Abstract
This paper derives closed-form solutions for the g-and-h 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.
Highlights
The Tukey g-and-h 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
To implement the method for simulating g-and-h, g, h distributions with specified γ3, γ4 and the Spearman correlations, we suggest the following six steps
+ ajTVj + ⋅ ⋅ ⋅ + aTTVT, where V1, . . . , VT are independent standard normal random variables and aij represents the element in the ith row and the jth column of the matrix associated with the Cholesky factorization performed in Step 3
Summary
The Tukey g-and-h 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. Are strictly monotone increasing functions with real-valued constants g and h that produce distributions defined as (i) asymmetric g-and-h (g ≠ 0, h > 0), (ii) log-normal (g ≠ 0, h = 0), and (iii) symmetric h (h ≥ 0), respectively. The constant ±g controls the skew of a distribution in terms of both direction and magnitude. The constant h controls the tailweight of a distribution where the function ehZ2/2 (i) preserves symmetry, (ii) is increasing for Z ≥ 0 and h ≥ 0, and (iii) produces increased tail-weight as the value of h becomes larger. (1)–(3) are computationally efficient for the purpose of generating nonormal distributions because they only require the specification of one or two shape parameters (g, h) and an algorithm that produces standard normal random deviates
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