Rates of convergence for an estimator of a density function based on jacobi polynomials
Julie A. Letellier · Communication in Statistics- Theory and Methods · 1997
The Empirical Bayes (EB) approach of Walter and Hamedani (1987) in estimating the prior g in a binomial mixture employs Legendre polynomials on (0,1). However, their estimator for the prior g can be negative, usually for values of the binomial probability close to 0 or 1. This motivates the study of the more general problem of obtaining a nonnegative estimator for a given density function f based on orthogonal polynomials. Certain summability methods lead to positive delta sequences which give rise to nonnegative polynomial estimators for f. These methods are used to construct a nonnegative estimator for f based on orthogonal polynomials which incorporate certain positive delta sequences. Several results on the convergence of the resulting estimator are given along with corresponding rates of convergence.