sn (version 0.4-0)

mst.mle: Maximum likelihood estimation for a (multivariate) skew-t distribution

Description

Fits a skew-t (ST) or multivariate skew-t (MST) distribution to data, or fits a linear regression model with (multivariate) skew-t errors, using maximum likelihood estimation.

Usage

mst.mle(X, y, freq, start, fixed.df=NA, trace=FALSE,
   algorithm = c("nlminb","Nelder-Mead", "BFGS", "CG", "SANN"), control=list())
st.mle(X, y, freq, start, fixed.df=NA, trace=FALSE,
   algorithm = c("nlminb","Nelder-Mead", "BFGS", "CG", "SANN"), control=list())

Arguments

y

a matrix (for mst.mle) or a vector (for st.mle). If y is a matrix, rows refer to observations, and columns to components of the multivariate distribution.

X

a matrix of covariate values. If missing, a one-column matrix of 1's is created; otherwise, it must have the same number of rows of y. If X is supplied, then it must include a column of 1's.

freq

a vector of weights. If missing, a one-column matrix of 1's is created; otherwise it must have the same number of rows of y.

start

a list contaning the components beta,Omega, alpha, df of the type described below. The dp component of the returned list from a previous call has the required format and it can be used as a

fixed.df

a scalar value containing the degrees of freedom (df), if these must be taked as fixed, or NA (default value) if df is a parameter to be estimated.

trace

logical value which controls printing of the algorithm convergence. If trace=TRUE, details are printed. Default value is FALSE.

algorithm

a character string which selects the numerical optimization procedure used to maximize the loglikelihood function. If this string is set equal to "nlminb", then this function is called; in all other cases, optim is called, with

control

this parameter is passed to the chose optimizer, either nlminb or optim; see the documentation of this function for its usage.

Value

• A list containing the following components:
• call
  a string containing the calling statement.
• dp
  a list containing the direct parameters beta, Omega, alpha. Here, beta is a matrix of regression coefficients with dim(beta)=c(ncol(X),ncol(y)), Omega is a covariance matrix of order ncol(y), alpha is a vector of shape parameters of length ncol(y). Notice that, if st.mle was called or equivalently mst.mle was called with y a vector, then Omega represents the square of the scale parameter.
• se
  a list containing the components beta, alpha, info. Here, beta and alpha are the standard errors for the corresponding point estimates; info is the observed information matrix for the working parameter, as explained below.
• algorithm
  the list returned by the chose optimizer, either nlminb or optim, plus an item with the name of the selected algorithm; see the documentation of either nlminb or optim for explanation of the other components.

Background

The family of multivariate skew-t distributions is an extension of the multivariate Student's t family, via the introduction of a shape parameter which regulates skewness; when shape=0, the skew-t distribution reduces to the usual t distribution. When df=Inf the distribution reduces to the multivariate skew-normal one; see dmsn. See the reference below for additional information.

Details

If y is a vector and it is supplied to mst.mle, then it is converted to a one-column matrix, and a scalar skew-t distribution is fitted. This is also the mechanism used by st.mle which is simply an interface to mst.mle.

The parameter freq is intended for use with grouped data, setting the values of y equal to the central values of the cells; in this case the resulting estimate is an approximation to the exact maximum likelihood estimate. If freq is not set, exact maximum likelihood estimation is performed. Numerical search of the maximum likelihood estimates is performed in a suitable re-parameterization of the original parameters with aid of the selected optimizer (nlminb or optim) which is supplied with the derivatives of the log-likelihood function. Notice that, in case the optimizer is optim), the gradient may or may not be used, depending on which specific method has been selected. On exit from the optimizer, an inverse transformation of the parameters is performed. For a specific description on the re-parametrization adopted, see Section 5.1 and Appendix B of Azzalini & Capitanio (2003).

References

Azzalini, A. and Capitanio, A. (2003). Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. The full version of the paper published in abriged form in J.Roy. Statist. Soc. B 65, 367--389, is available at http://azzalini.stat.unipd.it/SN/se-ext.ps

See Also

dmst,msn.mle,mst.fit, nlminb, optim

Examples

data(ais, package="sn")
attach(ais)
X.mat <- model.matrix(~lbm+sex)
b <- sn.mle(X.mat, bmi)
# 
b <- mst.mle(y=cbind(Ht,Wt))
#
# a multivariate regression case:
a <- mst.mle(X=cbind(1,Ht,Wt), y=bmi, control=list(x.tol=1e-6))
#
# refine the previous outcome
a1 <- mst.mle(X=cbind(1,Ht,Wt), y=bmi, control=list(x.tol=1e-9), start=a$dp)