Testing the regularity of a smooth signal
Abstract
We develop a test to determine whether a function lying in a fixed $L_2$Sobolevtype 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2013
 arXiv:
 arXiv:1304.2592
 Bibcode:
 2013arXiv1304.2592C
 Keywords:

 Mathematics  Statistics Theory
 EPrint:
 Published at http://dx.doi.org/10.3150/13BEJ575 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)