We consider the problem of testing whether pairs of univariate random variables are associated. Few tests of independence exist that are consistent against all dependent alternatives and are distribution free. We propose novel tests that are consistent, distribution free, and have excellent power properties. The tests have simple form, and are surprisingly computationally efficient thanks to accompanying innovative algorithms we develop. Moreover, we show that one of the test statistics is a consistent estimator of the mutual information. We demonstrate the good power properties in simulations, and apply the tests to a microarray study where many pairs of genes are examined simultaneously for co-dependence.
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