fMultivar (version 4031.84)

mvdist-msc: Multivariate Skew Cauchy Distribution

Description

Density, distribution function, and random number generation for the multivariate Cauchy distribution.

Details

The functions to compute densities dmsc, probabilities pmsc, and to generate random numbers rmsc for the multivariate skew Cauchy distribution are available in the contributed R package sn (note, they are no longer builtin in fMultivar). The reason is that the performance for these functions in package sn has superseeded those used before in the package fMultivar.

The usage of the sn functions is:

dmsc(x, xi, Omega, alpha, dp = NULL, log = FALSE)
pmsc(x, xi, Omega, alpha, dp = NULL, ...)
rmsc(n, xi, Omega, alpha, dp = NULL)

NOTE: 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. Be aware 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.

For further details we refer to the help page in the package sn.

References

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

Azzalini, A. and Capitanio, A. (1999), Statistical applications of the multivariate skew normal distribution, Journal 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.

Examples

if (FALSE) {
## grid2d - 
   # Make 2-D Grid Coordinates:
   N <- 101
   x <- y <- seq(-3, 3, l=N)
   X <- cbind(u=grid2d(x)$x, v=grid2d(x)$y)
   
## Set Parameters:
   xi <- c(0, 0) 
   Omega <- diag(2); Omega[2,1] <- Omega[1,2] <- 0.5
   alpha <- c(2, -6)
   
## dmsc - 
   # Compute skew Cauchy Density:
   z <- sn::dmsc(X, xi, Omega, alpha)
   Z <- list(x=x, y=x, z=matrix(z, ncol = length(x)))
   # Plot:   
   image(Z, main ="Skew Cauchy Density")
   contour(Z, add=TRUE)
   grid(col="red")

## pmsc - 
   # Compute skew Cauchy Probability:
   z <- NULL
   for (i in 1:nrow(X)) z <- c(z, sn::pmsc(X[i, ], xi, Omega, alpha)[[1]])
   Z <- list(x=x, y=x, z=matrix(z, ncol = length(x)))
   # Plot:   
   image(Z, main ="Skew Cauchy Probability")
   contour(Z, add=TRUE)
   grid(col="red")
   
## rmsc - 
   # Skew Cauchy Random Deviates:
   set.seed(4711)
   r <- sn::rmsc(10000, xi, Omega, alpha)
   plot(hexBinning(r[, 1], r[, 2]))
   # Note, we have fat tails ...
}