The smoothness test for a density function

The problem of testing hypothesis that a density function has no more than μμ derivatives versus it has more than μμ derivatives is considered. For a solution, the L2L2 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 behavior of the smoothness test is demonstrated in a numerical experiment in case of determination if a density function is continuous or discontinuous.