Statistical limits of sparse mixture detection

We consider the problem of detecting a general sparse mixture and obtain an explicit characterization phase transition under some conditions, generalizing univariate results Cai Wu. Additionally, we provide sufficient condition for adaptive optimality Higher Criticism type testing statistic formulated by Gao Ma. In course establishing these results, offer unified perspective through large deviations theory. The establish are direct consequences deviation principle normalized log-likelihood ratios between null signal distributions.