The smoothness test for a density function

TL;DR

This paper introduces a statistical test to determine the smoothness of a density function by analyzing wavelet projections and asymptotic distributions, with demonstrated finite sample performance.

Contribution

It proposes a new smoothness test based on wavelet projections and a novel enrichment technique for asymptotic analysis.

Findings

The test effectively distinguishes between different levels of smoothness.

Finite sample experiments show reliable performance.

The method can identify continuity versus discontinuity in density functions.

Abstract

The problem of testing hypothesis that a density function has no more than $\mu$ derivatives versus it has more than $\mu$ derivatives is considered. For a solution, the $L^2$ norms of wavelet orthogonal projections on some orthogonal "differences" of spaces from a multiresolution analysis is used. For the construction of the smoothness test an asymptotic distribution of a smoothness estimator is used. To analyze that asymptotic distribution, a new technique of enrichment procedure is proposed. The finite sample behaviour of the smoothness test is demonstrated in a numerical experiment in case of determination if a density function is continues or discontinues.