Wavelet Estimations For A Density And Its Derivatives With Size-Biased Noises

Density estimation for biased data plays important roles in practical appli-cations, because there exists the bias between observed data and real data which can’t be directly observed in many cases. Many existing studies consider density estimation with independent data by using the kernel method. Wavelet density estimation has made great achievements due to the advantages of wavelet ba-sis. Motivated by the work of Donoho, Doosti, Ramirez, Chaubey, Chesneau and etc, this dissertation studies some wavelet estimators for negatively associated stratified size-biased samples. More precisely, We discuss their Lp(1≤p≤∞) consistency and convergence rates of Lp(1≤p≤∞) risk in Besov space.Firstly, we show the LP (1≤p≤∞) consistency of two types of wavelet linear estimators for d dimensional independent samples as well as one dimensional nega-tively associated samples respectively without any assumptions of smoothness for density functions. A technique of Chacon et al plays a key role in our discussions. In addition, numerical simulations verify our theoretic results.Secondly, we define a linear wavelet estimator for density functions based on negatively associated stratified size-biased sample and investigate its convergence rate on LP (1≤p≤∞) risk in Besov space Br,qs(R). More precisely, if negatively associated random samples satisfy a technical condition, a risk upper bound is pro-vided by using Newman inequality. When the monotonicity of the biased function replaces that condition, we show a better result by using Rosenthal inequality. In addition, the nonlinear (hard thresholding) wavelet estimations are considered in the same space Br,qs(R). It turns out that those estimations are better than the linear ones for r<p. When the biased functions g=1 and the stratified size M=1, the biased model reduces to the classical one with free noise. In that case, our results are the exactly same as the theorems in Donoho et al. Numerical experiments are also presented in the end of that chapter.Finally, we study wavelet estimations for density derivatives. The Lp(1≤ p<∞) risk of a linear wavelet estimator is investigated for d-th derivative fx(d) of a density function fx. In particular, the results reduce to the above conclusions for a density estimation.