G-renewal process as a model for statistical warranty claim prediction

Kaminskiy, Mark P.; Krivtsov, Vasiliy

doi:10.1109/rams.2000.816321

Cited by 24 publications

(21 citation statements)

References 17 publications

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“…For virtual age models, some empirical studies on maximum likelihood estimators have been published [6,8,9,[17][18][19]. All these articles are based on simulation results.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic properties of imperfect repair models and estimation of repair efficiency

Doyen

2010

Naval Research Logistics

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International audienceThe aim of this articles is to study the asymptotic behavior of two imperfect repair models, called Arithmetic Reduction of Intensity and Arithmetic Reduction of Age models. These models have been proposed by Doyen and Gaudoin (Reliab Eng Syst Safe 84 (2004) 45-56) and include many usual virtual age models. First, it is proved that the failure intensity of these models is asymptotically almost surely equivalent to a deterministic increasing function with a cumulative error proportional to a logarithm. Second, the almost sure convergence and asymptotic normality of several estimators of repair efficiency are derived, when the wear-out process without repair is known. Finally, the coverage rate of the asymptotic confidence intervals issued from those estimators is empirically studied

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“…For virtual age models, some empirical studies on maximum likelihood estimators have been published [6,8,9,[17][18][19]. All these articles are based on simulation results.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic properties of imperfect repair models and estimation of repair efficiency

Doyen

2010

Naval Research Logistics

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“…By covering major repair assumptions encountered in practice, GRP provides more flexibility in modeling real life failure occurrence processes [9].…”

Section: The Proposed Q-weibull-grp and Q-exponential-grp Modelsmentioning

confidence: 99%

“…, n, respectively. Then, for each interval (t i−1 , t i ), mc replications are executed to obtain the corresponding mean number of failures (lines [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. Figure 2.…”

Section: Estimation Of the Expected Number Of Failures Via Monte Carlmentioning

confidence: 99%

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Combining Generalized Renewal Processes with Non-Extensive Entropy-Based q-Distributions for Reliability Applications

Lins

Moura

Droguett

et al. 2018

Entropy

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Abstract:The Generalized Renewal Process (GRP) is a probabilistic model for repairable systems that can represent the usual states of a system after a repair: as new, as old, or in a condition between new and old. It is often coupled with the Weibull distribution, widely used in the reliability context. In this paper, we develop novel GRP models based on probability distributions that stem from the Tsallis' non-extensive entropy, namely the q-Exponential and the q-Weibull distributions. The q-Exponential and Weibull distributions can model decreasing, constant or increasing failure intensity functions. However, the power law behavior of the q-Exponential probability density function for specific parameter values is an advantage over the Weibull distribution when adjusting data containing extreme values. The q-Weibull probability distribution, in turn, can also fit data with bathtub-shaped or unimodal failure intensities in addition to the behaviors already mentioned. Therefore, the q-Exponential-GRP is an alternative for the Weibull-GRP model and the q-Weibull-GRP generalizes both. The method of maximum likelihood is used for their parameters' estimation by means of a particle swarm optimization algorithm, and Monte Carlo simulations are performed for the sake of validation. The proposed models and algorithms are applied to examples involving reliability-related data of complex systems and the obtained results suggest GRP plus q-distributions are promising techniques for the analyses of repairable systems.

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“…For the BrownProschan model, Lim [15] has studied estimation with the EM-algorithm, Langseth and Lindqvist [13] have estimated parameters on real data set with maximum likelihood, and Lim, Lu and Park [14] have used Bayesian estimation. For virtual age models some studies on maximum likelihood estimators have been published: Shin, Lim and Lie [19], Yun and Choung [21], Kaminskiy and Krivtsov [9], Yanez, Joglar and Modarres [20], Gasmi, Love and Kahle [8], Doyen and Gaudoin [6]. But all these studies only proposed numerical results.…”

Section: Introductionmentioning

confidence: 99%

REPAIR EFFICIENCY ESTIMATION IN THE ARI₁ IMPERFECT REPAIR MODEL

Doyen¹

2005

Series on Quality, Reliability and Engineering Statistics

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The aim of this paper is to study the estimation of repair efficiency in an imperfect repair model, called Arithmetic Reduction of Intensity model with memory one (ARI 1 ). This model, first introduced by Doyen and Gaudoin, has a very simple failure intensity and repair efficiency is characterized by a single parameter. Thanks to that simplicity, the asymptotic almost sure behavior of the failure and cumulative failure intensities of ARI 1 model can be derived. Then, the almost sure convergence and asymptotic normality of several estimators (including maximum likelihood) of repair efficiency can be proved in the case where the wear out process without repair is known. The influence of the number of observed failures on the quality of the repair efficiency estimation is empirically studied. Finally, results are applied to a real maintenance data set.

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G-renewal process as a model for statistical warranty claim prediction

Cited by 24 publications

References 17 publications

Asymptotic properties of imperfect repair models and estimation of repair efficiency

Asymptotic properties of imperfect repair models and estimation of repair efficiency

Combining Generalized Renewal Processes with Non-Extensive Entropy-Based q-Distributions for Reliability Applications

REPAIR EFFICIENCY ESTIMATION IN THE ARI₁ IMPERFECT REPAIR MODEL

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G-renewal process as a model for statistical warranty claim prediction

Cited by 24 publications

References 17 publications

Asymptotic properties of imperfect repair models and estimation of repair efficiency

Asymptotic properties of imperfect repair models and estimation of repair efficiency

Combining Generalized Renewal Processes with Non-Extensive Entropy-Based q-Distributions for Reliability Applications

REPAIR EFFICIENCY ESTIMATION IN THE ARI1 IMPERFECT REPAIR MODEL

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REPAIR EFFICIENCY ESTIMATION IN THE ARI₁ IMPERFECT REPAIR MODEL