Minimax rates over Besov spaces in ill-conditioned mixture-models with varying mixing-weights

We consider ill-conditioned mixture-models with varying mixing-weights. We study the classical homogeneity testing problem in the minimax setup and try to push the model to its limits, that is to say to let the mixture model to be really ill-conditioned. We highlight the strong connection between the mixing-weights and the expected rate of testing. This link is characterized by the behavior of the smallest eigenvalue of a particular matrix computed from the varying mixing-weights. We provide optimal testing procedures and we exhibit a wide range of rates that are the minimax and minimax adaptive rates for Besov balls.