Probability Density Estimation Using Delta Sequences

Abstract

Let $X_1, X_2, \cdots, X_n$ be i.i.d. random variables with common density function $f$. A method of density estimation based on "delta sequences" is studied and mean square rates established. This method generalizes certain others including kernel estimators, orthogonal series estimators, Fourier transform estimators, and the histogram. Rates are obtained for densities in Sobolev spaces and for densities satisfying Lipschitz conditions. The former generalizes some results of Wahba who also showed the rates obtained are the best possible. The rates obtained in the latter case have been shown to be the best possible by Farrell. This is shown independently by giving examples for which the rates are exact. Finally, a necessary and sufficient condition for asymptotic unbiasedness for continuous densities is given.