Wavelet Estimation For Convolutions Of Densities

Nonparametric density estimation plays important roles in statistics. Many references deal with the density estimation of a random variable. However, people are more interested in the density function involving sum of random variables in the insurance claims and other applica-tions when the random variables are independent, the density function of the sum of random variables equals to the convolution of their density functions. Therefore, the density estimation about the sum of the random variables is often called convolution of density estimation.Motivated by Chesneau's work (Chesneau, C, Navarro. F.2014. On a Plug-In Wavelet Estimator for Convolutions of Densities. Journal of Statistical Theory and Practice. Vol.8, No. 4,653-673.), we apply the wavelet method to study the convolution of density estimation in Lp sense. Chapter 2 provides the Lp mean consistency of linear wavelet estimator. For 1? p?2, we don't assume any smoothness of density function while some smooth conditions of a density function is needed for p> 2. The first section of Chapter 3 gives a convergence rate of the above linear estimator in Besov spaces. Because the random samples have noises in many practical applications, we define a linear wavelet estimator for the model with additive noises and show its ?(1?P??) convergence rate in the second section. Chapter 4 discusses both the classical and additive noise model. We provide the convergence rates of nonlinear wavelet estimators in Besov spaces, the results improve the corresponding theorems in Chapter 3.