Hoeffding's independence test

In statistics, Hoeffding's test of independence, named after Wassily Hoeffding, is a test based on the population measure of deviation from independence


 * $$H = \int (F_{12}-F_1F_2)^2 \, dF_{12} $$

where $$F_{12}$$ is the joint distribution function of two random variables, and $$F_1$$ and $$F_2$$ are their marginal distribution functions. Hoeffding derived an unbiased estimator of $$H$$ that can be used to test for independence, and is consistent for any continuous alternative. The test should only be applied to data drawn from a continuous distribution, since $$H$$ has a defect for discontinuous $$F_{12}$$, namely that it is not necessarily zero when $$F_{12}=F_1F_2$$. This drawback can be overcome by taking an integration with respect to $$dF_1F_2$$. This modified measure is known as Blum–Kiefer–Rosenblatt coefficient.

A paper published in 2008 describes both the calculation of a sample based version of this measure for use as a test statistic, and calculation of the null distribution of this test statistic.