# Multivariate skew-t distribution and skew-Cauchy distribution

### Description

Probability density function, distribution function and random
number generation for the multivariate skew-*t* (ST) and
skew-Cauchy (SC) distributions.

### Usage

1 2 3 4 5 6 | ```
dmst(x, xi=rep(0,length(alpha)), Omega, alpha, nu=Inf, dp=NULL, log=FALSE)
pmst(x, xi=rep(0,length(alpha)), Omega, alpha, nu=Inf, dp=NULL, ...)
rmst(n=1, xi=rep(0,length(alpha)), Omega, alpha, nu=Inf, dp=NULL)
dmsc(x, xi=rep(0,length(alpha)), Omega, alpha, dp=NULL, log=FALSE)
pmsc(x, xi=rep(0,length(alpha)), Omega, alpha, dp=NULL, ...)
rmsc(n=1, xi=rep(0,length(alpha)), Omega, alpha, dp=NULL)
``` |

### Arguments

`x` |
for |

`xi` |
a numeric vector of length |

`Omega` |
a symmetric positive-definite matrix of dimension |

`alpha` |
a numeric vector of length |

`nu` |
a positive value representing the degrees of freedom of
ST distribution; does not need to be integer.
Default value is |

`dp` |
a list with three elements named |

`n` |
a numeric value which represents the number of random vectors to be
drawn; default value is |

`log` |
logical (default value: |

`...` |
additional parameters passed to |

### Details

Typical usages are

1 2 3 4 5 6 7 8 9 10 11 12 | ```
dmst(x, xi=rep(0,length(alpha)), Omega, alpha, nu=Inf, log=FALSE)
dmst(x, dp=, log=FALSE)
pmst(x, xi=rep(0,length(alpha)), Omega, alpha, nu=Inf, ...)
pmst(x, dp=, ...)
rmst(n=1, xi=rep(0,length(alpha)), Omega, alpha, nu=Inf)
rmst(n=1, dp=)
dmsc(x, xi=rep(0,length(alpha)), Omega, alpha, log=FALSE)
dmsc(x, dp=, log=FALSE)
pmsc(x, xi=rep(0,length(alpha)), Omega, alpha, ...)
pmsc(x, dp=, ...)
rmsc(n=1, xi=rep(0,length(alpha)), Omega, alpha)
rmsc(n=1, dp=)
``` |

Function `pmst`

requires `dmt`

from package
mnormt; the accuracy of its computation can be controlled via
argument `...`

.

### Value

A vector of density values (`dmst`

and `dmsc`

) or a single
probability (`pmst`

and `pmsc`

) or a matrix of random points
(`rmst`

and `rmsc`

).

### Background

The family of multivariate ST distributions is an extension of the
multivariate Student's *t* family, via the introduction of a `alpha`

parameter which regulates asymmetry; when `alpha=0`

, the skew-*t*
distribution reduces to the commonly used form of multivariate Student's
*t*. Further, location is regulated by `xi`

and scale by
`Omega`

, when its diagonal terms are not all 1's.
When `nu=Inf`

the distribution reduces to the multivariate skew-normal
one; see `dmsn`

. Notice that the location vector `xi`

does not represent the mean vector of the distribution (which in fact
may not even exist if `nu <= 1`

), and similarly `Omega`

is not
*the* covariance matrix of the distribution, although it is *a*
covariance matrix.
For additional information, see Section 6.2 of the reference below.

The family of multivariate SC distributions is the subset of the
ST family, obtained when `nu=1`

. While in the univariate case
there are specialized functions for the SC distribution,
`dmsc`

, `pmsc`

and `rmsc`

simply make a call to ```
dmst,
pmst, rmst
```

with argument `nu`

set equal to 1.

### References

Azzalini, A. with the collaboration of Capitanio, A. (2014).
*The Skew-Normal and Related Families*.
Cambridge University Press, IMS Monograph series.

### See Also

`dst`

, `dsc`

, `dmsn`

,
`dmt`

, `makeSECdistr`

### Examples

1 2 3 4 5 6 7 8 9 | ```
x <- seq(-4,4,length=15)
xi <- c(0.5, -1)
Omega <- diag(2)
Omega[2,1] <- Omega[1,2] <- 0.5
alpha <- c(2,2)
pdf <- dmst(cbind(x,2*x-1), xi, Omega, alpha, 5)
rnd <- rmst(10, xi, Omega, alpha, 6)
p1 <- pmst(c(2,1), xi, Omega, alpha, nu=5)
p2 <- pmst(c(2,1), xi, Omega, alpha, nu=5, abseps=1e-12, maxpts=10000)
``` |