Robustness properties of k means and trimmed k means

The generalized k means method is based on the minimization of the discrepa ncy between a random variable (or a sample of this random variable) and a s et with ii points measured through a penalty function Phi. As in the M esti mators setting (k = 1), a penalty function, Phi, with unbounded derivative, Psi, naturally leads to nonrobust generalized k means. However, surprising ly the lack of robustness extends also to the case of bounded Psi; that is, generalized k means do not inherit the robustness properties of the M esti mator from which they came. Attempting to robustify the generalized k means method, the generalized trimmed ic means method arises from combining fi m eans idea with a so-called impartial trimming procedure. In this article st udy generalized k means and generalized trimmed k means performance from th e viewpoint of Hampel's robustness criteria; that is, we investigate the in fluence function, breakdown point, and qualitative robustness, confirming t he superiority provided by the trimming. We include the study of two real d atasets to make clear the robustness of generalized trimmed k means.