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.