Block threshold rules for curve estimation using kernel and wavelet methods

Abstract

Motivated by recently developed threshold rules for wavelet estimators, we suggest threshold methods for general kernel density estimators, including those of classical Rosenblatt–Parzen type. Thresholding makes kernel methods competitive in terms of their adaptivity to a wide variety of aberrations in complex signals. It is argued that term-by-term thresholding does not always produce optimal performance, since individual coefficients cannot be estimated sufficiently accurately for reliable decisions to be made. Therefore, we suggest grouping coefficients into blocks and making simultaneous threshold decisions about all coefficients within a given block. It is argued that block thresholding has a number of advantages, including that it produces adaptive estimators which achieve minimax-optimal convergence rates without the logarithmic penalty that is sometimes associated with term-by-term thresholding. More than this, the convergence rates are achieved over large classes of functions with discontinuities, indeed with a number of discontinuities that diverges polynomially fast with sample size. These results are also established for block thresholded wavelet estimators, which, although they can be interpreted within the kernel framework, are often most conveniently constructed in a slightly different way.