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Wavelet estimators: Adapting to unknown smoothness. (English) Zbl 0871.62039

Summary: A wavelet thresholding algorithm is used to recover a function of unknown smoothness from noisy data. It is known that it can be tuned to be minimax in order over a wide range of Besov-type smoothness constraints and \(L_p\)-losses. We provide a method to estimate an adaptive threshold parameter for each resolution level. It is shown that the proposed algorithm is adaptive in order, i.e., it attains the rate of convergence which is minimax up to a constant over Besov regularity classes and \(L_p\)-error measures, \(1\leq p\leq\infty\). The algorithm is computationally straightforward: the whole effort to compute the threshold is of order \(N\log N\) for the sample size \(N\).

MSC:

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
65C99 Probabilistic methods, stochastic differential equations
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