# Multivariate skew-normal distribution

### Description

Probability density function, distribution function and random number generation for the multivariate skew-normal (SN) distribution.

### Usage

1 2 3 |

### Arguments

`x` |
either a vector of length |

`xi` |
a numeric vector of length |

`Omega` |
a symmetric positive-definite matrix of dimension |

`alpha` |
a numeric vector which regulates the slant of the density;
see ‘Background’. |

`tau` |
a single value representing the ‘hidden mean’ parameter
of the ESN distribution; |

`dp` |
a list with three elements, corresponding to |

`n` |
a numeric value which represents the number of random vectors to be drawn. |

`log` |
logical (default value: |

`...` |
additional parameters passed to |

### Details

Typical usages are

1 2 3 4 5 6 |

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. 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.

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

### See Also

`dsn`

, `dmst`

, `dmnorm`

,
`op2dp`

, `cp2dp`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | ```
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)
``` |