|Model Selection Based on Expected Squared Hellinger Distance
| Xiaofan Cao, Ph.D. Candidate
Department of Statistics, Colorado State University
Wednesday, October 3, 2007
This dissertation is motivated by a general model selection problem such that thetrue model is unknown and one or more approximating parametric families of models are given along with strategies for estimating the parameters using data. We develop model selection methods based on Hellinger distance that can be applied to a wide range of modeling problems without posing the typical assumptions for the true model to be within the approximating families or to come from a particular parametric family. We propose two estimators for the expected squared Hellinger distance as the model selection criteria.
In particular, the use of expected squared Hellinger distance is studied in ANOVA model selection problems where approximating models are typically sub-models of the full factorial model. The properties of the expected squared Hellinger distance are explored under balanced model structure assuming independent and identically distributed normal error terms. A model selection strategy specific to ANOVA model selection problems based on one of the estimated expected squared Hellinger distance is proposed. This strategy is illustrated using a real data set and its performance is tested by simulation studies. An example of ANOVA model selection problem with non-normal error terms that follow two-parameter exponential distribution is discussed.
Model selection method based on estimated expected squared Hellinger distance is also applied to modeling the p-values from the microarray data analysis. The problem of estimating false discovery rate (FDR) from the distribution of p-values arising from statistical tests of differential gene expression in a microarray experiment is considered. A finite mixture model is studied in which one component is uniform on [0,1] corresponding to equally expressed genes and one or more additional components correspond to differentially expressed genes. Two different mixture families are explicitly investigated for estimating false discovery rate – a mixture of Beta densities and a mixture of Uniform densities. In both cases, the Minimum Hellinger distance is used to provide robust estimates of the mixture components. For the Beta mixture model we choose the number of Beta components by comparing the estimated expected squared Hellinger distance. The performance of the proposed methods is illustrated through a case study involving data from a published microarray experiment.