Adaptive test on components of densities mixture

Abstract

We are interested in testing whether two independent samples of n independent random variables are based on the same mixing-components or not. We provide a test procedure to detect if at least two mixing-components are distinct when their difference is a smooth function. Our test procedure is proved to be optimal according to the minimax adaptive setting that differs from the minimax setting by the fact that the smoothness is not known. Moreover, we show that the adaptive minimax rate suffers from a loss of order ( $ (\sqrt {\log (\log (n))} )^{ - 1} $ compared to the minimax rate.