In the mathematical theory of probability, multivariate Laplace distributions are extensions of the Laplace distribution and the asymmetric Laplace distribution to multiple variables. The marginal distributions of symmetric multivariate Laplace distribution variables are Laplace distributions. The marginal distributions of asymmetric multivariate Laplace distribution variables are asymmetric Laplace distributions.[1]
A typical characterization of the asymmetric multivariate Laplace distribution has the characteristic function:
[1]
As with the symmetric multivariate Laplace distribution, the asymmetric multivariate Laplace distribution has mean , but the covariance becomes .[3] The asymmetric multivariate Laplace distribution is not elliptical unless , in which case the distribution reduces to the symmetric multivariate Laplace distribution with .[1]
The asymmetric Laplace distribution, including the special case of , is an example of a geometric stable distribution.[3] It represents the limiting distribution for a sum of independent, identically distributed random variables with finite variance and covariance where the number of elements to be summed is itself an independent random variable distributed according to a geometric distribution.[1] Such geometric sums can arise in practical applications within biology, economics and insurance.[1] The distribution may also be applicable in broader situations to model multivariate data with heavier tails than a normal distribution but finite moments.[1]
The relationship between the exponential distribution and the Laplace distribution allows for a simple method for simulating bivariate asymmetric Laplace variables (including for the case of ). Simulate a bivariate normal random variable vector from a distribution with and covariance matrix . Independently simulate an exponential random variables W from an Exp(1) distribution. will be distributed (asymmetric) bivariate Laplace with mean and covariance matrix .[1]
References
^ a b c d e f g h i j k l mKotz. Samuel; Kozubowski, Tomasz J.; Podgorski, Krzysztof (2001). The Laplace Distribution and Generalizations. Birkhauser. pp. 229–245. ISBN0817641661.
^Fragiadakis, Konstantinos & Meintanis, Simos G. (March 2011). "Goodness-of-fit tests for multivariate Laplace distributions". Mathematical and Computer Modelling. 53 (5–6): 769–779. doi:10.1016/j.mcm.2010.10.014.{{cite journal}}: CS1 maint: multiple names: authors list (link)
^ a bKozubowski, Tomasz J.; Podgorski, Krzysztof; Rychlik, Igor (2010). "Multivariate Generalize Laplace Distributions and Related Random Fields". Journal of Multivariate Analysis. 113. University of Gothenburg: 59–72. doi:10.1016/j.jmva.2012.02.010. S2CID 206252976.