Properties of Lifetime Estimators Based on Warranty Data Consisting only of Failures

Abstract

Nowadays, many consumer durable goods, such as, automobiles, appliances, and photocopiers etc., are sold with manufacturer’s warranty to insure product quality and reliability. Warranty claims contain a large amount of useful information about reliability of the products, such as, failure times, usage, and failure modes etc. For engineering purposes, usage is more relevant, and hence, modeling usage accumulation is of great interest in reliability analysis using warranty data. Such models are needed to manufacturers to evaluate reliability, predict warranty costs, and to assess design modification and customer satisfaction. Usually, warranty data consists of only failure information, and non-failure information is not obtainable which makes the reliability analysis difficult. The sales data is also important for reliability analysis as it contains time-in-service in calendar timescale for each non-failed product during the warranty plan. This chapter discusses maximum likelihood estimation of lifetime parameters using warranty data along with sales data and examines the precision of the estimators by the asymptotic variances obtained from Fisher Information Matrix. The practical consequence of this finding is that the proposed method produces estimators of the lifetime parameters with good precision for large sales amount.

1 Appendix

We know that integration and differentiation of a function are interchangeable provided that they exist and the function is continuous. As $g_t(\tau )\bar F_a(\tau )\bar F_b(\tau )d\tau$ and $\partial g_t(\tau )\bar F_a(\tau )\bar F_b(\tau )d\tau /\partial \theta$, where $\theta =(m_a,\eta_a,m_b, \eta_b ,\mu ,\sigma )'$, are continuous, the integration and differentiation are interchangeable. With this property, by differentiating $\Psi_t=\Psi_t(\theta)$ in (4.2) with respect to the parameter vector $\theta$, the first and second order derivatives, and Gradients and Hessians of log-likelihood function (4.3) can be obtained.

2 Gradients and Hessians of Log-Likelihood (4.3)

For the simplicity of the presentation of formulas, we denote $\frac{\log x_{ti}^{(a)}-\mu -\log t}{\sigma }$ as $z_{ti}^{(a)}$. $z_{tj}^{(b)}$ is defined similarly.

Gradients

$$\begin{aligned}\frac{\partial{\log L}}{\partial{m_a}} = & \frac{{r^{(a)}_t}}{m_a}-{r^{(a)}_t}\log \eta_a+ \sum_{i=1}^{{r^{(a)}_t}}\log {x^{(a)}_{ti}}-\sum_{i=1}^{{r^{(a)}_t}}\left(\frac{{x^{(a)}_{ti}}}{\eta_a}\right)^{m_a}\log\left(\frac{{x^{(a)}_{ti}}}{\eta_a}\right) \\ &-\sum_{j=1}^{{r^{(b)}_t}}\left(\frac{{x^{(b)}_{tj}}}{\eta_a}\right)^{m_a}\log\left(\frac{{x^{(b)}_{tj}}}{\eta_a}\right) + \left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)\frac{\Psi_{m_a}}{\Psi}, \\ \frac{\partial{\log L}}{\partial{m_b}}=&\frac{{r^{(b)}_t}}{m_b}-{r^{(b)}_t}\log \eta_b+ \sum_{j=1}^{{r^{(b)}_t}}\log {x^{(b)}_{tj}}-\sum_{j=1}^{{r^{(b)}_t}}\left(\frac{{x^{(b)}_{tj}}}{\eta_b}\right)^{m_b}\log\left(\frac{{x^{(b)}_{tj}}}{\eta_b}\right) \\ &-\sum_{i=1}^{{r^{(a)}_t}}\left(\frac{{x^{(a)}_{ti}}}{\eta_b}\right)^{m_b}\log\left(\frac{{x^{(a)}_{ti}}}{\eta_b}\right) + \left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)\frac{\Psi_{m_b}}{\Psi},\\[-24pt]\end{aligned}$$

$$\begin{aligned}\frac{\partial{\log L}}{\partial{\eta_a}} = & -\frac{{r^{(a)}_t} m_a}{\eta_a} + \frac{m_a}{\eta_a} \sum_{i=1}^{{r^{(a)}_t}}\left(\frac{{x^{(a)}_{ti}}}{\eta_a}\right)^{m_a} +\left(\frac{m_a}{\eta_a}\right)\sum_{j=1}^{{r^{(b)}_t}}\left(\frac{{x^{(b)}_{tj}}}{\eta_a}\right)^{m_a}\\ &+\left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)\frac{\Psi_{\eta_a} }{\Psi }, \\ \frac{\partial{\log L}}{\partial{\eta_b}}=& -\frac{{r^{(b)}_t} m_b}{\eta_b} + \frac{m_b}{\eta_b} \sum_{i=1}^{{r^{(a)}_t}}\left(\frac{{x^{(a)}_{ti}}}{\eta_b}\right)^{m_b} +\left(\frac{m_b}{\eta_b}\right)\sum_{j=1}^{{r^{(b)}_t}}\left(\frac{{x^{(b)}_{tj}}}{\eta_b}\right)^{m_b}\\ &+\left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)\frac{\Psi_{\eta_b} }{\Psi }, \\ \frac{\partial{\log L}}{\partial{\mu}}=&\sum_{i=1}^{{r^{(a)}_t}}\frac{\xi({z^{(a)}_{ti}})}{\sigma}+\sum_{j=1}^{{r^{(b)}_t}}\frac{\xi({z^{(b)}_{tj}})}{\sigma}+\left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)\frac{\Psi_\mu}{\Psi}, \\ \frac{\partial{\log L}}{\partial{\sigma}}=&\sum_{i=1}^{{r^{(a)}_t}}\frac{\xi\left({z^{(a)}_{ti}}\right){z^{(a)}_{ti}}}{\sigma}+\sum_{j=1}^{{r^{(b)}_t}}\frac{\xi\left({z^{(b)}_{tj}}\right){z^{(b)}_{tj}}}{\sigma}+\left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)\frac{\Psi_\sigma}{\Psi},\end{aligned}$$

where $\Psi_\theta := \frac{\partial \Psi}{\partial \theta}$. Here, we omit the suffix gth in $\Psi, \Psi_{\bullet}$, and $\Psi_{\bullet \bullet}$ throughout the Appendix.

Hessians

$$\begin{aligned}\frac{\partial^2{{\log}L}}{\partial{{m_a}^2}}=& -\frac{{r^{(a)}_t}}{{m_a}^2}-\sum_{i=1}^{{r^{(a)}_t}}\left(\frac{{x^{(a)}_{ti}}}{\eta_a}\right)^{m_a} \log^2 \left(\frac{{x^{(a)}_{ti}}}{\eta_a}\right) -\sum_{j=1}^{{r^{(b)}_t}}\left(\frac{{x^{(b)}_{tj}}}{\eta_a}\right)^{m_a}\log^2\left(\frac{{x^{(b)}_{tj}}}{\eta_a}\right) \\ &+\left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)[(\Psi_{m_a m_a} /\Psi)-(\Psi_{m_a} /\Psi)^2], \\ \frac{\partial^2{{\log}L}}{\partial{{\eta_a}^2}}=& \frac{{r^{(a)}_t} m_a}{{\eta_a}^2}-\frac{m_a(m_a+1)}{{\eta_a}^2} \sum_{i=1}^{{r^{(a)}_t}}\left(\frac{{x^{(a)}_{ti}}}{\eta_a}\right)^{m_a} -\frac{m_a(m_a+1)}{{\eta_a}^2} \sum_{j=1}^{{r^{(b)}_t}}\left(\frac{{x^{(b)}_{tj}}}{\eta_a}\right)^{m_a} \\ & +\left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)[(\Psi_{\eta_a \eta_a} /\Psi)-(\Psi_{\eta_a} /\Psi)^2], \end{aligned}$$

$$\begin{aligned}\frac{\partial^2{{\log}L}}{\partial{{m_b}^2}}=&-\frac{{r^{(b)}_t}}{{m_b}^2}-\sum_{j=1}^{{r^{(b)}_t}}\left(\frac{{x^{(b)}_{tj}}}{\eta_b}\right)^{m_b} \log^2 \left(\frac{{x^{(b)}_{tj}}}{\eta_b}\right) -\sum_{i=1}^{{r^{(a)}_t}}\left(\frac{{x^{(a)}_{ti}}}{\eta_b}\right)^{m_b} \log^2 \left(\frac{{x^{(a)}_{ti}}}{\eta_b}\right) \\ & +\left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)[(\Psi_{m_b m_b} /\Psi)-(\Psi_{m_b} /\Psi)^2], \\ \frac{\partial^2{{\log}L}}{\partial{{\eta_b}^2}}=&\frac{{r^{(b)}_t} m_b}{{\eta_b}^2}-\frac{m_b(m_b+1)}{{\eta_b}^2} \sum_{j=1}^{{r^{(b)}_t}}\left(\frac{{x^{(b)}_{tj}}}{\eta_b}\right)^{m_b} -\frac{m_b(m_b+1)}{{\eta_b}^2} \sum_{i=1}^{{r^{(a)}_t}}\left(\frac{{x^{(a)}_{ti}}}{\eta_b}\right)^{m_b} \\ & +\left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)[(\Psi_{\eta_b \eta_b} /\Psi)-(\Psi_{\eta_b} /\Psi)^2], \\ \frac{\partial^2{{\log}L}}{\partial{\mu^2}} =& -\sum_{i=1}^{{r^{(a)}_t}} \frac{A\left({z^{(a)}_{ti}}\right)}{\sigma^2}-\sum_{j=1}^{{r^{(b)}_t}} \frac{A\left({z^{(b)}_{tj}}\right)}{\sigma^2} +\left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)[(\Psi_{\mu\mu}/\Psi)-{(\Psi_\mu / \Psi)}^2], \\[-24pt]\end{aligned}$$

$$\begin{aligned}\frac{\partial^2{{\log}L}}{\partial{\sigma^2}} =& -\sum_{i=1}^{{r^{(a)}_t}} \frac{C\left({z^{(a)}_{ti}}\right)}{\sigma^2}-\sum_{j=1}^{{r^{(b)}_t}} \frac{C\left({z^{(b)}_{tj}}\right)}{\sigma^2} +\left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)[(\Psi_{\sigma\sigma}/\Psi)-{(\Psi_\sigma / \Psi)}^2], \\ \frac{\partial^2{{\log}L}}{\partial m_a \partial \eta_a} =& -\frac{{r^{(a)}_t}}{\eta_a}+\frac{1}{\eta_a}\sum_{i=1}^{{r^{(a)}_t}}\left(\frac{{x^{(a)}_{ti}}}{\eta_a}\right)^{m_a} +\frac{m_a}{\eta_a}\sum_{i=1}^{{r^{(a)}_t}}\left(\frac{{x^{(a)}_{ti}}}{\eta_a}\right)^{m_a} \log \left(\frac{{x^{(a)}_{ti}}}{\eta_a}\right) \\ &+\frac{1}{\eta_a}\sum_{j=1}^{{r^{(b)}_t}}\left(\frac{{x^{(b)}_{tj}}}{\eta_a}\right)^{m_a} +\frac{m_a}{\eta_a}\sum_{j=1}^{{r^{(b)}_t}}\left(\frac{{x^{(b)}_{tj}}}{\eta_a}\right)^{m_a} \log \left(\frac{{x^{(b)}_{tj}}}{\eta_a}\right) \\ &+\left({N_t} -{r^{(a)}_t}-{r^{(b)}_t}\right)[\Psi_{m_a \eta_a} /\Psi -\Psi_{m_a} \Psi_{\eta_a} / \Psi^2], \\ \frac{\partial^2{{\log}L}}{\partial m_a \partial m_b} =& \left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)[\Psi_{m_a m_b} /\Psi -\Psi_{m_a} \Psi_{m_b} / \Psi^2], \\ \frac{\partial^2{{\log}L}}{\partial m_a \partial \eta_b} =& \left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)[\Psi_{m_a \eta_b} /\Psi -\Psi_{m_a} \Psi_{\eta_b} / \Psi^2], \\ \frac{\partial^2{{\log}L}}{\partial m_a \partial \mu} =& \left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)[\Psi_{m_a \mu} /\Psi -\Psi_{m_a} \Psi_{\mu} / \Psi^2], \\ \frac{\partial^2{{\log}L}}{\partial m_a \partial \sigma} =& \left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)[\Psi_{m_a \sigma} /\Psi -\Psi_{m_a} \Psi_{\sigma} / \Psi^2], \\ \frac{\partial^2{{\log}L}}{\partial \eta_a \partial m_b} =& \left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)[\Psi_{\eta_a m_b} /\Psi -\Psi_{\eta_a} \Psi_{m_b} / \Psi^2], \\ \frac{\partial^2{{\log}L}}{\partial \eta_a \partial \eta_b} =& \left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)[\Psi_{\eta_a \eta_b} /\Psi -\Psi_{\eta_a} \Psi_{\eta_b} / \Psi^2], \\ \frac{\partial^2{{\log}L}}{\partial \eta_a \partial \mu} =& \left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)[\Psi_{\eta_a \mu} /\Psi -\Psi_{\eta_a} \Psi_{\mu} / \Psi^2], \\ \frac{\partial^2{{\log}L}}{\partial \eta_a \partial \sigma} =& \left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)[\Psi_{\eta_a \sigma} /\Psi -\Psi_{\eta_a} \Psi_{\sigma} / \Psi^2], \end{aligned}$$

$$\begin{aligned}\frac{\partial^2{{\log}L}}{\partial m_b \partial \eta_b} =& -\frac{{r^{(b)}_t}}{\eta_b} +\frac{1}{\eta_b}\sum_{j=1}^{{r^{(b)}_t}}\left(\frac{{x^{(b)}_{tj}}}{\eta_b}\right)^{m_b} +\frac{m_b}{\eta_b}\sum_{j=1}^{{r^{(b)}_t}}\left(\frac{{x^{(b)}_{tj}}}{\eta_b}\right)^{m_b} \log \left(\frac{{x^{(b)}_{tj}}}{\eta_b}\right) +\frac{1}{\eta_b}\sum_{i=1}^{{r^{(a)}_t}}\left(\frac{{x^{(a)}_{ti}}}{\eta_b}\right)^{m_b} \\ &+\frac{m_b}{\eta_b}\sum_{i=1}^{{r^{(a)}_t}}\left(\frac{{x^{(a)}_{ti}}}{\eta_b}\right)^{m_b} \log \left(\frac{{x^{(a)}_{ti}}}{\eta_b}\right) +\left({N_t} -{r^{(a)}_t}-{r^{(b)}_t}\right)[\Psi_{m_b \eta_b} /\Psi -\Psi_{m_b} \Psi_{\eta_b} / \Psi^2], \end{aligned}$$

$$\begin{aligned}\frac{\partial^2{{\log}L}}{\partial m_b \partial \mu} =& \left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)[\Psi_{m_b \mu} /\Psi -\Psi_{m_b} \Psi_{\mu} / \Psi^2], \\ \frac{\partial^2{{\log}L}}{\partial m_b \partial \sigma} =& \left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)[\Psi_{m_b \sigma} /\Psi -\Psi_{m_b} \Psi_{\sigma} / \Psi^2], \\ \frac{\partial^2{{\log}L}}{\partial \eta_b \partial \mu} =& \left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)[\Psi_{\eta_b \mu} /\Psi -\Psi_{\eta_b} \Psi_{\mu} / \Psi^2], \\ \frac{\partial^2{{\log}L}}{\partial \eta_b \partial \sigma} =& \left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)[\Psi_{\eta_b \sigma} /\Psi -\Psi_{\eta_b} \Psi_{\sigma} / \Psi^2], \\ \frac{\partial^2{{\log}L}}{\partial \mu \partial \sigma} =& -\sum_{i=1}^{{r^{(a)}_t}} \frac{B\left({z^{(a)}_{ti}}\right)}{\sigma^2}-\sum_{j=1}^{{r^{(b)}_t}} \frac{B\left({z^{(b)}_{tj}}\right)}{\sigma^2} +\left({N_t}-{r^{(a)}_t}-{r^{(b)}_t}\right)[\Psi_{\mu \sigma} /\Psi -\Psi_\mu \Psi_\sigma / \Psi^2], \end{aligned}$$

where

$$\begin{aligned}&\Psi_{\theta_1 \theta_2} := \frac{\partial^{2} \Psi}{\partial \theta_1 \partial \theta_2}\\ & {z^{(a)}_t} := \left(\log x^{(a)}_t - \mu - \log t\right)/\sigma \\ & {z^{(b)}_t} := \left(\log x^{(b)}_t - \mu - \log t\right)/\sigma \\ & \xi (z) := \phi(z)/\bar{\Phi}(z) \\ & A(z) := \xi(z)[\xi(z)-z] \\ & B(z) := \xi(z) +A(z)\cdot z \\ & C(z) := z[\xi(z) + B(z)].\end{aligned}$$

Here, $\phi(\cdot),\bar \Phi(\cdot)$ represent the pdf and survival function of the standard normal distribution.

3 Expectation of the Elements of Hessians

$$\begin{aligned}E \left[\frac{\partial^2{\log L}}{\partial{{m_a}^2}}\right] = & -\frac{{\rho^{(a)}_t}}{{m_a}^2}-\frac{1}{{\eta_a}^{m_a}}(E_{a3}-2\log \eta_a \cdot E_{a2}+\log^2 \eta_a \cdot E_{a1}) \\ & -\frac{1}{{\eta_a}^{m_a}}(E_{b3}-2\log \eta_a \cdot E_{b2}+\log^2 \eta_a \cdot E_{b1})\\ &+\left({N_t}-{\rho^{(a)}_t}-{\rho^{(b)}_t}\right)[(\Psi_{m_a m_a} /\Psi)-(\Psi_{m_a} /\Psi)^2], \end{aligned}$$

$$\begin{aligned}E \left[\frac{\partial^2{\log L}}{\partial{{\eta_a}^2}}\right] = & \frac{{\rho^{(a)}_t} m_a}{{\eta_a}^2}-\frac{m_a(m_a+1)}{{\eta_a}^{m_a+2}}E_{a1} -\frac{m_a(m_a+1)}{{\eta_a}^{m_a+2}}E_{b1} \\ &+ \left({N_t}-{\rho^{(a)}_t}-{\rho^{(b)}_t}\right)[(\Psi_{\eta_a \eta_a} /\Psi)-(\Psi_{\eta_a} /\Psi)^2], \\ E \left[\frac{\partial^2{\log L}}{\partial{{m_b}^2}}\right] = & -\frac{{\rho^{(b)}_t}}{{m_b}^2}-\frac{1}{{\eta_b}^{m_b}}(E_{b3}-2\log \eta_b \cdot E_{b2}+\log^2 \eta_b \cdot E_{b1}) \\ & -\frac{1}{{\eta_b}^{m_b}}(E_{a3}-2\log \eta_b \cdot E_{a2}+\log^2 \eta_b \cdot E_{a1})\\ & +\left({N_t}-{\rho^{(a)}_t}-{\rho^{(b)}_t}\right)[(\Psi_{m_b m_b} /\Psi)-(\Psi_{m_b} /\Psi)^2], \\ E \left[\frac{\partial^2{\log L}}{\partial{{\eta_b}^2}}\right] = &\frac{{\rho^{(b)}_t} m_b}{{\eta_b}^2}-\frac{m_b(m_b+1)}{{\eta_b}^{m_b+2}}E_{b1} -\frac{m_b(m_b+1)}{{\eta_b}^{m_b+2}}E_{a1}\\ & + \left({N_t}-{\rho^{(a)}_t}-{\rho^{(b)}_t}\right)[(\Psi_{\eta_b \eta_b} /\Psi)-(\Psi_{\eta_b} /\Psi)^2], \\ E \left[\frac{\partial^2{\log L}}{\partial{\mu^2}}\right] = & -\frac{E_{a4}}{\sigma^2}-\frac{E_{b4}}{\sigma^2} +\left({N_t}-{\rho^{(a)}_t}-{\rho^{(b)}_t}\right)[(\Psi_{\mu\mu}/\Psi) -{(\Psi_\mu / \Psi)}^2], \\ E \left[\frac{\partial^2{\log L}}{\partial{\sigma^2}}\right] = & -\frac{E_{a6}}{\sigma^2}-\frac{E_{b6}}{\sigma^2} + \left({N_t} -{\rho^{(a)}_t}-{\rho^{(b)}_t}\right)[(\Psi_{\sigma\sigma}/\Psi)-{(\Psi_\sigma / \Psi)}^2], \\ E \left[\frac{\partial^2{\log L}}{\partial m_a \partial m_b}\right] = & \left({N_t}-{\rho^{(a)}_t}-{\rho^{(b)}_t}\right)[\Psi_{m_a m_b} /\Psi -\Psi_{m_a} \Psi_{m_b} / \Psi^2], \\E \left[\frac{\partial^2{\log L}}{\partial m_a \partial \eta_a}\right] = & -\frac{{\rho^{(a)}_t}}{\eta_a}+\frac{1}{{\eta_a}^{m_a+1}}\{m_a E_{a2}+(1-m_a\log \eta_a) \cdot E_{a1}\}\\ &+\frac{1}{{\eta_a}^{m_a+1}}\{m_a E_{b2}+(1-m_a\log \eta_a )\cdot E_{b1}\} \\ & +\left({N_t}-{\rho^{(a)}_t}-{\rho^{(b)}_t}\right)[\Psi_{m_a \eta_a}/\Psi -\Psi_{m_a} \Psi_{\eta_a}/\Psi^2], \\ E \left[\frac{\partial^2{\log L}}{\partial m_a \partial \eta_b}\right] = &\left({N_t}-{\rho^{(a)}_t}-{\rho^{(b)}_t}\right)[\Psi_{m_a \eta_b}/\Psi -\Psi_{m_a} \Phi_{\eta_b}/\Psi^2], \\ E \left[\frac{\partial^2{\log L}}{\partial m_a \partial \mu}\right] = & \left({N_t}-{\rho^{(a)}_t}-{\rho^{(b)}_t}\right)[\Psi_{m_a \mu} /\Psi -\Psi_{m_a} \Psi_\mu / \Psi^2], \\ E \left[\frac{\partial^2{\log L}}{\partial m_a \partial \sigma}\right] = & \left({N_t}-{\rho^{(a)}_t}-{\rho^{(b)}_t}\right)[\Psi_{m_a \sigma} /\Psi -\Psi_{m_a} \Psi_\sigma / \Psi^2], \\ E \left[\frac{\partial^2{\log L}}{\partial \eta_a \partial m_b}\right] = &\left({N_t}-{\rho^{(a)}_t}-{\rho^{(b)}_t}\right)[\Psi_{\eta_a m_b}/\Psi -\Psi_{\eta_a} \Phi_{m_b}/\Psi^2], \\ E \left[\frac{\partial^2{\log L}}{\partial \eta_a \partial \eta_b}\right] = & \left({N_t}-{\rho^{(a)}_t}-{\rho^{(b)}_t}\right)[\Psi_{\eta_a \eta_b} /\Psi -\Psi_{\eta_a} \Psi_{\eta_b} / \Psi^2], \\ E \left[\frac{\partial^2{\log L}}{\partial \eta_a \partial \mu}\right] = & \left({N_t}-{\rho^{(a)}_t}-{\rho^{(b)}_t}\right)[\Psi_{\eta_a \mu} /\Psi -\Psi_{\eta_a} \Psi_\mu / \Psi^2], \\ E \left[\frac{\partial^2{\log L}}{\partial \eta_a \partial \sigma}\right] = & \left({N_t}-{\rho^{(a)}_t}-{\rho^{(b)}_t}\right)[\Psi_{\eta_a \sigma} /\Psi -\Psi_{\eta_a} \Psi_\sigma / \Psi^2], \\ E \left[\frac{\partial^2{\log L}}{\partial m_b \partial \eta_b}\right] = & -\frac{{\rho^{(b)}_t}}{\eta_b}+\frac{1}{{\eta_b}^{m_b+1}}\{m_b E_{b2}+(1-m_b\log \eta_b) \cdot E_{b1}\}\\ &+\frac{1}{{\eta_b}^{m_b+1}}\{m_b E_{a2}+(1-m_b\log \eta_b) \cdot E_{a1}\} \\ & +\left({N_t}-{\rho^{(a)}_t}-{\rho^{(b)}_t}\right)[\Psi_{m_b \eta_b}/\Psi -\Psi_{m_b} \Phi_{\eta_b}/\Psi^2], \end{aligned}$$

$$\begin{aligned}E \left[\frac{\partial^2{\log L}}{\partial m_b \partial \mu}\right] = & \left({N_t}-{\rho^{(a)}_t}-{\rho^{(b)}_t}\right)[\Psi_{m_b \mu} /\Psi -\Psi_{m_b} \Psi_\mu / \Psi^2], \\ E \left[\frac{\partial^2{\log L}}{\partial m_b \partial \sigma}\right] = & \left({N_t}-{\rho^{(a)}_t}-{\rho^{(b)}_t}\right)[\Psi_{m_b \sigma} /\Psi -\Psi_{m_b} \Psi_\sigma / \Psi^2], \\ E \left[\frac{\partial^2{\log L}}{\partial \eta_b \partial \mu}\right] = & \left({N_t}-{\rho^{(a)}_t}-{\rho^{(b)}_t}\right)[\Psi_{\eta_b \mu} /\Psi -\Psi_{\eta_b} \Psi_\mu / \Psi^2], \\ E \left[\frac{\partial^2{\log L}}{\partial \eta_b \partial \sigma}\right] = & \left({N_t}-{\rho^{(a)}_t}-{\rho^{(b)}_t}\right)[\Psi_{\eta_b \sigma} /\Psi -\Psi_{\eta_b} \Psi_\sigma / \Psi^2], \\ E \left[\frac{\partial^2{\log L}}{\partial \mu \partial \sigma}\right] = & -\frac{E_{a5}}{\sigma^2}-\frac{E_{b5}}{\sigma^2} +\left({N_t}-{\rho^{(a)}_t} -{\rho^{(b)}_t}\right)[\Psi_{\mu \sigma} /\Psi -\Psi_\mu \Psi_\sigma / \Psi^2], \end{aligned}$$

where $E_{a\cdot}$ and $E_{b\cdot}$ are defined as follows:

$$\begin{aligned}E_{a1} &= E[{x^{(a)}_t}^{m_a} | {X^{(a)}_{t}} < {Y_t}, {X^{(a)}_{t}} < {x^{(b)}_{t}}]\\ & = {\rho^{(a)}_t} \cdot \int_0^\infty {x}^{m_a} \cdot \frac{f_a\left(x^{(a)}_t\right)\bar F_b\left(x^{(a)}_t\right) \bar{G}\left(x^{(a)}_t\right)}{p^{(a)}_t}dx^{(a)}_t, \\ E_{a2} &= E[{x^{(a)}_t}^{m_a} \log x^{(a)}_t | {X^{(a)}_{t}} < {Y_t},{X^{(a)}_{t}} < {x^{(b)}_{t}}]\\ & = {\rho^{(a)}_t} \cdot \int_0^\infty {x}^{m_a} \log x^{(a)}_t \cdot \frac{f_a\left(x^{(a)}_t\right)\bar F_b\left(x^{(a)}_t\right) \bar{G}\left(x^{(a)}_t\right)}{p^{(a)}_t}dx^{(a)}_t, \\E_{a3} &= E[{x^{(a)}_t}^{m_a} \log^2 x^{(a)}_t | {X^{(a)}_{t}} < {Y_t},{X^{(a)}_{t}} < {x^{(b)}_{t}}]\\ & = {\rho^{(a)}_t} \cdot \int_0^\infty {x}^{m_a} \log^2 x^{(a)}_t \cdot \frac{f_a\left(x^{(a)}_t\right)\bar F_b\left(x^{(a)}_t\right) \bar{G}\left(x^{(a)}_t\right)}{p^{(a)}_t}dx^{(a)}_t, \\ E_{a4} & = E[A({z^{(a)}_t}) | {X^{(a)}_{t}} < {Y_t},{X^{(a)}_{t}} < {x^{(b)}_{t}}]\\ & = {\rho^{(a)}_t} \cdot \int_0^\infty A\left({z^{(a)}_t}\right) \cdot \frac{f_a\left(x^{(a)}_t\right)\bar F_b\left(x^{(a)}_t\right) \bar{G}\left(x^{(a)}_t\right)}{p^{(a)}_t}dx^{(a)}_t, \\ E_{a5}& = E[B({z^{(a)}_t}) | {X^{(a)}_{t}} < {Y_t},{X^{(a)}_{t}} < {x^{(b)}_{t}}]\\ & = {\rho^{(a)}_t} \cdot \int_0^\infty B\left({z^{(a)}_t}\right) \cdot \frac{f_a\left(x^{(a)}_t\right)\bar F_b\left(x^{(a)}_t\right) \bar{G}\left(x^{(a)}_t\right)}{p^{(a)}_t}dx^{(a)}_t, \\ E_{a6}& = E[C({z^{(a)}_t}) | {X^{(a)}_{t}} < {Y_t},{X^{(a)}_{t}} < {x^{(b)}_{t}}]\\ & = {\rho^{(a)}_t} \cdot \int_0^\infty C\left({z^{(a)}_t}\right) \cdot \frac{f_a\left(x^{(a)}_t\right)\bar F_b\left(x^{(a)}_t\right) \bar{G}\left(x^{(a)}_t\right)}{p^{(a)}_t}dx^{(a)}_t, \\E_{b1}& = E[{x^{(b)}_t}^{m_b} | {x^{(b)}_{t}} < {Y_t}, {x^{(b)}_{t}} < {X^{(a)}_{t}}]\\ & = {\rho^{(b)}_t} \cdot \int_0^\infty {x^{(b)}_t}^{m_b} \cdot \frac{f_b\left(x^{(b)}_t\right)\bar F_a\left(x^{(b)}_t\right) \bar{G}\left(x^{(b)}_t\right)}{p^{(b)}_t}dx^{(b)}_t, \end{aligned}$$

$$\begin{aligned}E_{b2} &= E[{x^{(b)}_t}^{m_b} \log x^{(b)}_t | {x^{(b)}_{t}} < {Y_t},{x^{(b)}_{t}} < {X^{(a)}_{t}}]\\ & = {\rho^{(b)}_t} \cdot \int_0^\infty {x^{(b)}_t}^{m_b} \log x^{(b)}_t \cdot \frac{f_b\left(x^{(b)}_t\right)\bar F_a\left(x^{(b)}_t\right) \bar{G}\left(x^{(b)}_t\right)}{p^{(b)}_t}dx^{(b)}_t, \\ E_{b3}& = E[{x^{(b)}_t}^{m_b} \log^2 x^{(b)}_t | {x^{(b)}_{t}} < {Y_t},{x^{(b)}_{t}} < {X^{(a)}_{t}}] \\ &= {\rho^{(b)}_t} \cdot \int_0^\infty {x^{(b)}_t}^{m_b} \log^2 x^{(b)}_t \cdot \frac{f_b\left(x^{(b)}_t\right)\bar F_a\left(x^{(b)}_t\right) \bar{G}\left(x^{(b)}_t\right)}{p^{(b)}_t}dx^{(b)}_t, \\ E_{b4} &= E[A({z^{(b)}_t}) | {x^{(b)}_{t}} < {Y_t},{x^{(b)}_{t}} < {X^{(a)}_{t}}]\\ & = {\rho^{(b)}_t} \cdot \int_0^\infty A\left({z^{(b)}_t}\right) \cdot \frac{f_b\left(x^{(b)}_t\right)\bar F_a\left(x^{(b)}_t\right) \bar{G}\left(x^{(b)}_t\right)}{p^{(b)}_t}dx^{(b)}_t, \\ E_{b5}& = E[B({z^{(b)}_t}) | {x^{(b)}_{t}} < {Y_t},{x^{(b)}_{t}} < {X^{(a)}_{t}}]\\ & = {\rho^{(b)}_t} \cdot \int_0^\infty B\left({z^{(b)}_t}\right) \cdot \frac{f_b\left(x^{(b)}_t\right)\bar F_a\left(x^{(b)}_t\right) \bar{G}\left(x^{(b)}_t\right)}{p^{(b)}_t}dx^{(b)}_t, \\ E_{b6} &= E[C({z^{(b)}_t}) | {x^{(b)}_{t}} < {Y_t},{x^{(b)}_{t}} < {X^{(a)}_{t}}] = {\rho^{(b)}_t} \cdot \int_0^\infty C\left({z^{(b)}_t}\right) \cdot \\ &\frac{f_b\left(x^{(b)}_t\right)\bar F_a\left(x^{(b)}_t\right) \bar{G}\left(x^{(b)}_t\right)}{p^{(b)}_t}dx^{(b)}_t, \end{aligned}$$

and

$$\begin{aligned}{\rho^{(a)}_t} := & p^{(a)}_t \cdot {N_t}, \\ {\rho^{(b)}_t} := & p^{(b)}_t \cdot {N_t},\end{aligned}$$

where $p^{(a)}_t$ and $p^{(b)}_t$ represent probabilities of failure modes (a) and (b), respectively;

$$\begin{aligned}p^{(a)}_t := & \int_0^\infty f_a(x)\bar F_b(x)\bar{G_t}(x)dx, \\ p^{(b)}_t := & \int_0^\infty f_b(x)\bar F_a(x)\bar{G_t}(x)dx, \end{aligned}$$

4 Gauss–Hermite Quadrature

The numerical integration, also called quadrature, is the study of how the numerical value of an integral can be found. The basic idea for the Gaussian quadrature rule is

$$\int_a^b w(v) dv = \int_a^b p(v)h(v)dv \cong \sum_{j=1}^q h(v_j),$$

(4.8)

where p(v) is a weight function and q is a Hermite integration order. Only h(v) needs to be a polynomial or close to polynomial, so that the weight function can be singular. The weights and nodes of the rule depend on the particular choice of the weight function. If the weight function, $p(v) = \exp(-v^2)$, $-\infty < v < \infty$, (4.8) is called Gauss–Hermite quadrature. By definition the Gauss–Hermite representation of $\Psi_t(\theta)$ in (4.2) can be given by

$$\Psi_t(\theta) \cong \sum_{j=1}^q \frac{p_j}{\sqrt{\pi}} e^{-{\eta_a}^{-m_a} t^{m_a} e^{m_a(\mu+\sqrt{2} \sigma v_j)}-{\eta_b}^{-m_b} t^{m_b} e^{m_b(\mu+\sqrt{2} \sigma v_j)}}.$$

(4.9)

The weights, p _j, and abscissas, v _j, are tabulated in Table 25.10 of Abramowitz and Stegun [1] for $q \leq 20$. In this study we use $q = 25$ for better accuracy. For this purpose, we use a program to generate p _j, and v _j.