The family of g-and-h transformations are popular algorithms used for simulating non-normal distributions because of their simplicity and ease of execution. In general, two limitations associated with g-and-h transformations are that their probability density functions (pdfs) and cumulative distribution functions (cdfs) are unknown. In view of this, the g-and-h transformations’ pdfs and cdfs are derived in general parametric form. Moments are also derived and it is subsequently shown how the g and h parameters can be determined for prespecified values of skew and kurtosis. Numerical examples and parametric plots of g-and-h pdfs and cdfs are provided to confirm and demonstrate the methodology. It is also shown how g-and-h distributions can be used in the context of distribution fitting using real data sets.
Headrick, Todd C., Kowalchuk, Rhonda K. and Sheng, Yanyan. "Parametric Probability Densities and Distribution Functions for Tukey g-and-h Transformations and their Use for Fitting Data." (Jan 2008).