Efficient Maximum Approximated Likelihood Inference for Tukey's g-and-h Distribution

TL;DR

This paper introduces a computationally efficient estimation method for Tukey's g-and-h distribution that achieves maximum likelihood efficiency and provides valid hypothesis testing procedures, demonstrated through simulations and real data analysis.

Contribution

It proposes a new approximated likelihood-based estimation method for Tukey's g-and-h distribution that is both efficient and computationally feasible, along with hypothesis testing techniques.

Findings

Estimation efficiency comparable to maximum likelihood estimators.

Development of asymptotic distribution for the proposed estimator.

Effective hypothesis tests for shape parameters demonstrated on data.

Abstract

Tukey's $g$-and-$h$ distribution has been a powerful tool for data exploration and modeling since its introduction. However, two long standing challenges associated with this distribution family have remained unsolved until this day: how to find an optimal estimation procedure and how to make valid statistical inference on unknown parameters. To overcome these two challenges, a computationally efficient estimation procedure based on maximizing an approximated likelihood function of the Tukey's $g$-and-$h$ distribution is proposed and is shown to have the same estimation efficiency as the maximum likelihood estimator under mild conditions. The asymptotic distribution of the proposed estimator is derived and a series of approximated likelihood ratio test statistics are developed to conduct hypothesis tests involving two shape parameters of Tukey's $g$-and-$h$ distribution. Simulation examples and an analysis of air pollution data are used to demonstrate the effectiveness of the proposed estimation and testing procedures.