New Highly Efficient High-Breakdown Estimator of Multivariate Scatter and Location for Elliptical Distributions

TL;DR

This paper introduces the S-q estimator, a high-breakdown, tunable estimator for multivariate elliptical distributions, offering higher efficiency and robustness compared to existing methods, with practical applications in financial portfolio optimization.

Contribution

The paper proposes the S-q estimator, a new tunable high-breakdown estimator for elliptical distributions, improving efficiency and robustness over existing estimators.

Findings

S-q estimator achieves higher maximum efficiency across elliptical distributions.

Maintains high breakdown point and robustness comparable to existing estimators.

Demonstrated effectiveness in financial portfolio optimization.

Abstract

High-breakdown-point estimators of multivariate location and shape matrices, such as the MM-estimator with smooth hard rejection and the Rocke S-estimator, are generally designed to have high efficiency at the Gaussian distribution. However, many phenomena are non-Gaussian, and these estimators can therefore have poor efficiency. This paper 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, its robustness 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 broadly applicable for practitioners. This is demonstrated with an example application -- the minimum-variance optimal allocation of financial portfolio investments.