Second-order Linearity of Wilcoxon Statistics *

The rank statistics Sn(t) = 1 n ∑n i=1 ciRi(t) (t ∈ R), with Ri(t) being the rank of ei−txi, i = 1, . . . , n and e1, . . . , en being the random sample from the basic distribution with the cdf F , are considered as a random process with t in the role of parameter. Under some assumptions on ci, xi and on the underlying distribution, it is proved that the process {Sn( t √n)− Sn(0)− ESn(t), |t|2 ≤M} converges weakly to the Gaussian process. This generalizes the existing results where the one-dimensional case t ∈ 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 βl.