Approximation of density functions by orthogonal series with grouped data

Let g(x) and f(x) be continuous density function on (a, b) and let {φ{symbol}j} be a complete orthonormal sequence of functions on L2(g), which is the set of squared integrable functions weighted by g on (a, b). Suppose that {Mathematical expression} over (a, b). Given a grouped sample of size n from f(x), the paper investigates the asymptotic properties of the restricted maximum likelihood estimator of density, obtained by setting all but the first m of the θ{symbol}j's equal to 0. Practical suggestions are given for performing estimation via the use of Fourier and Legendre polynomial series. © 1994 Società Italiana di Statistica.