Kokonendji, Célestin C.; Zarai, Mohammed Transorthogonal polynomials and simple cubic multivariate distributions. (English) Zbl 1130.62052 Far East J. Theor. Stat. 21, No. 2, 171-201 (2007). Summary: 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. Cited in 1 Document MSC: 62H05 Characterization and structure theory for multivariate probability distributions; copulas 60G50 Sums of independent random variables; random walks 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 62E10 Characterization and structure theory of statistical distributions Keywords:Bhattacharyya matrix; exponential generating function; Lévy process; natural exponential family; normal inverse Gaussian distribution; Sheffer polynomial; variance function; 2-pseudo-orthogonality PDFBibTeX XMLCite \textit{C. C. Kokonendji} and \textit{M. Zarai}, Far East J. Theor. Stat. 21, No. 2, 171--201 (2007; Zbl 1130.62052)