Testing the regularity of a smooth signal

TL;DR

This paper introduces a statistical test to determine if a noisy signal's underlying function has a certain smoothness level, effectively distinguishing functions of different smoothness within a specified region.

Contribution

It develops a new test that is consistent after removing a region of functions with lower smoothness close to the target smoothness, overcoming previous limitations.

Findings

The test is consistent after removing a region of size proportional to n^{-t/(2t+1/2)}.

The null hypothesis's complexity does not affect the size of the removable region.

The method applies to functions in a Sobolev-type ball with specified smoothness.

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.