Multivariate Power Exponential Distribution
These functions provide the density and random number generation for the multivariate power exponential distribution.
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)
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 |
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.
dmvpe gives the density and
rmvpe generates random deviates.
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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.
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)
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