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The complex behaviour of Galton rank-order statistic

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.

Otras publicaciones de la misma revista o congreso con autores/as de la Universidad de Cantabria

 Fuente: Bernoulli, 2022, 28(4), 2123-2150

Editorial: International Statistical Institute; Chapman and Hall

 Año de publicación: 2022

Nº de páginas: 28

Tipo de publicación: Artículo de Revista

 DOI: 10.3150/21-BEJ1406

ISSN: 1350-7265,1573-9759

Proyecto español: MTM2017-86061-C2-1-P

Url de la publicación: https://doi.org/10.3150/21-BEJ1406