On finite-sample robustness of directional location estimators

Robust location estimators for directional data are known for about 30 years. Scientific literature has focused on studying the asymptotic properties of these estimators like consistency and influence function. Apart from the finite-sample breakdown point, the finite-sample performance of robust directional location estimators has attracted less attention. Hence, it is discussed how the finite-sample max-bias of directional location estimators can be evaluated. Additionally, two new robust estimators of the mean direction are introduced: the spherical Minimum Covariance Determinant estimator (sMCD) and the spherical Minimum Spanning Tree estimator (sMST). The sMCD seeks to identify the densest subset of a given size while the sMST seeks for a well-separated subset. Finally, the robust estimators are compared with respect to the max-bias and to the bias under shift outlier scenarios by means of an extensive simulation study. The results indicate that –in contrast to linear data– the maximum likelihood estimator shows high robustness in terms of the finite-sample max-bias. However, robust estimators are clearly superior to the maximum likelihood estimator in shift outlier contamination schemes.