Statistics and Probability Letters 82 (2012) 1145–1150 Contents lists available at SciVerse ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro Strongly consistent nonparametric tests of conditional independence László Györfi a , Harro Walk b,∗ a Department of Computer Science and Information Theory, Budapest University of Technology and Economics, 1521 Stoczek u. 2, Budapest, Hungary b Department of Mathematics, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany article info Article history: Received 29 July 2011 Accepted 23 February 2012 Available online 13 March 2012 Keywords: Conditional independence Nonparametric test Partition Distribution-free strong consistency abstract A simple and explicit procedure for testing the conditional independence of two multi- dimensional random variables given a third random vector is described. The associated L 1 -based test statistic is defined for when the empirical distribution of the variables is restricted to finite partitions. Distribution-free strong consistency is proved. © 2012 Elsevier B.V. All rights reserved. 1. Introduction Consider an R d × R d ′ × R d ′′ -valued random vector (X , Y , Z ). We are interested in testing the null hypothesis that X and Y are conditionally independent given Z : H 0 : P{X ∈ A, Y ∈ B | Z }= P{X ∈ A | Z }P{Y ∈ B | Z } a.s. for arbitrary Borel sets A, B. There are many independence testing approaches in the statistics literature. For d = d ′ = 1, an early nonparametric test for independence, due to Hoeffding (1948), Blum et al. (1961), De Wet (1980), Cotterill and Csörgő (1985), is based on the notion of differences between the joint distribution function and the product of the marginals. The associated independence test is consistent under appropriate assumptions. Rosenblatt (1975) defined the statistic as the L 2 distance between the joint density estimate and the product of marginal density estimates. A second approach is to base the test on characteristic functions. One characteristic function-based independence test uses as its statistic the difference between the joint empirical characteristic function and the product of the marginal empirical characteristic functions at a particular point, chosen according to a variance maximization argument in Csörgő (1985). An alternative approach is to take a smoothed difference between the joint characteristic function and the product of the marginals, as proposed by Feuerverger (1993). Gretton and Györfi (2010) introduced nonparametric tests of independence of random vectors and proved their strong consistencies. As further literature we mention Bakirov et al. (2006), Beran et al. (2007), Dauxois and Nkiet (1998), Fukumizu et al. (2007), Gieser and Randles (1997), Gretton et al. (2005), Kankainen (1995), Genest et al. (2007) and Dette and Neumeyer (2000) with references. For nonparametric testing of conditional independence, in the context of index functions, Linton and Gozalo (1996) proposed and investigated tests of Kolmogorov–Smirnov and Cramér–von Mises type with a generalization of empirical distribution functions using rectangles, while Song (2009) used Rosenblatt transforms. Under a density assumption, Su and White (2008a,b) used Hellinger metric and characteristic functions, respectively, relaxing the independence assumption for ∗ Corresponding author. Tel.: +49 711 68565387; fax: +49 711 68565389. E-mail addresses: gyorfi@szit.bme.hu (L. Györfi), walk@mathematik.uni-stuttgart.de (H. Walk). 0167-7152/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2012.02.023