Construction of asymmetric multivariate copulas
نویسنده
چکیده
In this paper we introduce two methods for the construction of asymmetric multivariate copulas. The rst is connected with products of copulas. The second approach generalises the Archimedean copulas. The resulting copulas are asymmetric and may have more than two parameters in contrast to most of the parametric families of copulas described in the literature. We study the properties of the proposed families of copulas such as the dependence of two components (Kendalls tau, tail dependence), marginal distributions and the generation of random variates. key words: copula, Archimedean copula, tail dependence MSC 2000: 62H20 1. Introduction The aim of this paper is to construct multivariate families of asymmetric copulas. In the monographs by Nelsen (1999) and Joe (1997) the reader nds detailed accounts of the theory as well as surveys of commonly used copulas. Most of these copulas belong to Archimedean families with one or two parameters. So these copulas have a limited variety of shapes. Several authors have indicated that it is an open problem to nd appropriate families of multivariate copulas (dimension greater than 2) with a exible number of parameters which may be greater than two. Suitable families of copulas are also needed for parametric and semiparametric estimation methods for multivariate densities and distribution functions. Alfonsi and Brigo (2005) describe a new construction method for asymmetric copulas based on periodic functions. A transformation method for two-dimensional copulas is discussed in Durrleman et al. (2000). In the present paper we introduce two universal methods for developing parametric families of copulas. The rst one is connected with products of copulas and was proposed in a special case by Khoudraji (1995). The second approach generalises Archimedean families of copulas. The advantages of the families we propose are the following: (i) The families are exible in tting data with a number of parameters which may be greater than two. (ii) The one-dimensional and multivariate marginal distributions belong to the corresponding family of smaller dimension. (iii) Methods are available for the generation of random variates. (iv) The families are asymmetric and cover a wide range of dependencies. The latter property is shown to hold for some speci c families by means of the values of Kendalls tau. Moreover, in the present paper we study tail dependence properties of the proposed copulas and provide su¢ cient conditions for positive quadrant dependence in the case of product copulas. Appropriate families of copulas can be used for tting multivariate densities to datasets. The parametric estimation problem of copulas is discussed in several papers (see e.g. Genest et al. (1995)). An e¢ cient estimation method for parametric classes of copula densities is introduced and investigated in Chen et al. (2006). The asymptotic behaviour of two-stage estimation procedures is studied in Joe (2005). A
منابع مشابه
Erratum to "Construction of asymmetric multivariate copulas" [J. Multivariate Anal. 99(2008) 2234-2250]
Assume that C1, . . . , Ck : [0, 1]d → [0, 1] are copulas. Let gji : [0, 1] → [0, 1] for j = 1, . . . , k, i = 1, . . . , d be functions with the property that each of them is strictly increasing or is identically equal to 1. Suppose that ∏k j=1 gji(v) = v for v ∈ [0, 1], i = 1, . . . , d, and limv→0+0 gij(v) = gji(0) for j = 1, . . . , k, i = 1, . . . , d. Here we consider the two-dimensional ...
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