dist.Asymmetric.Multivariate.Laplace {LaplacesDemon} | R Documentation |
Asymmetric Multivariate Laplace Distribution
Description
These functions provide the density and random generation for the
asymmetric multivariate Laplace distribution with location and skew
parameter \mu
and covariance \Sigma
.
Usage
daml(x, mu, Sigma, log=FALSE)
raml(n, mu, Sigma)
Arguments
x |
This is a |
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 |
Sigma |
This is the |
log |
Logical. If |
Details
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.
Value
daml
gives the density, and
raml
generates random deviates.
References
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.
See Also
Examples
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)