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Hoeffding's independence test

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Statistical measure

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

H = ∫ ( F 12 − F 1 F 2 ) 2 d F 12 {\displaystyle H=\int (F_{12}-F_{1}F_{2})^{2}\,dF_{12}}

where F 12 {\displaystyle F_{12}} is the joint distribution function of two random variables, and F 1 {\displaystyle F_{1}} and F 2 {\displaystyle F_{2}} are their marginal distribution functions. Hoeffding derived an unbiased estimator of H {\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 H {\displaystyle H} has a defect for discontinuous F 12 {\displaystyle F_{12}} , namely that it is not necessarily zero when F 12 = F 1 F 2 {\displaystyle F_{12}=F_{1}F_{2}} . This drawback can be overcome by taking an integration with respect to d F 1 F 2 {\displaystyle 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.

See also

References

  1. Blum, J.R.; Kiefer, J.; Rosenblatt, M. (1961). "Distribution free tests of independence based on the sample distribution function" (PDF). The Annals of Mathematical Statistics. 32 (2): 485–498. doi:10.1214/aoms/1177705055. JSTOR 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 doi:10.1016/j.stamet.2007.07.002

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.


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