Recursive Kernel Density Estimators Under Censoring Verifying an <i>β</i>-mixing Dependence Structure

Abstract

In this paper, we consider the nonparametric recursive kernel density estimator on a compact ensemble when observations are censored and <i>β</i>-mixing. In this type of model, it is widely recognized that the traditional empirical distribution does not allow the densities <I>F</I> and <I>G</I> to be efficiently evaluated. Thus, Kaplan and Meier suggested a consistent estimator of Gnto properly estimate <I>G</I>. Let {<i>T<sub>k</sub></i>, <i>k</i> ≥ 1} be a strictly stationary sequence of random variables distributed as <I>T</I>. We aims to establish a strong uniform consistency on a compact set with a rate of recursive kernel estimator of the underlying density function <i>f</i> when the random variable of interest <I>T</I> is right censored by another <I>C</I> variable. In censoring, the observation is only partially known, which means that there are only the n pairs (<i>Y<sub>i</sub></i>, <i>δ<sub>i</sub></i>), <i>Y<sub>i</sub></i>= min(<i>T<sub>i</sub></i>, <i>C<sub>i</sub></i>) and <i>δ<sub>i</sub></i>= II<sub>{<i>T<sub>i</sub>≤C<sub>i</sub></i>}</sub>, where II<i><SUB>A</SUB></i>, where the indicator function for event <I>A</I>. Firstly, we propose the uniform convergence of this recursive estimator towards the density <i>f</i>. Then, we showed the veracity of our results by establishing all the necessary proofs. In other words we will prove our main result by establishing three lemmas. And finally we validated our theoretical results with a simulation study.