Testing Normality Against the Logistic Distribution Using Saddlepoint Approximation

Abstract

We consider the problem of testing normality against the logistic distribution, based on a random sample of observations. Since the two families are separate (non nested), the ratio of maximized likelihoods (RML) statistic does not have the usual asymptotic chi-square distribution. We derive the saddlepoint approximation to the distribution of the RML statistic and show that this approximation is more accurate than the normal and Edgeworth approximations, especially for tail probabilities that are the main values of interest in hypothesis testing. It is also shown that this test is almost identical to the most powerful invariant test.

Keywords:

• Edgeworth expansion
• Likelihood ratio test
• Most powerful invariant test
• Ratio of maximized likelihoods (RML)
• Tail probability approximation

Mathematics Subject Classification:

• 62F03
• 62F05

Acknowledgments

The authors are grateful to a referee for his comments on an earlier version of this article. This work was supported by the Research Council of Shiraz University.