Inference and expected total test time for step-stress life test in the presence of complementary risks and incomplete data

Abstract

The complementary risk is common and important in the engineering field. However, there is not much research about it because of its complex derivation compared with the competing risk model. In this paper, we concentrate on inference of step-stress partially accelerated life test in the presence of complementary risks under progressive type-II censoring scheme. The Weibull distribution is chosen as the baseline lifetime of the model. The tampered random variable model is adopted as the statistical acceleration model in the accelerated test. We apply both the classical and Bayesian methods to obtain the estimation of lifetime parameters and acceleration factors. The reliability and reversed hazard rate are estimated based on the parametric estimates. The computational formulae of expected total test time are creatively derived under the step-stress and censored setting. The theoretical calculations are compared with simulated values to verify the derivation. Also, numerical studies including the simulation study and real-data analysis in engineering background are conducted to compare and illustrate the performance of the approaches proposed in the paper. Some conclusions and suggestions for actual production are given at the end of the paper.

Appendices

Appendix

A Elements of observed Fisher information matrix $I(\hat{\theta })$

The second partial derivative of information matrix can be calculated as

$$\begin{aligned}&\frac{\partial ^{2} \ln L}{\partial \alpha _{j}^{2}}=-\frac{n_{1j}+n_{2j}}{\alpha _{j}^{2}} + \sum _{k=1}^{N_{1}} I({\delta }_{k}=j) \left[ -\lambda _{j} x_{k}^{\alpha _{j}}\left( \ln x_{k}\right) ^{2}\right] + \sum _{k=N_{1}+1}^{m} I({\delta }_{k}=j) \left[ -\lambda _{j} z_{k}^{\alpha _{j}}\left( \ln z_{k}\right) ^{2}\right] \\&+ \sum _{k=1}^{N_{1}} R_{k} \lambda _{j} x_{k}^{\alpha _j} \left[ \left( \ln x_{k}\right) ^{2} B_{kj(x)}+\ln x_{k} B'_{kj(x)1} \right] + \sum _{k=N_{1}+1}^{m} R_{k} \lambda _{j} z_{k}^{\alpha _j} \left[ \left( \ln z_{k}\right) ^{2} B_{kj(z)}+\ln z_{k} B'_{kj(z)1} \right] \\&+ \sum _{k=1}^{N_{1}} I({\delta }_{k}=3-j) \lambda _{j} x_{k}^{\alpha _{j}} \left[ \left( \ln x_{k}\right) ^{2}A_{kj(x)}+\ln x_k A'_{kj(x)1} \right] \\&+ \sum _{k=N_{1}+1}^{m} I({\delta }_{k}=3-j) \lambda _{j} z_{k}^{\alpha _{j}} \left[ \left( \ln z_{k}\right) ^{2}A_{kj(z)}+ \ln z_k A'_{kj(z)1} \right] , \\&\frac{\partial ^{2} \ln L}{\partial \lambda _{j}^{2}} = -\frac{n_{1j}+n_{2j}}{\lambda _{j}^{2}} + \sum _{k=1}^{N_{1}} I({\delta }_{k}=3-j) x_{k}^{\alpha _{j}} A'_{kj(x)2} + \sum _{k=1}^{N_{1}} R_{k} x_{k}^{\alpha _{j}} B'_{kj(x)2} \\&+ \sum _{k=1}^{N_{1}} I({\delta }_{k}=3-j) z_{k}^{\alpha _{j}} A'_{kj(z)2} + \sum _{k=1}^{N_{1}} R_{k} z_{k}^{\alpha _{j}} B'_{kj(z)2},\\&\frac{\partial ^{2} \ln L}{\partial \alpha _{j} \partial \lambda _{j}}=\frac{\partial ^{2} \ln L}{\partial \lambda _{j} \partial \alpha _{j} } = - \sum _{k=1}^{N_{1}} I({\delta }_{k}=j) x_{k}^{\alpha _j} \ln x_k + \sum _{k=1}^{N_{1}} I({\delta }_{k}=3-j) x_{k}^{\alpha _j} \left[ \ln x_k A_{kj(x)}+A'_{kj(x)2} \right] \\&- \sum _{k=N_{1}+1}^{m} I({\delta }_{k}=j) z_{k}^{\alpha _j} \ln z_k + \sum _{k=N_{1}+1}^{m} I({\delta }_{k}=3-j) z_{k}^{\alpha _j} \left[ \ln z_k A_{kj(z)}+A'_{kj(z)2} \right] \\&+ \sum _{k=1}^{N_{1}} R_{k} x_{k}^{\alpha _j} \left[ \ln x_k B_{kj(x)} + B'_{kj(x)2} \right] + \sum _{k=N_{1}+1}^{m} R_{k} z_{k}^{\alpha _j} \left[ \ln z_k B_{kj(z)} + B'_{kj(z)2} \right] , \\&\frac{\partial ^{2} \ln L}{\partial \alpha _{j} \partial \beta _{j}}=\frac{\partial ^{2} \ln L}{\partial \beta _{j} \partial \alpha _{j} } = \sum _{k=N_{1}+1}^{m} I({\delta }_{k}=j) \left[ \frac{z'_{kj}}{z_{kj}} - \lambda _{j}(\alpha _{j} z_{k}^{\alpha _j-1} z'_{kj} \ln z_{kj} + z_{k}^{\alpha _j} \frac{z'_{kj}}{z_{kj}}) \right] \\&+ \sum _{k=N_{1}+1}^{m} I({\delta }_{k}=3-j) \lambda _j \left[ \alpha _{j} z_{k}^{\alpha _j-1} z'_{kj} \ln z_{kj} A_{kj(z)} + z_{k}^{\alpha _j} \frac{z'_{kj}}{z_{kj}} A_{kj(z)} + z_{k}^{\alpha _j} \ln z_{kj} A'_{kj(z)3} \right] \\&+ \sum _{k=N_{1}+1}^{m} R_k \lambda _j \left[ \alpha _{j} z_{k}^{\alpha _j-1} z'_{kj} \ln z_{kj} B_{kj(z)} + z_{k}^{\alpha _j} \frac{z'_{kj}}{z_{kj}} B_{kj(z)} + z_{k}^{\alpha _j} \ln z_{kj} B'_{kj(z)3} \right] , \\&\frac{\partial ^{2} \ln L}{\partial \lambda _{j} \partial \beta _{j}}=\frac{\partial ^{2} \ln L}{\partial \beta _{j} \partial \lambda _{j} } = \sum _{k=N_{1}+1}^{m} I({\delta }_{k}=3-j) \left[ \alpha _{j} z_{k}^{\alpha _j-1} z'_{kj} A_{kj(z)} + z_{k}^{\alpha _j} A'_{kj(z)3} \right] \\&-\sum _{k=N_{1}+1}^{m} I({\delta }_{k}=j) \alpha _{j} z_{k}^{\alpha _j-1} z'_{kj} + \sum _{k=N_{1}+1}^{m} R_k \left[ \alpha _{j} z_{k}^{\alpha _j-1} z'_{kj} B_{kj(z)} + z_{k}^{\alpha _j} B'_{kj(z)3} \right] , \\&\frac{\partial ^{2} \ln L}{\partial \beta _{j}^{2}} = \frac{n_{2j}}{\beta _{j}^{2}} + \sum _{k=N_{1}+1}^{m} I({\delta }_{k}=j) \left[ \frac{\alpha _j-1}{\alpha _j \lambda _j} (\frac{z'_{kj}}{z_{kj}^{2}}C_{kj(x)}-\frac{C'_{kj(x)}}{z_{kj}}) - z_{k}^{\alpha _j-1}C'_{kj(x)} - (\alpha _j-1)z_{k}^{\alpha _j-2}z'_{kj}C_{kj(x)} \right] \\&- \sum _{k=N_{1}+1}^{m} I({\delta }_{k}=3-j) \left[ A'_{kj(z)3}C_{kj(x)}z_{k}^{\alpha _j-1} + A_{kj(z)}C'_{kj(x)}z_{k}^{\alpha _j-1} + (\alpha _j-1)A_{kj(z)}C_{kj(x)}z_{k}^{\alpha _j-2}z'_{kj} \right] \\&+ \sum _{k=N_{1}+1}^{m} R_k \left[ B'_{kj(z)3}C_{kj(x)}z_{k}^{\alpha _j-1} + B_{kj(z)}C'_{kj(x)}z_{k}^{\alpha _j-1} + (\alpha _j-1)B_{kj(z)}C_{kj(x)}z_{k}^{\alpha _j-2}z'_{kj} \right] , \end{aligned}$$

where $A'_{kj(x)o}$, $A'_{kj(z)o}$, $B'_{kj(x)o}$ and $B'_{kj(z)o}$ ($o=1,2,3$) are the derivatives of $A_{kj(x)}$, $A_{kj(z)}$, $B_{kj(x)}$ and $B_{kj(z)}$ over $\alpha _j$, $\lambda _j$ and $\beta _j$, respectively. $C'_{kj(x)}$ and $z'_{kj}$ are the derivatives of $C_{kj(x)}$ and $z_{kj}$ over $\beta _j$.

B Elements of derivative vectors $D_1$ and $D_2$

The first partial derivative vectors of R(x) and r(x) can be calculated as

$$\begin{aligned}{} & {} \dfrac{\partial R(x)}{\partial \alpha _{j}} = -\lambda _{j}(1-e^{-\lambda _{3-j}y^{\alpha _{3-j}}}) e^{-\lambda _{j}y^{\alpha _{j}}} y^{\alpha _{j}} \log y, \quad \dfrac{\partial R(x)}{\partial \lambda _{j}} = -(1-e^{-\lambda _{3-j}y^{\alpha _{3-j}}}) e^{-\lambda _{j}y^{\alpha _{j}}} y^{\alpha _{j}},\\{} & {} \quad \dfrac{\partial R(x)}{\partial \beta _{j}} = \alpha _j \lambda _j (1-e^{-\lambda _{3-j}z^{\alpha _{3-j}}}) e^{-\lambda _{j}z^{\alpha _{j}}} z^{\alpha _{j}} \frac{x-\tau }{\beta _j^2}, \\{} & {} \quad \dfrac{\partial r(x)}{\partial \alpha _{j}} = \frac{\lambda _j y^{\alpha _j-1} e^{-\lambda _j y^{\alpha _j}} + \alpha _j \lambda _j y^{\alpha _j-1} \ln y e^{-\lambda _j y^{\alpha _j}} \left( 1-e^{-\lambda _j y^{\alpha _j}}-\lambda _j y^{\alpha _j}\right) }{\left( 1-e^{ -\lambda _j y^{\alpha _j}}\right) ^{2}}, \\{} & {} \quad \dfrac{\partial r(x)}{\partial \lambda _{j}} = \frac{\alpha _j y^{\alpha _j-1} e^{-\lambda _j y^{\alpha _j}}\left( 1-e^{-\lambda _j y^{\alpha _j}}-\lambda _j y^{\alpha _j}\right) }{\left( 1-e^{-\lambda _j y^{\alpha _j}}\right) ^{2}}, \\{} & {} \qquad \dfrac{\partial r(x)}{\partial \beta _{j}} = \frac{\alpha _j \lambda _j z^{\alpha _j-1} e^{-\lambda _j z^{\alpha _j}} \left\{ \alpha _j \lambda _j(z-\tau ) z^{\alpha _j-1}-[1+(\alpha _j-1)(z-\tau )]\left( 1-e^{-\lambda _j z^{\alpha _j}}\right) \right\} }{\beta _j^{2} \left( 1-e^{-\lambda _j z^{\alpha _j}}\right) ^{2}}, \end{aligned}$$

where $z=\tau +\frac{x-\tau }{\beta _{j}}$ and $y=\left\{ \begin{array}{ll} x, &{} x < \tau \\ z, &{} x > \tau \end{array}\right.$.