title: Hoeffding's independence test
text: # 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
  
      $H=\int (F_{12}-F_{1}F_{2})^{2}\,dF_{12}$
  
  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 $F_{12}=F_{1}F_{2}$. This drawback can be overcome by taking an integration with respect to $dF_{1}F_{2}$. 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.
  
  