Statistical applications of wavelet theory

In this research, we explore the applications of wavelet theory in nonparametric regression and density estimation settings with special emphasis on B-wavelets. We first focus on estimating a Holder continuous function f from noisy, sampled data $\{y\sb{i}\} = \{f(x\sb{i} + \varepsilon\sb{i}\}$ using the wavelet decomposition and reconstruction methods of multiresolution analysis. The white noise $\{\varepsilon\sb{i}\}$ have mean zero and are uncorrelated. We study the behavior of the wavelet transformation of white noise, and use our understanding of the behavior to form a class of consistent estimators of the regression function f. We begin with a local optimal-order interpolatory scheme to get the empirical scaling function coefficients at the highest resolution. We then shrink and truncate the wavelet coefficients produced by the multiresolution decomposition so that the noise is reduced. The estimator is the function derived from the multiresolution reconstruction process based on these modified wavelet coefficients. Wavelet theory reduces the problem of estimating a density function to that of estimating its wavelet coefficients. We also studied this problem from the mean squared error point of view using both Daubechies\u27s orthogonal wavelet and Chui\u27s B-spline wavelet. An optimal choice of the resolution level is obtained by balancing the squared bias and the variance of the estimator. The histogram is a special case of the method if the Haar system of wavelets is employed. Based on the estimation of a density function, we also studied a regression problem where the observations are taken at random design points. They are analogues of some kernel and orthogonal series estimators. In studying the behavior of the wavelet transformation of white noise, we discovered some properties of spectral density functions in down sampling a stationary process. A version of Ruelle\u27s general Perron-Froebenius theorem is obtained. Passing a down sampled process through a linear filter, one still gets a stationary process. Its autocovariance function can be computed from the weights of the linear filter and the autocovariance function of the original stationary process. The relationship between the spectral density functions of the original process and of the down sampled, filtered process is obtained. It is proved that the corresponding spectral density functions decay exponentially. The proof is fitted in the framework of Ruelle\u27s general Perron-Froebenius theorem for sequence spaces. Computationally, we have implemented the statistical procedures developed in the theoretical results in S-plus