Testing the regularity of a smooth signal

We develop a test to determine whether a function lying in a fixed L2 Sobolev type ball of smoothness t, and generating a noisy signal, is in fact of a given smoothness s larger than 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.