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RESEARCH PRODUCT

Influence Functions and Efficiencies of k-Step Hettmansperger–Randles Estimators for Multivariate Location and Regression

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In Hettmansperger and Randles (Biometrika 89:851–860, 2002) spatial sign vectors were used to derive simultaneous estimators of multivariate location and shape. Oja (Multivariate nonparametric methods with R. Springer, New York, 2010) proposed a similar approach for the multivariate linear regression case. These estimators are highly robust and have under general assumptions a joint limiting multinormal distribution. The estimates are easy to compute using fixed-point algorithms. There are however no exact proofs for the convergence of these algorithms. The existence and uniqueness of the solutions also still remain unproven although we believe that they hold under general conditions. To circumvent these problems, we consider in this paper k-step versions of Hettmansperger and Randles (HR) location and shape estimators and their extensions to the linear regression problem. The influence functions, limiting distributions and asymptotical efficiencies of the estimators are derived at the multivariate elliptical case.