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

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.

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 $I+II+III$.