dist.Multivariate.Power.Exponential: Multivariate Power Exponential Distribution

In LaplacesDemon: Complete Environment for Bayesian Inference

Description

These functions provide the density and random number generation for the multivariate power exponential distribution.

Usage

dmvpe(x=c(0,0), mu=c(0,0), Sigma=diag(2), kappa=1, log=FALSE)
rmvpe(n, mu=c(0,0), Sigma=diag(2), kappa=1)

Arguments

x	This is data or parameters in the form of a vector of length k or a matrix with k columns.
n	This is the number of random draws.
mu	This is mean vector mu with length k or matrix with k columns.
Sigma	This is the k x k covariance matrix Sigma.
kappa	This is the kurtosis parameter, kappa, and must be positive.
log	Logical. If log=TRUE, then the logarithm of the density is returned.

Details

• Application: Continuous Multivariate

• Density:

  p(theta) = ((k*Gamma(k/2)) / (pi^(k/2) * sqrt(|Sigma|) * Gamma(1 + k/(2*kappa)) * 2^(1 + k/(2*kappa)))) * exp(-(1/2)*(theta-mu)^T Sigma (theta-mu))^kappa

• Inventor: Gomez, Gomez-Villegas, and Marin (1998)

• Notation 1: theta ~ MPE(mu, Sigma, kappa)

• Notation 2: theta ~ PE[k](mu, Sigma, kappa)

• Notation 3: p(theta) = MPE(theta | mu, Sigma, kappa)

• Notation 4: p(theta) = PE[k](theta | mu, Sigma, kappa)

• Parameter 1: location vector mu

• Parameter 2: positive-definite k x k covariance matrix Sigma

• Parameter 3: kurtosis parameter kappa

• Mean: E(theta) =

• Variance: var(theta) =

• Mode: mode(theta) =

The multivariate power exponential distribution, or multivariate exponential power distribution, is a multidimensional extension of the one-dimensional or univariate power exponential distribution. Gomez-Villegas (1998) and Sanchez-Manzano et al. (2002) proposed multivariate and matrix generalizations of the PE family of distributions and studied their properties in relation to multivariate Elliptically Contoured (EC) distributions.

The multivariate power exponential distribution includes the multivariate normal distribution (kappa = 1) and multivariate Laplace distribution (kappa = 0.5) as special cases, depending on the kurtosis or kappa parameter. A multivariate uniform occurs as kappa -> infinity.

If the goal is to use a multivariate Laplace distribution, the dmvl function will perform faster and more accurately.

The rmvpe function is a modified form of the rmvpowerexp function in the MNM package.

Value

dmvpe gives the density and rmvpe generates random deviates.

Author(s)

Statisticat, LLC. [e-mail]

References

Gomez, E., Gomez-Villegas, M.A., and Marin, J.M. (1998). "A Multivariate Generalization of the Power Exponential Family of Distributions". Communications in Statistics-Theory and Methods, 27(3), p. 589–600.

Sanchez-Manzano, E.G., Gomez-Villegas, M.A., and Marn-Diazaraque, J.M. (2002). "A Matrix Variate Generalization of the Power Exponential Family of Distributions". Communications in Statistics, Part A - Theory and Methods [Split from: J(CommStat)], 31(12), p. 2167–2182.

See Also

dlaplace, dmvl, dmvn, dmvnp, dnorm, dnormp, dnormv, and dpe.

Examples

library(LaplacesDemon)
n <- 100
k <- 3
x <- matrix(runif(n*k),n,k)
mu <- matrix(runif(n*k),n,k)
Sigma <- diag(k)
dmvpe(x, mu, Sigma, kappa=1)
X <- rmvpe(n, mu, Sigma, kappa=1)
joint.density.plot(X[,1], X[,2], color=TRUE)

LaplacesDemon documentation built on July 9, 2021, 5:07 p.m.