Parametric Bootstrap Goodness-of-Fit Tests for Imperfect Maintenance Models

The simultaneous modeling of ageing and maintenance efficiency of repairable systems is a major issue in reliability. Many imperfect maintenance models have been proposed. To analyze a dataset, it is necessary to check whether these models are adapted or not. In this paper, we propose a general methodology for testing the goodness of fit of any kind of imperfect maintenance model. Two families of tests are presented, based respectively on martingale residuals and probability integral transforms. The quantiles of the test statistics distributions under the null hypothesis are computed with parametric bootstrap methods. An extensive simulation study is provided, from which we recommend the use of two tests in practice, one from each family. Finally, the tests are applied to several real datasets.

[1]  M. Rosenblatt.  Remarks on a Multivariate Transformation , 1952 .

[2]  Lin Ye.  Geometric processes and replacement problem , 1988 .

[3]  M. Kijima.  SOME RESULTS FOR REPAIRABLE SYSTEMS WITH GENERAL REPAIR , 1989 .

[4]  N. Hjort.  Goodness of fit tests in models for life history data based on cumulative hazard rates , 1990 .

[5]  L. Yeh.  Nonparametric inference for geometric processes , 1992 .

[6]  Robert Tibshirani,et al.  An Introduction to the Bootstrap , 1994 .

[7]  W. Stute,et al.  Bootstrap based goodness-of-fit-tests , 1993 .

[8]  Z. Ying,et al.  Checking the Cox model with cumulative sums of martingale-based residuals , 1993 .

[9]  J. Klein,et al.  Statistical Models Based On Counting Process , 1994 .

[10]  Hoang Pham,et al.  A quasi renewal process and its applications in imperfect maintenance , 1996, Int. J. Syst. Sci..

[11]  Anthony C. Davison,et al.  Bootstrap Methods and Their Application , 1998 .

[12]  G. M. Clark,et al.  ESTIMATING PARAMETERS OF THE POWER LAW PROCESS WITH TWO MEASURES OF FAILURE TIME , 1998 .

[13]  O. Gaudoin.  CPIT goodness-of-fit tests for the power-law process , 1998 .

[14]  R. Szekli,et al.  Asymptotic and monotonicity properties of some repairable systems , 1998, Advances in Applied Probability.

[15]  O. Gaudoin,et al.  U-plot method for testing the goodness-of-fit of the power-law process , 1999 .

[16]  Bo Henry Lindqvist,et al.  The Trend-Renewal Process for Statistical Analysis of Repairable Systems , 2003, Technometrics.

[17]  R. Jiang,et al.  Type IV Weibull Models , 2003 .

[18]  Laurent Doyen,et al.  Classes of imperfect repair models based on reduction of failure intensity or virtual age , 2004, Reliab. Eng. Syst. Saf..

[19]  N. Henze,et al.  Recent and classical tests for exponentiality: a partial review with comparisons , 2005 .

[20]  B. Rémillard,et al.  Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models , 2005 .

[21]  Jin Zhao,et al.  A New Goodness-of-Fit Test Based on the Laplace Statistic for a Large Class of NHPP Models , 2005 .

[22]  Maurizio Guida,et al.  Bayesian analysis of repairable systems showing a bounded failure intensity , 2006, Reliab. Eng. Syst. Saf..

[23]  Richard J. Cook,et al.  The Statistical Analysis of Recurrent Events , 2007 .

[24]  David Hinkley,et al.  Bootstrap Methods: Another Look at the Jackknife , 2008 .

[25]  Aarnout Brombacher,et al.  Probability... , 2009, Qual. Reliab. Eng. Int..

[26]  L. Doyen.  Asymptotic properties of imperfect repair models and estimation of repair efficiency , 2010 .

[27]  B. Lindqvist,et al.  Monte Carlo exact goodness-of-fit tests for nonhomogeneous Poisson processes , 2011 .

[28]  L. Doyen.  On the Brown–Proschan model when repair effects are unknown , 2011 .

[29]  L. Doyen,et al.  Bayesian Analysis of ARA Imperfect Repair Models , 2012 .

[30]  Hong-Zhong Huang,et al.  A Data-Driven Approach to Selecting Imperfect Maintenance Models , 2012, IEEE Transactions on Reliability.

[31]  Sophie Mercier,et al.  Extended Geometric Processes: Semiparametric Estimation and Application to Reliability , 2013 .

[32]  Min Xie,et al.  Simplified Likelihood Based Goodness-of-fit Tests for the Weibull Distribution , 2016, Commun. Stat. Simul. Comput..