General Saddlepoint Approximations: Application to the Anderson-Darling Test Statistic

Abstract

We consider the relative merits of various saddlepoint approximations for the cumulative distribution function (cdf) of a statistic with a possibly non normal limit distribution. In addition to the usual Lugannani-Rice approximation, we also consider approximations based on higher-order expansions, including the case where the base distribution for the approximation is taken to be non normal. This extends earlier work by Wood et al. (1993). These approximations are applied to the distribution of the Anderson-Darling test statistic. While these generalizations perform well in the middle of the distribution's support, a conventional normal-based Lugannani-Rice approximation (Giles, 2001) is superior for conventional critical regions.

Keywords:

• Anderson-Darling test
• Goodness-of-fit
• Saddlepoint approximation

Mathematics Subject Classification:

• Primary 62E17, 62G10
• Secondary 62G30

Acknowledgments

We are grateful to Nilanjana Roy, Min Tsao, Graham Voss, Aman Ullah, and a referee for their very helpful comments. We also thank Andrew Wood for helpful correspondence, and for confirming that there is a type-setting error in Eq. (9) of Wood et al. (1993), and that our formula (11) is correct.

Notes

Note: “Low” represent the lower-order saddlepoint approximation; “High” represent the higher-order saddlepoint approximation. Chi-square (1), (2), and (3) use degrees of freedom, α, from Eqs. (26), (27), or α = 2, respectively.