Transorthogonal polynomials and simple cubic multivariate distributions

Célestin Kokonendji (1) , Mohamed Zarai

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

Transorthogonality for a sequence of polynomials on $\mathbb R^d$ 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.

Mots clés: Bhattacharyya matrix; exponential generating function; Lévy process; natural exponential family; normal inverse Gaussian distribution; Sheffer polynomial; variance function; 2-pseudo-orthogonality.; 2-pseudo-orthogonality

Domaines: Statistiques [math.ST]

Célestin Kokonendji, Mohamed Zarai. Transorthogonal polynomials and simple cubic multivariate distributions. 2006. ⟨hal-00222782⟩