MATRIX VARIATE KUMMER-DIRICHLET DISTRIBUTIONS

MATRIX VARIATE KUMMER-DIRICHLET DISTRIBUTIONS

ARJUN K. GUPTA, LILIAM CARDEN˜ O,AND DAYA K. NAGAR

Received 15 July 2000 and in revised form 8 June 2001

The multivariate Kummer-Beta and multivariate Kummer-Gamma families of distributions have been proposed and studied recently by Ng and Kotz. These distributions are extensions of Kummer-Beta and Kummer-Gamma distribu- tions. In this article we propose and study matrix variate generalizations of multivariate Kummer-Beta and multivariate Kummer-Gamma families of distributions.

1. Introduction

The Kummer-Beta and Kummer-Gamma families of distributions are defined by the density functions

Γ(α+β) Γ(α)Γ(β)

1F₁(α;α+β;−λ)₋₁

exp_(−λu)u^α−1_(1−u)^β−1, 0 < u < 1, (1.1) Γ(α)Ψ(α, α−γ+1;ξ)₋₁

exp₍₋ξv)v^α−1(1+v)^−γ, v > 0, (1.2) respectively, where α > 0, ^{β > 0}, ^{ξ > 0}, −∞< γ, λ <∞, 1F₁, and Ψ are confluent hypergeometric functions. These distributions are extensions of Gamma and Beta distributions, and for α < 1 (and certain values ofλ and γ) yield bimodal distributions on finite and infinite ranges, respectively.

These distributions are used (i) in the Bayesian analysis of queueing system where posterior distribution of certain basic parameters inM/M/∞queue- ing system is Kummer-Gamma and (ii) in common value auctions where the posterior distribution of “value of a single good” is Kummer-Beta. For prop- erties and applications of these distributions the reader is referred to Ng and Kotz [7],Armero and Bayarri [1],and Gordy [2].

Copyright^c 2001 Hindawi Publishing Corporation Journal of Applied Mathematics 1:3 (2001) 117–139 2000 Mathematics Subject Classification:62E15,62H99

URL:http://jam.hindawi.com/volume-1/S1110757X0100701X.html

As the corresponding multivariate generalization of these distributions, we have the followingn-dimensional densities:

Γ_n

i=1α_i+β _n

i=1Γ α_i

Γ(β)

1F_{1 n}

i=1

α_i; n i=1

α_i+β;−λ ₋₁

exp

−λ n i=1

u_i

× ⁿ

i=1

u^α_iⁱ⁻¹

1−ⁿ

i=1

u_{i β−1}

, 0 < u_i< 1, n i=1

u_i< 1,

(1.3) whereα_i> 0,ⁱ=1, . . . , n,^{β > 0},−∞< λ <∞,and

Γ

_n

i=1

α_i

Ψ _n

i=1

α_i, n i=1

α_i−γ+1;ξ ₋₁

exp

−ξ n i=1

v_i

× ⁿ

i=1

v^α_iⁱ⁻¹

1+ⁿ

i=1

v_{i −γ}

, v_i> 0,

(1.4)

where α_i > 0, ⁱ =1, . . . , n, ^{ξ > 0}, −∞< γ <∞, respectively. These dis- tributions have been considered by Ng and Kotz [7] who refer to (1.3) and (1.4) as multivariate Kummer-Beta and multivariate Kummer-Gamma distri- butions,respectively. Forλ=0,(1.1) and (1.3) reduce to Beta and Dirichlet distributions with probability density functions

Γ(α+β)

Γ(α)Γ(β)u^α−1(1−u)^β−1, 0 < u < 1, Γ

_n

i=1α_i+β _n

i=1Γ α_i

Γ(β)

n

i=1

u^α_iⁱ⁻¹

1−ⁿ

i=1

u_{i β−1}

, 0 < u_i< 1, n i=1

u_i< 1, (1.5) respectively. Since (1.3) is an extension of Dirichlet distribution and a multi- variate generalization of Kummer-Beta distribution,an appropriate nomen- clature for this distribution would beKummer-Dirichlet distribution. In the same vein,we may call (1.4) a Kummer-Dirichlet distribution. Further,in or- der to distinguish between these two distributions ((1.3) and (1.4)),we call them Kummer-Dirichlet type I and Kummer-Dirichlet type II distributions.

In this article we propose and study matrix variate generalizations of (1.3) and (1.4),respectively.

2. Matrix variate Kummer-Dirichlet distributions

We begin with a brief review of some definitions and notations. We adhere to standard notations (cf. Gupta and Nagar [3]). LetA= (aij)be a_p×pmatrix.

Then, A denotes the transpose of A; tr_{(A) =} a₁₁+···+a_pp; etr_{(A) =} exp(tr(A)); det(A) =determinant ofA; ^{A > 0}means that A is symmetric positive definite and A^1/2 denotes the unique symmetric positive definite square root ofA > 0. The multivariate gamma functionΓ_p(m)is defined as

Γ_p(m) =π^p(p−1)/4

p

j=1

Γ

m−j−1 2

, Re_(m)>p−1

2 , (2.1) where Re_(·)denotes the real part of_(·). It is straightforward to show that

Γ_p(m) =

R>0

det₍R)^m−(p+1)/2etr₍₋R)dR, Re₍m)> p−1

2 , (2.2) where the integral has been evaluated over the space of the p×psymme- tric positive definite matrices. The integral representation of the confluent hypergeometric function₁F₁is given by

1F₁(a;b;X) = Γ_p(b) Γ_p(a)Γ_p(b−a)

×

0<R<Ip

det₍R)^a−(p+1)/2detI_p−Rb−a−(p+1)/2

etr₍XR)dR, (2.3) where Re₍a)>(p−1)/2and Re₍b−a)>(p−1)/2. The confluent hypergeo- metric functionΨof ap×psymmetric matrixXis defined by

Ψ(a, c;X) = 1 Γ_p(a)

×

R>0

etr_(−XR)det_(R)^a−(p+1)/2detI_p+Rc−a−(p+1)/2

dR, (2.4) where Re_(X)> 0and Re_(a)>(p−1)/2.

Now we define the corresponding matrix variate generalizations of (1.3) and (1.4) as follows.

Definition 2.1. The_p×psymmetric positive definite random matricesU₁, . . . , U_n are said to have the matrix variate Kummer-Dirichlet type I distri- bution with parameters α₁, . . . , α_n, ^β and Λ, denoted by (U₁, . . . , U_n) ∼ KD^I_p(α₁, . . . , α_n, β, Λ), if their joint probability density function (pdf) is given by

K₁

α₁, . . . , α_n, β, Λ etr

−Λ n i=1

U_i

×

n

i=1

detU_{iαi−(p+1)/2} det

I_p−

n i=1

U_i

_β−(p+1)/2 ,

0 < U_i< I_p, 0 <

n i=1

U_i< I_p,

(2.5)

whereα_i>(p−1)/2,ⁱ=1, . . . , n,^{β >}(p−1)/2,Λ(p×p)is symmetric and K₁(α₁, . . . , α_n, β, Λ)is the normalizing constant.

Definition 2.2. Thep×psymmetric positive definite random matricesV₁, . . . , V_nare said to have the matrix variate Kummer-Dirichlet type II distribution with parametersα₁, . . . , α_n,^γandΞ,denoted by_(V1, . . . , V_n)∼KD^II_p(α₁, . . . , α_n, γ, Ξ),if their joint pdf is given by

K₂

α₁, . . . , α_n, γ, Ξ etr

−Ξ n i=1

V_i

× ⁿ

i=1

detV_{iαi−(p+1)/2} det

I_p+ⁿ

i=1

V_{i −γ}

, V_i> 0,

(2.6)

where α_i > (p−1)/2, ⁱ = 1, . . . , n, −∞ < γ < ∞, Ξ(p×p) > 0, and K₂(α1, . . . , α_n, γ, Ξ)is the normalizing constant.

The normalizing constants in (2.5) and (2.6) are given as K₁

α₁, . . . , α_n, β, Λ₋₁

=

···

0<_n

i=1Ui<Ip

Ui>0

etr

−Λ n i=1

U_i

×

n

i=1

detU_{iαi−(p+1)/2} det

I_p−

n i=1

U_i

_{β−(p+1)/2 n}

i=1

dU_i

= _n

i=1Γ_p α_i Γ_pn

i=1α_i

0<U<Ip

etr_(−ΛU)det_(U)^{ni=1αi−(p+1)/2}

×detI_p−U_β−(p+1)/2 dU

= _n

i=1Γ_p α_i

Γ_p(β) Γ_pn

i=1α_i+β ₁F_{1 n}

i=1

α_i;ⁿ

i=1

α_i+β;−Λ

,

(2.7) K₂

α₁, . . . , α_n, γ, Ξ₋₁

=

V1>0···

Vn>0

etr

−Ξ n i=1

V_i

× ⁿ

i=1

detV_{iαi−(p+1)/2} det

I_p+ⁿ

i=1

V_{i −γ n}

i=1

dV_i

= _n

i=1Γ_p α_i Γ_pn

i=1α_i

V>0

etr_(−ΞV)det_(V)^{ni=1αi−(p+1)/2}detI_p+V_−γ dV

= ⁿ

i=1

Γ_p α_i

Ψ _n

i=1

α_i, n i=1

α_i−γ+p+1 2 ;Ξ

,

(2.8) respectively, where ₁F₁ and Ψ are confluent hypergeometric functions of matrix argument.

ForΛ=0, the matrix variate Kummer-Dirichlet type I distribution col- lapses to an ordinary matrix variate Dirichlet type I distribution with pdf

Γ_pn

i=1α_i+β _n

i=1Γ_p α_i

Γ_p(β)

n

i=1

detU_{iαi−(p+1)/2} det

I_p−ⁿ

i=1

U_i

_β−(p+1)/2 ,

0 < U_i< I_p, 0 <

n i=1

U_i< I_p, (2.9) where α_i > (p−1)/2, ⁱ = 1, . . . , n, and β >(p−1)/2. A common nota- tion to designate that ₍U₁, . . . , U_n) has this density is ₍U₁, . . . , U_n) ∼ D^I_p(α₁, . . . , α_n;β). For γ=0, the matrix variate Kummer-Dirichlet type II density simplifies to the product ofnmatrix variate Gamma densities.

Forp=1, the densities in (2.5) and (2.6) simplify to Kummer-Dirichlet type I (multivariate Kummer-Beta) and Kummer-Dirichlet type II (multivari- ate Kummer-Gamma) densities, respectively. For n = 1, the matrix vari- ate Kummer-Dirichlet type I and matrix variate Kummer-Dirichlet type II distributions reduce to the matrix variate Kummer-Beta and matrix vari- ate Kummer-Gamma distributions,respectively. These two distributions have been studied by Nagar and Gupta [6] and Nagar and Carden˜o [5]. Substituting

n=1in (2.5) and (2.6),the matrix variate Kummer-Beta and matrix variate Kummer-Gamma densities are obtained as

K₁(α, β, Λ)etr_(−ΛU)det_(U)^α−(p+1)/2

×detI_p−U_β−(p+1)/2

, 0 < U < I_p, K₂(α, γ, Ξ)etr_(−ΞV)det_(V)^α−(p+1)/2detI_p+V_−γ

, V > 0,

(2.10)

respectively,whereα >(p−1)/2,β >(p−1)/2,−∞< γ <∞,Λ=Λ,and Ξ(p×p)> 0. These two distributions are designated byU∼KB_p(α, β, Λ)and V∼KG_p(α, γ, Ξ). It may be noted that the matrix variate Kummer-Dirichlet distributions are special cases of the matrix variate Liouville distribution.

Using certain transformations,generalized matrix variate Kummer-Dirich- let distributions are generated as given in the next two theorems.

Theorem 2.3. Let _(U1, . . . , U_n)∼ KD^I_p(α1, . . . , α_n, β, Λ) and Ψ₁, . . . , Ψ_n, Ω be symmetric matrices such that Ω > 0and Ω−_n

i=1Ψ_i> 0. Define Z_i=

Ω−ⁿ

i=1

Ψ_{i 1/2}

U_i

Ω−ⁿ

i=1

Ψ_{i 1/2}

+Ψ_i, i=1, . . . , n. (2.11)

T hen (Z₁, . . . , Z_n)have the generalized matrix variate Kummer-Dirichlet type I distribution with pdf

K₁

α₁, . . . , α_n, β, Λ detΩ−_n

i=1Ψ_i^r_{i=1αi+β−(p+1)/2}

× _n

i=1detZ_i−Ψ_{iαi−(p+1)/2}

detΩ−_n

i=1Z_{iβ−(p+1)/2} etrΩ−_n

i=1Ψ_i−1/2 Λ

Ω−_n

i=1Ψ_i−1/2n

i=1

Z_i−Ψ_i,

Ψ_i< Z_i< Ω, i=1, . . . , n, n i=1

Z_i< Ω.

(2.12) Proof. Making the transformation U_i = (Ω−_n

i=1Ψ_i)^−1/2(Z_i−Ψ_i)(Ω− _r

i=1Ψ_i)^−1/2, ⁱ = 1, . . . , n, with Jacobian J(U₁, . . . , U_n → Z₁, . . . , Z_n) = det_(Ω−ⁿ_i=1Ψ_i)^−n(p+1)/2in (2.5),we get (2.12).

If ₍Z₁, . . . , Z_n)has the pdf (2.12),then we write ₍Z₁, . . . , Z_n)∼GKD^I_p(α₁, . . . , α_n, β, Λ;Ω;Ψ₁, . . . , Ψ_n). Note that GKD^I_p(α1, . . . , α_n, β, Λ;I_p;0, . . . , 0)≡ KD^I_p(α1, . . . , α_n, β, Λ).

Theorem 2.4. Let_(V1, . . . , V_n)∼KD^II_p(α1, . . . , α_n, γ, Ξ)andΨ₁, . . . , Ψ_n, Ωbe symmetric matrices such thatΩ > 0andΩ+_n

i=1Ψ_i> 0. Define

Y_i=

Ω+ⁿ

i=1

Ψ_{i 1/2}

V_i

Ω+ⁿ

i=1

Ψ_{i 1/2}

+Ψ_i, i=1, . . . , n. (2.13)

T hen, (Y₁, . . . , Y_n)have the generalized matrix variate Kummer-Dirichlet type II distribution with pdf

K₂

α₁, . . . , α_n, γ, Ξ detΩ+_n

i=1Ψ_iⁿ_i=1αi−γ

×

_n

i=1detY_i−Ψ_{iαi−(p+1)/2}

detΩ+_n

i=1Y_i−γ etrΩ+_n

i=1Ψ_i−1/2 Ξ

Ω+_n

i=1Ψ_i−1/2n

i=1

Y_i−Ψ_i,

Y_i> Ψ_i, i=1, . . . , n.

(2.14) Proof. Making the transformation V_i = (Ω+_n

i=1Ψ_i)^−1/2(Yi−Ψ_i)(Ω+ _n

i=1Ψ_i)^−1/2, ⁱ=1, . . . , n, with the Jacobian _J(V1, . . . , V_n→Y₁, . . . , Y_n) = det(Ω+_n

i=1Ψ_i)^−n(p+1)/2in (2.6),we get (2.14).

If ₍Y₁, . . . , Y_n) has pdf (2.14), then we write ₍Y₁, . . . , Y_n)∼GKD^II_p(α₁, . . . , α_n, γ, Ξ;Ω;Ψ₁, . . . , Ψ_n). In this case GKD^II_p(α₁, . . . , α_n, γ;I_p;0, . . . , 0) ≡ KD^II_p(α₁, . . . , α_n;γ, Ξ).

3. Properties

In this section, we study certain properties of matrix variate Kummer- Dirichlet type I and II distributions. It may be noted that for Λ= λI_p, Ξ=ξI_pdensities (2.5) and (2.6) are orthogonally invariant. That is,for any fixed orthogonal matrixΓ(p×p),the distribution of(Γ U₁Γ, . . . , Γ U_nΓ)is the same as the distribution of_(U1, . . . , U_n),and similarly the distribution of (Γ V1Γ, . . . , Γ V_nΓ)is the same as that of_(V1, . . . , V_n). Our next two results give marginal and conditional distributions.

Theorem 3.1. If _(U1, . . . , U_n)∼KD^I_p(α₁, . . . , α_n, β, Λ), then the joint mar- ginal pdf ofU₁, . . . , U_m,m≤n,is given by

K₁

α₁, . . . , α_m, n i=m+1

α_i+β, Λ

etr

−Λ m i=1

U_i

× ^m

i=1

detU_{iαi−(p+1)/2} det

I_p−^m

i=1

U_i

ⁿ_{i=m+1αi+β−(p+1)/2}

×1F_{1 n}

i=m+1

α_i; n i=m+1

α_i+β;−Λ

I_p− m i=1

U_i

,

0 < U_i< I_p, 0 <

m i=1

U_i< I_p,

(3.1) and the conditional density of (U_m+1, . . . , U_n)|(U₁, . . . , U_m)is given by

K₁

α₁, . . . , α_n, β, Λ K₁

α₁, . . . , α_m,_n

i=m+1α_i+β, Λ

× etr₋Λ_n

i=m+1U_i detI_p−_m

i=1U_iⁿ_{i=m+1αi+β−(p+1)/2}

× _n

i=m+1detU_{iαi−(p+1)/2}

detI_p−_m

i=1U_i−_n

i=m+1U_{iβ−(p+1)/2}

1F_1n

i=m+1α_i;_n

i=m+1α_i+β;−Λ

I_p−_m

i=1U_i ,

0 < U_i< I_p−^m

i=1

U_i, i=m+1, . . . , n, ⁿ

i=m+1

U_i< I_p−^m

i=1

U_i. (3.2) Proof. First we find the marginal density ofU₁, . . . , U_n−1by integrating out U_nfrom the joint density ofU₁, . . . , U_n as

K₁

α₁, . . . , α_n, β, Λ

0<Un<Ip−_n−1

i=1Ui

etr

−Λ n i=1

U_i

× ⁿ

i=1

detU_{iαi−(p+1)/2} det

I_p−ⁿ

i=1

U_i

_β−(p+1)/2 dU_n.

(3.3)

Now, substituting Z_n = (Ip−_n−1

i=1 U_i)^−1/2U_n(Ip−_n−1

i=1 U_i)^−1/2 with JacobianJ(U_n→Z_n) =det(I_p−_n−1

i=1 U_i)^(p+1)/2in (3.2),we get

K₁

α₁, . . . , α_n, β, Λ etr

−Λ

n−1

i=1

U_i

×ⁿ⁻¹

i=1

detU_{iαi−(p+1)/2} det

I_p−ⁿ⁻¹

i=1

U_i

_{αn+β−(p+1)/2}

×

0<Zn<Ip

etr −

I_p−ⁿ⁻¹

i=1

U_{i 1/2}

Λ

I_p−ⁿ⁻¹

i=1

U_{i 1/2}

Z_n

×detZ_{nαn−(p+1)/2}

detI_p−Z_{nβ−(p+1)/2} dZ_n.

(3.4) But

K₁

α₁, . . . , α_n, β, Λ

×

0<Zr<Ip

etr ₋

I_p−ⁿ⁻¹

i=1

U_{i 1/2}

Λ

I_p−ⁿ⁻¹

i=1

U_{i 1/2}

Z_n

×detZ_{nαn−(p+1)/2}

detI_p−Z_{nβ−(p+1)/2} dZ_n

=K₁

α₁, . . . , α_n, β, ΛΓ_p α_n

Γ_p(β) Γ_p

α_n+β ₁F₁

α_n;α_n+β;−Λ

I_p−ⁿ⁻¹

i=1

U_i

=K₁

α₁, . . . , α_n−1, α_n+β, Λ

1F₁

α_n;αn+β;−Λ

I_p−

n−1

i=1

U_i

. (3.5) Hence,we get the joint density of₍U₁, . . . , U_n−1)as

K₁

α₁, . . . , α_n−1, α_n+β, Λ etr

−Λ

n−1

i=1

U_{i n−1}

i=1

detU_{iαi−(p+1)/2}

×det

I_p−ⁿ⁻¹

i=1

U_i

_{αn+β−(p+1)/2}

1F₁

α_n;α_n+β;−Λ

I_p−ⁿ⁻¹

i=1

U_i

. (3.6)

Repeating this proceduren−mtimes gives the marginal density of_(U1, . . . , U_m)as

K₁

α₁, . . . , α_m, n i=m+1

α_i+β, Λ

etr

−Λ m i=1

U_i

×

m

i=1

detU_{iαi−(p+1)/2} det

I_p−

m i=1

U_i

ⁿ_{i=m+1αi+β−(p+1)/2}

×1F_{1 n}

i=m+1

α_i; n i=m+1

α_i+β;−Λ

I_p− m i=1

U_i

.

(3.7) Now,the second part of the theorem follows immediately.

Corollary 3.2. If _(U1, . . . , U_n)∼KD^I_p(α₁, . . . , α_n, β, Λ), then the marginal pdf of U_i,i=1, . . . , nis given by

K₁



α_i, n j=1(=i)

α_j+β, Λ



etr_−ΛUidetU_{iαi−(p+1)/2}

×detI_p−U_iⁿ_{j=1(=i)αj+β−(p+1)/2}

×1F₁



 ⁿ

j=1(=i)

α_j; n j=1(=i)

α_j+β;−Λ

I_p−U_i

, 0 < U_i< I_p. (3.8) It is interesting to note that the marginal density ofU_i does not belong to the Kummer-Beta family and differs by an additional factor containing confluent hypergeometric function₁F₁.

In Theorem 3.3 we give results on marginal and conditional distributions for Kummer-Dirichlet type II distribution. Before doing so, we need to give an integral that will be used in the derivation of marginal distribution. From (2.6) and (2.8),we have

X>0

Y>0

etr_−Ξ(X+Y)det_(Y)^a1−(p+1)/2

×det_(X)^a2−(p+1)/2detI_p+X+Y_−b dX dY

=Γ_p a₁

Γ_p a₂

Ψ

a₁+a₂, a₁+a2−b+p+1 2 ;Ξ

,

(3.9)

where Re_(a1)>(p−1)/2, Re_(a2)>(p−1)/2and Re_(Ξ)> 0. Substituting

W= (Ip+X)^−1/2 Y(Ip+X)^−1/2 with the Jacobian _{J(Y →W) =}det_(Ip+ X)^(p+1)/2in (3.9) and integratingW,we obtain

X>0

etr_(−ΞX)det_(X)^a2−(p+1)/2detI_p+X_a1−b

×Ψ

a₁, a₁−b+p+1 2 ;Ξ

I_p+X dX

=Γ_p a₂

Ψ

a₁+a₂, a₁+a₂−b+p+1 2 ;Ξ

.

(3.10)

Now we turn to our problem of finding the marginal and conditional distri- butions.

Theorem 3.3. If _(V1, . . . , V_n)∼KD^II_p(α₁, . . . , α_n, γ, Ξ), then the joint mar- ginal pdf ofV₁, . . . , V_m,^m≤n,is given by

Γ_{p n}

i=m+1

α_i

K₂

α₁, . . . , α_m, n i=m+1

α_i, γ, Ξ

etr

−Ξ m i=1

V_i

×

m

i=1

detV_{iαi−(p+1)/2} det

I_p+

m i=1

V_i

_−γ+ⁿ_i=m+1αi

×Ψ _n

i=m+1

α_i, n i=m+1

α_i−γ+p+1 2 ;Ξ

I_p+

m j=1

V_j

, V_j> 0, j=1, . . . , m,

(3.11)

and the conditional density of _(Vm+1, . . . , V_n)|(V₁, . . . , V_m)is given by K₂

α₁, . . . , α_n, γ, Ξ Γ_pn

i=m+1α_i K₂

α₁, . . . , α_m,_n

i=m+1α_i, γ, Ξ

× etr₋Ξ_n

i=m+1V_i detI_p+_m

i=1V_i−γ+ⁿ_i=m+1αi

× _n

i=m+1detV_{iαi−(p+1)/2}

detI_p+_m

i=1V_i+_n

i=m+1V_i−γ

Ψ_n

i=m+1α_i,_n

i=m+1α_i−γ+(p+1)/2;Ξ

I_p+_m

j=1V_j , V_i> 0, i=m+1, . . . , n.

(3.12) Proof. In this case, to obtain the marginal density of V₁, . . . , V_n−1, we substitute W_n = (Ip+ _n−1

i=1V_i)^−1/2V_n(Ip+ _n−1

i=1V_i)^−1/2 with the

Jacobian _J(Vn→W_n) =det_(Ip+ⁿ⁻¹_i=1V_i)^(p+1)/2. Thus,the joint density ofV₁, . . . , V_n−1is obtained as

K₂

α₁, . . . , α_n, γ, Ξ etr

−Ξ

n−1

i=1

V_i

×

n−1

i=1

detV_{iαi−(p+1)/2} det

I_p+

n−1

i=1

V_i

_−γ+αn

×

Wn>0etr −

I_p+ⁿ⁻¹

i=1

V_{i 1/2}

Ξ

I_p+ⁿ⁻¹

i=1

V_{i 1/2}

W_n

×detW_{nαn−(p+1)/2}

detI_p+W_n−γ dW_n

=Γ_p α_n

K₂(α₁, . . . , α_n, γ, Ξ etr

−Ξ

n−1

i=1

V_i

×

n−1

i=1

detV_{iαi−(p+1)/2} det

I_p+

n−1

i=1

V_i

_−γ+αn

×Ψ

α_n, α_n−γ+p+1 2 ;Ξ

I_p+

n−1

i=1

V_i

.

(3.13)

Further,substitutingW_n−1= (I_p+_n−2

i=1 V_i)^−1/2V_n−1(I_p+_n−2

i=1V_i)^−1/2 with the Jacobian_J(Vn−1→W_n−1) =det_(Ip+ⁿ⁻²_i=1V_i)^(p+1)/2 in (3.13) and integrating W_n−1 using (3.10), we get the joint marginal density of V₁, . . . , V_n−2as

Γ_p α_n

K₂

α₁, . . . , α_n, γ, Ξ etr

−Ξ

n−2

i=1

V_i

×ⁿ⁻²

i=1

detV_{iαi−(p+1)/2} det

I_p+ⁿ⁻²

i=1

V_i

_{−γ+αn+αn−1}

×

Wn−1>0

etr ₋

I_p+ⁿ⁻²

i=1

V_{i 1/2}

Ξ

I_p+ⁿ⁻²

i=1

V_{i 1/2}

W_n−1

×detW_{n−1αn−1−(p+1)/2}

detI_p+W_{n−1−γ+αn}

×Ψ

α_n, α_n−γ+p+1 2 ;

I_p+ⁿ⁻²

i=1

V_{i 1/2}

×Ξ

I_p+ⁿ⁻²

i=1

V_{i 1/2}

W_n−1

dW_n−1

=Γ_p α_n

Γ_p α_n−1

K₂

α₁, . . . , α_n, γ, Ξ etr

−Ξ

n−2

i=1

V_i

×ⁿ⁻²

i=1

detV_{iαi−(p+1)/2} det

I_p+ⁿ⁻²

i=1

V_i

_{−γ+αn+αn−1}

×Ψ

α_n+α_n−1, α_n+α_n−1−γ+p+1 2 ;Ξ

I_p+ⁿ⁻²

i=1

V_i

.

(3.14) Integrating out V_n−2, . . . , V_m+1 similarly, we get the marginal density of V₁, . . . , V_mas

n

i=m+1

Γ_p α_i

K₂

α₁, . . . , α_n, γ, Ξ etr

−Ξ m i=1

V_i

×

m

i=1

detV_{iαi−(p+1)/2} det

I_p+

m i=1

V_i

_−γ+ⁿ_i=m+1αi

×Ψ _n

i=m+1

α_i, n i=m+1

α_i−γ+p+1 2 ;Ξ

I_p+

m i=1

V_i

.

(3.15)

The final expression of the marginal density of V₁, . . . , V_m is obtained by noting that

n

i=m+1

Γ_p α_i

K₂

α₁, . . . , α_n, γ, Ξ

=Γ_{p n}

i=m+1

α_i

K₂

α₁, . . . , α_m, n i=m+1

α_i, γ, Ξ

.

(3.16)

The derivation of the conditional density is now straightforward.

Corollary 3.4. If _(V1, . . . , V_n)∼KD^II_p(α1, . . . , α_n, γ, Ξ), then the density of V_i,ⁱ=1, . . . , n is given by

Γ_p



 ⁿ

j=1(=i)

α_j



K₂



α_i, n j=1(=i)

α_j, γ, Ξ



etr₋ΞV_i

×detV_{iαi−(p+1)/2}

detI_p+V_i−γ+ⁿ_j=1(=i)αj

×Ψ



 ⁿ

j=1(=i)

α_j, n j=1(=i)

α_j−γ+p+1 2 ;Ξ

I_p+V_i

, V_i> 0.

(3.17) Note that the marginal density of V_i differs from the Kummer-Gamma density. It is a pdf with an additional factor containing confluent hypergeo- metric functionΨ.

Theorem 3.5. Let_(U1, . . . , U_n)∼KD^I_p(α1, . . . , α_n, β, I_p)and define

W_i=

I_p−^m

i=1

U_{i −1/2}

U_i

I_p−^m

i=1

U_{i −1/2}

, i=m+1, . . . , n. (3.18) T hen the joint density of(W_m+1, . . . , W_n)is given by

Γ_pn

j=m+1α_j+β _n

i=m+1Γ_p α_i

Γ_p(β)

1F_{1 n}

i=1

α_i;ⁿ

i=1

α_i+β;−I_{p −1}

×etr

− n i=m+1

W_i

× ⁿ

i=m+1

detW_{iαi−(p+1)/2} det

I_p− ⁿ

i=m+1

W_i

_β−(p+1)/2

×₁F_{1 m}

i=1

α_i;ⁿ

j=1

α_j+β;−

I_p− ⁿ

i=m+1

W_i

,

0 < W_i< I_p, m i=1

W_i< I_p.

(3.19) Proof. Transforming W_i = (I_p−_m

i=1U_i)^−1/2U_i(I_p−_m

i=1U_i)^−1/2, ⁱ = m+1, . . . , n with Jacobian _J(Um+1, . . . , U_n →W_m+1, . . . , W_n) =det_(Ip−

_m

i=1U_i)(n−m)(p+1)/2,in the joint density of_(U1, . . . , U_n),we get K₁

α₁, . . . , α_n, β, I_p etr ₋

n i=m+1

W_i− _m

i=1

U_i

I_p− n i=m+1

W_i

× ^m

i=1

detU_{iαi−(p+1)/2} det

I_p−^m

i=1

U_i

ⁿ_{j=m+1αj+β−(p+1)/2}

× ⁿ

i=m+1

detW_{iαi−(p+1)/2} det

I_p− ⁿ

i=m+1

W_i

_β−(p+1)/2 ,

0 < U_i< I_p, i=m+1, . . . , n, n i=m+1

U_i< I_p,

0 < W_i< I_p, i=1, . . . , m, m i=1

W_i< I_p.

(3.20) Now,integratingU₁, . . . , U_m,

···

0<_m

i=1Ui<Ip

0<Ui<Ip

etr _{− m}

i=1

U_i

I_p− n i=m+1

W_i

×

m

i=1

detU_{iαi−(p+1)/2}

×det

I_p− m i=1

U_i

ⁿ_{i=m+1αi+β−(p+1)/2 m}

i=1

dU_i

= _m

i=1Γ_p α_i Γ_pm

i=1α_i

0<U<Ip

etr ₋

I_p− n i=m+1

W_i

U

×det_(U)^{mi=1αi−(p+1)/2}

×detI_p−Uⁿ_{i=m+1αi+β−(p+1)/2} dU

= _m

i=1Γ_p α_i

Γ_pn

i=m+1α_i+β Γ_pn

i=1α_i+β

×1F_{1 m}

i=1

α_i; n i=1

α_i+β;−

I_p−

n i=m+1

W_i

,

(3.21)

and using

K₁

α₁, . . . , α_n, β, I_pm

i=1Γ_p α_i

Γ_pn

i=m+1α_i+β Γ_pn

i=1α_i+β

= Γ_pn

i=m+1α_i+β _n

i=m+1Γ_p α_i

Γ_p(β)

1F_{1 n}

i=1

α_i;ⁿ

i=1

α_i+β;−I_{p −1}

,

(3.22)

we get the desired result.

Theorem 3.6. Let_(V1, . . . , V_n)∼KD^II_p(α1, . . . , α_n, γ, I_p)and define

Z_i=

I_p+^m

i=1

V_{i −1/2}

V_i

I_p+^m

i=1

V_{i −1/2}

, i=m+1, . . . , n. (3.23)

T hen the pdf of (Z_m+1, . . . , Z_n)is given by _n

i=m+1

Γ_p α_i

Ψ _n

i=1

α_i, n i=1

α_i−γ+p+1 2 ;I_p

₋₁

×etr

− n i=m+1

Z_{i n}

i=m+1

detZ_{iαi−(p+1)/2} det

I_p+

n i=m+1

Z_{i −γ}

×Ψ _m

i=1

α_i, n i=1

α_i−γ+p+1 2 ;

I_p+

n i=m+1

Z_i

, Z_i> 0.

(3.24) Proof. The proof is similar to the proof of Theorem 3.5.

Theorem 3.7. Let (U₁, . . . , U_n) ∼ KD^I_p(α₁, . . . , α_n, β, Λ) and define U = _n

i=1U_i and X_i=U^−1/2U_iU^−1/2,i=1, . . . , n−1. T hen (i)_(X1, . . . , X_n−1)andUare independent,

(ii)₍X₁, . . . , X_n−1)∼D^I_p(α₁, . . . , α_n−1;α_n),and (iii)U∼KB_p(_n

i=1α_i, β, Λ).

Proof. SubstitutingU_i=U^1/2X_iU^1/2,ⁱ=1, . . . , n−1and U_n=U^1/2(I_p− _n−1

i=1X_i)U^1/2 with the Jacobian_J(U1, . . . , U_n−1, U_n→X₁, . . . , X_n−1, U) =