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Fitting multivariate skew-t distributions
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Fits a multivariate skew-t (MST) distribution to data, or fits a
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linear regression model with multivariate skew-t errors,
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using maximum likelihood estimation. The outcome is then displayed
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mst.fit(X, y, freq, start, fixed.df=NA, plot.it=TRUE, trace=FALSE, ...)
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a matrix or a vector. If \code{y} is a matrix, its rows refer to
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observations, and its columns to components of the multivariate
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distribution. If \code{y} is a vector, it is converted to a one-column
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matrix, and a scalar skew-t distribution is fitted.
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a matrix of covariate values.
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If missing, a one-column matrix of 1's is created; otherwise,
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it must have the same number of rows of \code{y}.
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If missing, a vector of 1's is created; otherwise
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it must have the same number of rows of \code{y}.
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a scalar value containing the degrees of freedom (df), if these must
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be taken as fixed, or \code{NA} (default value) if df is a parameter
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a list containing the components \code{beta},\code{Omega}, \code{alpha},
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\code{df} of the type described below. The \code{dp} component of the returned
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list from a previous call has the required format.
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logical value which controls the graphical output (default=TRUE);
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see below for description.
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logical value which controls printing of the algorithm convergence.
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If \code{trace=TRUE}, details are printed. Default value is \code{FALSE}.
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additional parameters passed to \code{msn.mle}; in practice, the
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\code{start}, the \code{algorithm} and the \code{control} parameters
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A list containing the following components:
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a string containing the calling statement.
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a list containing the direct parameters \code{beta}, \code{Omega}, \code{alpha},
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\code{df}. Here, \code{beta} is a matrix of regression coefficients with
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\code{dim(beta)=c(nrow(X),ncol(y))}, \code{Omega} is a covariance matrix of
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order \code{ncol(y)}, \code{alpha} is a vector of shape parameters of length
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\code{ncol(y)}, \code{df} is a positive scalar.
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log-likelihood evaluated at \code{dp}.
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a list containing the components \code{beta}, \code{alpha}, \code{info}.
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Here, \code{beta} and \code{alpha} are the standard errors for the
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corresponding point estimates;
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\code{info} is the observed information matrix for the working parameter,
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see the documentation of \code{mst.mle} for its explanation
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\item{test.normality}{
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a list with elements \code{test} and \code{p.value}, which are the value
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of the likelihood ratio test statistic for normality (i.e. test that
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all components of the shape parameter are 0 and \code{df=Inf}),
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and the corresponding p-value.
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a list of with elements \code{distance}, \code{prob} and \code{df}, which are
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the Mahalanobis distances of the residuals from the origin, with respect
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to the metric associated to the matrix \code{Omega}, and the values
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\code{prob} of the associated probabilities computed from the Snedecor's F
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distribution with degrees of freedom given by the \code{df} vector of length
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two, whose first component equals \code{ncol(y)} and the second component is
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equal to the \code{df} parameter of fitted value ST distribution unless this
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value has been selected by the used via \code{fixed.df}.
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\section{Side Effects}{
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Graphical output is produced if \code{(plot.it & missing(freq))==TRUE}.
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Three plots are produced, and the programs pauses between each two of them,
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waiting for the <Enter> key to be pressed.
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The first plot uses the variable \code{y} if \code{X} is missing, otherwise
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it uses the residuals from the regression.
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The form of this plot depends on the value of \code{d=ncol(y)};
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if \code{d=1}, an histogram is plotted with the fitted distribution
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superimposed. If \code{d>1}, a matrix of scatter-plots is produced, with
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superimposed the corresponding bivariate densities of the fitted
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The second plot has two panels, each representing a QQ-plot of
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Mahalanobis distances. The first of these refers to the fitting of a
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multivariate normal distribution, a standard statistical procedure;
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the second panel gives the corresponding QQ-plot of suitable Mahalanobis
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distances for the multivariate skew-normal fit.
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The third plot is similar to the previous one, except that PP-plots
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For computing the maximum likelihood estimates, \code{mst.fit}
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invokes \code{mst.mle}, while \code{mst.fit} displays the results
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See the documentation of \code{mst.mle} for details of the numerical
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procedure for maximum likelihood estimation.
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This function may be removed in future versions of the package, and
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(some of) its functionality transferred somewhere else
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\section{Background}{
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The family of multivariate skew-t distributions is an extension of the
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multivariate Student's t family, via the introduction of a \code{shape}
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parameter which regulates skewness; when \code{shape=0}, the skew-t
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distribution reduces to the regular symmetric \emph{t}-distribution.
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When \code{df=Inf} the distribution reduces to the multivariate skew-normal
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one; see \code{dmsn}. See the reference below for additional information.
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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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\code{\link{mst.mle}}, \code{\link{msn.fit}}, \code{\link{dmst}}, \code{\link{dmsn}}
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data(ais, package="sn")
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# a simple-sample case
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b <- mst.fit(y=cbind(Ht,Wt))
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a <- mst.fit(X=cbind(1,Ht,Wt), y=bmi)
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# refine the previous outcome
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a1 <- mst.fit(X=cbind(1,Ht,Wt), y=bmi, start=a$dp)
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\keyword{distribution}
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man/mst.fit.Rd
