Strong Uniform Convergence Rates of Wavelet Density Estimators with Size-Biased Data

Abstract

This paper considers the strong uniform convergence of multivariate density estimators in Besov space Bp,qs(Rd) based on size-biased data. We provide convergence rates of wavelet estimators when the parametric μ is known or unknown, respectively. It turns out that the convergence rates coincide with that of Giné and Nickl’s (Uniform Limit Theorems for Wavelet Density Estimators, Ann. Probab., 37(4), 1605-1646, 2009), when the dimension d=1, p=q=∞, and ω(y)≡1.