Preservation of positive and negative orthant dependence concepts under mixtures and applications

In this paper we consider some dependence properties and orders among multivariate distributions, and we study their preservation under mixtures. Applications of these results in reliability, risk theory, and mixtures of discrete distributions are provided.

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Preservation of positive and negative orthant dependence concepts under mixtures and applications

Journal of Applied Probability ◽

10.1017/s002190020002074x ◽

2004 ◽

Vol 41 (04) ◽

pp. 961-974 ◽

Cited By ~ 1

Author(s):

Félix Belzunce ◽

Patrizia Semeraro

Keyword(s):

Discrete Distributions ◽

Risk Theory ◽

Multivariate Distributions ◽

Dependence Concepts ◽

Dependence Properties

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Relations between ageing and dependence for exchangeable lifetimes with an extension for the IFRA/DFRA property

Dependence Modeling ◽

10.1515/demo-2020-0001 ◽

2020 ◽

Vol 8 (1) ◽

pp. 1-33

Author(s):

Giovanna Nappo ◽

Fabio Spizzichino

Keyword(s):

Equivalence Classes ◽

Stochastic Dependence ◽

The Past ◽

Dependence Concepts ◽

Kendall Distribution ◽

Mutual Relations ◽

Bivariate Ageing ◽

Dependence Properties ◽

Ageing Notions

AbstractWe 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.

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The fundamental importance of differential equations with three singularities in Mathematical Statistics

Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie ◽

10.4102/satnt.v4i1.1012 ◽

1985 ◽

Vol 4 (1) ◽

pp. 18-24

Author(s):

H. S. Steyn

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Probability Distributions ◽

Mathematical Physics ◽

Second Order ◽

Discrete Distributions ◽

Mathematical Statistics ◽

Multivariate Distributions ◽

The Difference ◽

Linear Differential

It is well-known that the solution of a second order linear differential equation with at most five singularities plays a fundamental role in Mathematical Physics. In this paper it is shown that this statement also applies to Mathematical Statistics but with the difference that an equation with three singularities will suffice. Two wide classes of probability distributions are defined as solutions of such a differential equation, one for continuous distributions and one for discrete distributions. These two classes contain as members all the distributions which are normally considered as of importance in Mathematical Statistics. In the continuous case the probability functions are solutions of the relevant second order equation, while in the discrete case the probability generating functions are solutions there-of. By defining appropriate multidimentional extensions corresponding differential equations are obtained for continuous and discrete multivariate distributions.

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