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(2001) Hall, Peter; Penev, SpiridonJournal ArticleWe show that unless the target density is particularly smooth, cross-validation applied directly to nonlinear wavelet estimators produces an empirical value of primary resolution which fails, by an order of magnitude, to give asymptotic optimality. We note, too, that in the same setting, but for different reasons, cross-validation of the linear component of a wavelet estimator fails to give asymptotic optimality, if the primary resolution level that it suggests is applied to the nonlinear form of the estimator. We propose an alternative technique, based on multiple cross-validation of the linear component. Our method involves dividing the region of interest into a number of subregions, choosing a resolution level by cross-validation of the linear part of the estimator in each subregion, and taking the final empirically chosen level to be the minimum of the subregion values. This approach exploits the relative resistance of wavelet methods to over-smoothing: the final resolution level is too small in some parts of the main region, but that has a relatively minor effect on performance of the final estimator. The fact that we use the same resolution level throughout the region, rather than a different level in each subregion, means that we do not need to splice together different estimates and remove artificial jumps where the subregions abut.
(1997) Picard, Dominique; Hall, Peter; Penev, Spiridon; Kerkyacharian, GerardJournal ArticleUsually, methods for thresholding wavelet estimators are implemented term by term, with empirical coefficients included or excluded depending on whether their absolute values exceed a level that reflects plausible moderate deviations of the noise. We argue that performance may be improved by pooling coefficients into groups and thresholding them together. This procedure exploits the information that coefficients convey about the sizes of their neighbours. In the present paper we show that in the context of moderate to low signal-to-noise ratios, this lsquoblock thresholdingrsquo approach does indeed improve performance, by allowing greater adaptivity and reducing mean squared error. Block thresholded estimators are less biased than term-by-term thresholded ones, and so react more rapidly to sudden changes in the frequency of the underlying signal. They also suffer less from spurious aberrations of Gibbs type, produced by excessive bias. On the other hand, they are more susceptible to spurious features produced by noise, and are more sensitive to selection of the truncation parameter.