dist.Multivariate.Power.Exponential {LaplacesDemon} | R Documentation |
Multivariate Power Exponential Distribution
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
|
n |
This is the number of random draws. |
mu |
This is mean vector |
Sigma |
This is the |
kappa |
This is the kurtosis parameter, |
log |
Logical. If |
Details
Application: Continuous Multivariate
Density:
p(\theta) = \frac{k\Gamma(k/2)}{\pi^{k/2} \sqrt{|\Sigma|} \Gamma(1 + k/(2\kappa)) 2^{1 + k/(2\kappa)}} \exp(-\frac{1}{2}(\theta-\mu)^T \Sigma (\theta-\mu))^{\kappa}
Inventor: Gomez, Gomez-Villegas, and Marin (1998)
Notation 1:
\theta \sim \mathcal{MPE}(\mu, \Sigma, \kappa)
Notation 2:
\theta \sim \mathcal{PE}_k(\mu, \Sigma, \kappa)
Notation 3:
p(\theta) = \mathcal{MPE}(\theta | \mu, \Sigma, \kappa)
Notation 4:
p(\theta) = \mathcal{PE}_k(\theta | \mu, \Sigma, \kappa)
Parameter 1: location vector
\mu
Parameter 2: positive-definite
k \times 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 \rightarrow \infty
.
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. software@bayesian-inference.com
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)