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Estimation of component reliability from superposed renewal processes by means of latent variables

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A Correction to this article was published on 21 August 2021

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Abstract

We present a new way to estimate the lifetime distribution of a reparable system consisted of similar (equal) components. We consider as a reparable system, a system where we can replace a failed component by a new one. Assuming that the lifetime distribution of all components (originals and replaced ones) are the same, the position of a single component can be represented as a renewal process. There is a considerable amount of works related to estimation methods for this kind of problem. However, the data has information only about the time of replacement. It was not recorded which component was replaced. That is, the replacement data are available in an aggregate form. Using both Bayesian and a maximum likelihood function approaches, we propose an estimation procedure for the lifetime distribution of components in a repairable system with aggregate data. Based on a latent variables method, our proposed method out-perform the commonly used estimators for this problem. The proposed procedure is generic and can be used with any lifetime probability model. Aside from point estimates, interval estimates are presented for both approaches. The performances of the proposed methods are illustrated through several simulated data, and their efficiency and applicability are shown based on the so-called cylinder problem. The computational implementation is available in the R package srplv.

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References

  • Casella G, Berger RL (2002) Statistical inference, vol 2. Duxbury, Pacific Grove

    MATH  Google Scholar 

  • Crow LH (1990) Evaluating the reliability of repairable systems. In: Annual proceedings on reliability and maintainability symposium, pp 275–279. IEEE

  • Crowder MJ, Kimber A, Sweeting T, Smith R (1994) Statistical analysis of reliability data, vol 27. CRC Press, Boca Raton

    Google Scholar 

  • Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J Roy Stat Soc Ser B Methodol 39(1):1–22

    MathSciNet  MATH  Google Scholar 

  • Dewanji A, Kundu S, Nayak TK (2012) Nonparametric estimation of the number of components of a superposition of renewal processes. J Stat Plan Inference 142(9):2710–2718

    Article  MathSciNet  Google Scholar 

  • Gelman A, Rubin DB (1992) Inference from iterative simulation using multiple sequences. Stat Sci 7:457–472

    MATH  Google Scholar 

  • Gilks WR, Richardson S, Spiegelhalter DJ (1995) Introducing Markov chain Monte Carlo. In: Markov chain Monte Carlo in practice, p 1

  • Krivtsov V, Frankstein M, Yevkin O (2017) On reliability of a multi-socket repairable system. Qual Reliab Eng Int 33(5):1011–1017

    Article  Google Scholar 

  • Liu B, Shi Y, Cai J, Bai X, Zhang C (2017) Nonparametric Bayesian analysis for masked data from hybrid systems in accelerated lifetime tests. IEEE Trans Reliab 66(3):662–676

    Article  Google Scholar 

  • Louis TA (1982) Finding the observed information matrix when using the EM algorithm. J Roy Stat Soc Ser B Methodol 44(2):226–233

    MathSciNet  MATH  Google Scholar 

  • Meeker WQ, Escobar LA (2014) Statistical methods for reliability data. Wiley, Hoboken

    MATH  Google Scholar 

  • Miyakawa M (1984) Analysis of incomplete data in competing risks model. IEEE Trans Reliab 33(4):293–296

    Article  Google Scholar 

  • Nelder JA, Mead R (1965) A simplex method for function minimization. Comput J 7(4):308–313

    Article  MathSciNet  Google Scholar 

  • Nelson W, Doganaksoy N (1989) A computer program for an estimate and confidence limits for the mean cumulative function for cost or number of repairs of repairable products. GE Research & Development Center, Niskayuna

    Google Scholar 

  • Nelson WB (2003) Recurrent events data analysis for product repairs, disease recurrences, and other applications. SIAM, Seattle

    Book  Google Scholar 

  • Peixoto J (2009) Estimating lifetimes when several unidentified components are reported. In: Proceedings of the physical and engineering sciences section of the American statistical association, pp 5078–5092

  • R Core Team (2020) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna. https://www.R-project.org/

  • Rinne H (2008) The Weibull distribution: a handbook. CRC Press, Boca Raton

    Book  Google Scholar 

  • Robert CP, Casella G, Casella G (2010) Introducing Monte Carlo methods with R, vol 18. Springer, Berlin

    Book  Google Scholar 

  • Rodrigues AS, Bhering F, Carlos ADB, Polpo A (2018) Bayesian estimation of component reliability in coherent systems. IEEE Access 6:18520–18535

    Article  Google Scholar 

  • Rodrigues AS, Pereira CADB, Polpo A (2019) Estimation of component reliability in coherent systems with masked data. IEEE Access 7:57476–57487

    Article  Google Scholar 

  • Song S, Xie M (2018) An integrated method for estimation with superimposed failure data. In: 2018 IEEE international conference on prognostics and health management (ICPHM), pp 1–5. IEEE

  • Tierney L (1994) Markov chains for exploring posterior distributions. Ann Stat 22:1701–1728

    MathSciNet  MATH  Google Scholar 

  • Wang R, Sha N, Gu B, Xu X (2015) Parameter inference in a hybrid system with masked data. IEEE Trans Reliab 64(2):636–644

    Article  Google Scholar 

  • Zhang W, Tian Y, Escobar L, Meeker W (2015) SRPML: estimating a parametric component lifetime distribution from SRP data. R package version 1

  • Zhang W, Tian Y, Escobar LA, Meeker WQ (2017) Estimating a parametric component lifetime distribution from a collection of superimposed renewal processes. Technometrics 59(2):202–214

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was partially supported by the Brazilian agency CNPq: Grant 308776/2014-3. The agency had no role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript. This study was financed in part by CAPES (Brazil) - Finance Code 001 and Federal University of Mato Grosso do Sul. Pascal Kerschke, Heike Trautmann, Bernd Hellingrath and Carolin Wagner acknowledge support by the European Research Center for Information Systems (ERCIS).

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Correspondence to Agatha Rodrigues.

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The original online version of this article was revised as the affiliation of author Heike Trautmann was incorrectly published. The original article has been corrected.

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Appendix

Appendix

We can write the logarithm of the complete likelihood function of i-th system if Weibull distribution with parameter \(\beta \) (shape) and \(\eta \) (scale) is assumed, as

$$\begin{aligned}&l_{i}({\varvec{\theta }}\mid {\varvec{t}}_i,{\varvec{d}}_i) = \Big [1-\mathrm{I}(v_i=0)\Big ]\Bigg [\sum _{l=1}^{v_i}\sum _{k=1}^{n_l}\log f(x_{ilk}-x_{il(k-1)}) \\&\qquad + \sum _{l=1}^{v_i} \log R(\tau _i-x_{iln_l}) \Bigg ] + (m-v_i)\log R(\tau _i) \\&\quad =\Big [1-\mathrm{I}(v_i=0)\Big ]\sum _{l=1}^{v_i}\sum _{k=1}^{n_l} \Bigg \{\log (\beta )-\log (\eta )+(\beta -1)\Big [\log (x_{ilk}-x_{il(k-1)})\\&\qquad -\log (\eta )\Big ]-\Bigg (\frac{x_{ilk}-x_{il(k-1)}}{\eta }\Bigg )^{\beta }\Bigg \} \\&\qquad - \Big [1-\mathrm{I}(v_i=0)\Big ] \sum _{l=1}^{v_i} \Bigg (\frac{\tau _i-x_{iln_l}}{\eta }\Bigg )^{\beta } - (m-v_i)\Bigg (\frac{\tau _i}{\eta }\Bigg )^{\beta } \\&\quad = \Big [1-\mathrm{I}(v_i=0)\Big ]\Bigg \{r_i\log (\beta )-r_i \log (\eta )+(\beta -1)\sum _{l=1}^{v_i}\sum _{k=1}^{n_l}\log (x_{ilk}-x_{il(k-1)}) \\&\qquad -r_i(\beta -1)\log (\eta )-\sum _{l=1}^{v_i} \Bigg [\sum _{k=1}^{n_l}\Bigg (\frac{x_{ilk}-x_{il(k-1)}}{\eta }\Bigg )^{\beta } + \Bigg (\frac{\tau _i-x_{iln_l}}{\eta }\Bigg )^{\beta } \Bigg ]\Bigg \} \\&\qquad - (m-v_i)\Bigg (\frac{\tau _i}{\eta }\Bigg )^{\beta }. \end{aligned}$$

The first derivatives 0f \(l_{i}({\varvec{\theta }}\mid {\varvec{t}}_i,{\varvec{d}}_i)\) in relation to \(\beta \) and \(\eta \), respectively, are

$$\begin{aligned}&\frac{~\text {d}l_{i}({\varvec{\theta }}\mid {\varvec{t}}_i,{\varvec{d}}_i)}{~\text {d}\beta }= \Big [1-\mathrm{I}(v_i=0)\Big ]\Bigg \{\frac{r_i}{\beta } +\sum _{l=1}^{v_i}\sum _{k=1}^{n_l}\log (x_{ilk}-x_{il(k-1)}) - r_i\log (\eta )\\&\quad +\log (\eta )\Bigg (\frac{1}{\eta }\Bigg )^{\beta }\Bigg [\sum _{l=1}^{v_i}\Bigg (\sum _{k=1}^{n_l} (x_{ilk}-x_{il(k-1)})^{\beta } + (\tau _i-x_{iln_l})^{\beta }\Bigg ) \Bigg ] \\&\quad -\Bigg (\frac{1}{\eta }\Bigg )^{\beta }\Bigg [\sum _{l=1}^{v_i}\sum _{k=1}^{n_l} \log (x_{ilk}-x_{il(k-1)})(x_{ilk}-x_{il(k-1)})^{\beta }\\&\quad +\sum _{l=1}^{v_i}\log (\tau _i-x_{iln_l})(\tau _i-x_{iln_l})^{\beta }\Bigg ]\Bigg \} +\Bigg (\frac{1}{\eta }\Bigg )^{\beta }(m-v_i)\tau _i^{\beta }[\log (\eta )-\log (\tau _i)], \end{aligned}$$

and

$$\begin{aligned}&\frac{~\text {d}l_{i}({\varvec{\theta }}\mid {\varvec{t}}_i,{\varvec{d}}_i)}{~\text {d}\eta } =\Big [1-\mathrm{I}(v_i=0)\Big ]\Bigg \{-\frac{r_i}{\eta }-\frac{r_i(\beta -1)}{\eta } +\beta \Bigg (\frac{1}{\eta }\Bigg )^{\beta +1}\\&\quad \Bigg [\sum _{l=1}^{v_i}\sum _{k=1}^{n_l}(x_{ilk}-x_{il(k-1)})^{\beta } +\sum _{l=1}^{v_i}(\tau _i-x_{iln_l})^{\beta }\Bigg ]\Bigg \} +\beta \Bigg (\frac{1}{\eta }\Bigg )^{\beta +1}(m-v_i)\tau _i^{\beta }. \end{aligned}$$

The second derivatives are

$$\begin{aligned}&\frac{~\text {d}^2 l_{i}({\varvec{\theta }}\mid {\varvec{t}}_i,{\varvec{d}}_i)}{~\text {d}\beta ^2} =\Big [1-\mathrm{I}(v_i=0)\Big ]\Bigg \{-\frac{r_i}{\beta ^2} -[\log \eta ]^2\Bigg (\frac{1}{\eta }\Bigg )^{\beta } \\&\quad \times \Bigg [\sum _{l=1}^{v_i}\Bigg (\sum _{k=1}^{n_l} (x_{ilk}-x_{il(k-1)})^{\beta } + (\tau _i-x_{iln_l})^{\beta }\Bigg ) \Bigg ] +2\log (\eta )\Bigg (\frac{1}{\eta }\Bigg )^{\beta } \\&\quad \times \Bigg [\sum _{l=1}^{v_i}\sum _{k=1}^{n_l}\log (x_{ilk}-x_{il(k-1)})(x_{ilk} -x_{il(k-1)})^{\beta }+\sum _{l=1}^{v_i}\log (\tau _i-x_{iln_l})(\tau _i-x_{iln_l})^{\beta }\Bigg ]\\&\qquad -\Bigg (\frac{1}{\eta }\Bigg )^{\beta }\Bigg [\sum _{l=1}^{v_i}\sum _{k=1}^{n_l} [\log (x_{ilk}-x_{il(k-1)})]^2(x_{ilk}-x_{il(k-1)})^{\beta } \\&\qquad +\sum _{l=1}^{v_i}[\log (\tau _i-x_{iln_l})]^2(\tau _i-x_{iln_l})^{\beta }\Bigg ] \Bigg \}\\&\qquad +(m-v_i)\Bigg (\frac{1}{\eta }\Bigg )^{\beta }\tau _i^{\beta }\Bigg [-[\log (\tau _i)]^2 +2\log (\tau _i)\log (\eta )-[\log (\eta )]^2\Bigg ], \\&\frac{~\text {d}^2 l_{i}({\varvec{\theta }}\mid {\varvec{t}}_i,{\varvec{d}}_i)}{~\text {d}\beta ~\text {d}\eta } =\Big [1-\mathrm{I}(v_i=0)\Big ]\Bigg \{-\frac{r_i}{\eta } + \Bigg [\Bigg (\frac{1}{\eta }\Bigg )^{\beta +1}(1-\beta \log (\eta ))\Bigg ] \\&\quad \times \Bigg [\sum _{l=1}^{v_i}\Bigg (\sum _{k=1}^{n_l} (x_{ilk}-x_{il(k-1)})^{\beta }+ (\tau _i-x_{iln_l})^{\beta }\Bigg ) \Bigg ] \\&\qquad + \beta \Bigg (\frac{1}{\eta }\Bigg )^{\beta +1} \Bigg [\sum _{l=1}^{v_i}\sum _{k=1}^{n_l}\log (x_{ilk}-x_{il(k-1)})(x_{ilk}-x_{il(k-1)})^{\beta }\\&\qquad +\sum _{l=1}^{v_i}\log (\tau _i-x_{iln_l})(\tau _i-x_{iln_l})^{\beta }\Bigg ] \Bigg \} \\&\qquad +(m-v_i)\Bigg (\frac{1}{\eta }\Bigg )^{\beta +1}\tau _i^{\beta } [1-\beta \log (\eta )+\beta \log (\tau _i)], \end{aligned}$$

and

$$\begin{aligned}&\frac{~\text {d}^2 l_{i}({\varvec{\theta }}\mid {\varvec{t}}_i,{\varvec{d}}_i)}{~\text {d}\eta ^2 } =\Big [1-\mathrm{I}(v_i=0)\Big ]\Bigg \{\frac{\beta r_i}{\eta ^2}-\beta (\beta +1) \Bigg (\frac{1}{\eta }\Bigg )^{\beta +2} \\&\quad \Bigg [\sum _{l=1}^{v_i}\Bigg (\sum _{k=1}^{n_l} (x_{ilk}-x_{il(k-1)})^{\beta } + (\tau _i-x_{iln_l})^{\beta }\Bigg ) \Bigg ]\Bigg \} -\beta (\beta +1)\Bigg (\frac{1}{\eta }\Bigg )^{\beta +2}(m-v_i)\tau _i^{\beta }. \end{aligned}$$

Thus,

$$\begin{aligned} I= & {} -\frac{\partial ^2}{\partial {\varvec{\theta }}\partial {\varvec{\theta }}^\top } Q({\varvec{\theta }}\mid \widehat{{\varvec{\theta }}})=-\frac{1}{L} \sum _{i=1}^n\sum _{l=1}^L\frac{\partial ^2}{\partial {\varvec{\theta }} \partial {\varvec{\theta }}^\top }l_{i}\Big ({\varvec{\theta }}\mid {\varvec{t}}_i,{\varvec{d}}_{i}^{(l)}\Big ) \Bigg |_{{\varvec{\theta }}=\widehat{{\varvec{\theta }}}}\\= & {} \left[ {\begin{array}{cc} -\frac{1}{L}\sum _{i=1}^n\sum _{l=1}^L \frac{~\text {d}^2 l_{i}({\varvec{\theta }}\mid {\varvec{t}}_i,{\varvec{d}}_{i}^{(l)})}{~\text {d}\eta ^2 }\Bigg |_{{\varvec{\theta }} =\widehat{{\varvec{\theta }}}} &{} -\frac{1}{L}\sum _{i=1}^n\sum _{l=1}^L \frac{~\text {d}^2 l_{i}({\varvec{\theta }} \mid {\varvec{t}}_i,{\varvec{d}}_{i}^{(l)})}{~\text {d}\eta ~\text {d}\beta }\Bigg |_{{\varvec{\theta }}=\widehat{{\varvec{\theta }}}} \\ -\frac{1}{L}\sum _{i=1}^n\sum _{l=1}^L \frac{~\text {d}^2 l_{i}({\varvec{\theta }}\mid {\varvec{t}}_i,{\varvec{d}}_{i}^{(l)})}{~\text {d}\beta ~\text {d}\eta }\Bigg |_{{\varvec{\theta }} =\widehat{{\varvec{\theta }}}} &{} -\frac{1}{L}\sum _{i=1}^n\sum _{l=1}^L \frac{~\text {d}^2 l_{i}({\varvec{\theta }} \mid {\varvec{t}}_i,{\varvec{d}}_{i}^{(l)})}{~\text {d}\beta ^2 }\Bigg |_{{\varvec{\theta }}=\widehat{{\varvec{\theta }}}} \end{array} } \right] , \end{aligned}$$

in which \(\widehat{{\varvec{\theta }}}=(\widehat{\eta },\widehat{\beta })\). Besides,

$$\begin{aligned} II= & {} \sum _{i=1}^n\Bigg \{\frac{1}{L}\sum _{l=1}^L\frac{\partial }{\partial {\varvec{\theta }}}l_{i} \Big ({\varvec{\theta }}\mid {\varvec{t}}_i,{\varvec{d}}_{i}^{(l)}\Big )\Bigg |_{{\varvec{\theta }} =\widehat{{\varvec{\theta }}}}\Bigg \}\Bigg \{\frac{1}{L}\sum _{l=1}^L \frac{\partial }{\partial {\varvec{\theta }}}l_{i}\Big ({\varvec{\theta }}\mid {\varvec{t}}_i, {\varvec{d}}_{i}^{(l)}\Big )\Bigg |_{{\varvec{\theta }}=\widehat{{\varvec{\theta }}}}\Bigg \}^\top \\= & {} \sum _{i=1}^n\Bigg \{\frac{1}{L}\sum _{l=1}^L\Bigg ( \frac{~\text {d}l_{i}({\varvec{\theta }} \mid {\varvec{t}}_i,{\varvec{d}}_{i}^{(l)})}{~\text {d}\eta }\Bigg |_{{\varvec{\theta }}=\widehat{{\varvec{\theta }}}}, \frac{~\text {d}l_{i}({\varvec{\theta }}\mid {\varvec{t}}_i,{\varvec{d}}_{i}^{(l)})}{~\text {d}\beta } \Bigg |_{{\varvec{\theta }}=\widehat{{\varvec{\theta }}}}\Bigg )^\top \Bigg \} \\&\times \Bigg \{\frac{1}{L}\sum _{l=1}^L\Bigg ( \frac{~\text {d}l_{i} ({\varvec{\theta }}\mid {\varvec{t}}_i,{\varvec{d}}_{i}^{(l)})}{~\text {d}\eta }\Bigg |_{{\varvec{\theta }} =\widehat{{\varvec{\theta }}}}, \frac{~\text {d}l_{i}({\varvec{\theta }}\mid {\varvec{t}}_i,{\varvec{d}}_{i}^{(l)})}{~\text {d}\beta }\Bigg |_{{\varvec{\theta }}=\widehat{{\varvec{\theta }}}}\Bigg )^\top \Bigg \}^{\top } \end{aligned}$$

and

$$\begin{aligned} III= & {} -\frac{1}{L}\sum _{i=1}^n\sum _{l=1}^L\Bigg \{\frac{\partial }{\partial {\varvec{\theta }}}l_{i}\Big ({\varvec{\theta }}\mid {\varvec{t}}_i,{\varvec{d}}_{i}^{(l)}\Big )\Bigg \} \Bigg \{\frac{\partial }{\partial {\varvec{\theta }}}l_{i}\Big ({\varvec{\theta }} \mid {\varvec{t}}_i,{\varvec{d}}_{i}^{(l)}\Big )\Bigg \}^\top \Bigg |_{{\varvec{\theta }}=\widehat{{\varvec{\theta }}}}\\= & {} -\frac{1}{L}\sum _{i=1}^n\sum _{l=1}^L\Bigg \{\Bigg ( \frac{\mathrm{d} l_{i}({\varvec{\theta }} \mid {\varvec{t}}_i,{\varvec{d}}_{i}^{(l)})}{\mathrm{d}\eta }, \frac{\mathrm{d} l_{i}({\varvec{\theta }} \mid {\varvec{t}}_i,{\varvec{d}}_{i}^{(l)})}{\mathrm{d}\beta }\Bigg )^\top \Bigg \}\\&\quad \Bigg \{\Bigg ( \frac{\mathrm{d} l_{i}({\varvec{\theta }} \mid {\varvec{t}}_i,{\varvec{d}}_{i}^{(l)})}{\mathrm{d}\eta }, \frac{\mathrm{d} l_{i}({\varvec{\theta }} \mid {\varvec{t}}_i,{\varvec{d}}_{i}^{(l)})}{\mathrm{d}\beta }\Bigg )^\top \Bigg \}^\top \Bigg |_{{\varvec{\theta }} =\widehat{{\varvec{\theta }}}}. \end{aligned}$$

The quantity \(I_{{\varvec{\theta }}}(\widehat{{\varvec{\theta }}})\) can be estimated by \(I+II+III\).

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Rodrigues, A., Kerschke, P., Pereira, C.A.d.B. et al. Estimation of component reliability from superposed renewal processes by means of latent variables. Comput Stat 37, 355–379 (2022). https://doi.org/10.1007/s00180-021-01124-0

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