Residuals and robust link functions for ordinal regression models

Cumulative models are widespread in various fields to describe the dependence of an observed ordinal response on subjects' covariates. Outlying covariates as well as incoherent responses may affect the reliability of the Maximum Likelihood estimators and that of the derived testing procedures. However the various link functions which provide the relationship between the linear predictor and the cumulative probabilities differ in terms of sensitivity to outliers, and a suitable link function can limit the impact of anomalous data in the estimation process. Consequently conditions are derived which allow to evaluate the properties of the link functions in terms of robustness, either when the covariate are outliers free or when extreme design points may occur among regressors. The purpose is to carry out robust inference through the likelihood function by an appropriate choice of the link function.