Pointwise consistency of the hermite series density estimate

Abstract

The Hermite series estimate of a density f [epsilon] Lp, p> 1, convergessin the mean square to f (x) for almost all x [epsilon] R, if N (n) --> [infinity] and N (n) / n2 --> ) as n --> [infinity], where N is the number of the Hermite functions in the estimate while n is the number of observations. Moreover, the mean square and weak consistency are equivalent. For m times differentiable densities, the mean squares convergence rate is O(n-(2m-1)/2m). Results for complete convergence are also given.density estimate nonparametric orthogonal series Hermite series