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 and pmsn, 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 ‘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.
Function pmsn makes use of pmnorm from package mnormt;
the accuracy of its computation can be controlled via ...
Value
A vector of density values (dmsn) or of probabilities
(pmsn) or a matrix of random points (rmsn).
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.
Finally, the components of alpha are not equal to the slant parameters
of the marginal distributions; to fix the marginal parameters at prescribed
values, it is convenient to start from the OP parameterization, as illustrated
in the ‘Examples’ below. Another option is to start from the CP
parameterization, but notice that, at variance from the OP, not all
CP sets are invertible to lend a DP set.
References
Azzalini, A. and Capitanio, A. (1999).
Statistical applications of the multivariate skew normal distribution.
J.Roy.Statist.Soc. B61, 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.
Biometrika83, 715–726.
See Also
dsn, dmst, dmnorm,
op2dp, cp2dp
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)
#
# use OP parameters to fix marginal shapes at given lambda values:
op <- list(xi=c(0,1), Psi=matrix(c(2,2,2,3), 2, 2), lambda=c(5, -2))
rnd <- rmsn(10, dp=op2dp(op,"SN"))
#
# use CP parameters to fix mean vector, variance matrix and marginal skewness:
cp <- list(mean=c(0,0), var.cov=matrix(c(3,2,2,3)/3, 2, 2), gamma1=c(0.8, 0.4))
dp <- cp2dp(cp, "SN")
rnd <- rmsn(5, dp=dp)