A Non-Parametric Test of Independence

A test is proposed for the independence of two random variables with continuous distribution function (d. f. ). The test is consistent with respect to the class '' of d. f. 's with continuous joint and marginal probability densities (p. d. ). The test statistic D depends only on the rank order of the observations. The mean and variance of D are given and n (D - ED) is shown to have a normal limiting distribution for any parent distribution. In the case of independence this limiting distribution is degenerate, and nD has a non-normal limiting distribution whose characteristic function and cumulants are given. The exact distribution of D in the case of independence for samples of size n = 5, 6, 7 is tabulated. In the Appendix it is shown that there do not exist tests of independence based on ranks which are unbiased on any significance level with respect to the class ''. It is also shown that if the parent distribution belongs to '' and for some n 5 the probabilities of the n! rank permutations are equal, the random variables are independent.