On the fixed and evolving frailties in some survival models

Additive and multiplicative models for environmental impact on operating items are well-known in statistics for univariate and multivariate frailties. However, they are developed mostly for the fixed frailties assigned at the start of operation. In this paper, we suggest the unifying approach to modelling with fixed and evolving in time frailties. The latter can be done only for frailties defined by specific stochastic processes. Some results for additive models were reported in the literature. Therefore, the main focus in the paper is on the corresponding multiplicative models that were not studied so far, at least, in the presented context. The nonhomogeneous Poisson process and the generalized Pólya process are considered as the models for a random shock environment impacting directly the corresponding conditional failure rate. The relevant survival probabilities and failure rates are obtained and their properties are analyzed. Specifically, the principle “the weakest populations are dying out first” is discussed in the suggested framework. Some numerical examples illustrate our findings.