These functions provide the density and random generation for the asymmetric multivariate Laplace distribution with location and skew parameter \(\mu\) and covariance \(\Sigma\).

```
daml(x, mu, Sigma, log=FALSE)
raml(n, mu, Sigma)
```

x

This is a \(N \times K\) matrix of data, or a vector of length \(K\).

n

This is the number of observations, which must be a positive integer that has length 1.

mu

This is the location and skew parameter \(\mu\). This may be a \(N \times K\) matrix, or a vector of length \(K\).

Sigma

This is the \(K \times K\) positive-definite covariance matrix \(\Sigma\).

log

Logical. If `log=TRUE`

, then the logarithm of the
density is returned.

`daml`

gives the density, and
`raml`

generates random deviates.

Application: Continuous Multivariate

Density: \(p(\theta) = \frac{2\exp(\theta\Omega\theta)}{(2\pi)^{k/2}|\Sigma|^0.5} \frac{\theta\Omega\theta}{2 + \mu\Omega\mu}^{(2-k)/4} K_{(2-k)/2}(\sqrt{(2 + \mu\Omega\mu)(\theta\Omega\theta)})\)

Inventor: Kotz, Kozubowski, and Podgorski (2003)

Notation 1: \(\theta \sim \mathcal{AL}_K(\mu, \Sigma)\)

Notation 2: \(p(\theta) = \mathcal{AL}_K(\theta | \mu, \Sigma)\)

Parameter 1: location-skew parameter \(\mu\)

Parameter 2: positive-definite covariance matrix \(\Sigma\)

Mean: Unknown

Variance: Unknown

Mode: \(mode(\theta) = \mu\)

The asymmetric multivariate Laplace distribution of Kotz, Kozubowski, and Podgorski (2003) is a multivariate extension of the univariate, asymmetric Laplace distribution. It is parameterized according to two parameters: location-skew parameter \(\mu\) and positive-definite covariance matrix \(\Sigma\). Location and skew occur in the same parameter. When \(\mu=0\), the density is the (symmetric) multivariate Laplace of Anderson (1992). As each location deviates from zero, the marginal distribution becomes more skewed. Since location and skew are combined, it is appropriate for zero-centered variables, such as a matrix of centered and scaled dependent variables in cluster analysis, factor analysis, multivariate regression, or multivariate time-series.

The asymmetric multivariate Laplace distribution is also discussed earlier in Kozubowski and Podgorski (2001), and is well-suited for financial modeling via multivariate regression, specifically with currency exchange rates. Cajigas and Urga (2005) fit residuals in a multivariate GARCH model with the asymmetric multivariate Laplace distribution, regarding stocks and bonds. They find that it "overwhelmingly outperforms" normality.

Anderson, D.N. (1992). "A Multivariate Linnik Distribution".
*Statistical Probability Letters*, 14, p. 333--336.

Cajigas, J.P. and Urga, G. (2005) "Dynamic Conditional Correlation Models with Asymmetric Laplace Innovations". Centre for Economic Analysis: Cass Business School.

Kotz, S., Kozubowski, T.J., and Podgorski, K. (2003). "An Asymmetric Multivariate Laplace Distribution". Working Paper.

Kozubowski, T.J. and Podgorski, K. (2001). "Asymmetric Laplace
Laws and Modeling Financial Data". *Mathematical and Computer
Modelling*, 34, p. 1003--1021.

```
# NOT RUN {
library(LaplacesDemon)
x <- daml(c(1,2,3), c(0,1,2), diag(3))
X <- raml(1000, c(0,1,2), diag(3))
joint.density.plot(X[,1], X[,2], color=FALSE)
# }
```

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