑚−𝑋₂)^{𝛽−(𝑚+1)/2}det(𝐼_𝑚+(𝐼_𝑚+𝑊₁)⁻¹(𝐼_𝑚−𝑊₁)𝑋₂)^{𝛼1+𝛼2+𝛽},0<𝑊₁<𝐼_𝑚,0<𝑋₂<𝐼_𝑚. (6.10) 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 2^{𝑚(𝛼1+𝛼2)}det(𝑋₂)^{𝛼2−(𝑚+1)/2}det(𝐼_𝑚−𝑋₂)^{𝛽−(𝑚+1)/2}𝐵_𝑚(𝛼₁,𝛼₂,𝛽)det(𝐼_𝑚+𝑋₂)^{𝛼1+𝛼2+𝛽}×^𝐼𝑚₀det(𝑊₁)^{𝛼1−(𝑚+1)/2}det(𝐼_𝑚−𝑊₁)^{𝛼2+𝛽−(𝑚+1)/2}d𝑊₁det(𝐼_𝑚+(𝐼_𝑚+𝑋₂)⁻¹(𝐼_𝑚−𝑋₂)𝑊₁)^{𝛼1+𝛼2+𝛽},0<𝑋₂<𝐼_𝑚. (6.11) Now, by evaluating the above integral using results on Gauss hypergeometric function, we obtain ^𝐼𝑚₀det(𝑊₁)^{𝛼1−(𝑚+1)/2}det(𝐼_𝑚−𝑊₁)^{𝛼2+𝛽−(𝑚+1)/2}𝑑𝑊₁det(𝐼_𝑚+(𝐼_𝑚+𝑋₂)⁻¹(𝐼_𝑚−𝑋₂)𝑊₁)^{𝛼1+𝛼2+𝛽}=Γ_𝑚(𝛼₁)Γ_𝑚(𝛼₂+𝛽)Γ_𝑚(𝛼₁+𝛼₂+𝛽)²𝐹₁(𝛼₁,𝛼₁+𝛼₂+𝛽;𝛼₁+𝛼₂+𝛽;−(𝐼_𝑚+𝑋₂)⁻¹(𝐼_𝑚−𝑋₁=Γ))_𝑚(𝛼₁)Γ_𝑚(𝛼₂+𝛽)Γ_𝑚(𝛼₁+𝛼₂+𝛽)¹𝐹₀(𝛼₁;−(𝐼_𝑚+𝑋₂)⁻¹(𝐼_𝑚−𝑋₁=Γ))_𝑚(𝛼₁)Γ_𝑚(𝛼₂+𝛽)Γ_𝑚(𝛼₁+𝛼₂2+𝛽)^−𝑚𝛼1det(𝐼_𝑚+𝑋₂)^𝛼1. (6.12) Finally, substituting (6.12) in (6.11) and simplifying the resulting expression we obtain the desired result.

Using the result ₂𝐹₁(𝑎,𝑏;𝑐;𝑋)=det(𝐼_𝑚−𝑋)₂^−𝑏𝐹₁(𝑐−𝑎,𝑏;𝑐;−𝑋(𝐼_𝑚−𝑋)⁻¹), (6.13) the Gauss hypergeometric function given in (6.9) can be rewritten as ₂𝐹₁(𝛼₂,𝛼₁+𝛼₂+𝛽;𝛼₂+𝛽;−(𝐼_𝑚+𝑊₁)⁻¹(𝐼_𝑚−𝑊₁=))det(𝐼_𝑚+𝑊₁)^{𝛼1+𝛼2+𝛽}2^{𝑚(𝛼1+𝛼22+𝛽)}𝐹₁(𝛽,𝛼₁+𝛼₂+𝛽;𝛼₂𝐼+𝛽;^𝑚−𝑊₁2). (6.14) Hence, the density of 𝑊₁ can also be written as det(𝑊₁)^{𝛼1−(𝑚+1)/2}det(𝐼_𝑚−𝑊₁)^{𝛼2+𝛽−(𝑚+1)/2}2^𝑚𝛽𝐵_𝑚(𝛼₁,𝛼₂×+𝛽)₂𝐹₁(𝛽,𝛼₁+𝛼₂+𝛽;𝛼₂𝐼+𝛽;^𝑚−𝑊₁2),0<𝑊₁<𝐼_𝑚. (6.15) 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 transformation 𝑈=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 det(𝑈)^{𝛼1−(𝑚+1)/2}det(𝐼_𝑚−𝑈)^{𝛼2−(𝑚+1)/2}𝐵_𝑚(𝛼₁,𝛼₂)×det(𝑊)^{𝛼1+𝛼2−(𝑚+1)/2}det(𝐼_𝑚−𝑊)^{𝛽−(𝑚+1)/2}2^{−𝑚(𝛼1+𝛼2)}𝐵_𝑚(𝛼₁+𝛼₂,𝛽)det(𝐼_𝑚+𝑊)^{𝛼1+𝛼2+𝛽}, (6.16) where 0<𝑈<𝐼_𝑚 and 0<𝑊<𝐼_𝑚. 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 Theorem 6.6, the joint moments of det(𝑊₁) and det(𝑊₂) are given by 𝐸[det(𝑊₁)^𝑟1det(𝑊₂)^𝑟2]=𝐸[det(𝑈)^𝑟1det(𝐼_𝑚−𝑈)^𝑟2]𝐸[det(𝑊)^𝑟1+𝑟2], (6.17) 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(𝐼_𝑚−𝑈)^𝑟2] using Corollary 4.3 and (2.20) and simplifying the resulting expression, we obtain 𝐸[det(𝑊₁)^𝑟1det(𝑊₂)^𝑟2Γ]=^𝑚(𝛼₁+𝑟₁)Γ_𝑚(𝛼₂+𝑟₂)Γ_𝑚(𝛼₁+𝛼₂+𝛽)2^𝑚𝛽Γ_𝑚(𝛼₁)Γ_𝑚(𝛼₂)Γ_𝑚(𝛼₁+𝛼₂+𝛽+𝑟₁+𝑟₂)×₂𝐹₁(𝛽,𝛼₁+𝛼₂+𝛽;𝛼₁+𝛼₂+𝛽+𝑟₁+𝑟₂;𝐼_𝑚2). (6.18)

Acknowledgment

The research work of D. K. Nagar was supported by the Comité para el Desarrollo de la Investigación, Universidad de Antioquia research Grant no. IN550CE.