Second-order linearity of Wilcoxon statistics

The rank statistic S-n(t) = 1/n Sigma(n)(i = 1) c(i)R(i) (t) (t is an element of R-p), with R-i (t) being the rank of e(i) - t(inverted perpendicular)x(i), i = 1,..., 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(t/root(n) over bar) - S-n (0)- is an element of S-n (t), \t\(2) <= M} converges weakly to the Gaussian process. This generalizes the existing results where the one-dimensional case t is an element of 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 beta(l).