C. Matran | Searcharxiv

arXiv2021

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 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.

math.ST↗

arXiv2018

Invariant measures of disagreement with stochastic dominance

Stochastic dominance has not been too employed in practice due to its important limitations. To increase its versatility, the concept has recently been adapted by introducing various indices that measure the degree to which one probability distribution stochastically dominates another. In this paper, starting from the fundamentals and using very simple examples, we present and discuss some of these indices when one intends to maintain invariance through increasing functions. This naturally leads to consideration of the appealing common representation, $\theta(F,G)=P(X>Y)$, where $(X, Y)$ is a random vector with marginal distributions $F$ and $G$. The indices considered here arise from different dependencies between X and Y. This includes the case of independent marginals, as well as other indices related to a contamination model or to a joint quantile representation. We emphasize the complementary role of some of these indices, which, in addition to measuring disagreement with respect to stochastic dominance, enable us to describe the maximum possible difference in the status of a value $x\in \Rea$ under $F$ or $G$. We apply these indices to simulated and real-world datasets, exploring their practical advantages and limitations. The tour includes lesser-known facets of well-known statistics such as Mann-Whitney, one-tailed Kolmogorov-Smirnov and Galton's rank statistics, even providing additional theory for the latter.

stat.ME↗