ExtremalDep (version 1.0.0)

dmesn: Bivariate and Trivariate Extended Skew-Normal Distribution

Description

Density function and distribution function for the bivariate and trivariate extended skew-normal (ESN) distribution.

Usage

dmesn(
  x = c(0, 0),
  location = rep(0, length(x)),
  scale = diag(length(x)),
  shape = rep(0, length(x)),
  extended = 0
)
pmesn(
  x = c(0, 0),
  location = rep(0, length(x)),
  scale = diag(length(x)),
  shape = rep(0, length(x)),
  extended = 0
)

Value

dmesn returns the density, and pmesn returns the probability, of the bivariate or trivariate extended skew-normal distribution with the specified location, scale, shape, and extended

parameters. If extended = 0, the skew-normal distribution is obtained. If shape = 0 and extended = 0, the normal distribution is recovered.

Arguments

x

Quantile vector of length d = 2 or d = 3.

location

A numeric location vector of length d. Default is 0.

scale

A symmetric positive-definite scale matrix of dimension (d, d). Default is diag(d).

shape

A numeric skewness vector of length d. Default is 0.

extended

A single extension parameter. Default is 0.

Author

Simone Padoan, , https://faculty.unibocconi.it/simonepadoan/; Boris Beranger, , https://www.borisberanger.com.

References

Azzalini, A. and Capitanio, A. (1999). Statistical applications of the multivariate skew normal distribution. J. Roy. Statist. Soc. B 61, 579--602.

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.

Examples

sigma1 <- matrix(c(2, 1.5, 1.5, 3), ncol = 2)
sigma2 <- matrix(c(
  2, 1.5, 1.8,
  1.5, 3, 2.2,
  1.8, 2.2, 3.5
), ncol = 3)

shape1 <- c(1, 2)
shape2 <- c(1, 2, 1.5)

dens1 <- dmesn(x = c(1, 1), scale = sigma1, shape = shape1, extended = 2)
dens2 <- dmesn(x = c(1, 1), scale = sigma1)
dens3 <- dmesn(x = c(1, 1, 1), scale = sigma2, shape = shape2, extended = 2)
dens4 <- dmesn(x = c(1, 1, 1), scale = sigma2)

prob1 <- pmesn(x = c(1, 1), scale = sigma1, shape = shape1, extended = 2)
prob2 <- pmesn(x = c(1, 1), scale = sigma1)

# \donttest{
prob3 <- pmesn(x = c(1, 1, 1), scale = sigma2, shape = shape2, extended = 2)
prob4 <- pmesn(x = c(1, 1, 1), scale = sigma2)
# }