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

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.

💡 Research Summary

This paper addresses the problem of non‑parametric estimation of a multivariate regression function when the response variable is subject to random right‑censoring. The authors adopt the Inverse Probability of Censoring Weighted (I.P.C.W.) approach, which compensates for the missing information by weighting each observed pair ((X_i,Y_i)) with the inverse of an estimate of the censoring survival function (\hat G). The resulting estimator is
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