https://github.com/cran/sn
Tip revision: 10c14452f146bc608ecec1b2d2f07d074a864bf6 authored by Adelchi Azzalini on 06 January 2014, 00:00:00 UTC
version 1.0-0
version 1.0-0
Tip revision: 10c1445
sn-st.info.Rd
% file sn/man/sn-st.info.Rd
% This file is a component of the package 'sn' for R
% copyright (C) 2013 Adelchi Azzalini
%---------------------
\name{sn-st.info}
\alias{sn.infoUv}
\alias{sn.infoMv}
\alias{st.infoUv}
\alias{st.infoMv}
\title{Expected and observed Fisher information for \acronym{SN}
and \acronym{ST} distributions}
\description{
Computes Fisher information for parameters of simple sample having
skew-normal (\acronym{SN}) or skew-\eqn{t} (\acronym{ST}) distribution or
for a regression model with errors term having such distributions, in the
\acronym{DP} and \acronym{CP} parametrizations.
}
\usage{
sn.infoUv(dp=NULL, cp=NULL, x=NULL, y, w, penalty=NULL, norm2.tol=1e-06)
sn.infoMv(dp, x=NULL, y, w, norm2.tol=1e-06)
st.infoUv(dp=NULL, cp=NULL, x=NULL, y, fixed.nu=NULL, w, penalty=NULL,
norm2.tol=1e-06)
st.infoMv(dp, x=NULL, y, fixed.nu=NULL, w, penalty=NULL, norm2.tol=1e-06)
}
\arguments{
\item{dp, cp}{direct or centred parameters, respectively; one of the two
vectors must be supplied, but not both. For the univariate \acronym{SN}
distribution, \code{sn.infoUv} is to be used, and these arguments are
vectors. In the multivariate case, \code{sn.infoMv} is to be used and these
arguments are lists. See \code{\link{dp2cp}} for their description.}
\item{x}{an optional matrix which represents the design matrix of a
regression model}
\item{y}{a numeric vector (for \code{sn.infoUv} and \code{st.infoUv})
or a matrix (for \code{sn.infoMv} and \code{st.infoMv}) representing the
response. In the \acronym{SN} case ( \code{sn.infoUv} and
\code{sn.infoMv}), \code{y} can be missing, and in this case the observed
information matrix is computed; otherwise the observed information is
computed. In the \acronym{ST} case ( \code{st.infoUv} and \code{st.infoMv},
\code{y} is a required argument, since only the observed information matrix
for \acronym{ST} distributions is implemented. See \sQuote{Details} for
additional information.}
\item{w}{an optional vector of weights; if missing, a vector of 1's is
generated.}
\item{fixed.nu}{an optional numeric value which declared a fixed value of the
degrees of freedom, \code{nu}. If not \code{NULL}, the information matrix
has a dimension reduced by 1.}
\item{penalty}{a optional string?? with the same penalty function used in
the call to \code{\link{selm}}; see this function for its description;}
\item{norm2.tol}{for the observed information case, the Mahalanobis squared
distance of the score 0 is evaluated; if it exceeds \code{norm2.tol}, a
warning message is issued, since the \sQuote{information matrix} so
evaluated may be not positive-definite. See \sQuote{Details} for
additional information. }
}
\value{
a list containing the following components:
\item{dp, cp}{one of the two arguments is the one supplied on input;
the other one matches the previous one in the alternative parametrization.}
\item{type}{the type of information matrix: "observed" or "expected".}
\item{info.dp, info.cp}{matrices of Fisher (observed or expected)
information in the two parametrizations.}
\item{asyvar.dp, asyvar.cp}{inverse matrices of Fisher information in the two
parametrizations, when available; See \sQuote{Details} for additional
information. }
\item{aux}{a list containing auxiliary elements, depending of the selected
function and the type of computation.}
}
\section{Details}{
In the univariate case, when \code{x} is not set, then a simple random sample
is assumed and a matrix \code{x} with a single column of all 1's is
constructed; in this case, the supplied vector \code{dp} or \code{cp} must
have length 3. If \code{x} is set, then the supplied vector of parameters,
\code{dp} or \code{cp}, must have length \code{ncol(x)+2}.
In the multivariate case, a direct extension of this scheme applies.
If the observed information matrix is required, \code{dp} or \code{dp} should
represent the maximum likelihood estimates (MLE) for the given \code{y},
otherwise the information matrix may fail to be positive-definite. Therefore,
the squared Mahalobis norm of the score vector is evaluated and compared with
\code{norm2.tol}. If it exceeds this threshold, it is taken as an indication
that the parameter is not at the MLE and a warning message is issued. The
returned list still includes \code{info.dp} and \code{info.cp}, but in this
case these represent merely the matrices of second derivatives;
\code{asyvar.dp} and \code{asyvar.cp} are set to \code{NULL}.
}
\section{Background}{
The information matrix for the the univariate \acronym{SN} distribution in
the two stated parameterizations in discussed in Sections 3.1.3--4 of
Azzalini and Capitanio (2014). For the multivariate distribution,
Section 5.2.2 of this monograph summarizes briefly the findings of
Arellano-Valle and Azzalini (2008).
For \acronym{ST} ??
}
\references{
Arellano-Valle, R. B., and Azzalini, A. (2008).
The centred parametrization for the multivariate skew-normal distribution.
\emph{J.\ Multiv.\ Anal.} \bold{99}, 1362--1382.
Corrigendum: vol.\,100 (2009), p.\,816.
Azzalini, A. with the collaboration of Capitanio, A. (2014).
\emph{The Skew-Normal and Related Families}.
Cambridge University Press, IMS Monographs series.
}
\seealso{\code{\link{dsn}}, \code{\link{dmsn}}, \code{\link{dp2cp}}}
\examples{
infoE <- sn.infoUv(dp=c(0,1,5))
infoO <- sn.infoUv(cp=c(0,1,0.8), y=rsn(50, dp=c(0,1,5)))
#
data(wines)
X <- model.matrix(~ pH + wine, data=wines)
fit <- sn.mple(x=X, y=wines$alcohol)
infoE <- sn.infoUv(cp=fit$cp, x=X)
infoO <- sn.infoUv(cp=fit$cp, x=X, y=wines$alcohol)
}
\keyword{distribution}