Abstract
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 .
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Omelka, M. Second-order Linearity of Wilcoxon Statistics. AISM 59, 385–402 (2007). https://doi.org/10.1007/s10463-006-0054-8
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DOI: https://doi.org/10.1007/s10463-006-0054-8