Model Misspecification: Finite Mixture or Homogeneous?

A common problem in statistical modelling is to distinguish between finite mixture distribution and a homogeneous non-mixture distribution. Finite mixture models are widely used in practice and often mixtures of normal densities are indistinguishable from homogenous non-normal densities. This paper illustrates what happens when the EM algorithm for normal mixtures is applied to a distribution that is a homogeneous non-mixture distribution. In particular, a population-based EM algorithm for finite mixtures is introduced and applied directly to density functions instead of sample data. The population-based EM algorithm is used to find finite mixture approximations to common homogeneous distributions. An example regarding the nature of a placebo response in drug treated depressed subjects is used to illustrate ideas.