Virtual age models with time-dependent covariates: A framework for simulation, parametric inference and quality of estimation

Abstract

A repairable system faces some failures and imperfect maintenances throughout its lifetime. If several identical and independent systems are considered together, some differences may arise between the systems, such as the geographical location or the maintenance team for example, which are constant information, or the weather conditions, which vary with time. This observed heterogeneity will influence more or less the failure process. In this paper, we include these data in a generalized virtual age model with the use of covariates. Then we estimate simultaneously the effect of the maintenances, that of the covariates, and the intrinsic wear of the systems. We also propose two simulation methods as well as a numerical estimation procedure. Then we assess the quality of the estimation of the parameters with a thorough simulation study.

Two methods to approximate the superposition of imperfect failure processes

“…( 11) and in Eq. ( 20) are the same and the λ(t) in model (20), which is obtained from the masked failure data, and the λ(t) in model (11), which is obtained from the unmasked failure data are the same.…”

“…There is a bulk of research discussing different types of stochastic processes for modelling failure processes, or simply put, modelling interfailure times, see [1,2,3,4], for example. These models are also applied in maintenance policy optimisation, see [5,6,7,8,9,10,11,12], for example.…”

Virtual Age Models: Monitoring Information Level and Quality of Parametric Estimation

“…This is known as observed heterogeneity between the systems. All this information can be integrated into generalized virtual age models as covariates [3]. We may have constant covariates (such as the system manufacturer or the patient gender) or timedependent ones (like the weather conditions or the body temperature).…”

“…We may have constant covariates (such as the system manufacturer or the patient gender) or timedependent ones (like the weather conditions or the body temperature). A simulation method and an estimation procedure in this framework have been developed in [3], in the case of step-wise constant covariates. To the best of our knowledge, very few references include unobserved heterogeneity in virtual age models (see [4] and [5]).…”

Discussion of “Virtual age, is it real?”

“…Doyen et al 6 have studied the introduction of constant covariates in the initial intensity. Brenière et al 13 have generalized this approach to dynamic covariates. The FC paper deals with unobserved heterogeneity, with the use of frailties.…”