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

In many practical situations, we do not know the shape of the corresponding probability distributions and therefore, we need to use robust statistical techniques, i.e., techniques that are applicable to all possible distributions. Empirically, it turns out the the most efficient robust version of sample variance is the average value of the p-th powers of the deviations |xi − â| from the (estimated) mean â. In this paper, we use natural symmetries to provide a theoretical explanation for this empirical success, and to show how this optimal robust version of sample variance can be naturally extended to a robust version of sample covariance. 1 Formulation of the Problem Need to determine a parameter: traditional case. Often, we observe a sample of several instances x1, . . . , xn of a random variable X. In many practical situations, we know that the random variable X has a distribution with the probability density function (pdf) ρ(x) = ρ0(x−a), where ρ0(x) is a known function, and a is an unknown parameter. For example, X may be the measurement result, which can be represented as X = a+X0, where