# Consistent Nonparametric Tests of Independence

Arthur Gretton, László Györfi.

Year: 2010, Volume: 11, Issue: 46, Pages: 1391−1423

#### Abstract

Three simple and explicit procedures for testing the independence
of two multi-dimensional random variables are described. Two
of the associated test statistics (*L _{1}*,
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.
Two kinds of tests are provided.
Distribution-free strong consistent tests are derived
on the basis of large deviation bounds on the test statistics: these tests
make almost surely no Type I or Type II error
after a random sample size.
Asymptotically

*α*-level tests are obtained from the limiting distribution of the test statistics. For the latter tests, the Type I error converges to a fixed non-zero value

*α*, and the Type II error drops to zero, for increasing sample size. All tests reject the null hypothesis of independence if the test statistics become large. The performance of the tests is evaluated experimentally on benchmark data.