Studia Scientiarium Mathematicarum Hungarica 10. (1975)
1-2. szám - Govindarajulu Z.: Robustness of Mann-Whitney-Wilcoxon test to dependence in the variables
Studia Scientiarum Mathematicarum Hungarica 10 (1975) 39—45. ROBUSTNESS OF MANN-WHITNEY-WILCOXON TEST TO DEPENDENCE IN THE VARIABLES by Z. GOVINDARAJULU* Abstract. Let (A', Y) have an unknown bivarite distribution function H(x, y) having continuous marginals F(x) and G O'). The Mann-Whitney-Wilcoxon test statistic can be studentized so as to be asymptotically distribution-free for testing H0: F(x)—G(.x), for all x against the alternative Hx: FmG (with strict inequality for some *). The test is consistent and its asymptotic efficiency relative to the f-test is evaluated and an explicit form for it is obtained when H(x, y) is bivariate normal with correlation coefficient q. The relative efficiency is 3/я when q= — 1 or 0, is increasing for — lsp< — .5, decreasing for — .5<ßSl and is equal to 3/2=.866 when q = 1. 1. Introduction. Mann-Whitney [4] have proposed a distribution-free test for H0 against Hx when X and Y are independent. It is of much interest to study the sensitivity of the test when X and Y are dependent having an unknown bivariate distribution function H(x, y) with continuous marginals F(x) and G(y). Let (Хи Yt), i— = 1, ..., n denote a random sample of size n from H(x, y). Also, let Hn(x, y), Fn(x) and Gn(Y) respectively denote the empirical distribution functions (e.d.f/s) based on the samples (Xi, Yt) (i=l, ...,n), (Xl,...,Xn) and (Yx, ..., T„). Let Ztj = \ or 0 according as XxS Yj or Xt> Yj respectively for 1 2. An Asymptotically Distribution-free Test. Define (1) U= n~* 2 2 ZtJ = f Fn(x) dGn(x). i=iJ=1 Then, we have the following result pertaining to the asymptotic normality of U. Theorem 2.1. With the above notation, for all continuous F and G we have (2) lim P {n1,2(U—p)/(T S. z} = <P(z), where (3) = 2 ff F(*)[l -F(y)\dG(x)dG(y) x-cy + 2 ff G(x)[\-G(y)]dF(x)dF(y) x<y oo -2 ff [H(x,y)-F(x)G(y)]dG(x)dF(y), — oo p=f FdG and Ф denotes the standard normal distribution function. * Part of this research was conducted while the author was a visiting professor at the University of Michigan. This research was in part supported by the Navy under the Office of Naval Research Contract No. N00014—73—A—0385—0001, Task Order NR042—295. Studia Scientiarum Mathematicarum Hungarica 10 (1975)