Far East Journal of Theoretical Statistics
Volume 21, Issue 2, Pages 171 - 201
(March 2007)
|
|
TRANSORTHOGONAL POLYNOMIALS AND SIMPLE CUBIC MULTIVARIATE
DISTRIBUTIONS
Célestin C. Kokonendji (France) and Mohamed Zarai (Tunisie)
|
Abstract: Transorthogonality for a sequence of polynomials on
has been recently introduced in order to characterize the reference probability
measures, which are multivariate distributions of the natural exponential
families (NEFs) having a simple cubic variance function. The present paper
pursues this characterization of three various manners through exponential
generating functions, transdiagonality of Bhattacharyya matrices and
semigroup-Sheffer systems, respectively. The obtained results extend those
well-known of simple quadratic NEFs based on the classical orthogonality of
associated polynomials. The transorthogonality property is then compared to the
2-pseudo-orthogonality one which globaly characterizes the cubic NEFs. Finally,
some techniques of calculation of these polynomials are presented and then
illustrated on a multivariate normal inverse Gaussian Lévy process. |
Keywords and phrases: Bhattacharyya matrix, exponential generating function, Lévy
process, natural exponential family, normal inverse Gaussian distribution,
Sheffer polynomial, variance function, 2-pseudo-orthogonality. |
|
Number of Downloads: 264 | Number of Views: 1070 |
|