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On the sampling distribution of an $\ell^2$ distance between Empirical Distribution Functions with applications to nonparametric testing

hal.structure.identifierAdvanced Learning Evolutionary Algorithms [ALEA]
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorCARON, Francois
hal.structure.identifierDepartment of Statistics [Oxford]
dc.contributor.authorHOLMES, Chris
hal.structure.identifierLaboratoire de Mathématiques de Versailles [LMV]
dc.contributor.authorRIO, Emmanuel
dc.date.issued2012-04
dc.description.abstractEnWe consider a situation where two sample sets of independent real valued observations are obtained from unknown distributions. Under a null hypothesis that the distributions are equal, it is well known that the sample variation of the infinity norm, maximum, distance between the two empirical distribution functions has as asymptotic density of standard form independent of the unknown distribution. This result underpins the popular two-sample Kolmogorov-Smirnov test. In this article we show that other distance metrics exist for which the asymptotic sampling distribution is also available in standard form. In particular we describe a weighted squared-distance metric derived from a binary recursion of the real line which is shown to follow a sum of chi-squared random variables. This motivates a nonparametric test based on the average divergence rather than the maximum, which we demonstrate exhibits greater sensitivity to changes in scale and tail characteristics when the distributions are unequal, while maintaining power for changes in central location.
dc.language.isoen
dc.title.enOn the sampling distribution of an $\ell^2$ distance between Empirical Distribution Functions with applications to nonparametric testing
dc.typeRapport
dc.subject.halMathématiques [math]/Statistiques [math.ST]
dc.subject.halStatistiques [stat]/Théorie [stat.TH]
bordeaux.type.institutionINRIA
bordeaux.type.reportrr
hal.identifierhal-00688141
hal.version1
hal.audienceNon spécifiée
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-00688141v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.date=2012-04&rft.au=CARON,%20Francois&HOLMES,%20Chris&RIO,%20Emmanuel&rft.genre=unknown


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