## The Annals of Statistics

### Central Limit Theorem for Wilcoxon Rank Statistics Process

Jana Jureckova

#### Abstract

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

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

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176342556

Digital Object Identifier
doi:10.1214/aos/1176342556

Mathematical Reviews number (MathSciNet)
MR368257

Zentralblatt MATH identifier
0295.62017

JSTOR