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February 2015 Testing the regularity of a smooth signal
Alexandra Carpentier
Bernoulli 21(1): 465-488 (February 2015). DOI: 10.3150/13-BEJ575

Abstract

We develop a test to determine whether a function lying in a fixed $L_{2}$-Sobolev-type ball of smoothness $t$, and generating a noisy signal, is in fact of a given smoothness $s\geq t$ or not. While it is impossible to construct a uniformly consistent test for this problem on every function of smoothness $t$, it becomes possible if we remove a sufficiently large region of the set of functions of smoothness $t$. The functions that we remove are functions of smoothness strictly smaller than $s$, but that are very close to $s$-smooth functions. A lower bound on the size of this region has been proved to be of order $n^{-t/(2t+1/2)}$, and in this paper, we provide a test that is consistent after the removal of a region of such a size. Even though the null hypothesis is composite, the size of the region we remove does not depend on the complexity of the null hypothesis.

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Alexandra Carpentier. "Testing the regularity of a smooth signal." Bernoulli 21 (1) 465 - 488, February 2015. https://doi.org/10.3150/13-BEJ575

Information

Published: February 2015
First available in Project Euclid: 17 March 2015

zbMATH: 1320.94021
MathSciNet: MR3322327
Digital Object Identifier: 10.3150/13-BEJ575

Keywords: functional analysis , minimax bounds , non-parametric composite testing problem

Rights: Copyright © 2015 Bernoulli Society for Mathematical Statistics and Probability

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Vol.21 • No. 1 • February 2015
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