Block threshold rules for curve estimation using kernel and wavelet methods

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.

Abstract

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