Research Article
Simulating Univariate and Multivariate Tukey -and-ℎ
Distributions Based on the Method of Percentiles
Tzu Chun Kuo and Todd C. Headrick
Section on Statistics and Measurement, Department of EPSE, Southern Illinois University Carbondale, P.O. Box 4618,
222-J Wham Building, Carbondale, IL 62901-4618, USA
Correspondence should be addressed to Todd C. Headrick; headrick@siu.edu
Received 2 October 2013; Accepted 19 November 2013; Published 12 January 2014
Academic Editors: M. Campanino, X. Dang, and J. Villarroel
Copyright © 2014 T. C. Kuo and T. C. Headrick. is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
is paper derives closed-form solutions for the -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. e MOP methodology is
also extended from univariate to multivariate data generation. A procedure is described for simulating nonnormal distributions
with specified Spearman correlations. e 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
e Tukey -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. See [1–17].
e family of univariate -and-ℎ distributions can be sum-
marized as follows:
= () =
,ℎ
() =
−1
(
− 1)
ℎ
2
/2
, (1)
= () =
,0
() = lim
ℎ→0
,ℎ
() =
−1
(
− 1) ,
(2)
= () =
0,ℎ
() = lim
→0
,ℎ
() =
ℎ
2
/2
,
(3)
where is an i.i.d. standard normal random variable with
probability density function (pdf), (), and cumulative dis-
tribution function (cdf), Φ(). e transformations in (1)–(3)
are strictly monotone increasing functions with real-valued
constants and ℎ that produce distributions defined as (i)
asymmetric -and-ℎ ( ̸ =0, ℎ>0), (ii) log-normal ( ̸ =0,
ℎ = 0), and (iii) symmetric ℎ (ℎ ≥ 0), respectively. e
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,
−,ℎ
() = −
,ℎ
(−). e constant ℎ controls the tail-
weight of a distribution where the function
ℎ
2
/2
(i) preserves
symmetry, (ii) is increasing for ≥ 0 and ℎ ≥ 0, and
(iii) produces increased tail-weight as the value of ℎ 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 param-
eters (, ℎ) and an algorithm that produces standard normal
random deviates.
e values of and ℎ 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 disad-
vantages. Specifically, in the context of the MOM, the esti-
mates of conventional skew and kurtosis associated with
heavy-tailed or -skewed distributions can be substantially
biased, have high variance, or can be influenced by outliers;
see, for example, [19–24]. In terms of the MOP, the primary
Hindawi Publishing Corporation
ISRN Probability and Statistics
Volume 2014, Article ID 645823, 10 pages
http://dx.doi.org/10.1155/2014/645823