Ten lectures on wavelets
Block thresholding for density estimation: local and global adaptivity
Journal of Multivariate Analysis
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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,s1/p, p=2, q=1 and nearly optimal convergence rates when 1@?p