Sankhya: The Indian Journal of Statistics

2004, Volume 66, Pt. 1, 1--19

A Characterization of Lancaster Probabilities with Margins in a Multivariate Additive Class


D. Pommeret, Crest-Ensai, France

SUMMARY. Let $X$ and $Y$ be two random variables on $\R^d$ and let $(P_n)_{n\in\N^d}$ and $(Q_k)_{k\in\N^d}$ be two  basis of orthonormal polynomials with respect to the distributions of $X$ and $Y$, respectively. The joint distribution of $(X,Y)$ is called a {\it Lancaster  probability} if  the expectation $\E(P_n(X)Q_k(Y))$ vanishes for $ n \neq k$. This paper concerns the characterization of Lancaster probabilities for the particular case $X=U+V$ and $Y=V+W$, where $U,V,W$ are independent random variables on $\R^d$. It is shown that the distribution of $(X, Y)$ is a Lancaster probability if and only if  the natural exponential families generated by the random variables $U, V, W$ are simple quadratic (the variance is a specific quadratic function of the mean). This result is an extension of  Lancaster (1975). We also generalize the  definition of Lancaster probabilities that we relate to the more general class of quadratic natural exponential families on $\R^d$. Finally, two  tests for independence and a goodness of fit test for these multivariate joint distributions are outlined.

AMS (1991) subject classification}. 62H05, 62E10.

Key words and phrases. Lancaster probabilities, orthogonal polynomials, quadratic and simple quadratic natural exponential families.

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