sn (version 1.2-2)

dmsn: Multivariate skew-normal distribution

Description

Probability density function, distribution function and random number generation for the multivariate skew-normal (SN) distribution.

Usage

dmsn(x, xi=rep(0,length(alpha)), Omega, alpha, tau=0, dp=NULL, log=FALSE)
pmsn(x, xi=rep(0,length(alpha)), Omega, alpha,  tau=0, dp=NULL, ...)
rmsn(n=1, xi=rep(0,length(alpha)), Omega, alpha,  tau=0, dp=NULL)

Arguments

x

either a vector of length d, where d=length(alpha), or a matrix with d columns, giving the coordinates of the point(s) where the density or the distribution function must be evaluated.

xi

a numeric vector of length d representing the location parameter of the distribution; see Background. In a call to dmsn, xi can be a matrix; in this case, its dimensions must agree with th

Omega

a symmetric positive-definite matrix of dimension (d,d); see Background.

alpha

a numeric vector which regulates the slant of the density; see Background. Inf values in alpha are not allowed.

tau

a single value representing the `hidden mean' parameter of the ESN distribution; tau=0 (default) corresponds to a SN distribution.

dp

a list with three elements, corresponding to xi, Omega and alpha described above; default value FALSE. If dp is assigned, individual parameters must not be specified.

n

a numeric value which represents the number of random vectors to be drawn.

log

logical (default value: FALSE); if TRUE, log-densities are returned.

...

additional parameters passed to pmnorm

Value

• A vector of density values (dmsn) or of probabilities (pmsn) or a matrix of random points (rmsn).

concept

skew-normal distribution

Background

The multivariate skew-normal distribution is discussed by Azzalini and Dalla Valle (1996). The (Omega,alpha) parametrization adopted here is the one of Azzalini and Capitanio (1999). Chapter 5 of Azzalini and Capitanio (2014) provides an extensive account, including subsequent developments.

Notice that the location vector xi does not represent the mean vector of the distribution. Similarly, Omega is not the covariance matrix of the distribution, although it is a covariance matrix.

Details

Typical usages are dmsn(x, xi=rep(0,length(alpha)), Omega, alpha, log=FALSE) dmsn(x, dp=, log=FALSE) pmsn(x, xi=rep(0,length(alpha)), Omega, alpha, ...) pmsn(x, dp=) rmsn(n=1, xi=rep(0,length(alpha)), Omega, alpha) rmsn(n=1, dp=) Function pmsn makes use of pmnorm from package mnormt; the accuracy of its computation can be controlled via ...

References

Azzalini, A. and Capitanio, A. (1999). Statistical applications of the multivariate skew normal distribution. J.Roy.Statist.Soc. B 61, 579--602. Full-length version available at http://arXiv.org/abs/0911.2093

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

Azzalini, A. and Dalla Valle, A. (1996). The multivariate skew-normal distribution. Biometrika 83, 715--726.

See Also

dsn, dmst, dmnorm, op2dp

Examples

x <- seq(-3,3,length=15)
xi <- c(0.5, -1)
Omega <- diag(2)
Omega[2,1] <- Omega[1,2] <- 0.5
alpha <- c(2,-6)
pdf <- dmsn(cbind(x, 2*x-1), xi, Omega, alpha)
cdf <- pmsn(cbind(x, 2*x-1), xi, Omega, alpha)
p1 <- pmsn(c(2,1), xi, Omega, alpha)
p2 <- pmsn(c(2,1), xi, Omega, alpha, abseps=1e-12, maxpts=10000)
#
rnd <- rmsn(10, xi, Omega, alpha)
#
# Recall: the components of alpha are not the slant parameters of the marginal 
# distributions; to fix these parameters, it is convenient to start 
# from the OP parameterization, for instance as follows:
op <- list(xi=c(0,1), Psi=matrix(c(2,2,2,3), 2, 2), lambda=c(5, -2))
rnd2 <- rmsn(10, dp=op2dp(op,"SN"))