Optimal Detection of Heterogeneous and Heteroscedastic MixturesTony Cai, X. Jessie Jeng, and Jiashun Jin
Optimal Detection of Heterogeneous and Heteroscedastic MixturesTony Cai, X. Jessie Jeng, and Jiashun Jin
- Abstract: The problem of detecting heterogeneous and heteroscedastic Gaussian mixtures is considered. The focus is on how the parameters of heterogeneity, heteroscedasticity, and proportion of non-null component influence the difficulty of the problem. We establish an explicit detection boundary which separates the detectable region where the likelihood ratio test is shown to reliably detect the presence of non-null effect, from the undetectable region where no method can do so. In particular, the results show that the detection boundary changes dramatically when the proportion of non-null component shifts from the sparse regime to the dense regime. Furthermore, it is shown that the Higher Criticism test, which does not require the specific information of model parameters, is optimally adaptive to the unknown degrees of heterogeneity and heteroscedasticity in both the sparse and dense cases.
- Paper: pdf file.