From Multivariate Skewed Distributions to Copulas
In this paper, a methodology is presented for constructing skewed multivariate copulas to model data with possibly different marginal distributions. Multivariate skew elliptical distributions are transformed into corresponding copulas in the similar way a
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Abstract In this paper, a methodology is presented for constructing skewed multivariate copulas to model data with possibly different marginal distributions. Multivariate skew elliptical distributions are transformed into corresponding copulas in the similar way as the Gaussian copula and the multivariate t-copula are constructed. Three-parameter skew elliptical distributions are under consideration. For parameter estimation of the skewed distributions, the method of moments is used. To transform mixed third-order moments into a parameter vector, the star product of matrices is used; for star product and its applications, see, for example, Kollo (J. Multivar. Anal. 99:2328–2338, 2008) or Visk (Commun. Stat. 38:461–470, 2009). Results of the first applications are shortly described and referred to. Keywords Method of moments · Multivariate skewness · Skew normal copula · Skew normal distribution · Skew t-copula · Skew t-distribution Mathematics Subject Classification (2010) 62E17 · 62F10 · 62H12
1 Introduction In this paper, we consider a construction of copulas from multivariate skew symmetric elliptical distributions. Copula models have become extremely popular in applications, especially in financial and environmental studies. In applications, marginal distributions are often skewed, heavy tailed, and belong to different parametric families. In such situations, the copula approach gives us, in fact, the only way to model T. Kollo (B) · A. Selart Institute of Mathematical Statistics, University of Tartu, J. Liivi 2, Tartu 50409, Estonia e-mail: [email protected] A. Selart e-mail: [email protected] H. Visk Department of Public Health, University of Tartu, Ravila 19, Tartu 50411, Estonia e-mail: [email protected] R.B. Bapat et al. (eds.), Combinatorial Matrix Theory and Generalized Inverses of Matrices, DOI 10.1007/978-81-322-1053-5_6, © Springer India 2013
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the data with certain dependence structure. In Sect. 2, we consider three families of skew elliptical distributions and apply the method of moments to find estimates of the shifted asymmetric Laplace distribution, skew normal distribution, and skew t-distribution. The method of moments gives us biased estimates, and as these are easy to find, they can be used as initial values of parameters for further study when applying, for instance, maximum likelihood method. In Sect. 3, we define the skew normal copula and skew t-copula. Estimation of the copula parameters is considered, and results of some applications are discussed.
2 Skew-Symmetric Distributions: Notions, Notation, Estimation Since the multivariate skew normal distribution was introduced in Azzalini and Dalla Valle [2], the model has become popular, and their construction has been generalized in different ways. Several generalizations up to 2004 can be found in the collective monograph Genton [3]. The idea to transform a symmetric multivariate elliptical distribution by a distribution function or a function with similar mathematical properties has become extremely fruitful. There
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