We discuss a general approach to handling multiple hypotheses testing in the case when a particular hypothesis states that the vector of parameters identifying the distribution of observations belongs to a convex compact set associated with the hypot
hesis. With our approach, this problem reduces to testing the hypotheses pairwise. Our central result is a test for a pair of hypotheses of the outlined type which, under appropriate assumptions, is provably nearly optimal. The test is yielded by a solution to a convex programming problem, so that our construction admits computationally efficient implementation. We further demonstrate that our assumptions are satisfied in several important and interesting applications. Finally, we show how our approach can be applied to a rather general detection problem encompassing several classical statistical settings such as detection of abrupt signal changes, cusp detection and multi-sensor detection.
We address the problem of adaptive minimax density estimation on $bR^d$ with $bL_p$--loss on the anisotropic Nikolskii classes. We fully characterize behavior of the minimax risk for different relationships between regularity parameters and norm inde
xes in definitions of the functional class and of the risk. In particular, we show that there are four different regimes with respect to the behavior of the minimax risk. We develop a single estimator which is (nearly) optimal in orderover the complete scale of the anisotropic Nikolskii classes. Our estimation procedure is based on a data-driven selection of an estimator from a fixed family of kernel estimators.
