  In the last two decades, sparse regularization methods such as the LASSO have
been applied in various fields. Most of the regularization methods have one or
more regularization parameters, and to select the value of the regularization
parameter is essentially equal to select a model, thus we need to determine the
regularization parameter adequately. Regarding the determination of the
regularization parameter in the linear regression model, we often apply the
information criteria like the AIC and BIC, however, it has been pointed out
that these criteria are sensitive to outliers and tend not to perform well in
high-dimensional settings. Outliers generally have a negative influence on not
only estimation but also model selection, consequently, it is important to
employ a selection method that is robust against outliers. In addition, when
the number of explanatory variables is quite large, most conventional criteria
are prone to select unnecessary explanatory variables. In this paper, we
propose model evaluation criteria via the statistical divergence with
excellence in robustness in both of parametric estimation and model selection.
Furthermore, our proposed criteria simultaneously achieve the selection
consistency with the robustness even in high-dimensional settings. We also
report the results of some numerical examples to verify that the proposed
criteria perform robust and consistent variable selection compared with the
conventional selection methods.
