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% This file is a component of the package 'sn' for R
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% copyright (C) 2002-2013 Adelchi Azzalini
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Multivariate skew-\eqn{t} distribution
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Probability density function, distribution function and random number
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generation for the multivariate skew-\eqn{t} (MST) distribution.
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\title{Multivariate skew-\eqn{t} distribution and skew-Cauchy distribution}
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\description{Probability density function, distribution function and random
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number generation for the multivariate skew-\eqn{t} (\acronym{ST}) and
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skew-Cauchy (\acronym{SC}) distributions.}
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dmst(x, xi=rep(0,length(alpha)), Omega, alpha, df=Inf, dp = NULL, log=FALSE)
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pmst(x, xi=rep(0,length(alpha)), Omega, alpha, df=Inf, dp = NULL, ...)
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rmst(n=1, xi=rep(0,length(alpha)), Omega, alpha, df=Inf, dp = NULL)
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dmst(x, xi=rep(0,length(alpha)), Omega, alpha, nu=Inf, dp=NULL, log=FALSE)
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pmst(x, xi=rep(0,length(alpha)), Omega, alpha, nu=Inf, dp=NULL, ...)
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rmst(n=1, xi=rep(0,length(alpha)), Omega, alpha, nu=Inf, dp=NULL)
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dmsc(x, xi=rep(0,length(alpha)), Omega, alpha, dp=NULL, log=FALSE)
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pmsc(x, xi=rep(0,length(alpha)), Omega, alpha, dp=NULL, ...)
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rmsc(n=1, xi=rep(0,length(alpha)), Omega, alpha, dp=NULL)
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for \code{dmst}, this is either a vector of length \code{d},
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where \code{d=length(alpha)}, or a matrix with \code{d} columns,
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giving the coordinates of the point(s) where the density must be
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avaluated; for \code{pmst}, only a vector of length \code{d} is allowed.
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a numeric vector of lenght \code{d}, or a matrix with \code{d} columns,
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representing the location parameter of the distribution; see Background.
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If \code{xi} is a matrix, its dimensions must agree with those of \code{x}.
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a symmetric positive-definite matrix of dimension \code{(d,d)}; see Background.
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a numeric vector which regulates the shape of the density; see Background
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degrees of freedom (scalar); default is \code{df=Inf} which corresponds
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to the multivariate skew-normal distribution.
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a list with three elements named \code{xi}, \code{Omega}, \code{alpha}
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and \code{df}, containing quantities as described above.
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If \code{dp} is specified, this overrides the individual parameter
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a numeric value which represents the number of random vectors
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logical (default value: \code{FALSE}); if TRUE, log-densities are returned.
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additional parameters passed to \code{pmt}
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A vector of density values (\code{dmst}) or a single probability
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(\code{pmst}) or a matrix of random points (\code{rmst}).
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\item{x}{for \code{dmst} and \code{dmsc}, this is either a vector of length
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\code{d}, where \code{d=length(alpha)}, or a matrix with \code{d} columns,
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representing the coordinates of the point(s) where the density must be
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avaluated; for \code{pmst} and \code{pmsc}, only a vector of length
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\item{xi}{a numeric vector of lenght \code{d}, or a matrix with \code{d}
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columns, representing the location parameter of the distribution; see
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\sQuote{Background}. If \code{xi} is a matrix, its dimensions must agree
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with those of \code{x}.}
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\item{Omega}{a symmetric positive-definite matrix of dimension \code{(d,d)};
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see Section \sQuote{Background}.}
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\item{alpha}{a numeric vector of length \code{d} which regulates the slant
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of the density; see Section \sQuote{Background}.
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\code{Inf} values in \code{alpha} are not allowed.}
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\item{nu}{a positive value representing the degrees of freedom of
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\acronym{ST} distribution; default value is \code{nu=Inf} which corresponds
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to the multivariate skew-normal distribution.}
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\item{dp}{a list with three elements named \code{xi}, \code{Omega},
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\code{alpha} and \code{nu}, containing quantities as described above. If
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\code{dp} is specified, this prevents specification of the individual
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\item{n}{a numeric value which represents the number of random vectors to be
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drawn; default value is \code{1}.}
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\item{log}{logical (default value: \code{FALSE}); if \code{TRUE},
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log-densities are returned.}
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\item{...}{additional parameters passed to \code{pmt}.}
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\value{A vector of density values (\code{dmst} and \code{dmsc}) or a single
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probability (\code{pmst} and \code{pmsc}) or a matrix of random points
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(\code{rmst} and \code{rmst}).}
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\details{Typical usages are
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dmst(x, xi=rep(0,length(alpha)), Omega, alpha, df=Inf, log=FALSE)
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dmst(x, xi=rep(0,length(alpha)), Omega, alpha, nu=Inf, log=FALSE)
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dmst(x, dp=, log=FALSE)
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pmst(x, xi=rep(0,length(alpha)), Omega, alpha, df=Inf, ...)
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pmst(x, xi=rep(0,length(alpha)), Omega, alpha, nu=Inf, ...)
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rmst(n=1, xi=rep(0,length(alpha)), Omega, alpha, df=Inf)
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rmst(n=1, xi=rep(0,length(alpha)), Omega, alpha, nu=Inf)
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dmsc(x, xi=rep(0,length(alpha)), Omega, alpha, log=FALSE)
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dmsc(x, dp=, log=FALSE)
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pmsc(x, xi=rep(0,length(alpha)), Omega, alpha, ...)
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rmsc(n=1, xi=rep(0,length(alpha)), Omega, alpha)
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The positive-definiteness of \code{Omega} is not tested for efficiency
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reasons. Function \code{pmst} requires \code{pmt} from package \code{mnormt};
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the accuracy of its computation can be controlled via use of \code{...}
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Function \code{pmst} requires \code{\link[mnormt]{dmt}} from package
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\pkg{mnormt}; the accuracy of its computation can be controlled via
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argument \code{\dots}.}
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\section{Background}{
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The family of multivariate skew-\eqn{t} distributions is an extension of the
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multivariate Student's \eqn{t} family, via the introduction of a \code{shape}
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parameter which regulates skewness; when \code{shape=0}, the skew-\eqn{t}
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distribution reduces to the usual \eqn{t} distribution.
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When \code{df=Inf} the distribution reduces to the multivariate skew-normal
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The family of multivariate \acronym{ST} distributions is an extension of the
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multivariate Student's \eqn{t} family, via the introduction of a \code{alpha}
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parameter which regulates asymmetry; when \code{alpha=0}, the skew-\eqn{t}
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distribution reduces to the commonly used form of multivariate Student's
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\eqn{t}. Further, location is regulated by \code{xi} and scale by
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\code{Omega}, when its diagonal terms are not all 1's.
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When \code{nu=Inf} the distribution reduces to the multivariate skew-normal
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one; see \code{dmsn}. Notice that the location vector \code{xi}
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does not represent the mean vector of the distribution (which in fact
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may not even exist if \code{df <= 1}), and similarly
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\code{Omega} is not \emph{the} covariance matrix of the distribution,
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although it is \emph{a} covariance matrix.
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For additional information, see the reference below.
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may not even exist if \code{nu <= 1}), and similarly \code{Omega} is not
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\emph{the} covariance matrix of the distribution, although it is \emph{a}
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For additional information, see Section 6.2 of the reference below.
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The family of multivariate \acronym{SC} distributions is the subset of the
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\acronym{ST} family, obtained when \code{nu=1}. While in the univariate case
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there are specialized functions for the \acronym{SC} distribution,
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\code{dmsc}, \code{pmsc} and \code{rmsc} simply make a call to \code{dmst,
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pmst, rmst} with argument \code{nu} set equal to 1.}
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Azzalini, A. and Capitanio, A. (2003).
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Distributions generated by perturbation of symmetry
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with emphasis on a multivariate skew \emph{t} distribution.
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\emph{J.Roy. Statist. Soc. B} \bold{65}, 367--389.
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% Azzalini, A. and Capitanio, A. (2003).
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% Distributions generated by perturbation of symmetry
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% with emphasis on a multivariate skew \emph{t} distribution.
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% \emph{J.Roy. Statist. Soc. B} \bold{65}, 367--389.
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Azzalini, A. with the collaboration of Capitanio, A. (2014).
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\emph{The Skew-Normal and Related Families}.
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Cambridge University Press, IMS Monograph series.
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\code{\link{dst}}, \code{\link{dmsn}}, \code{\link[mnormt]{dmt}}
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\code{\link{dst}}, \code{\link{dsc}}, \code{\link{dmsn}},
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\code{\link[mnormt]{dmt}}, \code{\link{makeSECdistr}}
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x <- seq(-4,4,length=15)
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man/dmst.Rd
