| The theory of wavelets is a relatively recent development; in applied mathematics with a wide range of potential applications. Compactly supported wavelets are particularly interesting, because of their natural ability to represent data with intrinsically local properties. Donoho and Johnstone (1994) invented wavelet shrinkage to estimate functions. The wavelet estimates were near optimal over a wide range of spaces and enjoy spatial adaptivity.; In this dissertation, we start by reviewing nonparametric regression linear smoothers, and then introduce the wavelet analysis based on the multiresolution analysis. We use a cross-validation method suggested by Nason (1994) to select a wavelet basis which produces a wavelet estimator with minimum mean square error.; A bootstrap method is proposed for estimating the variance part of the MISE for the wavelet shrinkage estimator. Three theorems are proved deriving a useful error bound on the bootstrap estimate using the universal threshold when noise level is known. We also derive the rate of convergence {dollar}O(log n/n){dollar} for a piecewise polynomial regression function and obtain the rate of convergence {dollar}o(nsp{lcub}-2p/(2p+1){rcub}){dollar} when the regression function is piecewise p-th order differentiable and {dollar}fsp{lcub}(p){rcub}{dollar} is piecewise Holderian with exponential {dollar}delta>0.{dollar}; Bootstrap resampling is also used to develop pointwise confidence bands of the regression function, and a number of Monte Carlo simulation examples are discussed. |
