Wavelet Estimations For A Density With Multiplication Noises

In this paper, we study the optimal estimation of a density function with multipli-cation noises by wavelet methods. More precisely, we assume that our random samplesare independent and identically distributed, density functions are in the Sobolev spacesWNrwith integer exponents, and the biasing functions have positive supper and lowerbounds.Motivated by Tsybakov’s work (A. B. Tsybakov. Introduction to NonparametricEstimation. Springer-Verlag, Berlin,2009), Chapter2provides the lower bound of rateof convergence of all of the estimators over the Lp(p≥1) minimax risk. Then we definea linear wavelet estimator and show a rate of convergence of the linear wavelet estimatorover the Lprisk in the third Chapter. It turns out that when r≥p, our estimation isthe optimal; but when r <p, it doesn’t attain the optimal rate of convergence.Inspired by Donoho’s work (D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, D.Picard. Density estimation by wavelet thresholding. Ann. Statist.1996,24:508-539),we construct a nonlinear wavelet density estimator by wavelet thresholding method, andgive a rate of convergence over the Lprisk in the last part. When1<r, ournonlinear estimator attains the optimal rate of convergence; Whenp≤r <p, theconvergence rate is sub-optimal, i.e. it is optimal up to a ln n factor.