Abstract
A special case of the multivariate exponential power distribution is considered as a multivariate extension of the univariate symmetric Laplace distribution. In this paper, we focus on this multivariate symmetric Laplace distribution, and extend it to a multivariate skew distribution. We call this skew extension of the multivariate symmetric Laplace distribution the “multivariate skew Laplace (MSL) distribution” to distinguish between the asymmetric multivariate Laplace distribution proposed by Kozubowski and Podgórski (Comput Stat 15:531–540, 2000a) Kotz et al. (The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance, Chap. 6. Birkhäuser, Boston, 2001) and Kotz et al. (An asymmetric multivariate Laplace Distribution, Working paper, 2003). One of the advantages of (MSL) distribution is that it can handle both heavy tails and skewness and that it has a simple form compared to other multivariate skew distributions. Some fundamental properties of the multivariate skew Laplace distribution are discussed. A simple EM-based maximum likelihood estimation procedure to estimate the parameters of the multivariate skew Laplace distribution is given. Some examples are provided to demonstrate the modeling strength of the skew Laplace distribution.
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References
Anderson DN (1992) A multivariate linnik distribution. Stat Prob Lett 14: 333–336
Arslan O, Genc AI (2008) The skew generalized t distribution as a scale mixture of the skew exponential power distribution and its applications in robust statistical analysis. Statistics (in press). doi:10.1080/02331880802401241, http://www.informaworld.com
Azzalini A, Dalla Valle A (1996) The multivariate skew-normal distribution. Biometrika 83: 715–726
Azzalini A, Capitanio A (2003) Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. J Roy Stat Soc B 65: 367–389
Azzalini A, Genton MG (2008) Robust likelihood methods based on the skew-t and related distributions. Int Stat Rev 76(1): 106–129
Barndorff-Nielsen O (1977) Exponentially decreasing distributions for logarithm of particle size. Proc R Sac Land A 353: 401–419
Barndorff-Nielsen O (1978) Hyperbolic distributions and distributions on hyperbolae. Scand J Stat 9: 43–46
Barndorff-Nielsen O, Kent J, Sorensen M (1982) Normal variance–mean mixtures and z distributions. Int Stat Rev 50: 145–159
Blaesild P (1981) The two-dimensional hyperbolic distribution and related distributions, with an application to Johannsen’ bean data. Biometrika 68: 251–263
Blaesild P (1999) Generalized hyperbolic and generalized inverse gaussian distributions. Working Paper, University of Arhus, Denmark
Butler RJ, McDonald JB, Nelson RD, White SB (1990) Robust and partially adaptive estimation of regression models. Rev Econ Stat 72: 321–327
Devroye L (1997) Random variate generation for multivariate unimodal densities. Trans Model Comp Simulat 7: 447–477
DiCiccio TJ, Monti AC (2004) Inferential aspects of the skew exponential power distribution. J Am Stat Assoc 99: 439–450
Ernst MD (1998) A multivariate generalized Laplace distribution. Comput Stat 13: 227–232
Fang KT, Kotz S, Ng KW (1990) Symmetric multivariate and related distributions. Chapman and Hall, London
Genton MG (2004) Skew-elliptical distributions and their applications: a journey beyond normality, edited volume. Chapman & Hall/CRC, Boca Raton
Gómez-Sánchez-Manzano E, Gómez-Villegas MA, Marın JM (2008) Multivariate exponential power distributions as mixtures of normal distributions with Bayesian applications. Commun Stat Theory Methods 37: 972–985
Gómez E, Gómez-Villegas MA, Marın JM (1998) A multivariate generalization of the power exponential family of Distributions. Commun Stat Theory Methods 27: 589–600
Jones MC (2001) A skew t distribution. In: Charalambides CA, Koutras MV, Balakrishnan N (eds) Probability and Statistical Models with Applications: a volume in honor of Theophilos Cacoullos. Chapman and Hall, London, pp 269–278
Julià O, Vives-Rego J (2005) Skew-Laplace distribution in Gram-negative bacterial axenic cultures: new insights into intrinsic cellular heterogeneity. Microbiology 151: 749–755
Klein GE (1993) The sensitivity of cash-flow analysis to the choice of statistical model for interest rates changes (with discussions). Trans Soc Actuaries XLV: 79–186
Kollo T, Srivastava MS (2004) Estimation and testing of parameters in multivariate Laplace distribution. Commun Stat Theory Methods 33: 2363–2387
Kotz S, Kozubowski TJ, Podgórski K (2001) The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance. Birkhäuser, Boston
Kotz S, Kozubowski TJ, Podgórski K (2003) An asymmetric multivariate Laplace Distribution. Working paper
Kozubowski TJ, Podgórski K (1999) A class of asymmetric distributions. Actuarial Res Clearing House 1: 113–134
Kozubowski TJ, Podgórski K (2000a) A multivariate and asymmetric generalization of Laplace distribution. Comput Stat 15: 531–540
Kozubowski TJ, Podgórski K (2000b) Asymmetric Laplace distribution. Math Sci 25: 37–46
Künsch H (1984) Infinitesimal robustness for autoregressive processes. Ann Stat 12: 843–863
Lange K, Sinsheimer JS (1993) Normal/Independent distributions and their applications in robust regression. J Comput Graph Stat 2: 175–198
Lye JN, Martin VL (1993) Robust estimation, nonnormalities, and generalized exponential distributions. J Am Stat Assoc 88: 261–267
Ma Y, Genton MG (2000) Highly robust estimation of the autocovariance function. J Time Ser Anal 21: 663–681
McNeil A, Frey R, Embrechts P (2005) Quantitative risk management: concepts, techniques and tools. Princeton University Press, New Jersey
Nadarajah S (2003) The Kotz-type distribution with application. Statistics 37: 341–358
Naik DN, Plungpongpun K (2006) A Kotz type distribution for multivariate statistical inference. In: Balakrishnan N, Castillo E, Sarabia JM (eds) Advances in distribution theory, order statistics, and inference. Birkhauser, Boston, pp 111–124
Plungpongpun K (2003) Analysis of multivariate data using Kotz type distributions. PhD Dissertation, Computation and Applied Mathematics, Old Dominion University, USA. (ProQuest Information and Learning Company)
Purdom E, Holmes SP (2005) Error distribution for gene expression data. Stat Appl Genet Molec Biol 4(16). http://www.bepress.com/sagmb
Rao CR (1988) Methodology based on the L1-norm in statistical inference. Sankhyã Ind J Stat 50(Series A): 289–313
Roelant E, Van Aelst S (2007) An L1-type estimator of multivariate location and shape. Stat Methods Appl 15: 381–393
Wang J, Genton MG (2006) The multivariate skew-slash distribution. J Stat Plann Inference 136: 209–220
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Arslan, O. An alternative multivariate skew Laplace distribution: properties and estimation. Stat Papers 51, 865–887 (2010). https://doi.org/10.1007/s00362-008-0183-7
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DOI: https://doi.org/10.1007/s00362-008-0183-7
Keywords
- Normal variance–mean mixture distribution
- Heavy tailed distribution
- Laplace distribution
- Robust estimation
- Skewed distribution