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.
翻译:Galton的等级顺序统计是用于两样抽样比较的最古老的统计工具之一。 它也是一个非常自然的指数,用来衡量偏离随机支配地位的情况。 然而,根据限制性假设,对它的无药可治行为只进行了部分调查。 这项工作根据对所谓的接触组的分析(对量函数重合的数据集的修改),为这种行为提供了全面的{研究}。 我们显示,a.s.s.如果而且只有当[特 接触组具有零 Lebesgue 的测量值时,与人口对应方的趋同才会维持。 当设定为有限时,我们将显示,无药可治行为是由联络点附近对四分功能进行适当重新校准的当地行为所决定的。 定期过境导致标准比率和高斯限制分布,但更高排序的接触(在这项工作中引入的意义上)或最极端支持点的接触可能会导致不同的比率和非Gaussian限制。