Estimation of Multivariate Location and Covariance using the S -Hellinger Distance

TL;DR

This paper introduces the S-Hellinger distance, a flexible divergence measure for robustly estimating multivariate location and covariance, demonstrating its advantages over traditional methods through theoretical properties and simulations.

Contribution

It generalizes the Hellinger distance within the S-divergence family, providing a robust, affine-equivariant estimator with proven consistency and high breakdown point.

Findings

The S-Hellinger estimator is affine equivariant and asymptotically consistent.

It exhibits high robustness with a breakdown point under suitable conditions.

Simulation results show improved robustness over the minimum Hellinger distance estimator.

Abstract

This paper describes a generalization of the Hellinger distance which we call the S -Hellinger distance; this general family connects the Hellinger distance smoothly with the $L_2$-divergence by a tuning parameter $\alpha$ and is indeed a subfamily of the S -Divergence family of Ghosh et al. (2013 a, b). We use this general divergence in the context of estimating the location and covariances under (continuous) multivariate models and show that the proposed minimum S -Hellinger distance estimator is affine equivariant, asymptotically consistent and have high breakdown point under suitable conditions. We also illustrate its performance through an extensive simulation study which show that the proposed estimators give more robust estimator than the minimum Hellinger distance estimator for the location and correlation parameters under different types of contamination with the contamination proportion being as high as 20%.