This thesis discusses two topics related to robust statistics: robust polytomous regression and likelihood misspecification of Bayesian models.

The standard maximum likelihood estimation of polytomous regression is highly affected by outliers, especially when they are in the covariates space. Existing robust estimators protect against misclassification but are vulnerable to outlying covariates. To address this problem, two new robust M-estimators are introduced, along with the corresponding tests. Their properties are investigated through a theoretical analysis, a simulation study and applications to medical datasets.

Existing solutions to likelihood misspecification of Bayesian models are generally complex and hard to interpret. A simple, fully Bayesian solution to likelihood misspecification is suggested: computing the posterior distribution obtained by conditioning on a summary statistic instead of the full data. Properties of such models, called summary-models, are rigorously derived, and connections are established with the classical approach to robust statistics based on M-estimators and the influence function.