# 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

 ```1 2``` ```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. 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.

`dlaplace`, `dmvl`, `dmvn`, `dmvnp`, `dnorm`, `dnormp`, `dnormv`, and `dpe`.
 ```1 2 3 4 5 6 7 8 9``` ```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) ```