General Saddlepoint Approximation for Testing Separate Location-Scale Families of Hypotheses

Abstract

For testing separate families of hypotheses, the likelihood ratio test does not have the usual asymptotic properties. This paper considers the asymptotic distribution of the ratio of maximized likelihoods (RML) statistic in the special case of testing separate scale or location-scale families of distributions. We derive saddlepoint approximations to the density and tail probabilities of the log of the RML statistic. These approximations are based on the expansion of the log of the RML statistic up to the second order, which is shown not to depend on the location and scale parameters. The resulting approximations are applied in several cases, including normal versus Laplace, normal versus Cauchy, and Weibull versus log-normal. Our results show that the saddlepoint approximations are satisfactory, even for fairly small sample sizes, and are more accurate than normal approximations and Edgeworth approximations, especially for tail probabilities that are the values of main interest in hypothesis testing problems.

Keywords:

• Cumulant generating function
• Edgeworth expansion
• Extreme value distribution
• Likelihood ratio test
• Normal approximation
• Ratio of maximized likelihoods (RML)
• Tail probability approximation

Mathematics Subject Classification:

• 62F03
• 62F05

Acknowledgments

The authors are grateful to Prof. Nancy Reid and two anonymous referees for their valuable comments on earlier versions of this paper. This work was supported by the Research Council of Shiraz University.