Department of Mathematics - Special Colloquium - Recent developments on Finite mixture models

Finite mixture models are formed by convex combinations of distributions from classical distribution families. They are particularly well suited for modeling populations with homogeneous subpopulations and for accurately approximating a wide range of distributions. However, statistical inference for finite mixture models encounters numerous unexpected challenges. Given a set of independent and identically distributed observations of increasing size, consistent estimation of the mixing distribution can only be achieved at a much slower rate than usual. The likelihood function in normal mixture models is unbounded, which debilitates maximum likelihood estimation in a general sense. Additionally, the likelihood ratio statistic for testing hypotheses on the number of subpopulations may diverge to infinity, instead of having the usual chi-square limiting distribution. In this presentation, we will review various advances in the consistent estimation of mixing distributions and in testing hypotheses concerning the number of subpopulations. We will also briefly discuss some recent developments related to distributed learning and its robust version, where finite mixture ideas continue to play an important role.