Cramér-Von Mises Statistic for Repeated Measures

Pablo Martínez-Camblor,Carlos Carleos,Norberto Corral

doi:10.15446/rce.v37n1.44357

Abstract

The Cramer-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 Karhunen-Loeve 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 Cramer-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.

Highlights

The comparison of the equality among the marginal distribution functions of a k-dimensional random variable is a common problem in statistical inference (for example, in biomedicine, in problems of comparing diagnostic procedures or bioequivalence (Freitag, Czado & Munk 2007)
In a non exhaustive revision: Ciba-Geigy & Olsson (1982) developed a specific one for comparing dispersion in paired samples design; Lam & Longnecker (1983) introduced modifications which improve the power of the classical Wilcoxon rank sum test for this topic; Munzel (1999a) used the normalized version of distribution functions to derive an asymptotic theory for rank statistics including ties and considered a mixed model which permits almost arbitrary dependences; Munzel (1999b) studied different nonparametric permutation methods for repeated measures problems in a two sample framework; most recently, Freitag et al (2007) proposed a test based on the Mallows distance with this goal
Due to our objective is not to study the incomes distribution but the inequalities of these incomes, we have considered the relative Gross Domestic Product (GDP) per capita in Purchasing Power Standards (PPS) distribution i.e., the considered variable are 100 times the original values divided by the European Union one and the particular mean has been sustracted

Summary

Introduction

The comparison of the equality among the marginal distribution functions of a k-dimensional random variable is a common problem in statistical inference (for example, in biomedicine, in problems of comparing diagnostic procedures or bioequivalence (Freitag, Czado & Munk 2007). There exists a number of methods of comparing the equality among k-distributions from independent samples, the k-sample problem for dependent data has not been as widely studied and, the traditional parametric (ANOVA) and nonparametric (Friedman test) repeated measures procedures are the usual used techniques to solve these problems In this context, several rank tests have been proposed. The Donsker invariance principle and the classical Gaussian processes theory, in particular, the Karhunen-Loève expansion, are used in order to obtain (a not explicit version of) the asymptotic distribution for the Cramér-von Mises statistic when the samples are from the same subjects The properties of this statistic allow to develop a resampling procedure which does not need the (usual) interchangeability (or sphericity) assumption. In order to keep this independence and, in spite of several reported results are overlapping with the obtained by QE, we have maintained them in the appendix

Cramér-von Mises Statistic for Repeated Measures

Bootstrap Approximation

Simulation Study

Inequality Incomes Analysis

Main Conclusions