Nonparametric tests for independence: a review and comparative simulation study with an application to malnutrition data
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Nonparametric tests for independence: a review and comparative simulation study with an application to malnutrition data in India Helmut Herwartz1 · Simone Maxand2 Received: 4 January 2018 / Revised: 24 July 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract The detection of dependence structures within a set of random variables provides a valuable basis for a detailed subsequent investigation of their relationships. Nonparametric tests for independence require only basic assumptions on the marginal or joint distribution of the involved variables. In this paper, we review nonparametric tests of independence in bivariate as well as multivariate settings which are throughout ready-to-use, i.e., implemented in R packages. Highlighting their distinct empirical size and power properties in various small sample settings, our analysis supports an analyst in deciding for a most adequate test conditional on underlying distributional settings or data characteristics. Avoiding restrictive moment conditions, the copula based Cramér-von Mises distance of Genest and Rémillard (Test 13:335–370, 2004) is remarkably robust under the null hypothesis and powerful under diverse settings that are in line with the alternative hypothesis. Based on distinguished test outcomes in small samples, we detect nonlinear dependence structures between childhood malnutrition indices and possible determinants in an empirical application for India. Keywords Tests for independence · Nonparametric methods · Multivariate independence · Spatial ranks · Empirical copula · Distance covariance Mathematics Subject Classification 62G10 · 62H15 · 62P10
1 Introduction Statistical analyses mostly target at the identification and quantification of dependence structures between the variables of interest. Yet, dependence between random
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Simone Maxand [email protected]
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Department of Economics, University of Goettingen, Göttingen, Germany
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Department of Political and Economic Studies, University of Helsinki, P. O. Box 17, 00014 Helsinki, Finland
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H. Herwartz, S. Maxand
variables can be present in various (e.g., linear or nonlinear) forms. Most commonly, analysts apply standard linear regression models presuming linear dependence structures. Whereas classical procedures, such as Pearson’s correlation coefficient (e.g., Pearson 1920) or Wilks’ test (Wilks 1935), diagnose linear dependence in a parametric framework, they might fail to detect nonlinear and nonmonotone dependence structures. Therefore, nonparametric tests aim at keeping prior assumptions on the variables’ distribution under the null hypothesis and their relation under the alternative hypothesis at a minimum. Classical nonparametric approaches have been developed to test for monotone, but not necessarily linear, bivariate dependence structures by means of ranks. Popular representatives for rank based dependence measures are Kendall’s tau (Kendall 1938) and Spearman’s rho (Spearman 1904). Such bivariate dependence tests, however, might lack consistency unde
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