Continuity Corrections for Discrete Distributions Under the Edgeworth Expansion

Abstract

The approximation of discrete distributions by Edgeworth expansion series for continuity points of a discrete distribution F _n implies that if t is a support point of F _n, then the expansion should be performed at a continuity point $t + \theta {\text{,}}\theta \in \left( {0,1} \right)$. When a value $\theta$ is selected to improve the approximation of $P\left( {S_n \leqslant t} \right)$, and especially when a single term of the expansion is used, the selected $\theta ^*$ is defined to be a continuity correction. This paper investigates the properties of the approximations based on several terms of the expansion, when $\theta ^*$ is the value at which the infimum of a residual term is attained. Methods of selecting the estimation and the residual terms are investigated and the results are compared empirically for several discrete distributions. The results are also compared with the commonly used approximation based on the normal distribution with $\theta \equiv 0.5$. Some numerical comparisons show that the developed procedure gives better approximations than those obtained under the standard continuity correction technique, whenever $P\left( {S_n \leqslant t} \right)$ is close to 0 and 1. Thus, it is especially useful for p-value computations and for the evaluation of probabilities of rare events.