Some indices to measure departures from stochastic order
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This paper deals with three famous statistics involved on two-sample problems. The Mann-Whitney, the one-sided Kolmogorov-Smirnov, and the Galton rank order statistics are invoked here in an unusual way. Looking for indices to capture the disagreement of stochastic dominance of a distribution function G over another F, we resort to suitable couplings (X,Y) of random variables with marginal distribution functions F and G. We show as, the common representation, P(X>Y) under the independent, the contamination and the quantile frameworks give interpretable indices, whose plugin sample-based versions lead to these widely known statistics and can be used for statistical validation of approximate stochastic dominance. This supplies a workaround to the non-viable statistical problem of validating stochastic dominance on the basis of two samples. While the available literature on the asymptotics for the first and second statistics justifies their use for this task, for the Galton statistic the existent results just cover the case where both distributions coincide at a large extent or the case of distribution functions with only one crossing point. In the paper we provide new findings, giving a full picture of the complex behaviour of the Galton statistic: the time that a sample quantile function spent below another. We illustrate the performance of this index through simulations and discuss its application in a case study on the improvement of household wealth distribution in Spain over the period around the recent financial crisis
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