Estimation of Probability Density

Abstract

Assuming that the distribution being sampled is absolutely continous, Parzen [3] has established the consistency and asymptotic normality of a class of estimators $\{f_n(x)\}$ based on a random sample of size $n$, for estimating the probability density. In this paper, we relax the assumption of absolute continuity of the distribution $F(x)$ and show that the class of estimators $\{f_n(x)\}$ still consistently estimate the density at all points of continuity of the distribution $F(x)$ where the density $f(x)$ is also continuous. It is further shown that the sequence of estimators $\{f_n(x)\}$ are asymptotically normally distributed. The extension of these results to the bi-variate and essentially the multi-variate case with applications and a discussion on the construction of higher dimensional windows will be presented at the International Symposium in Multivariate Analysis to be held in Dayton, Ohio during June 1965.