Asymptotic properties of a kernel etimate of the conditional mode in the presence of doubly censored data

We introduce a double kernel conditional mode estimator when the response variable is doubly censored. In this situation, the variable of interest may be right censored by a variable R or left censored by a variable L and L is not greater than R almost surely. Self- consistent estimators of R and L are extensively studied in literature. As they fit in the expression of the conditional density estimator, using their known properties, among other tools, we establish the uniform almost sure convergence of the conditional density estimator and its derivatives. Then, we deduce the consistency and the asymptotic normality of our conditional mode estimator. For that, we use the theory of Vapnik-Cˇ ervonenkis classes of measurable functions. We also show the quality of this estimation and the Gaussian behavior of our proposed estimator through a simulation study in which, we consider various regression models with different sample sizes and censoring rates. Finally, we illustrate the performance of our estimator on a real data set which concerns the Stanford heart transplant data.