On a multivariate Pareto distribution

Abstract

This is the accepted version of the paper.This version of the publication may differ from the final published version. to allow for an adequate modeling of dependent heavy tailed risks with a non-zero probability of simultaneous loss. Numerous links to certain nowadays existing probabilistic models, as well as seemingly useful characteristic results are proved. Expressions for, e.g., decumulative distribution functions, densities, (joint) moments and regressions are developed. An application to the classical pricing problem is considered, and some formulas are derived using the recently introduced economic weighted premium calculation principles. Permanent repository link

Archimedean-based Marshall-Olkin Distributions and Related Dependence Structures

“…Clearly, we recover a copula of type (9) from a copula of type (8) In next result we will prove that for any suitable pair of functions D 1 and D 2 , the copula in (8) and (9) can be obtained starting from a three-variate distribution function of type (2) …”

“…The corresponding joint survival distribution of the random variables M 1 and M 2 isF (x 1 , x 2 ) = exp{−H 1 (x 1 ) − H 2 (x 2 ) − H 3 (max(x 1 , x 2 ))} that is called Generalized Marshall-Olkin distribution. These distributions incorporate, as special cases, the Marshall-Olkin type distributions introduced in Muliere and Scarsini (1987), the bivariate Weibull distributions introduced in Lu (1989) and the bivariate Pareto distributions introduced in Asimit et al (2010). The authors analyze the dependence structure implied by these distributions introducing the corresponding copula functions, that they call Generalized Marshall-Olkin copulas.…”

Independence and maximal volume of d-dimensional random convex hull

“…While the underlying population distributions are free of the assumption of symmetry, some distributions are also included in the MLS family. Multivariate Pareto of the second type (Asimit et al, 2010), a generalized multivariate gamma distributions (Carpenter and Diawara, 2007), and multivariate skew normal distributions and multivariate skew t-distributions (Azzalini and Capitanio, 2003) are examples of asymmetric distributions which can be classified as the MLS family. Some further extensions can also be found from Zhao and Kim (2016).…”

Evaluating Risk Measures and Capital Allocations Based on Multi-Losses Driven by a Heavy-Tailed Background Risk: The Multivariate Pareto-Ii Model

“…These prominently include copulas (e.g., Frees and Valdez [11]), multivariate distributions such as elliptical (e.g., Landsman and Valdez [12]), exponential dispersion (e.g., Landsman and Valdez [13]), Erlang mixtures (e.g., Lee and Lin [14]), Tweedie (e.g., Furman and Landsman [15]), phase-type (e.g., Cai and Li [16]), multivariate Gamma (e.g., Furman [17], Furman and Landsman [18]), Pareto (e.g., Asimit et al [19], Chiragiev and Landsman [20], Vernic [21]). Certainly, we have mentioned here only a few references, but they lead to other important references.…”

“…..,n}\{i} (σ i − σ j ) which is the same as the corresponding coefficient in Equation (19). Finally, the coefficient next to f Yn (y) in quantity (20) is equal to…”