A Sheet of Maple to Compute Second-Order Edgeworth Expansions and Related Quantities of any Function of the Mean of an iid Sample of an Absolutely Continuous Distribution

TL;DR

This paper introduces an automated Maple worksheet for deriving second-order Edgeworth and Cornish-Fisher expansions, applicable to various bootstrap methods and statistics, with tools for validation, correction, and visualization.

Contribution

It provides a novel, automated Maple tool for deriving and validating higher-order asymptotic expansions and related quantities for statistical estimators.

Findings

Identified an error in a classical second-order Cornish-Fisher expansion.

Highlighted the impact of normalizing constants on expansions of the t-distribution.

Successfully applied the worksheet to complex MLE estimators for improved confidence intervals.

Abstract

We designed a completely automated Maple ($\geqslant 15$) worksheet for deriving Edgeworth and Cornish-Fisher expansions as well as the acceleration constant of the bootstrap bias-corrected and accelerated technique. It is valid for non-parametric or parametric bootstrap, of any (studentized) statistics that is -a regular enough- function of the mean of an iid sample of an absolutely continuous distribution. This worksheet allowed us to point out one error in the second-order Cornish-Fisher expansion of the studentized mean stated in Theorem 13.5 by Das Gupta in [8, p. 194] as well as lay the stress on the influence of the slight change of the normalizing constant when computing the second-order Edgeworth and Cornish-Fisher expansions of the t-distribution as stated in Theorem 11.4.2 by Lehman and Romano in [14, p. 460]. In addition, we successfully applied the worksheet to a complex maximum likelihood estimator as a first step to derive more accurate confidence intervals in order to enhance quality controls. The worksheet also features export of Maple results into R code. In addition, we provide R code to plot these expansions as well as their increasing rearrangements. All these supplemental materials are available upon request.