## Annals of Statistics

### Large Deviations of Likelihood Ratio Statistics with Applications to Sequential Testing

Michael Woodroofe

#### Abstract

We study the tail of the null distribution of the $\log$ likelihood ratio statistic for testing sharp hypotheses about the parameters of an exponential family. We show that the classical chisquare approximation is of exactly the right order of magnitude, although it may be off by a constant factor. We then apply our results and techniques to find the error probabilities of a sequential version of the likelihood ratio test. The sequential version rejects if the likelihood ratio statistic crosses a given barrier by a given time. Our approach uses a local limit theorem which takes account of large deviations and integrates the local result by using the theory of Hausdorff measures.

#### Article information

Source
Ann. Statist., Volume 6, Number 1 (1978), 72-84.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176344066

Digital Object Identifier
doi:10.1214/aos/1176344066

Mathematical Reviews number (MathSciNet)
MR455183

Zentralblatt MATH identifier
0386.62019

JSTOR

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 62G10: Hypothesis testing

#### Citation

Woodroofe, Michael. Large Deviations of Likelihood Ratio Statistics with Applications to Sequential Testing. Ann. Statist. 6 (1978), no. 1, 72--84. doi:10.1214/aos/1176344066. https://projecteuclid.org/euclid.aos/1176344066