On Kendall's Tau for Order Statistics

TL;DR

This paper investigates the dependence structure of order statistics in multivariate distributions using Kendall's tau, establishing bounds and explicit formulas for the copula of order statistics, especially for the independent case.

Contribution

It proves that Kendall's tau of order statistic copulas is at least as large as that of the original copulas and provides explicit formulas for the independent case.

Findings

Kendall's tau of order statistic copulas is at least as large as that of the original copulas.

Explicit formula for Kendall's tau of the product copula's order statistic copula.

Kendall's tau for certain margins of the order statistic copula are computed.

Abstract

Every copula $ C $ for a random vector $ {\bf X}=(X_1,\dots,X_d) $ with identically distributed coordinates determines a unique copula $ C_{:d} $ for its order statistic $ {\bf X}_{:d}=(X_{1:d},\dots,X_{d:d}) $. In the present paper we study the dependence structure of $ C_{:d} $ via Kendall's tau, denoted by $ \kappa $. As a general result, we show that $ \kappa[C_{:d}] $ is at least as large as $ \kappa[C] $. For the product copula $ \Pi $, which corresponds to the case of independent coordinates of $ {\bf X} $, we provide an explicit formula for $ \kappa[\Pi_{:d}] $ showing that the inequality between $ \kappa[\Pi] $ and $ \kappa[\Pi_{:d}] $ is strict. We also compute Kendall's tau for certain multivariate margins of $ \Pi_{:d} $ corresponding to the lower or upper coordinates of $ {\bf X}_{:d} $.