## The Annals of Probability

- Ann. Probab.
- Volume 9, Number 6 (1981), 971-987.

### Large Deviations of Goodness of Fit Statistics and Linear Combinations of Order Statistics

Piet Groeneboom and Galen R. Shorack

#### Abstract

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.

#### Article information

**Source**

Ann. Probab., Volume 9, Number 6 (1981), 971-987.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994268

**Digital Object Identifier**

doi:10.1214/aop/1176994268

**Mathematical Reviews number (MathSciNet)**

MR632970

**Zentralblatt MATH identifier**

0473.60035

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F10: Large deviations

Secondary: 62G20: Asymptotic properties 62G30: Order statistics; empirical distribution functions

**Keywords**

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

#### Citation

Groeneboom, Piet; Shorack, Galen R. Large Deviations of Goodness of Fit Statistics and Linear Combinations of Order Statistics. Ann. Probab. 9 (1981), no. 6, 971--987. doi:10.1214/aop/1176994268. https://projecteuclid.org/euclid.aop/1176994268