Statistics Seminar: Michael Stewart - Sydney University -- Mixture Detection for Exponential Families

We discuss the deceptively rich parametric testing problem of
distinguishing between one-component and two-component mixtures. We
start with the simple case of normal location mixtures, tracing the
history of work on this problem from the discovery of a slowly
diverging log-likelihood ratio statistic in the 1980s, through to it’s
limiting extreme-value behaviour in the 1990s and early 2000s. We then
discuss some interesting applications related to signal detection,
multiple testing and variable selection in high-dimensional regression
and classification problems, including the introduction of the higher
criticism procedure. Finally we discuss recent work extending some
limiting distribution and local power results beyond normal location
mixtures to mixtures of general one-parameter exponential families,
including a special role played by variance-stabilising
transformations. We also indicate how certain results may be extended
to a more general class of models which include change-point and
hidden Markov models.