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December, 1981 Large Deviations of Goodness of Fit Statistics and Linear Combinations of Order Statistics
Piet Groeneboom, Galen R. Shorack
Ann. Probab. 9(6): 971-987 (December, 1981). DOI: 10.1214/aop/1176994268


Asymptotic behavior of large deviations of empirical distribution functions (df's) is considered. Borovkov (1967) and Hoadley (1967) obtained results for functionals continuous in the sup norm topology on the set of df's. Groeneboom, Oosterhoff, and Ruymgaart (1979) extended this to functionals continuous in a stronger $\tau$-topology. This result is now extended to functionals that are $\tau$-continuous only on a particular useful subset of df's. Applications to the Anderson-Darling statistic and linear combinations of order statistics are considered. We begin by correcting the work of Abrahamson (1967); from this the role of the key weight function $\psi(t) = -\log t(1 - t)$ is discovered. It is then exploited to the end indicated above, and it is considered as a weight function in tests of fit.


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Piet Groeneboom. Galen R. Shorack. "Large Deviations of Goodness of Fit Statistics and Linear Combinations of Order Statistics." Ann. Probab. 9 (6) 971 - 987, December, 1981.


Published: December, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0473.60035
MathSciNet: MR632970
Digital Object Identifier: 10.1214/aop/1176994268

Primary: 60F10
Secondary: 62G20 , 62G30

Keywords: $\tau$-topology , Bahadur efficiency , empirical measures , Kolmogorov-Smirnov tests , large deviations , linear combinations of order statistics , local efficiency , Sanov problem

Rights: Copyright © 1981 Institute of Mathematical Statistics

Vol.9 • No. 6 • December, 1981
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