These functions provide the density and random number generation for the multivariate power exponential distribution, given the Cholesky parameterization.

```
dmvpec(x=c(0,0), mu=c(0,0), U, kappa=1, log=FALSE)
rmvpec(n, mu=c(0,0), U, 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.

U

This is the \(k \times k\) upper-triangular matrix that is Cholesky factor \(\textbf{U}\) of 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.

`dmvpec`

gives the density and
`rmvpec`

generates random deviates.

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
`dmvlc`

function will perform faster and more accurately.

In practice, \(\textbf{U}\) is fully unconstrained for proposals
when its diagonal is log-transformed. 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
matrix-multiply the Cholesky back to the covariance matrix, but it
does not have to check for or correct the covariance matrix to
positive-definiteness, 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.

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.

`chol`

,
`dlaplace`

,
`dmvlc`

,
`dmvnc`

,
`dmvnpc`

,
`dnorm`

,
`dnormp`

,
`dnormv`

, and
`dpe`

.

```
# NOT RUN {
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)
# }
```

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