Beta and Gamma Families in Kummer Distribution … · Beta and Gamma Families in Kummer Distribution involving ... distributions, ... reduce to Beta and Dirichlet distributions with

Beta and Gamma Families in Kummer Distribution involving

Hermitian Positive Definite Matrix

By

1. Introduction

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

density functions involving hermition positive definite matrix

{1F1(; + ; - )}-1 ex(-u)u-1(1-u)-1, 0 < u < 1,

(1.1)

{()(, - + ; )}-1 exp(-v)v-1(1 + v)-, v > 0, (1.2)

respectively, where > 0, > 0, > 0, - < , < , 1F1, and are confluent

hypergeometric functions. These distributions are extensions of Gamma and Beta

International Journal of Advancements in Research & Technology, Volume 2, Issue 12, December-2013 ISSN 2278-7763

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distributions, and for < 1 (and certain values of and ) yield bimodal distributions on

finite and infinite ranges, respectively. These distributions are used (i) in the3 Bayesian

analysis of queueing system where posterior distribution of certain basic parameters in M

/ M / queueing system is Kummer-Gamma and (ii) in common value auctions where

the posterior distribution of “value of a single good” is Kummer-Beta. For properties and

applications of these distributions the reader is referred to NG and Kotz [7], Armero and

Bayarri [1], and Gordy [2].

As the corresponding multivariate generalization of these distributions, we have the

following n-dimensional densities:

exp

, 0 < ui < 1, < 1, (1.3)

Where i > 0, i = 1, ....., n, > 0, - < < , and

exp (1.4)

, vi > 0,

Where i > o, i = 1, ........, n, > 0, - < < , respectively. These distributions have

been considered by Ng and Kotz [7] who refer to (1.3) and (1.4) as multivariate Kummer-

Beta and multivariate Kummer-Gamma distributions, respectively. For = 0, (1.1) and

(1.3) reduce to Beta and Dirichlet distributions with probability density functions

u - 1(1 - u) -1, 0 < u < 1,

, 0 < ui < 1, < 1, (1.5)

respectively. Since (1.3) is an extension of Dirichlet distribution and a multivariate

generalization of Kummer-Beta distribution, an ppropriate nomenclature for this

distribution would be Kummer-Dirichlet distribution. In the same vein, we may call (1.4)

a Kummer-Dirichlet distribution. Further, in order 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 section, we study certain properties of matrix variate Kummer-Dirichlet type I and

II invariant. That is, for any fixed orthogonal matrix (p p), the distribution of ( u1 ,

..., un ) is the same as the distribution of (u1, ..., un). Our next two results give

marginal and conditional distributions.distributions.


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Theorem 1. Let (u1, ..., un) ~ K(1,...,n,, Ip) and define

Wi = ui = i = m + 1, ..., n. (1.6)

Then the joint density of (Wm+1, ..., Wn) is given by

etr

det()det

1F1

0 < < Ip, < Ip. (1.7)

Proof. Transforming Wi = (Ip - )-1/2 (Ip - )-1/2, i = m + 1,..., n with jacobian j (um+1, ..., un Wm +

1,..., Wn) = det (Ip -)(n - m) (p + 1)1/2, in the joint density of (u1, ..., un) we get

K1(1,...,n,, Ip)

det()det

det()det

0 < ui < Ip, i = m + 1, ..., n, < Ip, (1.8)

Now, integrating u1, ..., um,

etr

det()

det

=etr

det()

=

1F1 (1.9)

and using

K1(1,...,n,, Ip)

= (1.10)

we get the desired result.

Theorem 2. Let (V1, ..., Vn) - ~ K(1,...,n,, Ip) and define

Zi = ui = i = m + 1, ..., n. (1.11)

Then the pdf of (Zm+1,..., Zn) is given by


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etr det() det

Zi > 0 (1.12)

Proof. The proof is similar to the proof of theorem 1.

Theorem 3. Let (u1,..., un) K(1,...,n,, ) and define u = and Xi = u-1/2 ui u-1/2, i = 1, ..., n

– 1. Then

(i) (X1, ..., Xn-1) and u are independent,

(ii) (X1, ..., Xn-1) ~ (1,...,n-1; n) and

(iii) u ~ KBp (i, , ).

Proof. substituting ui = u1/2 Xi u1/2, i = 1, ..., n – 1 and un = u1/2 (Ip - Xi) 1/2 with the

Jacobian J (u1,..., un-1, u1 X1, ..., Xn-1, u) = det ()(n-1)p in the joint density of (u1,..., un), we

get the desired result.

Theorem 4. Let ((V1,..., Vn) K(1,...,n, , ) and define V = and Yi = V-1/2 Vi V-1/2, i =

1, ..., n – 1. Then

(i) (Y1, ..., Yn-1) and u are independent,

(ii) (Y1, ..., Yn-1) ~ (1,...,n-1; n) and

(iii) V ~ KGp (i, , ).

Proof. The proof is similar to the proof of Theorem 3.

In theorems 5 and 6, we derive the joint pdfs of partial sums of random matrices

distributed as matrix variate Kummer-dirichlet type I or II.

Theorem 5. Let (u1, ..., un) ~ K(1,...,n, , ) and define

u(i) = , (i) = , (i), = 0, nj, i = 1,...,.

Then (u(i),...,) ~ K((1),..., , , )

Proof. Make the transformation

u(i) = , Wj = uj, (1.13)

j = + 1, ..., - 1, i = 1, ..., . The Jacobian of this transformation is given by

J (u1, ..., un W1, ..., Wn1 – 1, u(1), ..., ,...Wn-1, )

= (, ..., , ..., ,...U(i))

= () (1.14)


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Now, substituting from (1.13) and (1.14) in the joint density of (u1, ..., un), we get the

joint density of ,..., U(i),

where i = 1, ..., , as

K1(1,...,n,, ) etr

()det

(1.15)

det ,

where 0 < u(i) < Ip, < Ip, 0 < Wj < Ip, j = +1, ..., - 1, < Ip, i =1, ..., . From (1.15), it is easly

to see that (u(1),..., ) and (,...W), i = 1, ...,, are independently distributed. Further, (u(1),..., )

~ K((1),..., , , ) and ,..., W) ~ (,...,), where i = 1, ...., .

When = 1 ~ KBp(i, , ).

Theorem 6. Let (V1, ..., Vn) ~ K(1,..., n,, , ) and define

V(i) = (i) = j = = nj, i = 1, ..., .

Then (V(1),..., ) ~ K((1),..., , , ). (1.16)

Proof. Make the transformation

V(i) = , Zj = Vj, (1.17)

where j = ,..., -1, i = 1, ... . The Jacobian of this transformation is given by

J(V1, ..., Vn Z1, ...., Zn1-1, V(1),..., ) , Zn-1, )

(,..., , ..., , V(i)) (1.18)

=

Now, substituting from (1.17) and (1.18) in the joint density of (V1, ..., Vn) , it can easily

be shown that (V(1), ..., ) and (, ..., ), i = 1, ... , are independently distributed. Further,

(V(1),..., ) ~ K((1),..., , , ) and (, ..., ) ~ (,...,;), i = 1, ..., .

When = 1, the distribution of is kummer-Gamma with parametersi, and . From the

joint density of u1, ..., un, we have

E

K1(1,...,n,, ) etr

det()det

= , Re (i + ri) > (p-1)


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=

Re (i + ri) > (p-1) (1.19)

E

K1(1,...,n,, ) etr

det()det

= , Re (h) > - (p-1)

=

Re (h) > - (p-1) (1.20)

Alternately, the above moment expression can be obtained by noting that ui has Kummer-

Beta distribution. Similarly, for Kummer-Dirichlet type II matrices.

E

=

Re(i + ri) > (p-1)

E (1.21)

Nex two results give certain asymptotic distributions for the Kummer-Dirichlet type I and

type II distributions (See Javier and Gupta [4]).

Theorem 7. Let (u1, ..., un) ~ K(1,..., n,, , ) and W = (W1,..., Wn) be defined by Wi

= ui, i = 1, ..., n. Then W is asymptotically distributed as a product of independent

matrix variate gamma densities;

f() = (1.22)

Where f(W) denotes the density of the matrix W.

Proof. In the joint density of (u1, ..., un) , transform Wi = ui, i = 1, ..., n with the Jacobian

J (u1, ..., un W1, ..., Wn), = -np(p+1)/2. The density of W = (W1, ..., Wn) is given by

f(W) =

etr

(1.23)


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The result follows,. Since

1F1

= 1F0 = det (Ip + ) (1.24)

det = etr .

An analogous result for Kummer-Dirchlet type II distribution is shown to be the

following.

Thoerem 8. Let (v1, ..., vN) ~ K (1,..., n,, ,| | ), 0, and W = (W1, ..., Wn) be

defined by Wi = || Vi, i = 1, ...., n. Then, W is asymptotically distributed as a product of

independent matrix variate gamma densities; more specifically

g() =

Where f(W) denotes the density of the matrix W.

Proof. We prove the result for > 0. The poof for < 0 follows similar steps.The density

of W=W1

g() =

etr (1.25)

det

The results follows,since

= det (Ip + ),

det = etr . (1.26)

References

[1] C. Armero and M. J. Bayarri, A Bayesian analysis of a queneing system with unlimited

service, J. Statist. Plann. Inference 58 (1997), no. 2, 241-261. CMP 1 450015. Zbl

880.62025.

[2] M. Gordy, Computationally convenient distributional assumptions for common-value

auctions, Comput. Econom. 12 (1998), 61-78. Zbl 912.90093.

[3] A. K. Gupta and D. K. Nagar, Matrix Variate Distributions, Chapman & Hall/ CRC

Monographs and Surveys in Pure and Applied Mathematics, vol. 104, Chapaman &

Hall/CRC, Florida, 2000. MR 2001d:62055. Zbl 935.62064.


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[4] W.R. Javier and A.K. Gupta, On generalized matric variate beta disgtributions, Statistics

16 (1985), no. 4, 549-557, MR 86m:62023. Zbl 602.62039.

[5] D.K. Nagar and L. Cardeno, Matrix variate Kummer – Gamma distribution, Random

Oper. Stochastic Equations 9 (2001), no. 3, 207-218.

[6] D.K. Nagar and A.K. Gupta, Matrix variate Kummer-Beta distribution, to appear in J.

Austral Math. Sec.

[7] K. W. Ng and S. Kotz, Kumer-Gamma and Kummer-Beta univariate and

multivariate distributions, Research report, Department of Statics, The University of

Hong Kong, Hong Kong, 1995.

[8] Arjun K. Gupta: Department of Mathematics and Statistics, Bowling Green State

University, Bowling Green, OH 43403-0221, USA.

[9] Liliam cardeno and Daya K. Nagar: Departmento de Mathematicas, Universidad de

Antioquia, Medellin, A. A. 1226, Colombia


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Beta and Gamma Families in Kummer Distribution … · Beta and Gamma Families in Kummer Distribution involving ... distributions, ... reduce to Beta and Dirichlet distributions with