Robustness against outliers in ordinal response model via divergence approach

This study deals with the problem of outliers in ordinal response model, which is a regression on ordered categorical data as the response variable. ``Outlier'' means that the combination of ordered categorical data and its covariates is heterogeneous compared to other pairs. Although the ordinal response model is important for data analysis in various fields such as medicine and social sciences, it is known that the maximum likelihood method with probit, logit, log-log and complementary log-log link functions, which are often used, is strongly affected by outliers, and statistical analysts are forced to limit their analysis when there may be outliers in the data. To solve this problem, this paper provides inference methods with two robust divergences (the density-power and $\gamma$-divergences). We also derive influence functions for the proposed methods and show conditions on the link function for them to be bounded and to redescendence. Since the commonly used link functions satisfy these conditions, the analyst can perform robust and flexible analysis with our methods. In addition, and this is a result that further highlights our contributions, we show that the influence function in the maximum likelihood method does not have redescendence for any link function in the ordinal response model. Through numerical experiments on artificial data, we show that the proposed methods perform better than the maximum likelihood method with and without outliers in the data for various link functions.