Solved – Bivariate Skewed Normal Distribution

What is the equation for a multivariate skewed normal distribution, specifically a two dimensional skewed normal distribution?

Bivariate (or multivariate) skew normal distributions can be constructed with the same methods that is used in the univariate case. The usual univariate skewnormal density (due to Azzalini https://en.wikipedia.org/wiki/Skew_normal_distribution) is given by $$ phi_{text{Skew}}(x;alpha) =2phi(x)Phi(alpha x) $$ where $phi$ is the usual standard normal density and $alpha$ is a new skewness parameter. $Phi$ is the standard normal cumulative distribution.

We can use the same construction in the multivariate case, introducing the covariance matrix $Omega$ but still keeping the center at zero. $$ phi_{d,text{Skew}}(x;Omega,alpha) = 2 phi_d(x;Omega)Phi(alpha^T x) $$ where $d$ is the dimension and $phi_d$ is the multinormal density with covariance matrix $Omega$ (and center zero), $Phi$ is still the univariate standard normal cumulative distribution.

A contour plot is shown below, the parameters used can be gleaned from the R code below it:

enter image description here

library(sn)    alpha <-  c(0.5, 1) Omega <-  matrix(c(1, 0.5, 0.5, 1), 2, 2) xran  <-  seq(-3, 3, length=101) yran  <-  seq(-3, 3, length=101) z     <-  outer(xran, yran, FUN=Vectorize( function(x, y) dmsn(c(x, y), c(0, 0),                                                     Omega, alpha) )  ) image(xran, yran, z) contour(xran, yran, z, ncontours=20, add=TRUE) title("bivariate skewnormal density") 

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