The Trend-Renewal Process for Statistical Analysis of Repairable Systems

Lindqvist, Bo Henry; Elvebakk, Georg; Heggland, Knut

doi:10.1198/004017002188618671

Cited by 123 publications

(133 citation statements)

References 21 publications

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“…Let us take into account some real data of failure times, namely the data set contained in the paper of Lindqvist et al (2003), given in Table 13. These data contain 41 failure times of a gas compressor with time censoring at time 7571 (days).…”

Section: Some Real Datamentioning

confidence: 99%

“…studied by Lindqvist et al (2003). The renewal distribution function F corresponds to the Weibull distribution We(γ , 1/ (1 + 1/γ )) with the parametrization resulting in the expectation 1.…”

Section: The Likelihood Functionmentioning

confidence: 99%

“…The TRP was introduced and investigated first by Lindqvist (1993) and by Lindqvist et al (1994) (see also Lindqvist and Doksum (2003), and Lindqvist (2006)). Parametric inference on the parameters of the TRP was considered in the paper of Lindqvist et al (2003), where the authors also proposed corresponding models, called heterogeneous trend-renewal processes, that extend the TRP to cases involving unobserved heterogeneity.…”

Section: Introductionmentioning

confidence: 99%

“…Parametric inference on the parameters of the TRP was considered in the paper of Lindqvist et al (2003), where the authors also proposed corresponding models, called heterogeneous trend-renewal processes, that extend the TRP to cases involving unobserved heterogeneity. Nonparametric maximum likelihood (ML) estimation of the trend function of a TRP under the often natural condition that λ(·) is monotone was considered by Heggland and Lindqvist (2007).…”

Section: Introductionmentioning

confidence: 99%

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Estimation of parameters for trend-renewal processes

Jokiel-Rokita

Magiera

2011

Stat Comput

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Methods of estimating unknown parameters of a trend function for trend-renewal processes are investigated in the case when the renewal distribution function is unknown. If the renewal distribution is unknown, then the likelihood function of the trend-renewal process is unknown and consequently the maximum likelihood method cannot be used. In such a situation we propose three other methods of estimating the trend parameters. The methods proposed can also be used to predict future occurrence times. The performance of the estimators based on these methods is illustrated numerically for some trend-renewal processes for which the statistical inference is analytically intractable.

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Section: Some Real Datamentioning

confidence: 99%

Section: The Likelihood Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Estimation of parameters for trend-renewal processes

Jokiel-Rokita

Magiera

2011

Stat Comput

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“…A class of models of this kind is the trend-renewal process (TRP), which was defined and studied by Lindqvist, Elvebakk and Heggland (2003), see also Lindqvist (2006). This model contains the RP and NHPP as special cases, and in a simple manner the TRP fills some of the gap between the two extreme repair models.…”

Section: Introductionmentioning

confidence: 99%

Nonparametric Estimation of Time Trend for Repairable Systems Data

Lindqvist

2010

Mathematical and Statistical Models and Methods in Reliability

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The trend-renewal-process (TRP) is defined to be a time-transformed renewal process, where the time transformation is given by a trend function λ(·) which is similar to the intensity of a nonhomogeneous Poisson process (NHPP). A nonparametric maximum likelihood estimator of the trend function of a TRP can be obtained much in the same manner as for the NHPP using kernel smoothing. But for a TRP one must consider the simultaneous estimation of the renewal distribution, which is here assumed to belong to a parametric class such as the Weibull-distribution. A weighted kernel estimator for λ(·) is derived using a general approach for kernel smoothing in counting processes.

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Space‐time model for repeating earthquakes and analysis of recurrence intervals on the San Andreas Fault near Parkfield, California

2014

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We propose a stochastic model for characteristically repeating earthquake sequences to estimate the spatiotemporal change in static stress loading rate. These earthquakes recur by a cyclic mechanism where stress at a hypocenter is accumulated by tectonic forces until an earthquake occurs that releases the accumulated stress to a basal level. Renewal processes are frequently used to describe this repeating earthquake mechanism. Variations in the rate of tectonic loading due to large earthquakes and aseismic slip transients, however, introduce nonstationary effects into the repeating mechanism that result in nonstationary trends in interevent times, particularly for smaller-magnitude repeating events which have shorter interevent times. These trends are also similar among repeating earthquake sites having similar hypocenters. Therefore, we incorporate space-time structure represented by cubic B-spline functions into the renewal model and estimate their coefficient parameters by maximizing the integrated likelihood in a Bayesian framework. We apply our model to 31 repeating earthquake sequences including 824 events on the Parkfield segment of the San Andreas Fault and estimate the spatiotemporal transition of the loading rate on this segment. The result gives us details of the change in tectonic loading caused by an aseismic slip transient in 1993, the 2004 Parkfield M6 earthquake, and other nearby or remote seismic activities. The degree of periodicity of repeating event recurrence intervals also shows spatial trends that are preserved in time even after the 2004 Parkfield earthquake when time scales are normalized with respect to the estimated loading rate.

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The Trend-Renewal Process for Statistical Analysis of Repairable Systems

Cited by 123 publications

References 21 publications

Estimation of parameters for trend-renewal processes

Estimation of parameters for trend-renewal processes

Nonparametric Estimation of Time Trend for Repairable Systems Data

Space‐time model for repeating earthquakes and analysis of recurrence intervals on the San Andreas Fault near Parkfield, California

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