Some stochastic properties of conditionally dependent frailty models

Abstract

The frailty approach is commonly used in reliability theory and survival analysis to model the dependence between lifetimes of individuals or components subject to common risk factors; according to this model the frailty (an unobservable random vector that describes environmental conditions) acts simultaneously on the hazard functions of the lifetimes. Some interesting conditions for stochastic comparisons between random vectors defined in accordance with these models have been described in the literature; in particular, comparisons between frailty models have been studied by assuming independence for the baseline survival functions and the corresponding environmental parameters. In this paper, a generalization of these models is developed, which assumes conditional dependence between the components of the random vector, and some conditions for stochastic comparisons are provided. Some examples of frailty models satisfying these conditions are also described.

Keywords:

• bivariate lifetimes
• dependence notions
• multivariate stochastic orders
• survival copulas

AMS Subject Classification:

• 62N99
• 60E15

Acknowledgements

The authors wish to thank the Editor and the two anonymous referees whose comments helped to improve a previous version of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partly supported by a grant from Junta de Andalucía (Spain) for research group (FQM-328).