Which Robust Versions of Sample Variance and Sample Covariance Are Most Appropriate for Econometrics: Symmetry-Based Analysis

Songsak Sriboonchitta, Ildar Batyrshin, Vladik Kreinovich

Abstract

In many practical situations, we do not know the shape of the corre-sponding probability distributions and therefore, we need to use robust statisticaltechniques, i.e., techniques that are applicable to all possible distributions. Em-pirically, it turns out the the most efficient robust version of sample variance isthe average value of the p-th powers of the deviations |x_i − â| from the (esti-mated) mean â. In this paper, we use natural symmetries to provide a theoreticalexplanation for this empirical success, and to show how this optimal robust ver-sion of sample variance can be naturally extended to a robust version of samplecovariance.