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Theory & Methods: A Construction of Lancaster Probabilities with Margins in the Multidimensional Meixner Class
Author(s) -
Koudou A.E.,
Pommeret D.
Publication year - 2000
Publication title -
australian and new zealand journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 1369-1473
DOI - 10.1111/1467-842x.00107
Subject(s) - mathematics , orthogonality , orthogonal polynomials , class (philosophy) , poisson distribution , exponential family , gaussian , combinatorics , discrete mathematics , pure mathematics , statistics , physics , geometry , quantum mechanics , artificial intelligence , computer science
The well‐known Meixner class (Meixner, 1934) of probabilities on R has been extended recently to R d (Pommeret, 1996). This generalized Meixner class corresponds to the simple quadratic natural exponential families characterized by Casalis (1996). Following Lancaster (1975), the present paper offers a characterization of the joint probability of a randomvector ( X, Y ) such that the two variables X and Y on R d belong to the multidimensional Meixner class and fulfil a bi‐orthogonality condition involving orthogonal polynomials. The joint probabilities, called Lancaster probabilities, are characterized by two sequences of orthogonal polynomials with respect to the margins and a sequence of expectations of products. Some multivariate probabilities are studied, namely the Poisson‐Gaussian and the gamma‐Gaussian.