Parametric Probability Densities and Distribution Functions for Tukey \u3ci\u3eg\u3c/i\u3e-and-\u3ci\u3eh\u3c/i\u3e Transformations and their Use for Fitting Data

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