Nonparametric regression and density estimation in Besov spaces via wavelets

Eric Karl Chicken, Purdue University

Abstract

For density estimation and nonparametric regression, block thresholding is very adaptive and efficient over a variety of general function spaces. By using block thresholding on kernel density estimators, the optimal minimax rates of convergence of the estimator to the true distribution are attained. This rate holds for large classes of densities residing in Besov spaces, including discontinuous functions with the number of discontinuities growing with sample size as well as functions with other types of irregularities. The results hold for both convolution and wavelet kernel methods. Additionally, the proposed wavelet estimator is an improvement on previous estimators in that it simultaneously achieves both local and global optimal rates through careful choice of block length and a truncation parameter for the estimate's orthogonal series expansion. The estimator is examined via simulations and compared against other kernel density estimators. In the case of nonparametric regression, most previous work has focused on data where the sampling points of the unknown function are equally spaced apart. If the design points occur as a Poisson process or have a uniform distribution rather than being equally spaced apart, the wavelet method of block thresholding can be applied directly to the data as though it was equispaced without sacrificing adaptivity or rates of convergence. When the underlying true function is in certain Besov and Hölder spaces, the resulting estimator achieves the optimal minimax rate of convergence. Simulations are run on this estimator and compared to previous estimators used for the same purpose.

Degree

Ph.D.

Advisors

Cai, Purdue University.

Subject Area

Statistics

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