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

${\displaystyle H=\int (F_{12}-F_{1}F_{2})^{2}\,dF_{12}}$

where ${\displaystyle F_{12}}$ is the joint distribution function of two random variables, and ${\displaystyle F_{1}}$ and ${\displaystyle F_{2}}$ are their marginal distribution functions. Hoeffding derived an unbiased estimator of ${\displaystyle H}$ 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 ${\displaystyle H}$ has a defect for discontinuous ${\displaystyle F_{12}}$, namely that it is not necessarily zero when ${\displaystyle F_{12}=F_{1}F_{2}}$.

A recent paper[1] 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.