Statistical Modelling 6 (2006), 209–229
Using the Box-Cox t distribution in GAMLSS to model
skewness and kurtosis
R.A. Rigby
Statistics, OR, and Mathematics (STORM) Research Center,
London Metropolitan University
U.K.
eMail:
r.rigby@londonmet.ac.uk
D. Mikis Stasinopoulos
STORM Research Center,
London Metropolitan University
U.K.
Abstract:
The Box-Cox t (BCT) distribution is presented as a
model for a dependent variable Y exhibiting both
skewness and leptokurtosis. The distribution is defined
by a power transformation Yν having
a shifted and scaled (truncated) t distribution
with degrees of freedom parameter τ. The
distribution has four parameters and is denoted by
BCT(μ, σ, ν, τ).
The parameters μ, σ, ν
and τ may be interpreted as relating to
location (median), scale (centile-based coefficient of
variation), skewness (power transformation to symmetry)
and kurtosis (degrees of freedom), respectively. The
generalized additive model for location, scale and shape
(GAMLSS) is extended to allow each of the parameters of
the distribution to be modelled as linear and/or non-linear
parametric and/or smooth non-parametric functions of
explanatory variables. A Fisher scoring algorithm is
used to fit the model by maximizing a (penalized) likelihood.
The first and expected second and cross derivatives of the
likelihood with respect to μ, σ,
μ and τ, required for the algorithm,
are provided. The use of the BCT distribution is illustrated
by two data applications.
Keywords:
centile estimation; cubic smoothing splines; generalized
additive models; LMS method; non-linear model;
penalized likelihood; reference curves; regression quantiles
Downloads:
Example
data and R code in zipped archive.
R and the GAMLLS packages are available for download from CRAN,
http://cran.r-project.org.
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