Some asymptotic properties for a smooth kernel estimator of the conditional mode under random censorship

Let (Ti)1≤i≤n be a sample of independent and identically distributed (iid) random variables (rv) of interest and (Xi)1≤i≤n be a corresponding sample of covariates. In censorship models the rv T is subject to random censoring by another rv C. Let θ(x) be the conditional mode function of the density of T given X=x. In this work we define a new smooth kernel estimator θˆn(x) of θ(x) and establish its almost sure convergence and asymptotic normality. An application to prediction and confidence bands is also given. Simulations are drawn to lend further support to our theoretical results for finite sample sizes.