Nonparametric Independence Testing for Small Sample Sizes

It is also useful for scientific discovery like in neuroscience, like correlation of X, Y only test for (univariate) to see if a stimulus X (say an image) is independent linear independence, natural alternatives like of the brain activity Y (say fMRI) in a relevant part of mutual information of X, Y are hard to estimate the brain. Since detecting nonlinear correlations is much easier due to a serious curse of dimensionality. A recent than estimating a nonparametric regression function (of approach, avoiding both issues, estimates norms of Y onto X), it can be done at smaller sample sizes, with further an operator in Reproducing Kernel Hilbert Spaces samples collected for estimation only if an effect is detected (RKHSs). Our main contribution is strong empirical by the hypothesis test. For such situations, correlation evidence that by employing shrunk operators only tests for univariate linear independence, while other when the sample size is small, one can attain an improvement statistics like mutual information that do characterize multivariate in power at low false positive rates. We independence are hard to estimate from data, suffering analyze the effects of Stein shrinkage on a popular from a serious curse of dimensionality. A recent popular test statistic called HSIC (Hilbert-Schmidt Independence approach for this problem (and a related two-sample testing Criterion). Our observations provide insights problem) involve the use of quantities defined in reproducing into two recently proposed shrinkage estimators, kernel Hilbert spaces (RKHSs) - see [Gretton et al., 2006; SCOSE and FCOSE - we prove that SCOSE Harchaoui et al., 2007; Gretton et al., 2005b; 2005a].