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hal-00688141, version 2

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

Francois Caron (, ) a12, Chris Holmes () b3, Emmanuel Rio c4

N° RR-7931 (2012)

Abstract: We 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.

• a –  INRIA
• b –  University of Oxford
• c –  Université de Versailles Saint-Quentin-en-Yvelines
• 1:  ALEA (INRIA Bordeaux - Sud-Ouest)
• INRIA – Université de Bordeaux – CNRS : UMR5251
• 2:  Institut de Mathématiques de Bordeaux (IMB)
• CNRS : UMR5251 – Université Sciences et Technologies - Bordeaux I – Université Victor Segalen - Bordeaux II
• 3:  Department of Statistics [Oxford]
• University of Oxford
• 4:  Laboratoire de Mathématiques de Versailles (LM-Versailles)
• CNRS : UMR8100 – Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)

• hal-00688141, version 2
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• Submitted on: Thursday, 22 November 2012 14:14:37
• Updated on: Friday, 23 November 2012 14:25:05