Wavelet Estimations Of Density Derivatives With A Class Of Multiplication Noises

Wavelet estimation of a density function for the classical model has made almost perfect achievements by Donoho and etc (D. L. Donoho, I. M. Johnstone, G. Kerky-acharian, D.Picard. Density estimation by wavelet thresholding. Ann. Statist.1996,24:508-539). Inspired by their work, we study the optimality of wavelet estimates of density derivatives for biased sample by wavelet methods. More precisely, we assume that the samples are independent and identically distributed (i.i.d.), the density deriva-tives belong to Besov spaces Brps, and the biasing function has both lower and upper bounds.Using the traditional method (A. B. Tsybakov. Introduction to Nonparametric Estimation. Springer-Verlag, Berlin,2009), we firstly provide a lower bound of Lp (p≥1) risk for an arbitrary estimator in Chapter2. Then we define a linear wavelet estimator and show a convergence rate for the Lp risk in the third Chapter. It turns out that when r≥p, our estimation is the optimal.Since the linear estimator doesn’t provide optimal estimate when r<p, we con-struct a nonlinear wavelet estimator for density derivatives by wavelet thresholding method in the last part, which attains the optimal convergence rate for1<r<(1-2d)p/2(s|d|1) When (1+2d)p/2(s+d)+1≤r<p,it is sub-optimal (optimal up to a In n factor).