Archimedean-based Marshall-Olkin distributions and related dependence structures. (English) Zbl 1392.62047
Summary: In this paper we study the dependence properties of a family of bivariate distributions (that we call Archimedean-based Marshall-Olkin distributions) that extends the class of the generalized Marshall-Olkin distributions of X. Li and F. Pellerey [J. Multivariate Anal. 102, No. 10, 1399–1409 (2011; Zbl 1221.60014)] in order to allow for an Archimedean type of dependence among the underlying shocks’ arrival times. The associated family of copulas (that we call Archimedean-based Marshall-Olkin copulas) includes several well known copula functions as specific cases for which we provide a different costruction and represents a particular case of implementation of P. M. Morillas [Metrika 61, No. 2, 169–184 (2005; Zbl 1079.62056)] construction. It is shown that Archimedean-based copulas are obtained through suitable transformations of bivariate Archimedean copulas: this induces asymmetry, and the corresponding Kendall’s function and Kendall’s tau as well as the tail dependence parameters are studied. The type of dependence so modeled is wide and illustrated through examples and the validity of the weak Lack of memory property (characterizing the Marshall-Olkin distribution) is also investigated and the sub-family of distributions satisfying it identified. Moreover, the main theoretical results are extended to the multidimensional version of the considered distributions and estimation issues discussed.
MSC:
| 62E15 | Exact distribution theory in statistics |
| 60E05 | Probability distributions: general theory |
| 62H20 | Measures of association (correlation, canonical correlation, etc.) |
Keywords:
Marshall-Olkin distribution; Marshall-Olkin copula; Kendall’s function; Kendall’s tau; lack of memory propertyReferences:
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