**Measuring and testing dependence by correlation of distances**

Distance correlation is a new measure of dependence between random vectors. Distance covariance and distance correlation are analogous to product-moment covariance and correlation, but unlike the classical definition of correlation, distance correlation is zero only if the random vectors are independent. The empirical distance dependence measures are based on certain Euclidean distances between sample elements rather than sample moments, yet have a compact representation analogous to the classical covariance and correlation. Asymptotic properties and applications in testing independence are discussed. Implementation of the test and Monte Carlo results are also presented.

R/MATLAB package: energy

http://arxiv.org/abs/1010.0297

**A smoothed bootstrap test for independence based on mutual information**

A test for independence of multivariate time series based on the mutual information measure is proposed. First of all, a test for independence between two variables based on i.i.d. (time-independent) data is constructed and is then extended to incorporate higher dimensions and strictly stationary time series data. The smoothed bootstrap method is used to estimate the null distribution of mutual information. The experimental results reveal that the proposed smoothed bootstrap test performs better than the existing tests and can achieve high powers even for moderate dependence structures. Finally, the proposed test is applied to assess the actual independence of components obtained from independent component analysis (ICA).

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