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

TL;DR

This paper extends the Smirnov-Bickel-Rosenblatt theorem to estimators using compactly-supported wavelets, providing theoretical insights into their asymptotic distribution in nonparametric statistics.

Contribution

It proves a version of the theorem for wavelet-based estimators under verifiable assumptions, broadening the understanding of their asymptotic behavior.

Findings

The theorem is validated for Daubechies wavelets with 6 to 20 vanishing moments.

The assumptions are checked through numerical approximations for specific wavelet bases.

The results suggest the theorem may hold for a wider class of wavelets, including larger N.

Abstract

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.