Multivariate failure time distributions derived from shared frailty and copulas

Copulas and frailty models have been two major tools for modeling dependence in multivariate failure time distributions. The objective of this paper is to investigate multivariate failure time models that include copula models and frailty models as special cases. To this end, we revisit a broad family of multivariate failure time models proposed by Marshall and Olkin (JASA 83:834–841, 1988). This family accommodates both frailty and copulas, unlike the models that accommodate only one of them. However, their work focused on very specific copulas and is limited to bivariate models. Instead, we focus more on popular members of copulas and some multivariate models. Another novel feature of our paper is to restrict our attention to the shared frailty model, and call our restricted class as the generic name “frailty-copula”. This name yields a taxonomic classification of all the members of distributions. We also consider somewhat complex frailty distributions (two-parameter gamma, lognormal, truncated-normal, and folded-normal), which were not considered in Marshall and Olkin (1988) and other papers of frailty models. To illustrate the usefulness of the proposed model, we briefly discuss maximum likelihood estimation methods with some numerical evaluations.