Journal of Mathematical Sciences, Vol. 99, No. 2, 2000 A GENERALIZATION OF KENDALL'S TAU AND THE ASYMPTOTIC EFFICIENCY OF THE CORRESPONDING INDEPENDENCE TEST Ya. Yu. Nikitin and N. A. Stepanova UDC 519.2 We consider a generalization of Kendall's rank correlation coefficient proposed by Kochar and Gupta in 1987. This generalization is a nondegenerate U-statistic with a special kernel. We calculate Bahadur's local efficiency for the independence test based on this statistic. It is shown that this test is locally efficient for alternatives described by the Woodworth dependence function. Bibligraphy: 29 titles. Kendall's rank correlation coefficient is a universally recognized measure of correlation between quali- tative and quantitative characteristics. Consider a sample (X1, ]I1),... , (Xn, Yn) from a two-dimensional distribution with an absolutely continuous distribution function (d.f.) F(x, y) and marginal d.f. G(x) and H(y). The statistic 4 ~-n- n(n-1) ~ I{(Xi-Xj)(Yi-Yj) >0}-1 (1) l~i<j~_n is called Kendall's tau. This statistic is a measure of the "concordance" of X and Y in the pairs (Xi, Yi), i = 1,..., n. It is normed so that its values lie between 1 (corresponding to the ideal concordance of rankings) and -1 (in the case of the complete discordance). Obviously, the statistic ~'n does not change if we replace the observations Xi and Yi by their ranks in the variational series of X and Y. This justifies the term "Kendall's rank correlation coefficient" applied to the statistic 7"n. The wide application of the statistic ~-n as a rank correlation coefficient was originated by the classical paper [1] of Kendall and his later monograph [2]. Later it became clear that measures of association based on (1) had already been considered by Esscher [3], Lindeberg [4, 5], and Jordan [6]. Since vn is a nondegenerate U-statistic with a bounded kernel, its properties were widely studied and are described in the majority of textbooks on nonparametric statistics. Consider the testing of the independence hypothesis Ho : F(x,y) = G(x) H(y), x, y E t~, against the alternative Hi: F(x, y) -- G(x)H(y) + O~(G(x), H(y)) (2) for all x and y and for some 0 > 0. We assume that the parameter ~ is small enough so that F(x, y) is a d.f. A nonnegative bounded function ~ defined on the unit square 12 is usually called a dependence func- tion. Alternative (2) is an important particular case of the so-called strictly positive quadrant dependence alternative. The latter term means that F(x, y) > G(x)H(y) for all x and y, and there exists at least one pair (x, y) for which the inequality is strict. The most well-known example of a dependence function is undoubtedly given by the function Q(x,y)=Cx(1-x)y(1-y), O<x,y<l, C>0, (3) usually called the Farlie-Gumbel-Morgenstern (FGM) function (see [7-9]). Johnson and Kotz [10-12] introduced and studied "generalized FGM-distributions." These investigations were continued in [13-15]. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 227-237. Original article submitted December 1, 1996. 1154 1072-3374/00/9902-1154 $25.00 (~) 2000 Kluwer Academic/Plenum Publishers