Nonrobustness of the Student's T Based Pseudo Maximum Likelihood Estimator

The maximum likelihood estimator based on Student's t distribution is generally thought to be robust to outliers in the regression errors. This paper shows that this is true if the degrees of freedom parameter is kept fixed. In 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 regression parameters remains bounded, but its change-of-variance function becomes unbounded. The rate at which both the influence functions and the change-of-variance function diverge to infinity, is, however, very slow. The implications of these findings are investigated by means of a small simulation experiment using several related competing estimators and several distributions for the error process. It appears that despite its nonrobustness properties, the Student's t based M-estimator performs well compared to the other estimators in a variety of settings