Signal detection via Phi-divergences for general mixtures

Abstract

The family of goodness-of-fit tests based on $\Phi$-divergences is known to be optimal for detecting signals hidden in high-dimensional noise data when the heterogeneous normal mixture model is underlying. This test family includes Tukey’s popular higher criticism test and the famous Berk–Jones test. In this paper we address the open question whether the tests’ optimality is still present beyond the prime normal mixture model. On the one hand, we transfer the known optimality of the higher criticism test for different models, for example, for the heteroscedastic normal, general Gaussian and exponential-$\chi^{2}$-mixture models, to the whole test family. On the other hand, we discuss the optimality for new model classes based on exponential families including the scale exponential, the scale Fréchet and the location Gumbel models. For all these examples we apply a general machinery which might be used to show the tests’ optimality for further models/model classes in future.