A0313
Title: Generalised covariances and correlations
Authors: Tobias Fissler - ETH Zurich (Switzerland)
Marc-Oliver Pohle - Heidelberg Institute for Theoretical Studies (Germany) [presenting]
Abstract: The covariance of two random variables measures the average joint deviations from their respective means. We generalise this well-known measure by replacing the means with other statistical functionals such as quantiles, expectiles, or thresholds. Deviations from these functionals are defined via generalised errors, often induced by identification or moment functions. As a normalised measure of dependence, a generalised correlation is constructed. Replacing the common Cauchy-Schwarz normalisation with a novel Fr\'echet-Hoeffding normalisation, we obtain the attainability of the entire interval $[-1, 1]$ for any given marginals. We uncover favourable properties of these new dependence measures and establish consistent estimators. The families of quantile and threshold correlations give rise to function-valued distributional correlations, exhibiting the entire dependence structure. They lead to tail correlations, which should arguably supersede the coefficients of tail dependence. Finally, we construct summary covariances (correlations), which arise as (normalised) weighted averages of distributional covariances. We retrieve Pearson covariance and Spearman correlation as special cases. The applicability and usefulness of our new dependence measures are illustrated by demographic data from the Panel Study of Income Dynamics.
