Bayesian accelerated life testing under competing log-location-scale family of causes of failure

Abstract

This article provides Bayesian analyses of data arising from multi-stress accelerated life testing of series systems. The component log-lifetimes are assumed to independently belong to some log-concave location-scale family of distributions. The location parameters are assumed to depend on the stress variables through a linear stress translation function. Bayesian analyses and associated predictive inference of reliability characteristics at usage stresses are performed using Gibbs sampling from the joint posterior. The developed methodology is numerically illustrated by analyzing a real data set through Bayesian model averaging of the two popular cases of Weibull and log-normal, with the later getting a special focus in this article as a slightly easier example of the log-location-scale family. A detailed simulation study is also carried out to compare the performance of various Bayesian point estimators for the log-normal case.

Appendices

Appendix 1: Proof of Theorem 1

It is enough to show that $\log L_j\left( \varvec{\psi }_j|{\mathscr {D}}\right)$ ($=\ell _j\left( \varvec{\psi }_j|{\mathscr {D}}\right)$, say) is a concave function of $\theta _{kj}$ and $\tau _j$. From Eq. (8), one gets

$$ \begin{aligned} \ell _j\left( \varvec{\psi }_j|{\mathscr {D}}\right)= & {} n_j \log \tau _j + \displaystyle \sum _{\begin{array}{c} 1 \le i \le n\\ \& I^i=j \end{array}} \log f_j\left( {\tau _j}\left( t^{i}-{\mathbf s_j^i}'\varvec{\theta }_{j}\right) \right) \\&+ \displaystyle \sum _{\begin{array}{c} 1 \le i \le n\\ \& I^i \ne j \end{array}}\log \overline{F}_j\left( {\tau _{j}}\left( t^i-{\mathbf s_j^i}'\varvec{\theta }_j\right) \right) \\&+ \sum _{i=n+1}^N \log \overline{F}_j\left( {\tau _{j}}\left( t^i-{\mathbf s_j^i}'\varvec{\theta }_j\right) \right) , \end{aligned}$$

which implies

$$ \begin{aligned} \frac{\partial ^{2} \ell _j\left( \varvec{\psi }_j|{\mathscr {D}}\right) }{\partial \theta _{kj}^2}= & {} \displaystyle \sum _{\begin{array}{c} 1 \le i \le n\\ \& I^i=j \end{array}} \left( -\tau _j s_{kj}^i\right) ^2 \,\,\frac{\partial ^{2} \log f_j(u_j^i)}{\partial {u_{j}^i}^2} + \displaystyle \sum _{\begin{array}{c} 1 \le i \le n\\ \& I^i \ne j \end{array}} \left( -\tau _j s_{kj}^i\right) ^2 \,\,\frac{\partial ^{2} \log \overline{F}_j(u_j^i)}{\partial {u_{j}^i}^2}\\&+ \sum _{i=n+1}^N \left( -\tau _j s_{kj}^i\right) ^2 \,\,\frac{\partial ^{2} \log \overline{F}_j(u_j^i)}{\partial {u_{j}^i}^2}, \end{aligned}$$

and

$$ \begin{aligned} \frac{\partial ^{2} \ell _j\left( \varvec{\psi }_j|{\mathscr {D}}\right) }{\partial \tau _j^2}= & {} -\frac{n_j}{\tau _j^2} + \displaystyle \sum _{\begin{array}{c} 1 \le i \le n\\ \& I^i=j \end{array}} \left( \frac{u_j^i}{\tau _j}\right) ^2 \,\,\frac{\partial ^{2} \log f_j(u_j^i)}{\partial {u_{j}^i}^2} + \displaystyle \sum _{\begin{array}{c} 1 \le i \le n\\ \& I^i \ne j \end{array}} \left( \frac{u_j^i}{\tau _j}\right) ^2 \,\,\frac{\partial ^{2} \log \overline{F}_j(u_j^i)}{\partial {u_{j}^i}^2} \\&+ \sum _{i=n+1}^N \left( \frac{u_j^i}{\tau _j}\right) ^2 \,\,\frac{\partial ^{2} \log \overline{F}_j(u_j^i)}{\partial {u_{j}^i}^2}, \end{aligned}$$

where $u_j^i=\tau _j\left( t^i-{\mathbf s_j^i}'\varvec{\theta }_j\right)$. Note that $\frac{\partial ^{2} \log f_j(u_j^i)}{\partial {u_{j}^i}^2}<0$, since $f_j(\cdot )$ is log-concave. Furthermore, the log-concavity of $f_j(\cdot )$ implies the same for $\overline{F}_j(\cdot )$ (see Basu et al. 2003). Thus, $\frac{\partial ^{2} \log \overline{F}_j(u_j^i)}{\partial {u_{j}^i}^2}<0$. Therefore, $\frac{\partial ^{2} \ell _j\left( \varvec{\psi }_j|{\mathscr {D}}\right) }{\partial \theta _{kj}^2}< 0$ and $\frac{\partial ^{2} \ell _j\left( \varvec{\psi }_j|{\mathscr {D}}\right) }{\partial \tau _j^2}<0$, which completes the proof.

Appendix 2: Acronyms and notations

BF:

Bayes’ factor

BMA:

Bayesian Model Averaging

h.r.:

Hazard rate

MCMC:

Markov Chain Monte Carlo

MLE:

Maximum likelihood estimate

MTTF:

Mean time to failure

p.d.f.:

Probability density function

SD:

Standard deviation

SE:

Standard error

s.f.:

Survival function

J, j :

Number of components in series system and dummy in $\{1,\ldots ,J\}$

K, k :

Number of stress variables and and dummy in $\{1,\ldots ,K\}$

N, i :

Number of systems put on an ALT and dummy in $\{1,\ldots ,N (n)\}$

n :

Number of failed systems in an ALT

$X_j$, $Y_j$ :

Lifetime and log-lifetime of the jth component

T, I :

System log-lifetime and cause of failure

${\mathbf Z}=(Z_1,\ldots ,Z_K)'$ :

$K\times 1$ vector of stress variables

U, u :

A variable U and its observed value with or without an i in the super-script (like $U^i$ or $u^i$) for the ith system

$f(\cdot |\cdot )$, $\overline{F}\left( \cdot |\cdot \right)$, $h\left( \cdot |\cdot \right)$ :

p.d.f., s.f., and h.r. of random variables

$g_{kj}(\cdot )$ :

Known transformation of the kth stress variable acting on the jth component of the system

${\mathbf s_j}=(s_{1j},\ldots ,s_{Kj})'\, \text {with}\,\, s_{kj}=g_{kj}(z_k)$ :

$K\times 1$ vector of transformed stress variables for the jth component

$\mathbf S_j={[\mathbf s_j^1,\ldots ,\mathbf s_j^N]}'$ :

$N \times K$ stress matrix corresponding to component j

$\phi (\cdot )$, $\overline{\varPhi }(\cdot )$ :

p.d.f. and s.f. of the standard normal distribution

$\varvec{\theta }_j=(\theta _{1j},\ldots ,\theta _{Kj})'$, $\varvec{\theta }=(\theta _1',\ldots ,\theta _J')'$ :

$K\times 1$ and $KJ\times 1$ vectors of STF parameters for the jth component and system

$\mu _{j}(\varvec{\theta }_{j}, \mathbf {z})$, $\tau _j$, $\sigma _j^2$ :

STF (location) and scale parameters of $Y_j$ and its variance when normal

$\varvec{\psi }_j=(\varvec{\theta }_j',\tau _j)'$, $\varvec{\psi }=(\varvec{\psi }_1',\ldots ,\varvec{\psi }_J')'$ :

$(K+1)\times 1$ and $(KJ+J)\times 1$ vectors of lifetime parameters for the jth component and system

$\triangle$ :

A model-independent quantity

${\mathscr {D}}$, $\mathscr {Y}_j$, ${\mathscr {Y}}$ :

Observed, augmented and complete data

$L_j\left( \varvec{\psi }_j|{\mathscr {D}}\right)$, $L_j(\varvec{\psi }_j|{\mathscr {Y}})$, $L\left( \varvec{\psi }|{\mathscr {D}}\right)$ :

Likelihood function of $\varvec{\psi }_j$ and $\varvec{\psi }$ given ${\mathscr {D}}$ or ${\mathscr {Y}}$

$\pi (\cdot ), \pi (\cdot |{\mathscr {D}}), \pi (\cdot |{\mathscr {Y}})$ :

Prior and posterior of relevant quantities usually clear from the context