  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.
