On the Asymptotic Distribution of the Wilcoxon Signed Rank Test Statistic

Abstract

The Wilcoxon signed rank test statistic (T⁺) is one of the popular nonparametric test for one sample and paired sample data. As the sample size getting large, it is well known that this test statistic can be approximated by a normal distribution [1]. The Gibbons and Chakraborti assumes the independence between Z_i and r(D_i) and T⁺=sum(sum_1<=i<=j<=N(T_ij)) in their proof of asymptotic normality of Wilcoxon signed rank test statistic T⁺ , however these assumptions are not proved. In this work, we demonstrate the rigorous proof of both of these assumptions as two theorems to complete Gibbons and Chakraborti’s proof of asymptotic distribution of the Wilcoxon signed rank test statistics T⁺ and provide applications for these two theorems.