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.

Stochastic comparisons of multivariate reversed frailty model

“…The following two results describe the conditions for stochastic order and likelihood ratio order between the random vectors X 1 and X 2 . The latter result is similar to Theorem 3.2 of Fernández‐Ponce et al (), but the ordering between the lifetime vectors is reversed.…”

“…The following theorem is similar to Theorem 3.12 of Fernández‐Ponce et al () except that the stochastic ordering inequality between the baseline distribution is reversed.…”

“…Fernández‐Ponce et al () gave stochastic comparison between generalized bivariate frailty models defined by Here, the components in the vectors are dependent and the results for this model can easily be extended to higher dimensions. They also considered two particular cases of this model, namely, the bivariate exponential frailty model and the bivariate model with marginal frailty distributions.…”

“…We also give a necessary and sufficient condition under which multivariate weak reversed hazard rate ordering holds between two lifetime vectors with different frailties. Next, analogous to results for the frailty model by Fernández‐Ponce et al (), we study the following multivariate case of the reversed frailty model in which independence among the components of the random vector is not assumed: …”

“…Bivariate frailty models with conditional dependence among the random vectors (bivariate) and univariate frailty have been defined and studied by Gupta and Gupta (). Later, Fernández‐Ponce, Pellerey, and Rodríguez‐Griñolo () generalized this model for multivariate frailty models and multivariate frailty. They examined two particular cases of it, that is, bivariate exponential frailty model and bivariate model with marginal frailty distributions (i.e., representation of baseline distribution in terms of the copula).…”