The t-transformed power method distributions for simulating univariate and multivariate non-normal distributions

Abstract

This paper introduces a family of $t$-transformed power method distributions for fitting empirical and theoretical distributions with a wide range of skew and kurtosis values. Equations associated with the first four product moments (mean, variance, skew, and kurtosis) are derived and a methodology is subsequently illustrated for simulating univariate non-normal distributions based on the moment matching approach. Also included is a methodology for generating multivariate non-normal distributions correlated at the user-specified Pearson correlation matrix. The Monte Carlo simulation results indicate that the proposed methodology produces excellent agreement between estimates of Pearson correlations, skew, and kurtosis and their corresponding parameter values.

Keywords:

• Monte Carlo
• Simulation
• Moments
• Non-normal distributions
• Correlated distributions

Acknowledgments

The author would like to express sincere thanks to anonymous reviewers for reviewing this manuscript.

Conflict of interest

The author would like to state that there are no conflicts of interest regarding the publication of this article.

Data availability statement

The following are the steps (websites) employed for obataining data associated with Figures 1 and 5 of this study:

1. The data used in Figure 1 were obtained by drawing a random sample of $n=2,000$ observations from the $t_{df=4}$ distribution using Mathematica (version 9) (Wolfram Research, Inc. 2012) function RandomVariate with random seed = 232.

2. The three datasets (i.e., Hip data, Chest data, and Thigh data) used in Figure 5 were downloaded from the website http://lib.stat.cmu.edu/datasets/bodyfat.

Supplementary material

Provided in Appendixes A and B are two R (version 3.4.2) (R Core Team 2017) algorithms for generating estimates of Pearson correlations and estimates (${\widehat{\gamma }}_3$$\mathrm{ }$and ${\widehat{\gamma }}_4$) of skew and kurtosis, respectively, associated with the four distributions in Figure 7. These Appendixes will appear online only. Using these two algorithms, a user can generate corresponding estimates shown in Tables 8 and 9, respectively.