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August, 1982 Limit Theorems and Inequalities for the Uniform Empirical Process Indexed by Intervals
Galen R. Shorack, Jon A. Wellner
Ann. Probab. 10(3): 639-652 (August, 1982). DOI: 10.1214/aop/1176993773


The uniform empirical process $U_n$ is considered as a process indexed by intervals. Powerful new exponential bounds are established for the process indexed by both "points" and intervals. These bounds trivialize the proof of the Chibisov-O'Reilly theorem concerning the convergence of the process with respect to $\|\cdot/q\|$-metrics and are used to prove an interval analogue of the Chibisov-O'Reilly theorem. A strong limit theorem related to the well-known Holder condition for Brownian bridge $U$ is also proved. Connections with related work of Csaki, Eicker, Jaeschke, and Stute are mentioned. As an application we introduce a new statistic for testing uniformity which is the natural interval analogue of the classical Anderson-Darling statistic.


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Galen R. Shorack. Jon A. Wellner. "Limit Theorems and Inequalities for the Uniform Empirical Process Indexed by Intervals." Ann. Probab. 10 (3) 639 - 652, August, 1982.


Published: August, 1982
First available in Project Euclid: 19 April 2007

zbMATH: 0497.60026
MathSciNet: MR659534
Digital Object Identifier: 10.1214/aop/1176993773

Primary: 60F05
Secondary: 60B10 , 60G17 , 62E20

Keywords: $||cdot/q||$-metrics , Anderson-Darling statistics , Exponential bounds , process convergence , Watson's statistic

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 3 • August, 1982
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