Block-dependent thresholding in wavelet regression

Nonparametric regression via wavelets, is usually implemented under the assumptions of dyadic sample size, equally spaced and fixed sample points, and independent and identically distributed normal errors. An estimator is proposed which, through the use of linear transformations and block thresholding, can simultaneously achieve both global and local optimal rates of convergence even for data that does not possess the above three assumptions. Additionally, the estimator exhibits fast computation time and is spatially adaptive over large classes of Besov and Holder functions. The thresholds are dependent on the varying levels of noise in each block of wavelet coefficients, rather than on a single estimate of the noise as is usually done. This block-dependent method is compared against term-by-term wavelet methods with noise-dependent thresholding via theoretical asymptotic convergence rates as well as by simulations and comparisons on a well-known data set. Simulation results show that this blockdependent estimator is superior in terms of reconstruction error to term-by-term wavelet estimators and universal-type block estimators.