Estimating Functionals Related to a Density by a Class of Statistics Based on Spacings

Abstract

We consider estimation of functionals of a probability density of the elements of a sample. We discuss a class of statistics based on spacings of increasing order and show that these statistics are almost surely consistent. Special attention is paid to entropy estimation. In particular we derive asymptotic normality in that case. It turns out that the entropy estimator is efficient under certain conditions on the unknown density.