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Tukey's $g$-and-$h$ distribution has been a powerful tool for dataexploration and modeling since its introduction. However, two long standingchallenges associated with this distribution family have remained unsolveduntil this day: how to find an optimal estimation procedure and how to makevalid statistical inference on unknown parameters. To overcome these twochallenges, a computationally efficient estimation procedure based onmaximizing an approximated likelihood function of the Tukey's $g$-and-$h$distribution is proposed and is shown to have the same estimation efficiency asthe maximum likelihood estimator under mild conditions. The asymptoticdistribution of the proposed estimator is derived and a series of approximatedlikelihood ratio test statistics are developed to conduct hypothesis testsinvolving two shape parameters of Tukey's $g$-and-$h$ distribution. Simulationexamples and an analysis of air pollution data are used to demonstrate theeffectiveness of the proposed estimation and testing procedures.

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