Nonparametric Regression with Sample Design Following a Random Process

Nonparametric regression is considered where the sample point placement is not fixed and equispaced, but generated by a random process with rate n. Conditions are found for the random processes that result in optimal rates of convergence for nonparametric regression when using a block thresholded wavelet estimator. Previous results on nonparametric regression via wavelets on both fixed and random sample point placement are shown to be special cases of the general result given here. The estimator is adaptive over a large range of Hölder function spaces and the convergence rate exhibited is an improvement over term-by-term wavelet estimators. Threshold selection is implemented in a data-adaptive fashion, rather than using a fixed threshold as is usually done in block thresholding. This estimator, BlockSure, is compared against fixed-threshold block estimators and the more traditional term-by-term threshold wavelet estimators on several random design schemes via simulations.