The package provides facilities to build and manipulate probability distributions of the skew-normal (SN) and some related families, notably the skew-\(t\) (ST) and the ‘unified skew-normal’ (SUN) families. For the SN, ST and skew-Cauchy (SC) families, statistical methods are made available for data fitting and model diagnostics, in the univariate and the multivariate case.
The package comprises two main sides: one side provides facilities for the pertaining probability distributions; the other one deals with related statistical methods. Underlying formulation, parameterizations of distributions and terminology are in agreement with the monograph of Azzalini and Capitanio (2014).
There are two layers of support for the probability distributions of interest. At the basic level, there exist functions which follow the classical R scheme for distributions. In addition, there exists facilities to build an object which incapsulates a probability distribution and then certain operations can be be performed on such an object; these probability objects operate according to the S4 protocol. The two schemes are described next.
The following functions work similary to {d,p,q,r}norm and other R
functions for probability distributions:
skew-normal (SN): functions {d,p,q,r}sn for the
univariate case, functions {d,p,r}msn for the multivariate case,
where in both cases the ‘Extended skew-normal’ (ESN)
variant form is included;
skew-\(t\) (ST): functions {d,p,q,r}st for the
univariate case, functions {d,p,r}mst for the multivariate case;
skew-Cauchy (SC): functions {d,p,q,r}sc for the
univariate case, functions {d,p,r}msc for the multivariate case.
In addition to the usual specification of their parameters as a sequence of
individual components, a parameter set can be specified as a single dp
entity, namely a vector in the univariate case, a list in the multivariate
case; dp stands for ‘Direct Parameters’ (DP).
Another set of parameters is in use, denoted Centred Parameters (CP),
which are more convenient for interpretability, since they correspond to
familiar quantifies, such as the mean and the standard deviation.
Conversion from the dp parameter set to the corresponding CP
parameters can be accomplished using the function dp2cp,
while function cp2dp performs the inverse transformation.
The SUN family is mainly targeted to the multivariate context,
and this is reflected in the organization of the pertaining functions,
although univariate SUN distributions are supported.
Density, distribution function and random numbers are handled by
{d,p,r}sun. Mean value, variance matrix and Mardia's measures
of multivariate skewness and kurtosis are computed by
sun{Mean,Vcov,Mardia}.
In addition, one can introduce a user-specified density function using
dSymmModulated and dmSymmModulated, in the univariate and the
multivariate case, respectively. These densities are of the
‘symmetry-modulated’ type, also called ‘skew-symmetric’, where
one can specify the base density and the modulation factor with high degree of
flexibility. Random numbers can be sampled using the corresponding functions
rSymmModulated and rmSymmModulated. In the bivariate case,
a dedicated plotting function exists.
Function makeSECdistr can be used to build a ‘SEC
distribution’ object representing a member of a specified parametric family
(among the types SN, ESN, ST, SC) with a given dp parameter
set. This object can be used for various operations such as plotting or
extraction of moments and other summary quantities.
Another way of constructing a SEC distribution object is via
extractSECdistr which extracts suitable components of an object
produced by function selm to be described below.
Additional operations on these objects are possible in the multivariate case,
namely marginalSECdistr and affineTransSECdistr
for marginalization and affine trasformations.
For the multivariate SN family only (but including ESN),
conditionalSECdistr performs a conditioning on the values taken
on by some components of the multivariate variable.
Function makeSUNdistr can be used to build a SUN
distribution object representing a member of the SUN
parametric family.
This object can be used for various operations such as plotting or
extraction of moments and other summary quantities.
Moreover there are several trasformation operations which can be performed on a SUN distribution object, or two such objects in some cases: computing a (multivariate) marginal distribution, a conditional distribution (on given values of some components or on one-sided intervals), an affine trasformation, a convolution (that is, the distribution of the sum of two independent variables), and joining two distributions under assumption of independence.
The main function for data fitting is represented by selm, which allows
to specify a linear regression model for the location parameter, similarly
to function lm, but assuming a skew-elliptical distribution
of the random component;
this explains the name selm=(se+lm). Allowed types of distributions
are SN (but not ESN), ST and SC.
The fitted distribution is univariate or multivariate, depending on the nature
of the response variable of the posited regression model. The model fitting
method is either maximum likelihood or maximum penalized likelihood;
the latter option effectively allows the introduction of a prior distribution
on the slant parameter of the error distribution, hence leading to a
‘maximum a posteriori’ estimate.
Once the fitting process has been accomplished, an object of class either
selm (for univariate response) or mselm (for multivariate
response) is produced.
A number of ‘methods’ are available for these objects: show,
plot, summary, coef, residuals, logLik
and others.
For univariate selm-class objects, univariate and bivariate profile
log-likelihood functions can be obtained; a predict method also exists.
These methods are built following the S4 protocol; however, the user must not
be concerned with the choice of the adopted protocol (unless this is wished).
The actual fitting process invoked via selm is actually performed by a
set of lower-level procedures. These are accessible for direct call,
if so wished, typically for improved efficiency, at the expense of a little
additional programming effort. Similarly, functions to compute the Fisher
information matrix are available, in the expected and the observed form
(with some restrictions depending on the selected distribution).
The extractSECdistr function extracts the fitted SEC
distribution from selm-class and mselm-class objects, hence
providing a bridge with the probability side of the package.
The facilities for statistical work do not support the SUN family.
Additional material is available in the section
‘User guides, package vignettes and other documentation’
accessible from the front page of the documentation.
See especially the document pkg_sn-intro.pdf
Adelchi Azzalini. Please send comments, error reports et cetera to the author, whose web page is http://azzalini.stat.unipd.it/.
Azzalini, A. with the collaboration of Capitanio, A. (2014). The Skew-Normal and Related Families. Cambridge University Press, IMS Monographs series.