1. Introduction
Thomas and George [
1] introduced a generalized type-1 Dirichlet model having several mathematical and statistical properties. The density has been derived from the property of the following ratios:
Let
be such that
and let
be independently distributed as type-1 beta with parameters
respectively. Then
has the density function of the following form:
Obviously it is a generalization of type-1 Dirichlet probability model. The normalizing constant
can be evaluated as
for
where
ℜ denotes the real part of
. For more properties of the Model (1) one may refer Thomas and George [
1]. Note that
is structurally a product of
k independent real variables and its density can be written in terms of a
G-function of the type
The majority of the established special functions can be represented in terms of the
G-function. A notable property of
G-functions is the closure property. The closure property implies that whenever a function is expressible as a
G-function of a constant multiple of some constant power of the function argument, the derivative and the antiderivative of this function are expressible so too. A general
G-function is defined as the following Mellin-Barnes integral:
where
and
is a suitable contour.
The existence of different types of contours, properties and applications of
G-functions are available in Mathai and Haubold [
2].
2. Integral Representations
All the random variables considered above take values in
and hence the density functions can be uniquely determined by their moments. For arbitary
t, we have
where
Note that the moments of the product of independent random variables are the products of the respective moments. Treating Equation (3) as a Mellin transform of the density of
, the density is available by the inverse Mellin transform. Thus, the density of
is the following:
for
and zero elsewhere.
Proof.
The result follows by equating Equation (4) with the marginal density of obtained by integrating out from the joint density of given in Equation (1). ☐
Let
be an ordered set of points in the Euclidean
n-space
,
. Let
O denotes the origin of a rectangular coordinate system. Now the
vector
can be considered as a point in
. If
are linearly independent then the convex hull generated by these
k-points almost surely determine a
k-parallelotope in
with the sides
The random volume or k-content
of this random parallelotope is given by
where
is a matrix of order
is the transpose of
X and
denotes the determinant of
. The classical approach to random points and random volumes consists in looking at independently distributed isotropic random points and dealing with random geometric configurations with the help of techniques from differential and integral geometry. Mathai [
3] looked into random volumes under a more general structure by deleting the assumptions of independence and isotropy. Mathai [
3] has shown that if the
, real random matrix
X of full rank
k has the density:
for
, then the probability distribution of
has the following structure:
Thus, it is possible to express the density of
as a marginal density of
obtained from the joint density given in Equation (1) with specific set of parameters. The notion
has application in the study of variance of multivariate distributions. More details on random volumes may be seen from Mathai [
3]. Thomas and Mathai [
4] expressed the density of
as a marginal density of
in the Model (1) with parameters as
Now let us consider the Gaussian or ordinary hypergeometric function
which is a special function represented by the hypergeometric series:
where
when
is defined.
Proof.
Let us consider the model (1) for the case when
and take the parameters as
and
. Now the density of
as a marginal density of
, can be obtained as the following:
for
and zero elsewhere, where
is the beta function. ☐
Alternatively, we can obtain the density of
by using Meijer’s
G-function given in Equation (4). Then the density function obtained has the form:
and zero elsewhere.
Since the density function is unique, Equations (5) and (6) must be equal. Hence the result follows.
Since Equation (5) is a probability density function we obtain the following relation:
Many multivariate procedures based on random samples from multivariate normal populations can be interpreted as the study of the distribution of
. The exact distribution of likelihood ratio criteria for testing hypothesis in MANOVA, MANCOVA, multivariate regression analysis etc can be obtained as a special case of distribution of
. Thomas and Thannippara [
5,
6] expressed the density of the above mentioned likelihood ratio criteria in terms of the marginal distribution of the generalized type-1 Dirichlet model given in Equation (1) with specific set of parameter values. The density of the likelihood ratio criterion
for
is obtained to be the following:
for
and zero elsewhere. Since Equation (7) is a probability density function we obtain the following relation:
Note that the evaluation of a G-function involves evaluation of residues at poles of different orders. Hence in such cases we may end up with psi, gamma or zeta functions. The above results are useful in evaluating the definite integrals involving G-functions of the type .