[en] We present, compare and classify popular families of flexible multivariate distributions. Our classification is based on the type of symmetry (spherical, elliptical, central symmetry or asymmetry) and the tail behaviour (a single tail weight parameter or multiple tail weight parameters). We compare the families both theoretically (relevant properties and distinctive features) and with a Monte Carlo study (comparing the fitting abilities in finite samples).
Disciplines :
Mathematics Engineering, computing & technology: Multidisciplinary, general & others
Author, co-author :
Babić, Sladana; Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Gent, Belgium ; Vlerick Business School, Brussels, Belgium
LEY, Christophe ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH) ; Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Gent, Belgium
Veredas, David; Vlerick Business School, Brussels, Belgium ; Department of Economics, Ghent University, Gent, Belgium
External co-authors :
yes
Language :
English
Title :
Comparison and classification of flexible distributions for multivariate skew and heavy-tailed data
Genton, M.G.; Thompson, K.R. Skew-elliptical time series with application to flooding risk. In Time Series Analysis and Applications to Geophysical Systems; Springer: New York, NY, USA, 2004; pp. 169-185
Lambert, P.; Vandenhende, F. A copula-based model for multivariate non-normal longitudinal data: Analysis of a dose titration safety study on a new antidepressant. Stat. Med. 2002, 21, 3197-3217
Kotz, S.; Balakrishnan, N.; Johnson, N.L. Continuous Multivariate Distributions: Models and Applications, Volume 1; Wiley: New York, NY, USA, 2004
Genton, M.G. Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality; CRC Press: Boca Raton, FL, USA; New York, NY, USA, 2004
Nelsen, R.B. An Introduction to Copulas, 2nd ed.; Springer Science Business Media: New York, NY, USA, 2006
Sarabia Alegría, J.M.; Gómez Déniz, E. Construction of multivariate distributions: A review of some recent results. SORT 2008, 32, 3-36
Jones, M. On families of distributions with shape parameters (with discussion). Int. Stat. Rev. 2015, 83, 175-192
Kelker, D. Distribution theory of spherical distributions and a location-scale parameter generalization. Sankhya Indian J. Stat. Ser. A 1970, 32, 419-430
Cambanis, S.; Huang, S.; Simons, G. On the theory of elliptically contoured distributions. J. Multivar. Anal. 1981, 11, 368-385
Fang, K.; Kotz, S.; Ng, K. Symmetric Multivariate and Related Distributions; CRC Monographs on Statistics & Applied Probability; Chapman and Hall/CRC: London, UK, 1990.13. Hallin, M.; Paindaveine, D. Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks. Ann. Stat. 2002, 30, 1103-1133
Liebscher, E. A semiparametric density estimator based on elliptical distributions. J. Multivar. Anal. 2005, 92, 205-225
Hallin, M.; Paindaveine, D. Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity. Ann. Stat. 2006, 34, 2707-2756
Hallin, M.; Oja, H.; Paindaveine, D. Semiparametrically efficient rank-based inference for shape. II. Optimal R-estimation of shape. Ann. Stat. 2006, 34, 2757-2789
Hallin, M.; Paindaveine, D.; Verdebout, T. Optimal rank-based testing for principal components. Ann. Stat. 2010, 38, 3245-3299
Dominicy, Y.; Ilmonen, P.; Veredas, D. Multivariate Hill estimators. Int. Stat. Rev. 2017, 85, 108-142
Paindaveine, D. Elliptical symmetry. In Encyclopedia of Environmetrics, 2nd ed.; El-Shaarawi, A.H., Piegorsch, W., Eds.; JohnWiley and Sons Ltd.: Chichester, UK, 2012; pp. 802-807
Lombardi, M.J.; Veredas, D. Indirect estimation of elliptical stable distributions. Comput. Stat. Data Anal. 2009, 53, 2309-2324
Nolan, J.P. Multivariate elliptically contoured stable distributions: Theory and estimation. Comput. Stat. 2013, 28, 2067-2089
Tyler, D.E. A distribution-free M-estimator of multivariate scatter. Ann. Stat. 1987, 15, 234-251
Klüppelberg, C.; Kuhn, G.; Peng, L. Estimating the tail dependence function of an elliptical distribution. Bernoulli 2007, 13, 229-251
Schmidt, R. Tail dependence for elliptically contoured distributions. Math. Methods Oper. Res. 2002, 55, 301-327
Genton, M.G.; Loperfido, N.M. Generalized skew-elliptical distributions and their quadratic forms. Ann. Inst. Stat. Math. 2005, 57, 389-401
Azzalini, A.; Capitanio, A. Statistical applications of the multivariate skew normal distribution. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 1999, 61, 579-602
Branco, M.D.; Dey, D.K. A general class of multivariate skew-elliptical distributions. J. Multivar. Anal. 2001, 79, 99-113
Azzalini, A.; Capitanio, A. Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 2003, 65, 367-389
Wang, J.; Boyer, J.; Genton, M.G. A skew-symmetric representation of multivariate distributions. Stat. Sin. 2004, 14, 1259-1270
Shushi, T. Generalized skew-elliptical distributions are closed under affine transformations. Stat. Prob. Lett. 2018, 134, 1-4
Adcock, C.; Eling, M.; Loperfido, N. Skewed distributions in finance and actuarial science: A review. Eur. J. Financ. 2015, 21, 1253-1281
Azzalini, A. A class of distributions which includes the normal ones. Scand. J. Stat. 1985, 12, 171-178
Azzalini, A.; Dalla Valle, A. The multivariate skew-normal distribution. Biometrika 1996, 83, 715-726
Hallin, M.; Ley, C. Skew-symmetric distributions and Fisher information-A tale of two densities. Bernoulli 2012, 18, 747-763
Hallin, M.; Ley, C. Skew-symmetric distributions and Fisher information: The double sin of the skew-normal. Bernoulli 2014, 20, 1432-1453
Ley, C.; Paindaveine, D. On the singularity of multivariate skew-symmetric models. J. Multivar. Anal. 2010, 101, 1434-1444
Jones, M.C.; Faddy, M.J. A skew extension of the t-distribution, with applications. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 2003, 65, 159-174
Arellano-Valle, R.B.; Genton, M.G. Multivariate extended skew-t distributions and related families. Metron 2010, 68, 201-234
McNeil, A.J.; Frey, R.; Embrechts, P. Quantitative Risk Management: Concepts, Techniques and Tools; Princeton University Press: Princeton, NJ, USA, 2005; Volume 3
Forbes, F.; Wraith, D. A new family of multivariate heavy-tailed distributions with variable marginal amounts of tailweight: Application to robust clustering. Stat. Comput. 2014, 24, 971-984
Wraith, D.; Forbes, F. Location and scale mixtures of Gaussians with flexible tail behaviour: Properties, inference and application to multivariate clustering. Comput. Stat. Data Anal. 2015, 90, 61-73
Tukey, J.W. Modern techniques in data analysis. In Proceedings of the NSF-Sponsored Regional Research Conference; Southeastern Massachusetts University: North Dartmouth, MA, USA, 1977; Volume 7
Hoaglin, D.C. Summarizing shape numerically: The g-and-h distributions. In Exploring Data Tables, Trends, and Shapes; Wiley: Hoboken, NJ, USA, 2006; pp. 461-513
Jones, M.; Pewsey, A. Sinh-arcsinh distributions. Biometrika 2009, 96, 761-780
Sklar, M. Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 1959, 8, 229-231
Fréchet, M. Sur les tableaux de corrélation dont les marges sont données. Ann. Univ. Lyon 3e Sér. Sci. Sect. A 1951, 14, 53-77
Joe, H. Multivariate Models and Multivariate Dependence Concepts; CRC Press: London, UK; New York, NY, USA, 1997
Joe, H. Families of m-variate distributions with given margins and m(m-1)/2 bivariate dependence parameters. In Lecture Notes-Monograph Series; Institute of Mathematical Statistics: Hayward, CA, USA, 1996; pp. 120-141
Aas, K.; Czado, C.; Frigessi, A.; Bakken, H. Pair-copula constructions of multiple dependence. Insur. Math. Econ. 2009, 44, 182-198
Bedford, T.; Cooke, R.M. Vines: A new graphical model for dependent random variables. Ann. Stat. 2002, 30, 1031-1068
Bedford, T.; Cooke, R.M. Probability density decomposition for conditionally dependent random variables modeled by vines. Ann. Math. Artif. Intell. 2001, 32, 245-268
Kurowicka, D.; Joe, H. Dependence Modeling: Vine Copula Handbook; World Scientific: Singapore, 2010
Oh, D.H.; Patton, A.J. Modeling dependence in high dimensions with factor copulas. J. Bus. Econ. Stat. 2017, 35, 139-154
Embrechts, P.; Hofert, M. Statistical inference for copulas in high dimensions: A simulation study. ASTIN Bull. J. IAA 2013, 43, 81-95
Fang, H.B.; Fang, K.T.; Kotz, S. The meta-elliptical distributions with given marginals. J. Multivar. Anal. 2002, 82, 1-16
Krzysztofowicz, R.; Kelly, K. A Meta-Gaussian Distribution with Specified Marginals; Technical Document; University of Virginia: Charlottesville, VA, USA, 1996
Ley, C. Flexible modelling in statistics: Past, present and future. J. Soc. Fr. Stat. 2015, 156, 76-96
Zhang, R.; Czado, C.; Min, A. Efficient maximum likelihood estimation of copula based meta t-distributions. Comput. Stat. Data Anal. 2011, 55, 1196-1214
R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2017
Azzalini, A. The R Package sn: The Skew-Normal and Related Distributions such as the Skew-t (Version 1.5-1); Università di Padova: Padova, Italia, 2017
Hofert, M.; Kojadinovic, I.; Maechler, M.; Yan, J. Copula: Multivariate Dependence with Copulas; R package version 0.999-18, 2017. Available online: https://CRAN.R-project.org/package=copula (accessed on 29 September 2019)
Pfaff, B.; McNeil, A. QRM: Provides R-Language Code to Examine Quantitative Risk Management Concepts; R package version 0.4-13, 2016. Available online: https://CRAN.R-project.org/package=QRM (accessed on 29 September 2019)
Delignette-Muller, M.L.; Dutang, C. fitdistrplus: An R Package for Fitting Distributions. J. Stat. Softw. 2015, 64, 1-34
Arellano-Valle, R.B.; Ferreira, C.S.; Genton, M.G. Scale and shape mixtures of multivariate skew-normal distributions. J. Multivar. Anal. 2018, 166, 98-110
Ley, C.; Verdebout, T. Modern Directional Statistics; Chapman and Hall/CRC: Boca Raton, FL, USA, 2017
