Asymptotic Behaviors of Nearest Neighbor Kernel Density Estimator in Left-truncated Data

Kernel density estimators are the basic tools for density estimation in non-parametric statistics. The k-nearest neighbor kernel estimators represent a special form of kernel density estimators, in which the bandwidth is varied depending on the location of the sample points. In this paper‎, we initially introduce the k-nearest neighbor kernel density estimator in the random left-truncation model, ‎ and then prove some of its asymptotic behaviors, such as strong uniform consistency and asymptotic normality. ‎In particular‎, ‎we show that the proposed estimator has truncation-free variance‎. ‎Simulations are presented to illustrate the results and show how the estimator behaves for finite samples‎. Moreover, the proposed estimator is used to estimate the density function of a real data set.