Distance-based Model-Selection with application to the Analysis of Gene Expression Data

Multivariate mixture models provide a convenient method of densityestimation and model based clustering as well as providing possible explanations for the actual data generationprocess. But the problem of choosing the number of components ($g$) in a statistically meaningful way is still asubject of considerable research . Available methods for estimating $g$include, optimizing AIC and BIC, estimating the number through nonparametricmaximum likelihood, hypothesis testing and Bayesian approaches with entropydistances. In our current research we present several rules for selecting afinite mixture model, and hence $g$, based on estimation and inference usinga quadratic distance measure.In one methodology the goal is to find the minimalnumber of components that are needed to adequately describe the truedistribution based on a nonparametric confidence set for the true distribution. We also present results for selecting $g$ based on a riskanalysis that includes a penalty for overfitting. Another less formalmethodology is based on the concordance measure which is analogous to $R^2$in regression. Moreover, we find develop diagnostics for purposes ofoutlier detection. These diagnostics help to distinguish between outliersand true clusters, and they provide insight into the initial values foriterative estimation of additional components.In this dissertation we also develop tools for determining the number of modesin a mixture of multivariate normal densities. We use these criterion to selectclusters which display distinct modes. Finally we fine tune our methods to analyze gene-expression data frommicro-arrays, and compare them with other competitive methods.