Semiparametric estimate of the efficiency of imperfect maintenance actions for a gamma deteriorating system

Abstract

A system is considered, which is deteriorating over time according to a non homogeneous gamma process with unknown parameters. The system is subject to periodic and instantaneous imperfect maintenance actions (repairs). Each imperfect repair removes a proportion ρ of the accumulated degradation since the previous repair. The parameter ρ hence appears as a measure for the maintenance efficiency. This model is called arithmetic reduction of degradation of order 1. The system is inspected right before each maintenance action, thus providing some multivariate measurement of the successively observed deterioration levels. Based on these data, a semiparametric estimator of ρ is proposed, considering the parameters of the underlying gamma process as nuisance parameters. This estimator is mainly based on the range of admissible ρ's, which depends on the data. Under technical assumptions, consistency results are obtained, with surprisingly high convergence rates (up to exponential). The case where several i.i.d. systems are observed is next envisioned. Consistency results are obtained for the efficiency estimator, as the number of systems tends to infinity, with a convergence rate that can be higher or lower than the classical square root rate. Finally, the performances of the estimators are illustrated on a few numerical examples.

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

“…Other models were designed, generally with an estimation procedure, for example by Dauxois and Maalouf [6], Dauxois et al [7], Dijoux et al [8], Doyen et al [9], and Peng et al [10]. Some models also include preventive maintenances, as those exposed by Peng et al [11], Said and Taghipour [12] and Salles et al [13]. Preventive maintenances are carried out without the occurence of a failure.…”

“…With the hypothesis set in Section 2.2, the conditional cumulative distribution function is decomposed along the discontinuity points of the covariates in Equation (13). There are three cases: 1.…”

Modeling multivariate degradation processes with time‐variant covariates and imperfect maintenance effects

“…In principle, we might have the following two assumptions on the maintenance efficiency—following the ideas in Doyen and Gaudoin, 1 Kahle, 6 Mercier and Castro, 7 and Salles et al, 50 which lead to two different imperfect maintenance models:…”

“…The above‐summarized studies focus predominately on inspection or preventive replacement strategies. Imperfect maintenance, however, receives very few investigations in the context of multivariate degradation processes, though there are indeed some studies dealing with univariate degradation processes (see, e.g., Castanier et al 46 ; Mercier and Castro 47 ; Zhang et al 48 ; Kahle 6 ; Mercier and Castro 7 ; Zhao et al 49 ; Salles et al 50 ). Basically, the extant degradation‐based imperfect maintenance models, for example, (virtual) age reduction 6,47 and degradation level reduction, 6,7,49 are created by mimicking the ideas of associated lifetime‐based imperfect maintenance models.…”

Imperfect condition-based maintenance for a gamma degradation process in presence of unknown parameters

“…The case of gamma process and imperfect maintenance is scarcely studied, refer to (36; 37; 38). Authors in (36) focus on semi-parametric estimation of the maintenance efficiency. In (37) an exponential distributed degradation reduction is considered and the parameter of the exponential distribution is estimated via MLE method based on degradation data just after maintenance.…”