Nonparametric relative error estimation of the regression function for censored data

TL;DR

This paper introduces a new kernel estimator for the regression function under censored data, focusing on relative error, and provides theoretical convergence results, asymptotic normality, and confidence bands supported by simulations.

Contribution

The paper develops a novel kernel estimator for the mean squared relative error in censored regression models, with proven convergence, asymptotic normality, and explicit variance formulas.

Findings

Estimator converges uniformly almost surely with a specified rate.

Asymptotic normality of the estimator is established.

Confidence bands are derived from the asymptotic variance.

Abstract

Let $ (T_i)_i$ be a sequence of independent identically distributed (i.i.d.) random variables (r.v.) of interest distributed as $ T$ and $(X_i)_i$ be a corresponding vector of covariates taking values on $ \mathbb{R}^d$. In censorship models the r.v. $T$ is subject to random censoring by another r.v. $C$. In this paper we built a new kernel estimator based on the so-called synthetic data of the mean squared relative error for the regression function. We establish the uniform almost sure convergence with rate over a compact set and its asymptotic normality. The asymptotic variance is explicitly given and as product we give a confidence bands. A simulation study has been conducted to comfort our theoretical results