Estimation of Probability Density

Assuming that the distribution being sampled is absolutely continous, Parzen 3 has established the consistency and asymptotic normality of a class of estimators \fₙ (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 \fₙ (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 \fₙ (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.