Multivariate skew-t distribution and skew-Cauchy distribution
Probability density function, distribution function and random number generation for the multivariate skew-t (ST) and skew-Cauchy (SC) distributions.
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)
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 |
Typical usages are
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 ....
A vector of density values (dmst and dmsc) or a single
probability (pmst and pmsc) or a matrix of random points
(rmst and rmsc).
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.
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
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)
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