Robust M-Estimators on Spheres

We introduce M-estimators for location and concentration parameters of von Mises-Fisher distributions on unit spheres. These include the directional mean, normalized spatial median, spherical median, and the mle of the concentration parameter. We find the influence functions and asymptotic distributions of such estimators and give necessary and sufficient conditions under which the M-estimators become SB-robust. SB-robust M-estimators, which are optimal in a sense similar to Hampel, are proposed. We discuss both simultaneous estimation of location and concentration and estimation of one parameter when the other is known. The behavior of the optimal estimators, together with several alternatives, under extreme contamination and moderate sample sizes is studied using simulation. An example, previously studied by Fisher, Lewis, and Embleton (1987, Analysis of Spherical Data, Cambridge Univ. Press, Cambridge, UK), dealing with remanent magnetization is reanalyzed using these techniques.