# 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. Functions copied from `sn`

CRAN library v0.4.18 because they were later deprecated in that library.

### Usage

1 2 3 4 |

### Arguments

`y` |
a matrix (for |

`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 |

`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 |

`start` |
for |

`fixed.df` |
a scalar value containing the degrees of freedom (df), if these must
be taked as fixed, or |

`trace` |
logical value which controls printing of the algorithm convergence.
If |

`algorithm` |
a character string which selects the numerical optimization procedure
used to maximize the loglikelihood function. If this string is set
equal to |

`control` |
this parameter is passed to the chose optimizer, either |

### 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).

### Value

A list containing the following components:

`call` |
a string containing the calling statement. |

`dp` |
for |

`se` |
a list containing the components |

`algorithm` |
the list returned by the chose optimizer, either |

### 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

### See Also

`dst`

### Examples

1 2 3 |

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