Optimal Detection of Heterogeneous and Heteroscedastic Mixtures
Tony Cai, X. Jessie Jeng, and Jiashun Jin
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.