**Sankhya: The Indian Journal of Statistics**

1995, Volume 57, Series A, Pt. 3, pp. 410--423

** ASYMPTOTIC EXPANSIONS OF THE LIKELIHOOD RATIO STATISTIC WITH ORDERED HYPOTHESES**

By

YAZHEN WANG, *University of Missouri-Columbia*

SUMMARY. Likelihood ratio tests are used to test ordered hypotheses involving the parameters of k independent samples from an exponential family. For the test of constancy of the parameters versus an ordered alternative, the likelihood ratio test statistic is asymptotically distributed as a mixture of k chi-squared distributions under the null hypothesis. This paper derives an asymptotic expansion of the null distribution of the likelihood ratio test statistic in powers of n^{-1/2}, where n is the average sample size. In particular, it is shown that the null distribution of the likelihood ratio test statistic is generally order n^{-1/2} away from the limiting distribution. For the equal sample size case, however, all odd powers of n^{-1/2} vanish, and, similar to the Bartlett adjustment, the likelihood ratio test statistic can be adjusted through multiplication by scale factors such that the null distribution of the adjusted likelihood ratio test statistic is n^{-2} away from the limiting distribution.

*AMS (1991) subject classification.* Primary 62F05, secondary 62F30.

*Key words and phrases.* Asymptotic expansion, Bartlett adjustment, chi-bar-squared distribution, Edgeworth expansion, isotonic estimation, likelihood ratio statistic, ordered hypothesis.