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Estimation of a log-linear model for the reliability assessment of products under two stress variables

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Abstract

In this article, different models for reliability inference of devices affected by more than one accelerating variable in accelerated life tests are presented. General log-linear relationship is modeled with the lognormal and Weibull distributions considering the effect of two accelerating variables. Estimation of the parameters is performed via maximum likelihood estimation using the Newton–Raphson algorithm and through a Bayesian approach defining conjugate prior and initial non-informative distributions. In order to illustrate these models, an example is presented based on an accelerated life test applied to resistances. Obtained results show that although there are slight differences in the estimates of the parameters based on the two models and approaches, it can be noted that they have an important impact in the reliability inference. The best model and estimation approach is selected via information criteria. In addition, reliability information is obtained from the device under study.

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Authors and Affiliations

  1. Department of Industrial Engineering and Manufacturing, Institute of Engineering and Technology, Autonomous University of Ciudad Juárez, Av. del Charro 450 north, 32310, Ciudad Juárez, Chihuahua, México

    Luis Alberto Rodríguez-Picón

  2. Division of Post Graduate and Research Studies, Technological Institute of Ciudad Juárez, Av. Tecnológico No. 1340, C.P. 32500, Ciudad Juárez, Chihuahua, México

    Víctor Hugo Flores-Ochoa

Authors
  1. Luis Alberto Rodríguez-Picón
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  2. Víctor Hugo Flores-Ochoa
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Correspondence to Luis Alberto Rodríguez-Picón.

Appendices

Appendix 1: Hessian and gradient for lognormal-GLL

Gradient of \(l\left( \theta \right)\) is defined as follows

$$l^{\prime}\left( \theta \right) = \frac{\partial l\left( \theta \right)}{{\partial \theta_{i} }}, \quad 1 \le i \le 4$$
$$\frac{\partial l\left( \theta \right)}{{\partial \alpha_{0} }} = s\mathop \sum \limits_{i = 1}^{n} \left( {\ln \left( {t_{i} } \right) - \alpha_{0} - \alpha_{1} \frac{1}{T} - \alpha_{2} \ln \left( V \right)} \right)$$
$$\frac{\partial l\left( \theta \right)}{{\partial \alpha_{1} }} = \frac{s}{T}\mathop \sum \limits_{i = 1}^{n} \left( {\ln \left( {t_{i} } \right) - \alpha_{0} - \alpha_{1} \frac{1}{T} - \alpha_{2} \ln \left( V \right)} \right)$$
$$\frac{\partial l\left( \theta \right)}{{\partial \alpha_{2} }} = \ln \left( V \right)s\mathop \sum \limits_{i = 1}^{n} \left( {\ln \left( {t_{i} } \right) - \alpha_{0} - \alpha_{1} \frac{1}{T} - \alpha_{2} \ln \left( V \right)} \right)$$
$$\frac{\partial l\left( \theta \right)}{\partial s} = \frac{n}{2s} - \frac{1}{2}\mathop \sum \limits_{i = 1}^{n} \left( {\ln \left( {t_{i} } \right) - \alpha_{0} - \alpha_{1} \frac{1}{T} - \alpha_{2} \ln \left( V \right)} \right)^{2}$$
$$l^{\prime}\left( \theta \right) = \left[ {\begin{array}{*{20}c} {s\mathop \sum \limits_{i = 1}^{n} \left( {\ln \left( {t_{i} } \right) - \alpha_{0} - \alpha_{1} \frac{1}{T} - \alpha_{2} \ln \left( V \right)} \right)} \\ {\frac{s}{T}\mathop \sum \limits_{i = 1}^{n} \left( {\ln \left( {t_{i} } \right) - \alpha_{0} - \alpha_{1} \frac{1}{T} - \alpha_{2} \ln \left( V \right)} \right)} \\ {\begin{array}{*{20}c} {\ln \left( V \right)s\mathop \sum \limits_{i = 1}^{n} \left( {\ln \left( {t_{i} } \right) - \alpha_{0} - \alpha_{1} \frac{1}{T} - \alpha_{2} \ln \left( V \right)} \right)} \\ {\frac{n}{2s} - \frac{1}{2}\mathop \sum \limits_{i = 1}^{n} \left( {\ln \left( {t_{i} } \right) - \alpha_{0} - \alpha_{1} \frac{1}{T} - \alpha_{2} \ln \left( V \right)} \right)^{2} } \\ \end{array} } \\ \end{array} } \right]$$

Transpose of this vector is defined as follows,

$$\left[ {l^{'} \left( \theta \right)} \right]^{T} = \left[ {\begin{array}{*{20}c} {s\gamma } & {\frac{s}{T}\gamma } & {\begin{array}{*{20}c} {\ln \left( V \right)s\gamma } & {\frac{n}{2s} - \frac{1}{2}\mathop \sum \limits_{i = 1}^{n} \gamma^{2} } \\ \end{array} } \\ \end{array} } \right]$$

where,

$$\gamma = \mathop \sum \limits_{i = 1}^{n} \left( {\ln \left( {t_{i} } \right) - \alpha_{0} - \alpha_{1} \frac{1}{T} - \alpha_{2} \ln \left( V \right)} \right).$$

The Hessian of \(l\left( \theta \right)\) is defined as,

$$H\left( \theta \right) = \frac{{\partial^{2} l\left( \theta \right)}}{{\partial \theta_{i} \partial \theta_{j} }}, \quad 1 \le i, \, j \le 4$$

Thus, the construction of the Hessian is a \(4 \times 4\) matrix,

$$H\left( \theta \right) = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{0}^{2} }}} & {\frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{0} \partial \alpha_{1} }}} & {\frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{0} \partial \alpha_{2} }}} & {\frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{0} \partial s}}} \\ {\frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{1} \partial \alpha_{0} }}} & {\frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{1}^{2} }}} & {\frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{1} \partial \alpha_{2} }}} & {\frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{1} \partial s}}} \\ {\frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{2} \partial \alpha_{0} }}} & {\frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{2} \partial \alpha_{1} }}} & {\frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{2}^{2} }}} & {\frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{2} \partial s}}} \\ {\frac{{\partial^{2} l\left( \theta \right)}}{{\partial s\partial \alpha_{0} }}} & {\frac{{\partial^{2} l\left( \theta \right)}}{{\partial s\partial \alpha_{1} }}} & {\frac{{\partial^{2} l\left( \theta \right)}}{{\partial s\partial \alpha_{2} }}} & {\frac{{\partial^{2} l\left( \theta \right)}}{{\partial s^{2} }}} \\ \end{array} } \right]$$
$$\frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{0}^{2} }} = - ns$$
$$\frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{0} \partial \alpha_{1} }} = \frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{1} \partial \alpha_{0} }} = - \frac{ns}{T}$$
$$\frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{0} \partial \alpha_{2} }} = \frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{2} \partial \alpha_{0} }} = - \ln \left( V \right)ns$$
$$\frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{0} \partial s}} = \frac{{\partial^{2} l\left( \theta \right)}}{{\partial s\partial \alpha_{0} }} = \mathop \sum \limits_{i = 1}^{n} \left( {\ln \left( {t_{i} } \right) - \alpha_{0} - \alpha_{1} \frac{1}{T} - \alpha_{2} \ln \left( V \right)} \right)$$
$$\frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{1}^{2} }} = - \frac{ns}{{T^{2} }}$$
$$\frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{1} \partial \alpha_{2} }} = \frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{2} \partial \alpha_{1} }} = - \frac{\ln \left( V \right)ns}{T}$$
$$\frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{1} \partial s}} = \frac{{\partial^{2} l\left( \theta \right)}}{{\partial s\partial \alpha_{1} }} = \frac{1}{T}\mathop \sum \limits_{i = 1}^{n} \left( {\ln \left( {t_{i} } \right) - \alpha_{0} - \alpha_{1} \frac{1}{T} - \alpha_{2} \ln \left( V \right)} \right)$$
$$\frac{{\partial^{2} l\left( \theta \right)}}{{\partial \alpha_{2}^{2} }} = - \left( {\ln \left( V \right)} \right)^{2} ns$$
$$\frac{{\partial ^{2} l\left( \theta \right)}}{{\partial \alpha _{2} \partial s}} = \frac{{\partial ^{2} l\left( \theta \right)}}{{\partial s\partial \alpha _{2} }} = \ln \left( V \right)\mathop \sum \limits_{{i = 1}}^{n} \left( {\ln \left( {t_{i} } \right) - \alpha _{0} - \alpha _{1} \frac{1}{T} - \alpha _{2} \ln \left( V \right)} \right)$$
$$\frac{{\partial^{2} l\left( \theta \right)}}{{\partial s^{2} }} = - \frac{n}{{2s^{2} }}$$
$$H\left( \theta \right) = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - ns} \\ {\begin{array}{*{20}c} { - ns/T} \\ { - \ln \left( V \right)ns} \\ \gamma \\ \end{array} } \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - ns/T} \\ {\begin{array}{*{20}c} { - ns/T^{2} } \\ { - \ln \left( V \right)ns/T} \\ {\gamma /T} \\ \end{array} } \\ \end{array} } & {\begin{array}{*{20}c} { - \ln \left( V \right)ns} \\ {\begin{array}{*{20}c} { - \ln \left( V \right)ns/T} \\ { - \left( {\ln \left( V \right)} \right)^{2} ns} \\ {\ln \left( V \right)\gamma } \\ \end{array} } \\ \end{array} } & {\begin{array}{*{20}c} \gamma \\ {\begin{array}{*{20}c} {\gamma /T} \\ {\ln \left( V \right)\gamma } \\ { - \frac{n}{{2s^{2} }}} \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right]$$

The Fisher information matrix \(\left[ {I\left( \theta \right)} \right]\) evaluated at the MLE contain the entries,

$$I\left( \theta \right) = - \frac{{\partial^{2} l\left( \theta \right)}}{{\partial \theta_{i} \partial \theta_{j} }}, \quad 1 \le i, j \le 4$$

It is easy to see that next equality holds,

$$I\left( \theta \right) = - H\left( \theta \right)$$

Thus, we have,

$$I\left( \theta \right) = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {ns} \\ {\begin{array}{*{20}c} {ns/T} \\ {\ln \left( V \right)ns} \\ { - \gamma } \\ \end{array} } \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {ns/T} \\ {\begin{array}{*{20}c} {ns/T^{2} } \\ {\ln \left( V \right)ns/T} \\ { - \gamma /T} \\ \end{array} } \\ \end{array} } & {\begin{array}{*{20}c} {\ln \left( V \right)ns} \\ {\begin{array}{*{20}c} {\ln \left( V \right)ns/T} \\ {\left( {\ln \left( V \right)} \right)^{2} ns} \\ { - \ln \left( V \right)\gamma } \\ \end{array} } \\ \end{array} } & {\begin{array}{*{20}c} { - \gamma } \\ {\begin{array}{*{20}c} { - \gamma /T} \\ { - \ln \left( V \right)\gamma } \\ {\frac{n}{{2s^{2} }}} \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right]$$

Furthermore, the inverse of the Fisher information is an estimator of the asymptotic variance and covariance matrix,

$$Var\left( {\hat{\theta }} \right) = \left[ {I\left( \theta \right)} \right]^{ - 1}$$

Appendix 2: R code for Newton–Raphson process

figure afigure afigure a

Appendix 3: OpenBUGS code for lognormal-GLL and Weibull-GLL Bayesian estimation approach

figure bfigure b

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Rodríguez-Picón, L.A., Flores-Ochoa, V.H. Estimation of a log-linear model for the reliability assessment of products under two stress variables. Int J Syst Assur Eng Manag 8 (Suppl 2), 1026–1040 (2017). https://doi.org/10.1007/s13198-016-0564-6

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