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An alternative multivariate skew Laplace distribution: properties and estimation. (English) Zbl 1247.60015
Summary: 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 T. Kozubowski and K. Podgórski [Comput. Stat. 15, No. 4, 531–540 (2000; Zbl 1035.60010)] and S. Kotz, T. Kozubowski and K. Podgórski [The Laplace distribution and generalizations (Birkhäuser, Boston) (2001; Zbl 0977.62003)]. 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.

60E05 Probability distributions: general theory
62H10 Multivariate distribution of statistics
62H12 Estimation in multivariate analysis
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[1] Anderson DN (1992) A multivariate linnik distribution. Stat Prob Lett 14: 333–336 · Zbl 0754.60022
[2] 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
[3] Azzalini A, Dalla Valle A (1996) The multivariate skew-normal distribution. Biometrika 83: 715–726 · Zbl 0885.62062
[4] 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 · Zbl 1065.62094
[5] Azzalini A, Genton MG (2008) Robust likelihood methods based on the skew-t and related distributions. Int Stat Rev 76(1): 106–129 · Zbl 1206.62102
[6] Barndorff-Nielsen O (1977) Exponentially decreasing distributions for logarithm of particle size. Proc R Sac Land A 353: 401–419
[7] Barndorff-Nielsen O (1978) Hyperbolic distributions and distributions on hyperbolae. Scand J Stat 9: 43–46 · Zbl 0386.60018
[8] Barndorff-Nielsen O, Kent J, Sorensen M (1982) Normal variance–mean mixtures and z distributions. Int Stat Rev 50: 145–159 · Zbl 0497.62019
[9] Blaesild P (1981) The two-dimensional hyperbolic distribution and related distributions, with an application to Johannsen’ bean data. Biometrika 68: 251–263 · Zbl 0463.62048
[10] Blaesild P (1999) Generalized hyperbolic and generalized inverse gaussian distributions. Working Paper, University of Arhus, Denmark
[11] Butler RJ, McDonald JB, Nelson RD, White SB (1990) Robust and partially adaptive estimation of regression models. Rev Econ Stat 72: 321–327
[12] Devroye L (1997) Random variate generation for multivariate unimodal densities. Trans Model Comp Simulat 7: 447–477 · Zbl 0917.65004
[13] DiCiccio TJ, Monti AC (2004) Inferential aspects of the skew exponential power distribution. J Am Stat Assoc 99: 439–450 · Zbl 1117.62318
[14] Ernst MD (1998) A multivariate generalized Laplace distribution. Comput Stat 13: 227–232 · Zbl 0922.62043
[15] Fang KT, Kotz S, Ng KW (1990) Symmetric multivariate and related distributions. Chapman and Hall, London · Zbl 0699.62048
[16] Genton MG (2004) Skew-elliptical distributions and their applications: a journey beyond normality, edited volume. Chapman & Hall/CRC, Boca Raton
[17] 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 · Zbl 1135.62041
[18] 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 · Zbl 0895.62053
[19] 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
[20] 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
[21] 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
[22] Kollo T, Srivastava MS (2004) Estimation and testing of parameters in multivariate Laplace distribution. Commun Stat Theory Methods 33: 2363–2387 · Zbl 1217.62080
[23] Kotz S, Kozubowski TJ, Podgórski K (2001) The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance. Birkhäuser, Boston · Zbl 0977.62003
[24] Kotz S, Kozubowski TJ, Podgórski K (2003) An asymmetric multivariate Laplace Distribution. Working paper
[25] Kozubowski TJ, Podgórski K (1999) A class of asymmetric distributions. Actuarial Res Clearing House 1: 113–134
[26] Kozubowski TJ, Podgórski K (2000a) A multivariate and asymmetric generalization of Laplace distribution. Comput Stat 15: 531–540 · Zbl 1035.60010
[27] Kozubowski TJ, Podgórski K (2000b) Asymmetric Laplace distribution. Math Sci 25: 37–46 · Zbl 0961.60026
[28] Künsch H (1984) Infinitesimal robustness for autoregressive processes. Ann Stat 12: 843–863 · Zbl 0587.62077
[29] Lange K, Sinsheimer JS (1993) Normal/Independent distributions and their applications in robust regression. J Comput Graph Stat 2: 175–198
[30] Lye JN, Martin VL (1993) Robust estimation, nonnormalities, and generalized exponential distributions. J Am Stat Assoc 88: 261–267 · Zbl 0775.62081
[31] Ma Y, Genton MG (2000) Highly robust estimation of the autocovariance function. J Time Ser Anal 21: 663–681 · Zbl 0970.62056
[32] McNeil A, Frey R, Embrechts P (2005) Quantitative risk management: concepts, techniques and tools. Princeton University Press, New Jersey · Zbl 1089.91037
[33] Nadarajah S (2003) The Kotz-type distribution with application. Statistics 37: 341–358 · Zbl 1037.62048
[34] 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 · Zbl 05196666
[35] 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)
[36] Purdom E, Holmes SP (2005) Error distribution for gene expression data. Stat Appl Genet Molec Biol 4(16). http://www.bepress.com/sagmb · Zbl 1083.62114
[37] Rao CR (1988) Methodology based on the L1-norm in statistical inference. Sankhyã Ind J Stat 50(Series A): 289–313 · Zbl 0677.62058
[38] Roelant E, Van Aelst S (2007) An L1-type estimator of multivariate location and shape. Stat Methods Appl 15: 381–393 · Zbl 1187.62104
[39] Wang J, Genton MG (2006) The multivariate skew-slash distribution. J Stat Plann Inference 136: 209–220 · Zbl 1081.60013
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