Empirical Bayesian test for the smoothness

In the context of adaptive nonparametric curve estimation problem, a common assumption is that the function (signal) to estimate belongs to a nested family of functional classes, parameterized by a quantity which often has a meaning of smoothness amount. It has already been realized that the problem of estimating the smoothness is meaningless. What can then be inferred about the smoothness? We try to answer this question and discuss the implications of our results for the hypothesis testing problem for the smoothness parameter. The imbedded model structure (nested family of classes) accounts for the fact that a consistent test can be constructed only for the one-sided hypothesis. The test statistic is based on the marginalized maximum likelihood estimator of the smoothness for the appropriate choice of the prior distribution on the unknown signal.