ASYMPTOTIC-DISTRIBUTION OF RANK STATISTICS UNDER DEPENDENCIES WITH MULTIVARIATE APPLICATION

By modifying the method of projection, the results of Hajek and Huskova are extended to show the asymptotic normality of signed and linear rank statistics under general alternatives for dependent random variables that can be expressed as independent vectors of fixed equal length. The score function is twice differentiable; the regression constants are arbitrary; and the distribution functions are continuous, but arbitrary. As an application, a rank transform statistic is proposed for the one-sample multivariate location model. The ranks of the absolute values of the observations are calculated without regard to component membership, and the scored ranks are substituted in place of the observed values. The limiting distribution of the proposed test statistic is shown to be χ2 divided by the degrees of freedom under the null hypothesis, and noncentral χ2 divided by the degrees of freedom under the sequence of Pitman alternatives. © 1990.