Cramér-Von Mises Statistic for Repeated Measures

Abstract

The Cramér-von Mises criterion is employed to compare whether the marginal distribution functions of a k-dimensional random variable are equal or not. The well-known Donsker invariance principle and the KarhunenLoéve expansion is used in order to derive its asymptotic distribution. Two different resampling plans (one based on permutations and the other one based on the general bootstrap algorithm, gBA) are also considered to approximate its distribution. The practical behaviour of the proposed test is studied from a Monte Carlo simulation study. The statistical power of the test based on the Cramér-von Mises criterion is competitive when the underlying distributions are different in location and is clearly better than the Friedman one when the sole difference among the involved distributions is the spread or the shape. Both resampling plans lead to similar results although the gBA avoids the usual required interchangeability assumption. Finally, the method is applied on the study of the evolution inequality incomes distribution between some European countries along the years 2000 and 2011.Key words: Asymptotic Distribution, Bootstrap, Cramér-von Mises statistic, Hypothesis testing, Permutation test, Repeated Measures. ResumenEl criterio de Cramér-von Mises es empleado para comparar la igualdad entre las distribuciones marginales de una variable aleatoria k-dimensional. El conocido principio de invaranza de Donsker y la expansión de KarhunenLoéve se usan para derivar su distribución asintótica. Dos planes de remuestreo diferentes (uno basado en permutaciones y el otro basado en el algoritmo bootstrap general, gBA) son usados para aproximar su distribución. El comportamiento práctico del test propuesto es estudiado mediante simulaciones de Monte Carlo. La potencia estadística del test basado en el criterio

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