Abstract: Wavelet-based curve estimators have received considerable recent attention, particularly in terms of their ability to adapt to irregularities in a curve. Nevertheless, the threshold rules on which wavelet estimators are based are not well understood, and indeed some contemporary workers employ rules that are suboptimal by an order of magnitude. In this paper we give necessary conditions and sufficient conditions on the form of the threshold for the resulting curve estimator to achieve optimal convergence rate s in the case of smooth and piecewise-smooth functions. We discuss practical threshold rules that achieve optimal rates and study spatially adaptive rules that permit a degree of local smoothing. We address the important statistical problem of which tuning parameters in threshold rules produce genuine statistical smoothing in the sense of allowing adjustment of variance against bias in a first-order sense. Some tuning parameters affect only bias while others influence neither bias nor variance to first order.
Key words and phrases: Bias, convergence rate, mean squared error, smoothing parameter, threshold, variance, wavelet.