On the block thresholding wavelet estimators with censored data

Abstract

We consider block thresholding wavelet-based density estimators with randomly right-censored data and investigate their asymptotic convergence rates. Unlike for the complete data case, the empirical wavelet coefficients are constructed through the Kaplan–Meier estimators of the distribution functions in the censored data case. On the basis of a result of Stute [W. Stute, The central limit theorem under random censorship, Ann. Statist. 23 (1995) 422–439] that approximates the Kaplan–Meier integrals as averages of i.i.d. random variables with a certain rate in probability, we can show that these wavelet empirical coefficients can be approximated by averages of i.i.d. random variables with a certain error rate in L 2 . Therefore we can show that these estimators, based on block thresholding of empirical wavelet coefficients, achieve optimal convergence rates over a large range of Besov function classes B p , q s , s > 1 / p , p ≥ 2 , q ≥ 1 and nearly optimal convergence rates when 1 ≤ p < 2 . We also show that these estimators achieve optimal convergence rates over a large class of functions that involve many irregularities of a wide variety of types, including chirp and Doppler functions, and jump discontinuities. Therefore, in the presence of random censoring, wavelet estimators still provide extensive adaptivity to many irregularities of large function classes. The performance of the estimators is tested via a modest simulation study.