Kernel Density Estimation:the General Case

Given a sequence of independent identically distributed random variablesX1, X2, ..., Xn, ... with common probability density function p(x), how can we estimate? Parzen discusses the problem of estimation of a probability density function and the problem of determining the mode of a probability density function and show how one may construct a family of estimates of p(x), and of the mode, which have weak consistent and asymptotic normality. Ryzin states conditions under which strong consistency of estimates obtained. Wang and VanRzin presents a class of smooth weight function estimators for discrete distribution. The resulting estimators are strongly consistent and asymptotically normal under mild regularity conditions. Campos and Dorea present a general class of kernel estimates for p(x) and prove the stong consistency and asymptotic normality, while there are several wrongs in its process.In our paper, we propose a general class of kernel estimates for p(x). And it is shown that our results on strong consistency and asymptotic normality include the classical results for continuous densities and extend some results of kernel estimators for discrete distributions. At last, we point the wrongs in Dorea's paper and present corresponding correct results.