Laurent Bordes

Condition-based inspection/replacement policies for non-monotone deteriorating systems with environmental covariates

The aim of this paper is to discuss the problem of modelling and optimising condition-based maintenance policies for a deteriorating system in presence of covariates. The deterioration is modelled by a non-monotone stochastic process. The covariates process is assumed to be a time-homogenous Markov chain with finite state space. A model similar to the proportional hazards model is used to show the influence of covariates on the deterioration. In the framework of the system under consideration, an appropriate inspection/replacement policy which minimises the expected average maintenance cost is derived. The average cost under different conditions of covariates and different maintenance policies is analysed through simulation experiments to compare the policies performances.

A stochastic EM algorithm for a semiparametric mixture model

Recently, there has been a considerable interest in finite mixture models with semi-/non-parametric component distributions. Identifiability of such model parameters is generally not obvious, and when it occurs, inference methods are rather specific to the mixture model under consideration. Hence, a generalization of the EM algorithm to semiparametric mixture models is proposed. The approach is methodological and can be applied to a wide class of semiparametric mixture models. The behavior of the proposed EM type estimators is studied numerically not only through several Monte-Carlo experiments but also through comparison with alternative methods existing in the literature. In addition to these numerical experiments, applications to real data are provided, showing that the estimation method behaves well, that it is fast and easy to be implemented.

Semiparametric mixtures of regressions

We present an algorithm for estimating parameters in a mixture-of-regressions model in which the errors are assumed to be independent and identically distributed but no other assumption is made. This model is introduced as one of several recent generalizations of the standard fully parametric mixture of linear regressions in the literature. A sufficient condition for the identifiability of the parameters is stated and proved. Several different versions of the algorithm, including one that has a provable ascent property, are introduced. Numerical tests indicate the effectiveness of some of these algorithms.

Semiparametric two-component mixture model with a known component: An asymptotically normal estimator

In this paper we consider a two-component mixture model, one component of which has a known distribution while the other is only known to be symmetric. The mixture proportion is also an unknown parameter of the model. This mixture model class has proved to be useful to analyze gene expression data coming from microarray analysis. In this paper a general estimation method is proposed leading to a joint central limit result for all the estimators. Applications to basic testing problems related to this class of models are proposed, and the corresponding inference procedures are illustrated through some simulation studies.

Parametric inference in a perturbed gamma degradation process

Abstract We consider a degradation model which is the sum of two independent processes: an homogeneous gamma process and a Brownian motion. This model is called perturbed gamma process. Based on independent copies of the perturbed gamma process observed at irregular instants we propose to estimate the unknown parameters of the model using the moment method. Some general conditions allow to derive the asymptotic behavior of the estimators. We also show that these general conditions are fulfilled for some specific observation schemes. Finally, we illustrate our method by a numerical study and an application to a real data set.

Minimum-Distance Parametric Estimation Under Progressive Type-I Censoring

The objective of this paper is to provide a new estimation method for parametric models under progressive Type-I censoring. First, we propose a Kaplan-Meier nonparametric estimator of the reliability function taken at the censoring times. It is based on the observable number of failures, and the number of censored units occurring from the progressive censoring scheme at the censoring times. This estimator is then shown to asymptotically follow a normal distribution. Next, we propose a minimum-distance method to estimate the unknown Euclidean parameter of a given parametric model. This method leads to consistent, asymptotically normal estimators. The maximum likelihood estimation method based on group-censored samples is discussed next, and the efficiencies of these two methods are compared numerically. Then, based on the established results, we derive a method to obtain the optimal Type-I progressive censoring scheme, Finally we illustrate all these results through a Monte Carlo simulation study, and an illustrative example.

Acceleration methods for Monte Carlo simulation of rare events

In the oil and gas industry, obtaining accurate reliability estimators on safety barriers is an important issue that can lead to very long computing times. To address this issue, we propose an extension of a truncation method and we introduce a new computational technique called Dissociation method. Through different numerical examples, we observe a significant improvement of the results obtained on simple Petri net models when applying these Monte Carlo acceleration methods.

Some acceleration methods for Monte Carlo simulation of rare events

In the oil and gas industry, obtaining accurate reliability estimators on safety barriers is an important issue that can lead to very long computing times. To address this issue, we propose an extension of a truncation method and we introduce a new computational technique called Dissociation method. Through different numerical examples, we observe a significant improvement of the results obtained on simple Petri net models when applying these Monte Carlo acceleration methods.

Nonperiodic Inspection/Replacement Policy for Monotone Deteriorating System with Covariates

Abstract This paper discusses the condition-based non-periodic maintenance policy for a stochastic deteriorating system influenced by a dynamic environment which is described by a covariates process. The deterioration is modelled by a stochastic univariate process. The process of covariates is assumed to be a time homogeneous finite state space Markov chain. A model similar to the proportional hazards model is used to represent the influence of the covariates. In the framework of a monotone deteriorating system, we derive the optimal maintenance threshold, optimal inspection sequence to minimise the expected maintenance cost per time unit. An adequate maintenance policy in which the inspection schemes depend both on the level of degradation and on the state of covariates is studied. Comparison of the expected costs per time unit under different conditions of covariates and different maintenance policies is given by numerical results of Monte Carlo simulation.

Some Algorithms to Fit some Reliability Mixture Models under Censoring

Estimating the unknown parameters of a reliability mixture model may be a more or less intricate problem, especially if durations are censored. We present several iterative methods based on Monte Carlo simulation that allow to fit parametric or semiparametric mixture models provided they are identifiable. We show for example that the well-known data augmentation algorithm may be used successfully to fit semiparametric mixture models under right censoring. Our methods are illustrated by a reliability example.