Hoeffding's independence test explained

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.^[1]

A paper published in 2008^[2] 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.

See also

• Correlation
• Kendall's tau
• Spearman's rank correlation coefficient
• Distance correlation

Primary sources

• Wassily Hoeffding, A non-parametric test of independence, Annals of Mathematical Statistics 19: 293 - 325, 1948. (JSTOR)
• Hollander and Wolfe, Non-parametric statistical methods (Section 8.7), 1999. Wiley.

Notes and References

1. Blum. J.R.. Kiefer. J.. Rosenblatt. M.. Distribution free tests of independence based on the sample distribution function. The Annals of Mathematical Statistics. 1961 . 32. 485–498. 2. 10.1214/aoms/1177705055 . 2237758 .
2. Wilding, G.E., Mudholkar, G.S. (2008) "Empirical approximations for Hoeffding's test of bivariate independence using two Weibull extensions", Statistical Methodology, 5 (2), 160- - 170