A Proof of the Asymptotic Equivalence of Two-Tail Probability Approximations

Abstract

This article considers asymptotic approximations to tail probabilities of a random variable whose distribution depends on a parameter n heuristically representing sample size. Random variables considered have cumulant generating functions with properties similar to that of sums of independent and identically distributed random variables. Probability approximations of Robinson (1982) and Lugannani and Rice (1980) are shown to be equivalent to a relative size O(1/n), under regularity conditions no stronger than the weaker of those necessary to prove either of the two approximations. Applications to permutation testing are discussed.

Keywords:

• Asymptotics
• Saddlepoint

Mathematics Subject Classification:

• Primary 41A60
• Secondary 41A55

Acknowledgment

This research was supported by NSF DMS 0092659.