Orthogonal polynomials generated by random vectors

Abstract

Every random q-vector with finite moments generates a set of orthonormal polynomials.These are generated from the basis functions x n = x n 1 1 • • • x n q q using Gram-Schmidt orthogonalization. One can cycle through these basis functions using any number of ways. Here, we give results using minimum cycling. The polynomials look simpler when centered about the mean of X, and still simpler form when X is symmetric about zero. This leads to an extension of the multivariate Hermite polynomial for a general random vector symmetric about zero. As an example, the results are applied to the multivariate normal distribution.

Some uses of orthogonal polynomials in statistical inference

“…More recent results include among others: applications in correspondence analysis Beh, 9 D'Ambra et al, 10 and Beh and Lombard, 11 birth and death processes Guillemin and Pinchon, 12 goodness‐of‐fit tests for parametric regression models Bar‐Hen and Daudin, 13 inference in the exponential distribution based on k ‐sample doubly type‐II censored data Sanjel and Balakrishnan, 14 Gibbs sampling Diaconis et al, 15 multiple correspondence analysis for ordinal‐scale variables Lombardo and Beh, 16,17 reweighted smooth tests of goodness of fit De Boeck et al, 18 study of dependence between ordinal‐nominal categorical variables in Lombardo et al, 19 canonical correlations for Dirichlet measures in Griffiths and Spano, 20 to adjust the hyperbolic secant and logistic distributions to analyze financial asset returns in Bagnato et al, 21 a general method of calculus in Withers and Nadarajah 22 …”