Central limit theorem for the variable bandwidth kernel density estimators

In this paper we study the ideal variable bandwidth kernel density estimator introduced by McKay (1993a, b) and Jones et al. (1994) and the plug-in practical version of the variable bandwidth kernel estimator with two sequences of bandwidths as in Giné and Sang (2013). Based on the bias and variance analysis of the ideal and plug-in variable bandwidth kernel density estimators, we study the central limit theorems for each of them. The simulation study confirms the central limit theorem and demonstrates the advantage of the plug-in variable bandwidth kernel method over the classical kernel method.