Model choice and variable selection in mixed & semiparametric models

Abstract

Semiparametric and mixed models allow different kinds of data structures and \ndata types to be considered in regression models. Spatial and temporal \nstructures of discrete or spatial data can be treated as flexibly as, for \ninstance, functional data. This growing flexibility increasingly requires a \nstatistician to make choices between competing models. \nIn model selection the degrees of freedom play an important role as a measure of \nmodel complexity. In this thesis three approaches for the estimation of the \ndegrees of freedom in mixed and semiparametric models are developed, each for \ndifferent distributions of the (conditional) responses. The interpretation of \nsemiparametric models as mixed models justifies using the same model selection \ntechniques for both model classes. \nBy using Steinian methods, the degrees of freedom can be determined for a \ngroup of distributions belonging to the exponential family. The developed \nmethods for determining the degrees of freedom are illustrated by an example of \ntree growth data. \nFor a larger class of distributions the degrees of freedom can be determined by \ncross-validation and bootstrap methods. Additionally, an approximate Steinian \nmethod can be adapted for further distributions. \nBased on the implicit function theorem the degrees of freedom of a variance or \nsmoothing parameter can de derived analytically if the response is normally \ndistributed. Failure to take these degrees of freedom into account can lead to \nbiased model selection. In addition to the methodological derivation, the \ngeometrical properties of the degrees of freedom of the variance and smoothing \nparameters are analysed. Furthermore, numerical problems in the computation of \nthe degrees of freedom are considered.