Bernoulli

  • Bernoulli
  • Volume 21, Number 1 (2015), 465-488.

Testing the regularity of a smooth signal

Alexandra Carpentier

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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.

Article information

Source
Bernoulli, Volume 21, Number 1 (2015), 465-488.

Dates
First available in Project Euclid: 17 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1426597078

Digital Object Identifier
doi:10.3150/13-BEJ575

Mathematical Reviews number (MathSciNet)
MR3322327

Zentralblatt MATH identifier
1320.94021

Keywords
functional analysis minimax bounds non-parametric composite testing problem

Citation

Carpentier, Alexandra. Testing the regularity of a smooth signal. Bernoulli 21 (2015), no. 1, 465--488. doi:10.3150/13-BEJ575. https://projecteuclid.org/euclid.bj/1426597078


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