A note on the Cook's distance

A modification of the classical Cook's distance is proposed, providing us with a generalized Mahalanobis distance in the context of multivariate elliptical linear regression models. We establish the exact distribution of a pivotal type statistics based on this generalized Mahalanobis distance, which provides critical points for the identification of outlier data points. Based on the equivalence between the modified Cook's distance and what is called the mean-shift multivariate outlier elliptical model, twelve new modifications are proposed for the Cook's distance. We also describe the explicit relationship between the Cook's distance and the likelihood displacement with the modified Cook's distance. We illustrate the procedure with some examples, in the context of multiple and multivariate linear regression.