The complex behaviour of Galton rank order statistic

TL;DR

This paper provides a comprehensive analysis of the asymptotic behaviour of Galton's rank order statistic, revealing conditions for convergence and characterizing the effects of contact set properties on its limiting distribution.

Contribution

It introduces a detailed study of the contact set and its influence on the asymptotic properties of the rank order statistic, including new insights into higher order contacts and extreme support contacts.

Findings

Convergence occurs if and only if the contact set has zero Lebesgue measure.

Regular crossings lead to standard rates and Gaussian limits.

Higher order contacts or extreme support contacts can cause different rates and non-Gaussian limits.

Abstract

Galton's rank order statistic is one of the oldest statistical tools for two-sample comparisons. It is also a very natural index to measure departures from stochastic dominance. Yet, its asymptotic behaviour has been investigated only partially, under restrictive assumptions. This work provides a comprehensive {study} of this behaviour, based on the analysis of the so-called contact set (a modification of the set in which the quantile functions coincide). We show that a.s. convergence to the population counterpart holds if and only if {the} contact set has zero Lebesgue measure. When this set is finite we show that the asymptotic behaviour is determined by the local behaviour of a suitable reparameterization of the quantile functions in a neighbourhood of the contact points. Regular crossings result in standard rates and Gaussian limiting distributions, but higher order contacts (in the sense introduced in this work) or contacts at the extremes of the supports may result in different rates and non-Gaussian limits.