Robustness in ordinal regression

Logistic regression is a widely used tool designed to model the success probability of a Bernoulli
random variable depending on some explanatory variables. A generalization of this bimodal model is the multinomial case where the dependent variable has more than two categories. When these categories are naturally ordered (e.g. in questionnaires where individuals are asked whether they strongly disagree, disagree, are indifferent, agree or strongly agree with a given statement), one speaks about ordered or ordinal regression. The classical technique for estimating the unknown parameters is based on Maximum Likelihood estimation (e.g. Powers and Xie, 2008 or Agresti, 2002).
However, as Albert and Anderson (1984) showed in the binary context, Maximum Likelihood
estimates sometimes do not exist. Existence conditions in the ordinal setting, derived by Haberman in a discussion of McCullagh’s paper (1980), as well as a procedure to verify that they are fulfilled on a particular dataset will be presented.
On the other hand, Maximum Likelihood procedures are known to be vulnerable to contamination in the data. The lack of robustness of this technique in the simple logistic regression setting has already been investigated in the literature (e.g. Croux et al., 2002 or Croux et al., 2008). The breakdown behaviour of the ML-estimation procedure will be considered in the context of ordinal logistic regression. A robust alternative based on a weighting idea will then be suggested and compared to the classical one by means of their influence functions. Influence functions can be used to construct a diagnostic plot allowing to detect influential observation for the classical ML procedure (Pison and Van Aelst, 2004).