Simulating Univariate and Multivariate Tukey -and- Distributions Based on the Method of Percentiles

This paper derives closed-form solutions for the -and- shape parameters associated with the Tukey family of distributions based on the method of percentiles (MOP). A proposed MOP univariate procedure is described and compared with the method of moments (MOM) in the context of distribution fitting and estimating skew and kurtosis functions. The MOP methodology is also extended from univariate to multivariate data generation. A procedure is described for simulating nonnormal distributions with specified Spearman correlations. The MOP procedure has an advantage over the MOM because it does not require numerical integration to compute intermediate correlations. Simulation results demonstrate that the proposed MOP procedure is superior to the MOM in terms of distribution fitting, estimation, relative bias, and relative error. 1. Introduction The Tukey -and- families of univariate and multivariate nonormal distributions are commonly used for distribution fitting, modeling events, random variable generation, and other applied mathematical contexts such as operational risk, extreme oceanic wind speeds, common stock returns, and solar flare data. See [1–17]. The family of univariate -and- distributions can be summarized as follows: where is an i.i.d. standard normal random variable with probability density function (pdf), , and cumulative distribution function (cdf), . The transformations in (1)–(3) are strictly monotone increasing functions with real-valued constants and that produce distributions defined as (i) asymmetric -and- ( , ), (ii) log-normal ( , ), and (iii) symmetric ( ), respectively. The constant ± controls the skew of a distribution in terms of both direction and magnitude. Taking the negative of will change the direction of the skew but not its magnitude; that is, . The constant controls the tail-weight of a distribution where the function (i) preserves symmetry, (ii) is increasing for and , and (iii) produces increased tail-weight as the value of becomes larger. In summary, (1)–(3) are computationally efficient for the purpose of generating nonormal distributions because they only require the specification of one or two shape parameters ( ) and an algorithm that produces standard normal random deviates. The values of and associated with (1)–(3) can be determined from either the method of moments (MOM), for example, [8, 10, 13], or the method of percentiles (MOP), for example, [9, 18]. However, these two methods have disadvantages. Specifically, in the context of the MOM, the estimates of conventional skew and kurtosis associated with