The Annals of Statistics

Central Limit Theorem for Wilcoxon Rank Statistics Process

Jana Jureckova

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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 $S_{\Delta N}$.

Article information

Ann. Statist., Volume 1, Number 6 (1973), 1046-1060.

First available in Project Euclid: 12 April 2007

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


Jureckova, Jana. Central Limit Theorem for Wilcoxon Rank Statistics Process. Ann. Statist. 1 (1973), no. 6, 1046--1060. doi:10.1214/aos/1176342556.

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