# dmst: Multivariate skew-t distribution and skew-Cauchy distribution In sn: The Skew-Normal and Related Distributions Such as the Skew-t

## 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 `dmst` and `dmsc`, this is either a vector of length `d`, where `d=length(alpha)`, or a matrix with `d` columns, representing the coordinates of the point(s) where the density must be avaluated; for `pmst` and `pmsc`, only a vector of length `d` is allowed. `xi` a numeric vector of length `d` representing the location parameter of the distribution; see ‘Background’. In a call to `dmst` or `dmsc`, `xi` can be a matrix, whose rows represent a set of location parameters; in this case, its dimensions must match those of `x`. `Omega` a symmetric positive-definite matrix of dimension `(d,d)`; see Section ‘Background’. `alpha` a numeric vector of length `d` which regulates the slant of the density; see Section ‘Background’. `Inf` values in `alpha` are not allowed. `nu` a positive value representing the degrees of freedom of ST distribution; does not need to be integer. Default value is `nu=Inf` which corresponds to the multivariate skew-normal distribution. `dp` a list with three elements named `xi`, `Omega`, `alpha` and `nu`, containing quantities as described above. If `dp` is specified, this prevents specification of the individual parameters. `n` a numeric value which represents the number of random vectors to be drawn; default value is `1`. `log` logical (default value: `FALSE`); if `TRUE`, log-densities are returned. `...` additional parameters passed to `pmt`.

## 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.

`dst`, `dsc`, `dmsn`, `dmt`, `makeSECdistr`
 ```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) ```