Weighted Probability Density Estimator with Updated Bandwidths

In this paper, the authors study a class of weighted version of probability density estimator. It is shown that the weighted estimator contains some existing estimators of probability density, no matter they are recursive or non-recursive. Some statistical results including weak consistency, strong consistency, rate of strong consistency, and asymptotic normality are established under some mild conditions. Moreover, the random weighted estimator is also investigated. Some numerical simulations and a real data analysis are presented to study the numerical performances of the estimators.