On robustness of model selection criteria based on divergence measures: Generalizations of BHHJ divergence-based method and comparison

In model selection problems, robustness is one important feature for selecting an adequate model from the candidates. We focus on statistical divergence-based selection criteria and investigate their robustness. We mainly consider BHHJ divergence and related classes of divergence measures. BHHJ divergence is a representative robust divergence measure that has been utilized in, for example, parametric estimation, hypothesis testing, and model selection. We measure the robustness against outliers of a selection criterion by approximating the difference of values of the criterion between the population with outliers and the non-contaminated one. We derive and compare the conditions to guarantee robustness for model selection criteria based on BHHJ and related divergence measures. From the results, we find that conditions for robust selection differ depending on the divergence families, and that some expanded classes of divergence measures require stricter conditions for robust model selection. Moreover, we prove that robustness in estimation does not always guarantee robustness in model selection. Through numerical experiments, we confirm the advantages and disadvantages of each divergence family, asymptotic behavior, and the validity for employing criteria on the basis of robust divergence. Especially, we reveal the superiority of BHHJ divergence in robust model selection for extensive cases.