Almost sure convergence with rate of a robust regression estimator for twice-censored data

Abstract

In this paper, we propose a new kernel M-estimator of the regression function using nonparametric methods for twice-censored data. The almost sure (a.s.) convergence with rate over a compact set of the estimator is established under appropriate conditions using an unbounded score function. As a by-product, we build a Nadaraya-Watson-type estimator in case of the twice-censoring and derive its strong uniform consistency with rate. Furthermore, knowing that the choice of the smoothing parameter is a difficult and important question, and based on cross-validation ideas, we construct some data-driven criterion for choosing a reasonable bandwidth. In the presence of censoring and/or outliers, a simulation study is carried out to illustrate the finite-sample behavior of the proposed estimator and to evaluate the effectiveness of the proposed cross-validation criterion through comparisons with the classical cross-validation criterion. Numerical comparisons with classical estimators are also presented. Finally, a real-data application is provided to demonstrate the practical value of the proposed estimator.