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

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 is known or unknown, respectively. It turns out that the convergence rates coincide with that of Gine 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 omega(y) 1.