On the Use of a Statistic Based on Sequential Ranks to Prove Limit Theorems for Simple Linear Rank Statistics

Abstract

A technique is introduced to prove limit theorems for simple linear rank statistics by means of an approximating statistic based on sequential ranks. This approximation is shown to be close enough to prove asymptotic normality of simple linear rank statistics under the null hypothesis and to obtain a bound on the rate of convergence to normality when the score function is unbounded. In addition, a law of the iterated logarithm and an invariance principle are given for simple linear rank statistics.