Robustness of the student t based M-estimator

This paper considers the maximum likelihood type (M) estimator based on Student's t distribution for the location/scale model. The Student t M-estimator is generally thought to be robust to outliers. This paper shows that this is only true if the degrees of freedom parameter is kept fixed. By contrast, if the degrees of freedom parameter is also estimated from the data, the influence functions for the scale and degrees of freedom parameter become unbounded. Moreover, the influence function of the location parameter remains bounded, but its change-of-variance function is unbounded. The intuition behind these results is explained in the paper. The rates at which both the influence functions and the change-of-variance function diverge to infinity, are very slow. This implies that outliers have to be extremely large in order to become detrimental to the performance of the Student t based M-estimator with estimated degrees of freedom. The theoretical results are illustrated in a a simulation experiment using several related competing estimators and several distributions for the error process.