Fitting functions for selm models
Description
A 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 lowerlevel functions can be
called directly for increased efficiency, at the expense of some more
programming effort and lack of methods for the returned object.
Usage
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selm.fit(x, y, family = "SN", start = NULL, w, fixed.param = list(),
offset = NULL, selm.control)
sn.mple(x, y, cp = NULL, w, penalty = NULL, trace = FALSE, opt.method =
c("nlminb", "NelderMead", "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", "NelderMead", "BFGS", "CG", "SANN"),
control = list())
msn.mle(x, y, start = NULL, w, trace = FALSE, opt.method = c("nlminb",
"NelderMead", "BFGS", "CG", "SANN"), control = list())
msn.mple(x, y, start = NULL, w, trace = FALSE, penalty = NULL,
opt.method = c("nlminb", "NelderMead", "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", "NelderMead", "BFGS", "CG", "SANN"),
control = list())

Arguments
x 
a fullrank design matrix with the first column of all 1's. 
y 
a vector or a matrix of response values such that

family 
a character string which selects the parametric family of
distributions assumed for the error term of the regression model.
It must one of 
start, dp, cp 
a vector or a list of initial parameter values,
depeding whether 
w 
a vector of nonnegative integer weights of length equal to

fixed.param 
a list of assignments of parameter values to be kept
fixed during the optimization process. Currently, there is only one such
option, namely 
penalty 
an optional character string with the name of the penalty
function of the loglikelihood;
default value 
offset 
this can be used to specify an a priori known
component to be included in the linear predictor during fitting. This
should be 
trace 
a logical value which regulates printing of successive calls
to the target function; default value is 
fixed.nu 
a positive value to keep fixed the parameter 
symmetr 
a logical flag indicating whether a contraint of symmetry is
imposed on the slant parameter; default is 
opt.method 
a character string which selects the optimization method
within the set 
selm.control 
a list whose components regulate the working of

control 
a list of control items passed to the optimization function. 
Details
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 skewnormal distribution and
mst
refers to a multivariate skewt 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 ofopt.method
.
Function msn.mle
, for MLE estimation of linear models with
SN errors, is unchanged from version 0.4x of the package.
Function msn.mple
is similar to msn.mle
but allows to introduce
a penalization of the loglikelihood; 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 0.4x 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).
Value
A list whose specific components depend on the named function. Typical components are:
call 
the calling statement 
dp 
vector or list of estimated DP parameters 
cp 
vector or list of estimated CP parameters 
logL 
the maximized (penalized) loglikelihood 
aux 
a list with auxiliary output values, depending on the function 
opt.method 
a list produced by the numerical 
Background
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 loglikelihood.
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.
Maximization of the univariate ST loglikelihood is speededup 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 reparameterization described
in Section 5.1 of Azzalini and Capitanio (2003). The expressions of the
corresponding loglikelihood derivatives are given in Appendix B of the full
version of the paper.
Author(s)
Adelchi Azzalini
References
Azzalini, A. and Capitanio, A. (1999). Statistical applications of the multivariate skew normal distribution. J.Roy.Statist.Soc. B 61, 579–602. Fulllength version available at http://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. Fulllength version available at http://arXiv.org/abs/0911.2342
Azzalini, A. with the collaboration of Capitanio, A. (2014). The SkewNormal and Related Families. Cambridge University Press, IMS Monographs series.
DiCiccio, T. J. and Monti, A. C. (2011). Inferential aspects of the skew tdistribution. Quaderni di Statistica 13, 1–21.
See Also
selm
for a comprehensive higher level fitting function,
Qpenalty
for specification of a penalty function
Examples
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