Series Estimation of a Probability Density Function

Abstract

A class of nonparametric estimators of f(x) based on a set of n observations has been proved by Parzen [l] to be consistent and asymptotically normal subject to certain conditions. Although quite useful for a wide variety of practical problems, these estimators have two serious disadvantages when n is large: 1. All n observations must be stored in rapid-access storage.2. Evaluation of f(x) for a particular value of x requires a long computation involving each of the observations. The Parzen estimator, which has n terms, can be replaced by a series approximation which has a number of terms determined by the accuracy required in the estimate rather than by the number of observations in the sample. The summation over all of the observations is performed only to establish the value of the coefficients in the series. Although no member of the class of estimators has been proved “best” for estimating an unknown density from a finite sample, a power series expansion for a particular member of the class was singled out because of computational simplicity. A comparison is made between the proposed estimator and the Gram–Charlier Series of Type A. A multidimensional counterpart of the proposed estimator has also been derived.