Nonparametric Independence Tests: Space Partitioning and Kernel Approaches

Abstract. Three simple and explicit procedures for testing the independence of two multi-dimensional random variables are described. Two of the associated test statistics (L1, log-likelihood) are defined when the empirical distribution of the variables is restricted to finite partitions. A third test statistic is defined as a kernel-based independence measure. All tests reject the null hypothesis of independence if the test statistics become large. The large deviation and limit distribution properties of all three test statistics are given. Following from these results, distributionfree strong consistent tests of independence are derived, as are asymptotically α-level tests. The performance of the tests is evaluated experimentally on benchmark data. Consider a sample of Rd × Rd′ -valued random vectors (X1, Y1), . . . , (Xn, Yn) with independent and identically distributed (i.i.d.) pairs defined on the same probability space. The distribution of (X, Y ) is denoted by ν, while �...

• Type: Conference
• Year: 2008
• Where: ALT
• Authors: Arthur Gretton, László Györfi
• Keywords: ALT 2008, Machine Learning, Multi-dimensional Random Variables, Multivariate Hypothesis Tests, Test Statistic