The complex behaviour of Galton rank-order statistic

Galton’s rank-order statistic is one of the oldest statistical tools for two-sample comparisons. It 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 comprehensive study this behaviour, based on analysis so-called contact set (a modification in which quantile functions coincide). We show that a.s. convergence population counterpart holds if and zero Lebesgue measure. When finite we determined by local suitable reparameterization neighbourhood points. Regular crossings result standard rates Gaussian limiting distributions, but higher order contacts (in sense introduced work) or at extremes supports may different non-Gaussian limits.