Expansions about the gamma for the distribution and quantiles of a standard estimate

TL;DR

This paper develops advanced asymptotic expansions for the distribution, density, and quantiles of a standard estimate, utilizing skewness matching and Hermite polynomial generalizations to improve accuracy.

Contribution

It introduces a novel approach to expand about skew variables with matched skewness, reducing complexity in distribution approximations.

Findings

Expanded formulas for distribution, density, and quantiles of estimates

Reduced number of terms needed for accurate approximations

Demonstrated effectiveness with gamma distribution expansions

Abstract

We give expansions for the distribution, density, and quantiles of an estimate, building on results of Cornish, Fisher, Hill, Davis and the authors. The estimate is assumed to be non-lattice with the standard expansions for its cumulants. By expanding about a skew variable with matched skewness, one can drastically reduce the number of terms needed for a given level of accuracy. The building blocks generalize the Hermite polynomials. We demonstrate with expansions about the gamma.