On efficiency and robustness of estimators for a spherical location

ABSTRACT

We study M-estimators (sample mean direction and normalized spatial median), restricted M-estimators (maximum likelihood estimator (MLE), Watson estimator and $L_1$-estimator) and R-estimators (spherical median and spherical Wilcoxon estimator) for the location of a rotationally symmetric distribution on the unit hypersphere. The influence function and asymptotic distribution of an R-estimator are derived for a general density. Asymptotically most efficient estimators are obtained in classes of restricted M-estimators and R-estimators. In terms of gross error sensitivity, the spherical median is shown to dominate over all other estimators mentioned above under certain conditions. Explicit expressions for asymptotic relative efficiencies and gross error sensitivities of various estimators are derived for Langevin and mixture Langevin models. As a consequence the trade-off between robustness and efficiency amongst various estimators has been explored.

KEYWORDS:

• Asymptotic relative efficiency
• directional data
• gross error sensitivity
• M-estimator
• R-estimator

AMS SUBJECT CLASSIFICATIONS:

• 62H11
• 62F20
• 62F35

Acknowledgments

We thank an associate editor and referees for constructive comments and suggestions which have considerably improved the presentation and organization of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.