On the minimax optimality of wavelet estimators with censored data

Abstract

Wavelet-based density estimators with randomly right-censored data are considered. We investigate the asymptotic rates of convergence of estimators based on thresholding of empirical wavelet coefficients. Unlike the complete data case, the empirical wavelet coefficients are constructed through the Kaplan–Meier estimators of the distribution functions. It turns out that these coefficients can be approximated by an average of i.i.d. random variables with a certain error rate. We show that the estimators achieve nearly optimal minimax convergence rates within logarithmic terms over a large range of Besov function classes B pq α , α > 1 / p , p ⩾ 1 , q ⩾ 1 , a feature not available for linear estimators when p < 2 .