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  • MATRIX VARIATE KUMMER-DIRICHLET DISTRIBUTIONS

    ARJUN K. GUPTA, LILIAM CARDEÑO, 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)uα−1(1−u)β−1, 0 < u < 1,

    (1.1){ Γ(α)Ψ(α,α−γ+1;ξ)

    }−1 exp(−ξv)vα−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 of λ 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 in M/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

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

  • 118 Matrix variate Kummer-Dirichlet distributions

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

    Γ (∑n

    i=1 αi +β )∏n

    i=1 Γ ( αi

    ) Γ(β)

    { 1F1

    ( n∑

    i=1

    αi;

    n∑ i=1

    αi +β;−λ

    )}−1 exp

    ( −λ

    n∑ i=1

    ui

    )

    × n∏

    i=1

    uαi−1i

    ( 1−

    n∑ i=1

    ui

    )β−1 , 0 < ui < 1,

    n∑ i=1

    ui < 1,

    (1.3)

    where αi > 0, i = 1, . . . ,n, β > 0, −∞ < λ < ∞, and{ Γ

    ( n∑

    i=1

    αi

    ) Ψ

    ( n∑

    i=1

    αi,

    n∑ i=1

    αi −γ+1;ξ

    )}−1 exp

    ( −ξ

    n∑ i=1

    vi

    )

    × n∏

    i=1

    vαi−1i

    ( 1+

    n∑ i=1

    vi

    )−γ , vi > 0,

    (1.4)

    where αi > 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. For λ = 0, (1.1) and (1.3) reduce to Beta and Dirichlet distributions with probability density functions

    Γ(α+β)

    Γ(α)Γ(β) uα−1(1−u)β−1, 0 < u < 1,

    Γ (∑n

    i=1 αi +β )

    ∏n i=1 Γ

    ( αi

    ) Γ(β)

    n∏ i=1

    uαi−1i

    ( 1−

    n∑ i=1

    ui

    )β−1 , 0 < ui < 1,

    n∑ i=1

    ui < 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 be Kummer-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]). Let A = (aij) be a p×p matrix.

  • Arjun K. Gupta et al. 119

    Then, A ′ denotes the transpose of A; tr(A) = a11 + · · · + app; etr(A) = exp(tr(A)); det(A) = determinant 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 Γp(m) is defined as

    Γp(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

    Γp(m) =

    ∫ R>0

    det(R)m−(p+1)/2 etr(−R)dR, Re(m) > p−1

    2 , (2.2)

    where the integral has been evaluated over the space of the p×p symme- tric positive definite matrices. The integral representation of the confluent hypergeometric function 1F1 is given by

    1F1(a;b;X) = Γp(b)

    Γp(a)Γp(b−a)

    × ∫ 0 (p−1)/2. The confluent hypergeo- metric function Ψ of a p×p symmetric matrix X is defined by

    Ψ(a,c;X) = 1

    Γp(a)

    × ∫ R>0

    etr(−XR)det(R)a−(p+1)/2 det ( Ip+R

    )c−a−(p+1)/2 dR,

    (2.4)

    where Re(X) > 0 and 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×p symmetric positive definite random matrices U1, . . . , Un are said to have the matrix variate Kummer-Dirichlet type I distri- bution with parameters α1, . . . ,αn, β and Λ, denoted by (U1, . . . ,Un) ∼ KDIp(α1, . . . ,αn,β,Λ), if their joint probability density function (pdf) is given by

  • 120 Matrix variate Kummer-Dirichlet distributions

    K1 ( α1, . . . ,αn,β,Λ

    ) etr

    ( −Λ

    n∑ i=1

    Ui

    )

    × n∏

    i=1

    det ( Ui

    )αi−(p+1)/2 det (

    Ip −

    n∑ i=1

    Ui

    )β−(p+1)/2 ,

    0 < Ui < Ip, 0 <

    n∑ i=1

    Ui < Ip,

    (2.5)

    where αi > (p−1)/2, i = 1, . . . ,n, β > (p−1)/2, Λ(p×p) is symmetric and K1(α1, . . . ,αn,β,Λ) is the normalizing constant.

    Definition 2.2. The p×p symmetric positive definite random matrices V1, . . . , Vn are said to have the matrix variate Kummer-Dirichlet type II distribution with parameters α1, . . . ,αn, γ and Ξ, denoted by (V1, . . . ,Vn) ∼ KDIIp (α1, . . . , αn,γ,Ξ), if their joint pdf is given by

    K2 ( α1, . . . ,αn,γ,Ξ

    ) etr

    ( −Ξ

    n∑ i=1

    Vi

    )

    × n∏

    i=1

    det ( Vi

    )αi−(p+1)/2 det (

    Ip +

    n∑ i=1

    Vi

    )−γ , Vi > 0,

    (2.6)

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

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

    { K1

    ( α1, . . . ,αn,β,Λ

    )}−1 =

    ∫ · · ·

    ∫ 0<

    ∑n i=1 Ui0

    etr

    ( −Λ

    n∑ i=1

    Ui

    )

    × n∏

    i=1

    det ( Ui

    )αi−(p+1)/2 det (

    Ip −

    n∑ i=1

    Ui

    )β−(p+1)/2 n∏ i=1

    dUi

    =

    ∏n i=1 Γp

    ( αi

    ) Γp

    (∑n i=1 αi

    ) ∫ 0

  • Arjun K. Gupta et al. 121

    =

    ∏n i=1 Γp

    ( αi

    ) Γp(β)

    Γp (∑n

    i=1 αi +β ) 1F1

    ( n∑

    i=1

    αi;

    n∑ i=1

    αi +β;−Λ

    ) ,

    (2.7)

    { K2

    ( α1, . . . ,αn,γ,Ξ

    )}−1 =

    ∫ V1>0

    · · · ∫ Vn>0

    etr

    ( −Ξ

    n∑ i=1

    Vi

    )

    × n∏

    i=1

    det ( Vi

    )αi−(p+1)/2 det (

    Ip +

    n∑ i=1

    Vi

    )−γ n∏ i=1

    dVi

    =

    ∏n i=1 Γp

    ( αi

    ) Γp

    (∑n i=1 αi

    )∫ V>0

    etr(−ΞV)det(V) ∑n

    i=1 αi−(p+1)/2 det ( Ip +V

    )−γ dV

    =

    n∏ i=1

    Γp ( αi

    ) Ψ

    ( n∑

    i=1

    αi,

    n∑ i=1

    αi −γ+ p+1

    2 ;Ξ

    ) ,

    (2.8)

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

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

    Γp (∑n

    i=1 αi +β )∏n

    i=1 Γp ( αi

    ) Γp(β)

    n∏ i=1

    det ( Ui

    )αi−(p+1)/2 det (

    Ip −

    n∑ i=1

    Ui

    )β−(p+1)/2 ,

    0 < Ui < Ip, 0 <

    n∑ i=1

    Ui < Ip,

    (2.9)

    where αi > (p − 1)/2, i = 1, . . . ,n, and β > (p − 1)/2. A common nota- tion to designate that (U1, . . . ,Un) has this density is (U1, . . . ,Un) ∼ DIp(α1, . . . ,αn;β). For γ = 0, the matrix variate Kummer-Dirichlet type II density simplifies to the product of n matrix variate Gamma densities.

    For p = 1, the densities in (2.5) and (2.6) simplify to Kummer-Dirichlet type I (multivariate Kummer-Beta) and Kummer-Dirichlet type