dist.Multivariate.Power.Exponential.Cholesky {LaplacesDemon}  R Documentation 
Multivariate Power Exponential Distribution: Cholesky Parameterization
Description
These functions provide the density and random number generation for the multivariate power exponential distribution, given the Cholesky parameterization.
Usage
dmvpec(x=c(0,0), mu=c(0,0), U, kappa=1, log=FALSE)
rmvpec(n, mu=c(0,0), U, 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 
U 
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, GomezVillegas, 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: positivedefinite
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 onedimensional or univariate power exponential distribution. GomezVillegas (1998) and SanchezManzano 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
dmvlc
function will perform faster and more accurately.
In practice, \textbf{U}
is fully unconstrained for proposals
when its diagonal is logtransformed. The diagonal is exponentiated
after a proposal and before other calculations. Overall, the Cholesky
parameterization is faster than the traditional parameterization.
Compared with dmvpe
, dmvpec
must additionally
matrixmultiply the Cholesky back to the covariance matrix, but it
does not have to check for or correct the covariance matrix to
positivedefiniteness, which overall is slower. Compared with
rmvpe
, rmvpec
is faster because the Cholesky decomposition
has already been performed.
The rmvpec
function is a modified form of the rmvpowerexp function
in the MNM package.
Value
dmvpec
gives the density and
rmvpec
generates random deviates.
Author(s)
Statisticat, LLC. software@bayesianinference.com
References
Gomez, E., GomezVillegas, M.A., and Marin, J.M. (1998). "A Multivariate Generalization of the Power Exponential Family of Distributions". Communications in StatisticsTheory and Methods, 27(3), p. 589–600.
SanchezManzano, E.G., GomezVillegas, M.A., and MarnDiazaraque, 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
chol
,
dlaplace
,
dmvlc
,
dmvnc
,
dmvnpc
,
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)
U < chol(Sigma)
dmvpec(x, mu, U, kappa=1)
X < rmvpec(n, mu, U, kappa=1)
joint.density.plot(X[,1], X[,2], color=TRUE)