selm modelsA call to selm activates a call to selm.fit and
from here to some other function which actually performs the parameter
search, among those listed below. These lower-level functions can be
called directly for increased efficiency, at the expense of some more
programming effort and lack of methods for the returned object.
selm.fit(x, y, family = "SN", start = NULL, w, fixed.param = list(),
offset = NULL, selm.control=list())sn.mple(x, y, cp = NULL, w, penalty = NULL, trace = FALSE, opt.method =
c("nlminb", "Nelder-Mead", "BFGS", "CG", "SANN"), control = list())
st.mple(x, y, dp = NULL, w, fixed.nu = NULL, symmetr = FALSE, penalty = NULL,
trace = FALSE, opt.method = c("nlminb", "Nelder-Mead", "BFGS", "CG", "SANN"),
control = list())
msn.mle(x, y, start = NULL, w, trace = FALSE, opt.method = c("nlminb",
"Nelder-Mead", "BFGS", "CG", "SANN"), control = list())
msn.mple(x, y, start = NULL, w, trace = FALSE, penalty = NULL,
opt.method = c("nlminb", "Nelder-Mead", "BFGS", "CG", "SANN"),
control = list())
mst.mple(x, y, start = NULL, w, fixed.nu = NULL, symmetr=FALSE,
penalty = NULL, trace = FALSE,
opt.method = c("nlminb", "Nelder-Mead", "BFGS", "CG", "SANN"),
control = list())
A list whose specific components depend on the named function. Typical components are:
the calling statement
vector or list of estimated DP parameters
vector or list of estimated CP parameters
the maximized (penalized) log-likelihood
a list with auxiliary output values, depending on the function
a list produced by the numerical opt.method
a full-rank design matrix with the first column of all 1's.
a vector or a matrix of response values such that
NROW(y)=nrow(x).
a character string which selects the parametric family of
distributions assumed for the error term of the regression model.
It must one of "SN" (default), "ST" or "SC", which
correspond to the skew-normal, the skew-t and the skew-Cauchy
family, respectively.
See makeSECdistr for more information on these families and
the skew-elliptically contoured (SEC) distributions; notice that
family "ESN" is not allowed here.
a vector or a list of initial parameter values,
depeding whether y is a vector or a matrix. It is assumed that
cp is given in the CP parameterization, dp and
start in the DP parameterization.
For st.mple and mst.mple, see also the paragraph about
start in the documentation ‘Details’ of selm.
a vector of non-negative integer weights of length equal to
NROW(y); if missing, a vector of all 1's is generated.
a list of assignments of parameter values to be kept
fixed during the optimization process. Currently, there is only one such
option, namely fixed.param=list(nu='value'), to fix the degrees
of freedom at the named 'value' when family="ST", for instance
list(nu=3). Setting fixed.param=list(nu=1) is equivalent to
select family="SC".
an optional character string with the name of the penalty
function of the log-likelihood;
default value NULL corresponds to no penalty.
this can be used to specify an a priori known
component to be included in the linear predictor during fitting. This
should be NULL or a numeric vector of length equal to the number of
cases. One or more offset terms can be included in the
formula instead or as well, and if more than one are specified their sum is
used.
a logical value which regulates printing of successive calls
to the target function; default value is FALSE which suppresses
printing.
a positive value to keep fixed the parameter nu
of the ST distribution in the optimization process; with default
value NULL, nu is estimated like the other parameters.
a logical flag indicating whether a contraint of symmetry is
imposed on the slant parameter; default is symmetr=FALSE.
a character string which selects the optimization method
within the set c("nlminb", "Nelder-Mead", "BFGS", "CG", "SANN");
the last four of these are "methods" of function optim.
a list whose components regulate the working of
selm.fit; see ‘Details’ for their description;
a list of control items passed to the optimization function.
Computational aspects of maximum likelihood estimation for univariate
SN distributions are discussed in Section 3.1.7 of Azzalini and
Capitanio (2014). The working of sn.mple follows these lines;
maximization is performed in the CP space. All other functions
operate on the DP space.
The technique underlying msn.mle is based on a partial analytical
maximization, leading implicitly to a form of profile log-likelihood.
This scheme is formulated in detail in Section 6.1 of Azzalini and Capitanio
(1999) and summarized in Section 5.2.1 of Azzalini and Capitanio (2014).
The same procedure is not feasible when one adopts MPLE;
hence function msn.mple has to maximize over a larger parameter space.
When the SN family is fitted with the constraint alpha=0, this amounts
to adopt a classical linear model with Gaussian distributional assumption.
The corresponding MLE's are the same as those produced by lm,
except that the denominator the of the MLE variance (matrix) has the
`uncorrected' form.
In the multivariate case, the covariance matrix of MLE is computed
using expression (10) in Section 15.8 of Magnus and Neudecker (2007).
Maximization of the univariate ST log-likelihood is speeded-up by using the expressions of the gradient given by DiCiccio and Monti (2011), reproduced with inessential variants in Section 4.3.3 of Azzalini and Capitanio (2014).
The working of mst.mple is based on a re-parameterization described
in Section 5.1 of Azzalini and Capitanio (2003). The expressions of the
corresponding log-likelihood derivatives are given in Appendix B of the full
version of the paper.
Adelchi Azzalini
A call to selm produces a call to selm.fit which
selects the appropriate function among sn.mple, st.mple,
msn.mle, msn.mple, mst.mple, depending on the
arguments of the calling statement. In the adopted scheme for function names,
msn refers to a multivariate skew-normal distribution and
mst refers to a multivariate skew-\(t\) distribution, while
mle and mple refers to maximum likelihood and maximum
penalized likelihood estimation, respectively.
Of these functions, sn.mple works in CP space; the others
in the DP space. In all cases, a correspondig mapping to the
alternative parameter space is performed before exiting selm.fit,
in addition to the selected parameter set.
The components of selm.control are as follows:
method: the estimation method, "MLE" or "MPLE".
penalty: a string with the name of the penalty function.
info.type: a string with the name of the information matrix,
"observed" or "expected"; currently fixed at "observed".
opt.method: a character string which selects the optimization
method.
opt.control: a list of control parameters of opt.method.
Function msn.mle, for MLE estimation of linear models with
SN errors, is unchanged from version 0.4-x of the package.
Function msn.mple is similar to msn.mle but allows to introduce
a penalization of the log-likelihood; when penalty=NULL, a call to
msn.mle is more efficient.
Functions sn.mple and mst.mple work like sn.mle and
mst.mle in version 0.4-x if the argument penalty is not
set or it is set to NULL, except that mst.mple does not
handle a univariate response (use st.mple for that).
Azzalini, A. and Capitanio, A. (1999). Statistical applications of the multivariate skew normal distribution. J.Roy.Statist.Soc. B 61, 579--602. Full-length version available at https://arXiv.org/abs/0911.2093
Azzalini, A. and Capitanio, A. (2003). Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. J.Roy. Statist. Soc. B 65, 367--389. Full-length version available at https://arXiv.org/abs/0911.2342
Azzalini, A. with the collaboration of Capitanio, A. (2014). The Skew-Normal and Related Families. Cambridge University Press, IMS Monographs series.
DiCiccio, T. J. and Monti, A. C. (2011). Inferential aspects of the skew \(t\)-distribution. Quaderni di Statistica 13, 1--21.
Magnus, J. R. and Neudecker, H. (2007). Matrix Differential Calculus with Applications in Statistics and Econometrics, third edition. John Wiley & Sons.
selm for a comprehensive higher level fitting function,
Qpenalty for specification of a penalty function
data(wines, package="sn")
X <- model.matrix(~ phenols + wine, data=wines)
fit <- msn.mle(x=X, y=cbind(wines$acidity, wines$alcohol), opt.method="BFGS")
fit <- st.mple(x=X, y = wines$acidity, fixed.nu=4, penalty="Qpenalty")
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