New highly efficient high-breakdown estimator of multivariate scatter and location for elliptical distributions

High-breakdown-point estimators of multivariate location and shape matrices, such as the $\text{MM}$-estimator with smoothed hard rejection and the Rocke $S$-estimator, are generally designed to have high efficiency for Gaussian data. However, many phenomena are non-Gaussian, and these estimators can therefore have poor efficiency. This article proposes a new tunable $S$-estimator, termed the $S_q$-estimator, for the general class of symmetric elliptical distributions, a class containing many common families such as the multivariate Gaussian, $t$-, Cauchy, Laplace, hyperbolic, and normal inverse Gaussian distributions. Across this class, the $S_q$-estimator is shown to generally provide higher maximum efficiency than other leading high-breakdown estimators while maintaining the maximum breakdown point. Furthermore, the $S_q$-estimator is demonstrated to be distributionally robust, and its robustness to outliers is demonstrated to be on par with these leading estimators while also being more stable with respect to initial conditions. From a practical viewpoint, these properties make the $S_q$-estimator broadly applicable for practitioners. These advantages are demonstrated with an example application—the minimum-variance optimal allocation of financial portfolio investments.