Second-order Linearity of Wilcoxon Statistics

The rank statistic $$S_n({\bf t}) = 1 / n \sum^{n}_{i=1} c_i R_{i}({\bf t}) ({\bf t} \in \mathbb{R}^{p})$$ , with R i (t) being the rank of $$e_{i}-{\bf t}^{\hbox{T}}{\bf x}_i, i=1,\ldots,n$$ and e 1 , . . . , e n being the random sample from a distribution with a cdf F, is considered as a random process with t in the role of parameter. Under some assumptions on c i , x i and on the underlying distribution, it is proved that the process $$\{S_n(\frac{{\bf t}}{\sqrt{n}})-S_n(\mathbf{0}) - {\rm E} S_{n}({\bf t}), |{\bf t}|_2 \leq M\}$$ converges weakly to the Gaussian process. This generalizes the existing results where the one-dimensional case $${\bf t} \in \mathbb{R}$$ was considered. We believe our method of the proof can be easily modified for the signed-rank statistics of Wilcoxon type. Finally, we use our results to find the second order asymptotic distribution of the R-estimator based on the Wilcoxon scores and also to investigate the length of the confidence interval for a single parameter β l .