Numéro spécial : fiabilité
Testing an “Exponential Delay Time model” against a “Random Sign Censoring model” in Reliability
[Test du modèle “Delay Time” exponentiel contre le modèle “Random Sign Censoring” en Fiabilité]
Journal de la société française de statistique, Tome 155 (2014) no. 3, pp. 104-119.

Nous nous intéressons dans cet article à un système industriel sujet à différentes causes de pannes et sur lequel deux types de maintenance peuvent être effectuées : soit une maintenance corrective dans le cas d’une panne critique, soit une maintenance préventive afin de réduire le risque d’une panne critique. Des modélisations basées sur la notion de risques concurrents ont été proposées dans la littérature. Mais peu d’inférence statistique a été menée sur ces modèles. En particulier, il existe très peu de tests statistiques permettant de décider quel modèle pourrait le mieux s’ajuster à un jeu de données précis.

L’objectif de cet article est justement d’introduire un test non-paramétrique permettant de décider entre deux de ces modèles de fiabilité : le modèle “Delay Time” avec loi exponentielle et le modèle “Random Sign Censoring”. Nous introduisons une statistique de test et prouvons sa normalité asymptotique. Nous terminons l’article en étudiant par simulation le comportement de notre procédure dans le cas de petits échantillons et en présentant une application du test sur un jeu de données réelles.

In this paper we consider an industrial system subject to different causes of failure and different types of maintenance: a corrective maintenance is performed after a critical failure and a preventive maintenance can be performed in order to decrease the risk of critical failure. The recurrence of these types of maintenance has been often modeled in a competing risks framework.

However rather few statistical inference has been carried out in these models. In particular, there is a need to introduce statistical tests in order to help the engineers to select the model which better fits their data. Thus, in this paper, we introduce a nonparametric test with aim to decide between a Delay Time model with exponential distribution and a Random Sign model. We prove the asymptotic normality of our test statistic and we carry out a Monte Carlo simulation to learn how works our test on finite sample sizes. An application on a real dataset is also given.

Keywords: Asymptotic distribution, Censoring, Competing risks, Corrective Maintenance, Delay Time model, Nonparametric test, Preventive Maintenance, Random Sign Censoring Model
Mot clés : Censure, Loi asymptotique, Maintenance Corrective, Maintenance Préventive, Modèle “Delay Time”, Modèle “Random Sign Censoring”, Risques concurrents, Test non-paramétrique 
@article{JSFS_2014__155_3_104_0,
     author = {Dauxois, Jean-Yves and Jomhoori, Sarah and Yousefzadeh, Fatemeh},
     title = {Testing an {{\textquotedblleft}Exponential} {Delay} {Time} model{\textquotedblright} against a {{\textquotedblleft}Random} {Sign} {Censoring} model{\textquotedblright} in {Reliability}},
     journal = {Journal de la soci\'et\'e fran\c{c}aise de statistique},
     pages = {104--119},
     publisher = {Soci\'et\'e fran\c{c}aise de statistique},
     volume = {155},
     number = {3},
     year = {2014},
     mrnumber = {3272714},
     zbl = {1316.62146},
     language = {en},
     url = {http://www.numdam.org/item/JSFS_2014__155_3_104_0/}
}
TY  - JOUR
AU  - Dauxois, Jean-Yves
AU  - Jomhoori, Sarah
AU  - Yousefzadeh, Fatemeh
TI  - Testing an “Exponential Delay Time model” against a “Random Sign Censoring model” in Reliability
JO  - Journal de la société française de statistique
PY  - 2014
SP  - 104
EP  - 119
VL  - 155
IS  - 3
PB  - Société française de statistique
UR  - http://www.numdam.org/item/JSFS_2014__155_3_104_0/
LA  - en
ID  - JSFS_2014__155_3_104_0
ER  - 
%0 Journal Article
%A Dauxois, Jean-Yves
%A Jomhoori, Sarah
%A Yousefzadeh, Fatemeh
%T Testing an “Exponential Delay Time model” against a “Random Sign Censoring model” in Reliability
%J Journal de la société française de statistique
%D 2014
%P 104-119
%V 155
%N 3
%I Société française de statistique
%U http://www.numdam.org/item/JSFS_2014__155_3_104_0/
%G en
%F JSFS_2014__155_3_104_0
Dauxois, Jean-Yves; Jomhoori, Sarah; Yousefzadeh, Fatemeh. Testing an “Exponential Delay Time model” against a “Random Sign Censoring model” in Reliability. Journal de la société française de statistique, Tome 155 (2014) no. 3, pp. 104-119. http://www.numdam.org/item/JSFS_2014__155_3_104_0/

[1] Andersen, Per Kragh; Borgan, Ørnulf; Gill, Richard D.; Keiding, Niels Statistical models based on counting processes, Springer Series in Statistics, Springer-Verlag, New York, 1993, xii+767 pages | DOI | MR | Zbl

[2] Bunea, Cornel; Bedford, Tim The effect of model uncertainty on maintenance optimization, IEEE Transactions on Reliability, Volume 51 (2002) no. 4, pp. 486-493

[3] Block, Henry W.; Basu, A. P. A continuous bivariate exponential extension, J. Amer. Statist. Assoc., Volume 69 (1974), pp. 1031-1037 | MR | Zbl

[4] Bunea, Cornel; Cooke, Roger; Lindqvist, Bo Henry Competing risk perspective on reliability databases, Mathematical and statistical methods in reliability (Trondheim, 2002) (Ser. Qual. Reliab. Eng. Stat.), Volume 7, World Sci. Publ., River Edge, NJ, 2003, pp. 355-370 | DOI | MR

[5] Christer, Antony H. A review of delay time analysis for modelling plant maintenance, Stochastic models in reliability and maintenance, Springer, Berlin, 2002, pp. 89-123 | MR

[6] Cooke, Roger M. The total time on test statistic and age-dependent censoring, Statist. Probab. Lett., Volume 18 (1993) no. 4, pp. 307-312 | DOI | MR | Zbl

[7] Cooke, Roger M. The design of reliability databases, Part I and II., Reliability Eng. System safety, Volume 51 (1996), p. 137-146 and 209–223

[8] Cooke, Roger M.; Paulsen, J. Concepts for measuring maintenance performance and methods for analysing competing failures modes, Reliability Eng. System safety, Volume 55 (1997), pp. 135-141

[9] Crowder, Martin Classical Competing Risks, Springer Series in Statistics, Chapman & Hall/CRC, Boca Raton, FL, 2001 | Zbl

[10] Deloux, Estelle; Dijoux, Yann; Fouladirad, Mitra Generalization of the proportional hazards model for maintenance modelling and optimization, J. Risk Reliab., Volume 226 (2012) no. 5, pp. 439-447

[11] Dijoux, Yann; Doyen, Laurent; Gaudoin, Olivier Conditionally independent generalized competing risks for maintenance analysis, Advances in mathematical modeling for reliability, IOS, Amsterdam, 2008, pp. 88-95 | MR

[12] Dewan, Isha; Deshpande, J. V.; Kulathinal, S. B. On testing dependence between time to failure and cause of failure via conditional probabilities, Scand. J. Statist., Volume 31 (2004) no. 1, pp. 79-91 | DOI | MR | Zbl

[13] Doyen, Laurent; Gaudoin, Olivier Imperfect maintenance in a generalized competing risks framework, J. Appl. Probab., Volume 43 (2006) no. 3, pp. 825-839 | DOI | MR | Zbl

[14] Dauxois, Jean-Yves; Guilloux, Agathe Nonparametric inference under competing risks and selection-biased sampling, J. Multivariate Anal., Volume 99 (2008) no. 4, pp. 589-605 | DOI | MR | Zbl

[15] Dijoux, Yann; Gaudoin, Olivier The alert-delay competing risks model for maintenance analysis, J. Statist. Plann. Inference, Volume 139 (2009) no. 5, pp. 1587-1603 | DOI | MR | Zbl

[16] Dijoux, Yann; Gaudoin, Olivier Generalized Random Sign and Alert Delay models for imperfect maintenance, Lifetime Data Anal., Volume 20 (2014) no. 2, pp. 185-209 | DOI | MR | Zbl

[17] Dauxois, Jean-Yves; Kirmani, Syed N. U. A. On testing the proportionality of two cumulative incidence functions in a competing risks setup, J. Nonparametr. Stat., Volume 16 (2004) no. 3-4, pp. 479-491 | DOI | MR | Zbl

[18] Fleming, Thomas R.; Harrington, David P. A class of hypothesis tests for one- and two-sample censored survival data, Comm. Statist. A—Theory Methods, Volume 10 (1981) no. 8, pp. 763-794 | MR | Zbl

[19] Harrington, David P.; Fleming, Thomas R. A class of rank test procedures for censored survival data, Biometrika, Volume 69 (1982) no. 3, pp. 553-566 | DOI | MR | Zbl

[20] Hokstad, Per; Jensen, R. Predicting the failure rate for components that go through a degradation state, Safety and Reliability, S. Lydersen, G.K. Hansen and H.A. Sandtorv (eds), Balkema, Rotterdam, 1998, pp. 389-396

[21] Langseth, Helge; Lindqvist, Bo Henry A maintenance model for components exposed to several failure mechanisms and imperfect repair, Mathematical and statistical methods in reliability (Trondheim, 2002) (Ser. Qual. Reliab. Eng. Stat.), Volume 7, World Sci. Publ., River Edge, NJ, 2003, pp. 415-430 | DOI | MR

[22] Langseth, Helge; Lindqvist, Bo Henry Competing risks for repairable systems: a data study, J. Statist. Plann. Inference, Volume 136 (2006) no. 5, pp. 1687-1700 | DOI | MR | Zbl

[23] Li, Wenjian; Pham, Hoang Reliability Modeling of Multi-State Degraded Systems with Multi-Competing Failures and Random Shocks, IEEE Transactions on Reliability, Volume 54 (2005) no. 2, pp. 297-301

[24] Lindqvist, Bo H.; Støve, Bård; Langseth, Helge Modelling of dependence between critical failure and preventive maintenance: the repair alert model, J. Statist. Plann. Inference, Volume 136 (2006) no. 5, pp. 1701-1717 | DOI | MR | Zbl

[25] Tsiatis, Anastasios A nonidentifiability aspect of the problem of competing risks, Proc. Nat. Acad. Sci. U.S.A., Volume 72 (1975), pp. 20-22 | MR | Zbl

[26] van der Vaart, Aad W.; Wellner, Jon A. Weak convergence and empirical processes, Springer Series in Statistics, Springer-Verlag, New York, 1996, xvi+508 pages (With applications to statistics) | MR | Zbl