# Multivariate skew-normal distribution Probability density function, distribution function and random number generation for the multivariate skew-normal (SN ) distribution. ```r 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` 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. - `...`: additional parameters passed to `pmnorm`. ## Returns A vector of density values (`dmsn`) or of probabilities (`pmsn`) or a matrix of random points (`rmsn`). ## 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=) ``` For efficiency reasons, `rmsn` makes very limited checks on the validity of the arguments. For instance, failure to positive definiteness of `Omega` would not be detected, and an uncontrolled crash occurs. Function `pmsn` makes use of `pmnorm` from package `mnormt`; the accuracy of its computation can be controlled via `...` ## 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. B** 61 , 579--602. Full-length version available at [https://arXiv.org/abs/0911.2093](https://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`, `pmnorm`, `op2dp`, `cp2dp` ## Examples ```r 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) ```

dmsn function