MultivariateGenLaplace: The Multivariate Asymmetric Generalized Laplace Distribution

Description

Density and random generation for the multivariate asymmetric generalized Laplace distribution.

Usage

dmgl(x, mu = rep(0, d), sigma = diag(d), shape = 1, log = FALSE)
rmgl(n, mu, sigma, shape = 1)

Value

dmgl gives the GL density of a $d$-dimensional vector x. rmgl generates a sample of size n of $d$-dimensional random GL variables.

Arguments

x: vector of quantiles.
n: number of observations.
mu: simmetry parameter.
sigma: scale parameter -- positive-definite matrix.
shape: shape parameter.
log: logical; if TRUE, probabilities are log--transformed.

Author

Marco Geraci

Details

This is the distribution described by Kozubowski et al (2013) and has density $$ f(x) = \frac{2\exp(\mu'\Sigma^{-1}x)}{(2\pi)^{d/2}\Gamma(s)|\Sigma|^{1/2}}(\frac{Q(x)}{C(\Sigma,\mu)})^{\omega}B_{\omega}(Q(x)C(\Sigma,\mu)) $$ where $\mu$ is the symmetry parameter, $\Sigma$ is the scale parameter, $Q(x)=\sqrt{x'\Sigma^{-1}x}$, $C(\Sigma,\mu)=\sqrt{2+\mu'\Sigma^{-1}\mu}$, $\omega = s - d/2$, $d$ is the dimension of $x$, and $s$ is the shape parameter (note that the parameterization in nlmm is $\alpha = \frac{1}{s}$). $\Gamma$ denotes the Gamma function and $B_{u}$ the modified Bessel function of the third kind with index $u$. The parameter $\mu$ is related to the skewness of the distribution (symmetric if $\mu = 0$). The variance-covariance matrix is $s(\Sigma + \mu\mu')$. The multivariate asymmetric Laplace is obtained when $s = 1$ (see MultivariateLaplace).

In the symmetric case ($\mu = 0$), the multivariate GL distribution has two special cases: multivariate normal for $s \rightarrow \infty$ and multivariate symmetric Laplace for $s = 1$.

The univariate symmetric GL distribution is provided via GenLaplace, which gives the distribution and quantile functions in addition to the density and random generation functions.

References

Geraci, M. and Farcomeni A. (2020). A family of linear mixed-effects models using the generalized Laplace distribution. Statistical Methods in Medical Research, 29(9), 2665-2682.

Kozubowski, T. J., K. Podgorski, and I. Rychlik (2013). Multivariate generalized Laplace distribution and related random fields. Journal of Multivariate Analysis 113, 59-72.