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A common method for deriving non-parametric tests is to reformulate aparametric test in terms of sample ranks. Despite being distribution free (evenin finite samples), the resulting tests often display remarkable asymptoticpower properties, typically matching the efficiency of their parametriccounterpart. Empirically, these favorable power properties have been shown topersist in non-asymptotic regimes as well, prompting the need for finite-samplecharacterizations of the corresponding rank-based statistics. Here, we providesuch characterization for the family of weighted $p$-norms of rank spacings,which includes the classical tests of Mann-Whitney, Dixon, and variousgeneralizations thereof. For $p=1$, we provide exact expressions for theinvolved distributions, while for $p>1$ we describe the associated momentsequences and derive an algorithm to recover the distributions of interest fromthese sequences in a fast and stable manner. We use this framework to develop anew family of non-parametric tests mirroring properties of generalizedlikelihood-ratios, prove new tail bounds for Dixon's and Greenwood'sstatistics, and prove a previously formulated conjecture regarding the globalefficiency of rank-based tests against the $F$-test in the context ofscale-families.

Exact and arbitrarily accurate non-parametric two-sample tests based on rank spacings