Uniform limit laws of the logarithm for nonparametric estimators of the regression function in presence of censored data

TL;DR

This paper derives uniform limit laws of the logarithm for nonparametric censored data estimators, enabling the construction of almost sure asymptotic confidence bands for regression and related functions.

Contribution

It establishes new uniform-in-bandwidth limit laws of the logarithm for I.P.C.W. estimators under censorship, extending to confidence band construction.

Findings

Uniform limit laws of the logarithm are proven for multivariate regression estimators.

Almost sure asymptotic confidence bands are constructed from these laws.

Simulation examples demonstrate the practical applicability of the confidence bands.

Abstract

In this paper, we establish uniform-in-bandwidth limit laws of the logarithm for nonparametric Inverse Probability of Censoring Weighted (I.P.C.W.) estimators of the multivariate regression function under random censorship. A similar result is deduced for estimators of the conditional distribution function. The uniform-in-bandwidth consistency for estimators of the conditional density and the conditional hazard rate functions are also derived from our main result. Moreover, the logarithm laws we establish are shown to yield almost sure simultaneous asymptotic confidence bands for the functions we consider. Examples of confidence bands obtained from simulated data are displayed.