Relations between ageing and dependence for exchangeable lifetimes with an extension for the IFRA/DFRA property (original) (raw)
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arXiv: Probability, 2019
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...
Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes
Journal of Multivariate Analysis, 2005
For a couple of lifetimes ðX 1 ; X 2 Þ with an exchangeable joint survival function % F; attention is focused on notions of bivariate aging that can be described in terms of properties of the level curves of % F: We analyze the relations existing among those notions of bivariate aging, univariate aging, and dependence. A goal and, at the same time, a method to this purpose is to define axiomatically a correspondence among those objects; in fact, we characterize notions of univariate and bivariate aging in terms of properties of dependence. Dependence between two lifetimes will be described in terms of their survival copula. The language of copulae turns out to be generally useful for our purposes; in particular, we shall introduce the more general notion of semicopula. It will be seen that this is a natural object for our analysis. Our definitions and subsequent results will be illustrated by considering a few remarkable cases; in particular, we find some necessary or sufficient conditions for Schur-concavity of % F; or for IFR properties of the one-dimensional marginals. The case characterized by the condition that the survival copula of ðX 1 ; X 2 Þ is Archimedean will be considered in some detail. For most of our arguments, the extension to the case of n42 is straightforward. r
Semi-copulas and Interpretations of Coincidences Between Stochastic Dependence and Ageing
Lecture Notes in Statistics, 2010
We aim at providing probabilistic explanations of equivalences, between conditions of positive dependence and of univariate ageing, that have been pointed out in the literature. To this purpose we consider bivariate survival functions F(x, y) and properties of them that are respectively invariant under transformations of the type F (ϕ(x), ϕ(y)) and ψ F(x, y) , for ϕ, ψ : [0, 1] → [0, 1] increasing bijections. Bivariate Schur-constant survival models will have a central role in our discussion.
Dependence for Archimedean copulas and aging properties of their generating functions
Sankhyā: The Indian Journal of Statistics ( …, 2004
This paper details the correspondence between various dependence concepts and stochastic orderings for an Archimedean copula C φ (x, y) = φ −1 {φ(x) + φ(y)} and the aging properties of the corresponding life distribution F φ (t) = 1 − φ −1 (t). Various applications of the results are given.
On dynamic mutual information for bivariate lifetimes
Advances in Applied Probability, 2015
We consider dynamic versions of the mutual information of lifetime distributions, with a focus on past lifetimes, residual lifetimes, and mixed lifetimes evaluated at different instants. This allows us to study multicomponent systems, by measuring the dependence in conditional lifetimes of two components having possibly different ages. We provide some bounds, and investigate the mutual information of residual lifetimes within the time-transformed exponential model (under both the assumptions of unbounded and truncated lifetimes). Moreover, with reference to the order statistics of a random sample, we evaluate explicitly the mutual information between the minimum and the maximum, conditional on inspection at different times, and show that it is distribution-free in a special case. Finally, we develop a copula-based approach aiming to express the dynamic mutual information for past and residual bivariate lifetimes in an alternative way.
A family of bivariate exponential distributions and their copulas
Sankhya B, 2013
In the present paper we derive a family of bivariate exponential distributions based on an extended lack of memory property of a class of univariate distributions. These bivariate exponential laws are time transformed exponential models possessing Archimedean copulas. The bivariate aging properties and various dependence relationships are characterized in terms of the univariate aging concepts of the baseline distributions from which the bivariate models are generated.
STOCHASTIC DEPENDENCES. FUTURE RESEARCH ANNOUNCEMENT.
Letter, 2022
The topics of various forms of stochastic dependences (see for example [1]) is widely explored in literature from elementary approaches to highly advanced. In [2] we try to grasp the subject of dependence in as wide as possible framework as well as indicate its association with a wider framework [3, 5, 6, 7] of logical (nonclassical) conjunction in terms of t-norms [3,5]. General theory of distributions of random vectors has, as its typical general tool, concept of copula [4, 8, 9]. Relatively recently we proposed in [10-13] an alternative concept of joiner for that theory. The latter concept has emerged from use of Cox and Aalen modeling [14-17]. Since the concept of copula is widely elaborated in literature, here we rather concentrate on the more recent notion of joiners as they are less known. This short communication only signalizes future work [2] on the subject. Presently, this work is in preparation. More on the very general approach that puts together stochastic and logical dependences into one common framework will be found in Appendix to [2].