Maximum likelihood estimation for a (multivariate) skew-t distribution

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. Functions copied from sn CRAN library v0.4.18 because they were later deprecated in that library.

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

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.
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.
freq: a vector of weights. If missing, a vector of 1's is created; otherwise it must have length equal to the number of rows of y.
start: for mst.mle, a list contaning the components beta,Omega, alpha, df of the type described below; for st.mle, a vector whose components contain analogous ingredients as before, with the exception that the scale parameter is the square root of Omega. In both cases, the dp component of the returned list from a previous call has the required format and it can be used as a new start. If the start parameter is missing, initial values are selected by the function.
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 method set equal to the given string. Default value is "nlminb".
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: for mst.mle, this is 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). For st.mle, dp is a vector of length ncol(X)+3, containing c(beta, omega, alpha, df), where omega is the square root of Omega.
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.

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.

likelihood estimation, use st.mle.grouped.

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).

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.

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

Examples


dat <- rt(100, df=5, ncp=100)
fit <- st.mle(y=dat)
fit
#> $call
#> st.mle(y = dat)
#> 
#> $dp
#>  location     scale     shape        df 
#> 69.507859 57.744264 14.520298  5.980555 
#> 
#> $se
#>             scale      dat          
#> 2.060842 8.046592 6.243089 3.892236 
#> 
#> $logL
#> [1] -500.9718
#> 
#> $algorithm
#> $algorithm$par
#>   location                 shape            
#> 69.5078594  4.0560240  0.2514587  1.7885133 
#> 
#> $algorithm$objective
#> [1] 1001.944
#> 
#> $algorithm$convergence
#> [1] 0
#> 
#> $algorithm$iterations
#> [1] 14
#> 
#> $algorithm$evaluations
#> function gradient 
#>       24       15 
#> 
#> $algorithm$message
#> [1] "relative convergence (4)"
#> 
#> $algorithm$value
#> [1] 1001.944
#> 
#> $algorithm$name
#> [1] "nlminb"
#> 
#>