# Multivariate Skew Cauchy Distribution 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 ```r ## Not run: ## 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 ... ## End(Not run) ```

mvdist-msc function