# Revisiting Relations between Stochastic Ageing and Dependence for Exchangeable Lifetimes with an Extension for the IFRA/DFRA Property

## TL;DR

This paper revisits the relationship between stochastic ageing and dependence in exchangeable lifetimes, extending concepts to semi-copulas and exploring implications for IFRA/DFRA properties within a Bayesian framework.

## Contribution

It introduces extensions of dependence properties to semi-copulas and analyzes their impact on ageing notions like IFRA/DFRA, including new classes like Pseudo-Archimedean models.

## Key findings

- Extended Kendall distribution and equivalence classes to semi-copulas.
- Analyzed relations between IFRA/DFRA ageing and PKD/NKD dependence concepts.
- Provided insights from a Bayesian perspective on dependence extensions.

## 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.