On the Validity of Edgeworth and Saddlepoint Approximations

In the classical theory of Edgeworth expansion for the sample mean, it is typically assumed that the sampling distribution is either lattice valued or is sufficiently smooth to satisfy Cramér′s regularity condition. However, applications of Edgeworth expansions to problems involving the bootstrap require regularity conditions which fail into the poorly understood grey area between these two cases. In the past, a limited amount of theory has been developed to take care of this problem, but it is restricted to the special case of the "classical" bootstrap, where the resample size is equal to the sample size and the resampling probabilities are all identical, In the present paper we extend this theory by developing Edgeworth expansions for general discrete distributions where the number of atoms is either fixed or increasing with sample size at an arbitrary rate. Implications of this result for the Lugannani-Rice tail area approximation are discussed, and it is established that this approximation is a large deviation formula in the present context. Our results shed light on much older work about the validity of Edgeworth expansions in the absence of Cramer′s condition, despite being motivated by very recent developments in bootstrap theory, for example, to contexts where the sample size and resample size are different or where the resampling probabilities are unequal.