Asymptotic optimality of the Wilson-Hilferty cube-root transformation on the gamma distribution for higher-order odd central moments

The Wilson-Hilferty cube-root transformation is often applied to the gamma distribution to achieve approximate normality. Its asymptotic optimality was originally derived by expanding the third central moment of the scaled and power-transformed chi-squared distribution as the degrees of freedom approaches infinity. However, the original approach does not easily generalize to higher-order odd central moments of the power-transformed gamma distribution due to the increasing complexity of the resulting mathematical expression. To overcome the difficulty, we provide a novel efficient quantile-based approach to demonstrate that the cube-root transformation remains asymptotically optimal when any odd central moment of order three or higher is used as a criterion for approximate symmetry of the power-transformed gamma distribution. Unlike the previous approaches, the quantile-based approach provides a theoretical justification for the cube-root transformation on the tails of the gamma distribution, which is crucial for the normal-based inference on gamma-distributed data.