Constructing a Three-Stage Asymptotic Coverage Probability for the Mean Using Edgeworth Second-Order Approximation

In this paper we consider a three-stage procedure that was presented by Hall (Ann Stat 9(6):1229–1238, 1981) to yield a fixed-width confidence interval for the mean with a precise confidence level using Edgeworth second-order expansion assuming the underlying continuous distribution has finite but unknown six moments. The procedure is based on expanding an asymptotic second order approximation of a differentiable and bounded function of the final stage stopping rule found in Yousef et al. (J Stat Plan Inference 143(9):1606–1618, 2013) by Edgeworth expansion. The performance of the asymptotic coverage was shown to be controlled by the performance of the Edgeworth approximation for the standardized underlying density and thus sensitive to the skewness and kurtosis of the underlying standardized distribution. The impact of several parameters on the asymptotic coverage is explored under continuous classes of distributions; normal, student’s t-distribution, uniform, beta and chi-squared. For brevity, simulation results are given for three types of underlying distributions: standard uniform, standard normal and standard exponential.