A Smirnov-Bickel-Rosenblatt theorem for compactly-supported wavelets

Details A Smirnov-Bickel-Rosenblatt theorem for compactly-supported wavelets A Smirnov-Bickel-Rosenblatt theorem for compactly-supported wavelets

2011-10-22

arxiv.org/abs/1110.4961v2

Mathematics

Statistics Theory

Scientific paper

In nonparametric statistical problems, we wish to find an estimator of an unknown function f. We can split its error into bias and variance terms; Smirnov, Bickel and Rosenblatt have shown that, for a histogram or kernel estimate, the supremum norm of the variance term is asymptotically distributed as a Gumbel random variable. In the following, we prove a version of this result for estimators using compactly-supported wavelets, a popular tool in nonparametric statistics. Our result relies on an assumption on the nature of the wavelet, which must be verified by provably-good numerical approximations. We verify our assumption for Daubechies wavelets and symlets, with N = 6, ..., 20 vanishing moments; larger values of N, and other wavelet bases, are easily checked, and we conjecture that our assumption holds also in those cases.

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