A Robust Similarity Estimator

Abstract

We analyze a measure of statistical association based on the similarity of the outcomes of random variables, in both sign and magnitude. Motivated by its attractive properties, we propose a class of estimators for the linear correlation coefficient, a sample-average and a maximum-likelihood version, that operate directly on the Fisher scale and possess a robust sampling distribution that is invariant over the entire class of elliptical distributions. Under scale homogeneity, the finite-sample distribution of the estimator is available in exact form, facilitating robust inference for correlations even in small samples. The similarity measure extends naturally to higher dimensions, where it admits an interpretation as an indicator of joint similarity among multiple random variables. As empirical applications, we construct robust confidence intervals for financial correlations using intraday returns and develop a new specification of a multivariate GARCH model with robust correlation dynamics.