A Moment-Targeting Normality Transformation Based on Simultaneous Optimization of Tukey g-h Distribution Parameters

Abstract

This study proposes Optimized Skewness and Kurtosis Transformation (OSKT), a novel moment-targeting normality transformation that corrects asymmetry and peakedness in non-normal data. OSKT employs a transformation function derived from the Tukey g-h distribution, incorporating skewness and kurtosis parameters, and is optimized by minimizing a single objective function based on the Anderson-Darling test statistic. The optimization process uses L-BFGS-B to tune the transformation parameters to find the best fit for the standard normal distribution. OSKT ensures a balance between symmetry and tail behavior by minimizing deviations from theoretical normality. It has highly competitive performance compared to the alternative, Box-Cox, Yeo-Johnson transformations, including their robust variants and moment-matching Lambert W method, for normalizing complex distributions. According to our analysis, OSKT also achieves superior normalization for highly non-Gaussian data, successfully transforming highly resistant distributions, including approximately symmetric bimodal datasets, where other methods fail.