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
- [math]\displaystyle{ H = \int (F_{12}-F_1F_2)^2 \, dF_{12} }[/math]
where [math]\displaystyle{ F_{12} }[/math] is the joint distribution function of two random variables, and [math]\displaystyle{ F_1 }[/math] and [math]\displaystyle{ F_2 }[/math] are their marginal distribution functions. Hoeffding derived an unbiased estimator of [math]\displaystyle{ H }[/math] 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 [math]\displaystyle{ H }[/math] has a defect for discontinuous [math]\displaystyle{ F_{12} }[/math], namely that it is not necessarily zero when [math]\displaystyle{ F_{12}=F_1F_2 }[/math]. This drawback can be overcome by taking an integration with respect to [math]\displaystyle{ dF_1F_2 }[/math]. 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
References
- ↑ Blum, J.R.; Kiefer, J.; Rosenblatt, M. (1961). "Distribution free tests of independence based on the sample distribution function". The Annals of Mathematical Statistics 32 (2): 485–498. doi:10.1214/aoms/1177705055. https://www.jstor.org/stable/pdf/2237758.pdf.
- ↑ 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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