Empirical Bayesian Test of the Smoothness

In the context of adaptive nonparametric curve estimation a common assumption is that a function (signal) to estimate belongs to a nested family of functional classes. These classes are often parametrized by a quantity representing the smoothness of the signal. It has already been realized by many that the problem of estimating the smoothness is not sensible. What can then be inferred about the smoothness? The paper attempts to answer this question. We consider implications of our results to hypothesis testing about the smoothness and smoothness classification problem. The test statistic is based on the empirical Bayes approach, i.e., it is the marginalized maximum likelihood estimator of the smoothness parameter for an appropriate prior distribution on the unknown signal.