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

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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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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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Authors and Affiliations

  1. Department of Statistics, Federal University of Espírito Santo, Vitória, ES, Brazil

    Agatha Rodrigues

  2. Big Data Analytics in Transportation, Technical University Dresden, 01062, Dresden, Germany

    Pascal Kerschke

  3. Institute of Mathematics and Statistics, University of São Paulo, São Paulo, SP, Brazil

    Carlos Alberto de B. Pereira

  4. Institute of Mathematics, Federal University of Mato Grosso do Sul, Campo Grande, MS, Brazil

    Carlos Alberto de B. Pereira

  5. Information Systems and Statistics, University of Münster, 48149, Münster, Germany

    Heike Trautmann, Carolin Wagner & Bernd Hellingrath

  6. Western Australia University, Perth, Australia

    Adriano Polpo

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  1. Agatha Rodrigues
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  2. Pascal Kerschke
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  3. Carlos Alberto de B. Pereira
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  4. Heike Trautmann
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  5. Carolin Wagner
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  6. Bernd Hellingrath
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  7. Adriano Polpo
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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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