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Abstract

We first review an approach that had been developed in the past years to introduce concepts of “bivariate ageing” for exchangeable lifetimes and to analyze mutual relations among stochastic dependence, univariate ageing, and bivariate ageing.

A specific feature of such an approach dwells on the concept of semi-copula and in the extension, from copulas to semi-copulas, of properties of stochastic dependence. In this perspective, we aim to discuss some intricate aspects of conceptual character and to provide the readers with pertinent remarks from a Bayesian Statistics standpoint. In particular we will discuss the role of extensions of dependence properties. “Archimedean” models have an important role in the present framework.

In the second part of the paper, the definitions of Kendall distribution and of Kendall equivalence classes will be extended to semi-copulas and related properties will be analyzed. On such a basis, we will consider the notion of “Pseudo-Archimedean” models and extend to them the analysis of the relations between the ageing notions of IFRA/DFRA-type and the dependence concepts of PKD/NKD.

conditional, This research was partially supported by the KBN 511/2 /91 grant. Key words and phrases: conditional specification, conditional distribution, regression function, conditional Poisson law, bivariate Poisson conditionals distribution. A MS (1980) subject classification: 62H05, 62E10. 238 J. W e s o l o w s k i Ahsanullah and Wesolowski [1] character izat ion of the bivariate normal i ty by the normal conditional distr ibution and the linear m, Wesolowski [1*2] uniqueness theorems for power series conditional distribution and a con- sistent m. In this

the multivariate distribution function with standard uniform margins. However not this definition made copula so attractive, but rather the Sklar’s theorem, where the very importance of copulae in the area of multivariate distributions has been restated in an elegant way. AMS 2010 Subject Classification: Primary 62-00; Secondary 62H05 Keywords and Phrases: Copula, multivariate distribution 282 Okhrin Theorem 1.2 (Sklar (1959)) Let F be a multivariate distribution function with margins F1; : : : ;Fd , then there exists the copula C./ such that F.x1; : : : ;xd / D C

function H is best characterized by a unique function -a bivariate copula C, defined everywhere on the unit square 12 by the following relation H(x,y) = C(F(x),G(y)), x)2/GR (see Sklar's Theorem, Sklar (1959)). A bivariate copula is a bivariate cu- mulative distribution function with uniform margins on the unit interval I. 2000 Mathematics Subject Classification: 62E10, 62H05. Key words and phrases: Copulas, positive quadrant dependence, regression, charac- terizing theorems, Lancaster probabilities, diagonal expansion of bivariate density, or- thonormal polynomials

(identifiablity), others with characterization of classes of distributions μγ for a given class of posterior means, often assumed to be linear functions (identification). See for instance: Krishnaji (1974), Korwar (1975, 1977), Xekalaki (1983), Cacoullos and Papageorgiou (1983, 1984), Papageorgiou (1984, 1985), Kyriakoussis and Papageorgiou (1991), Johnson and Kotz (1992), Arnold et al. (1993), Sapatinas (1995), Wesolowski AMS (1991) subject classification: Primary 62H05; Secondary 62E10, 62F15. Key words and phrases: mixtures, posterior mean, identifiability

conditional volatili- ties are modelled using, e.g., GARCH-type processes. For a recent review of multivariate GARCH processes, including Dynamic Conditional Correlation (DCC), Constant Condi- tional Correlation (CCC), Baba, Engle, Kraft, and Kroner (1990) (BEKK), and others, we refer to Silvennoinen and Teräsvirta (2009). These models still assume that the parameters for the process are constant over an entire estimation period. Such an approach has been Corresponding author: Ostap Okhrin AMS 2010 subject classification: Primary 62H12, Secondary 62H05 Keywords and

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