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November, 1973 Central Limit Theorem for Wilcoxon Rank Statistics Process
Jana Jureckova
Ann. Statist. 1(6): 1046-1060 (November, 1973). DOI: 10.1214/aos/1176342556


The rank statistics $S_{\Delta N} = N^{-1} \sum^N_{i=1} c_{Ni} R^\Delta_{Ni}$, with $R^Delta_{Ni}$ being the rank of $X_{Ni} + \Delta d_{Ni}, i = 1, 2, \cdots, N$ and $X_{N1}, \cdots, X_{NN}$ being the random sample from the basic distribution with density function $f$, are considered as a random process with $\Delta$ in the role of parameter. Under some assumptions on $C_{Ni}$'s, $d_{Ni}$'s and on the underlying distribution, it is proved that the process $\{S_{\Delta N} - S_{0N} - ES_{\Delta N}; 0 \leqq \Delta \leqq 1\}$, being properly standardized, converges weakly to the Gaussian process with covariances proportional to the product of parameter values. Under additional assumptions, $\Delta b_N$ can be written instead of $ES_{\Delta N}$, where $b_N = \sum^N_{i=1} C_{Ni}d_{Ni}\int f^2(x) dx$. As an application, this result yields the asymptotic normality of the standardized form of the length of a confidence interval for regression coefficient based on statistic $S_{\Delta N}$.


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Jana Jureckova. "Central Limit Theorem for Wilcoxon Rank Statistics Process." Ann. Statist. 1 (6) 1046 - 1060, November, 1973.


Published: November, 1973
First available in Project Euclid: 12 April 2007

zbMATH: 0295.62017
MathSciNet: MR368257
Digital Object Identifier: 10.1214/aos/1176342556

Keywords: asymptotic behavior of rank test statistics as a function of regression parameter , asymptotic distribution of nonparametric estimate of regression coefficient , nonparametrics , space $D\lbrack 0,1\rbrack$ of right-continuous functions , Wilcoxon rank test statistic against regression alternatives

Rights: Copyright © 1973 Institute of Mathematical Statistics


Vol.1 • No. 6 • November, 1973
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