Proposal of a Modulated Extended Cumulative Exposure Model for the Step-Up Voltage Test

Abstract

The extended cumulative exposure model (ECEM) includes features of the cumulative exposure model (CEM) and the memoryless model (MM). These often used to express the failure probability model in step-stress accelerated life test (SSALT). The CEM is widely accepted in reliability fields because accumulation of fatigue is considered to be reasonable. The MM is also used in electrical engineering because accumulation of fatigue is not observed in some cases. The ECEM includes features of both models. In this paper, we propose a modulated ECEM model based on the time-scale. We show the estimability of the parameters using simulation study. In addition, we apply the proposed model to a step-stress test result as an experimental case.

Appendix

25.1.1 Parameter Estimation

Figure 25.8 shows a profile log-likelihood parameterized by four parameter in the modulated ECEM. By the figure, the optimum point can be found. In searching for the optimum point of the log-likelihood function, it is necessary to use iterative methods because of its nonlinearity. Optimization schemes without derivatives are many proposed, but the Newton-Raphson method is finally indispensable for problems appeared in this paper. Because, the likelihood function is extraordinary flat.

Fig. 25.8

Profile log-likelihood function for the modulated ECEM; model parameters are $\gamma =0.8, \ p=10, \ k=0.05234677$, and $\beta =1$, the sample size $N=100$, and condition values $\varDelta t=1,\ \varDelta v=0.1$, and $v_{th}= 0$

25.1.2 Likelihood Function

Here, N is the sample size, and i(j) denotes that sample j is broken at level i. To obtain the parameters, we pursue the maximizer of the parameters in the likelihood function,

$$\begin{aligned} L=\prod ^N _{j=1} g(t_{i(j)})^{\delta i(j)} \times \left\{ 1-G(t_T)\right\} ^{1-\delta i(j)}, \end{aligned}$$

(25.26)

when time of breakdown is continuously observed or unobserved by censoring to the right (type II); in the latter case, $t_T$ is the truncation time. When we only observe the number of failures at each stress level, the likelihood function becomes to

$$\begin{aligned} L=\prod ^N _{j=1} \left[ G(t_{i(j)})-G(t_{i(j)-1}) \right] , \end{aligned}$$

(25.27)

which provides the case for grouped data. Furthermore, if the truncation time $t_T(j)$ is provided with r failures, the likelihood function is

$$\begin{aligned} L=\prod ^N _{j=1} \left[ G(t_{i(j)})-G(t_{i(j)-1}) \right] / G(t_T(j)), \end{aligned}$$

(25.28)

In this paper, we deal with data from step-stress test for Eq. (25.27).

25.1.3 The 1st and 2nd Derivatives

We denote Eq. (25.27) as follows,

$$\begin{aligned} L= & {} \prod ^N _{j=1} \left[ G(t_{i(j)})-G(t_{i(j)-1}) \right] =\prod ^N _{j=1} \left[ \exp \{-\varepsilon (t_{i(j)-1})^\beta \}-\exp \{-\varepsilon (t_{i(j)})^\beta \} \right] \nonumber \\= & {} \prod ^N_{j=1} x_j, \end{aligned}$$

(25.29)

then the derivatives of the log-likelihood function are,

(25.30)

where, $\theta _1$ denotes $\gamma$, p, k, and $\beta$. Note that $k=K^{-1/p}$. And,

(25.31)

where, $\varepsilon _{i(j)}=\varepsilon (t_{i(j)})$. The second derivatives are,

(25.32)

where,

(25.33)

Here, we express $\varepsilon _{i(j)}^\beta$ and $\varepsilon _{i(j)-1}^\beta$ as $\varepsilon ^\beta$ for simplicity and its derivatives for parameter $\theta$ as follows;

then, the 1st derivatives are

and the 2nd derivatives are

Now, $\xi _{ij}=k(v_{i-j}-v_{th})$ and Eq. (25.23), then

$$\begin{aligned} \varepsilon (t_i)=\sum _{j=0}^{i-1}\left[ \gamma ^j \xi _{ij}^p\varDelta t_{i-j}\right] .\nonumber \end{aligned}$$

Finally, we can obtain the 1st and 2nd derivatives as follows;

$$\begin{aligned}&\varepsilon _\gamma =\sum _{j=0}^{i-1}\left[ j\gamma ^{j-1} \xi _{ij}^p\varDelta t_{i-j}\right] ,\quad \varepsilon _p=\sum _{j=0}^{i-1}\left[ \gamma ^{j} \xi _{ij}^p (\log \xi _{ij})\varDelta t_{i-j}\right] ,\nonumber \\&\varepsilon _k=\sum _{j=0}^{i-1}\left[ \gamma ^{j} pk^{n-1}\xi _{ij}^p\varDelta t_{i-j}\right] ,\quad \varepsilon _{\gamma \gamma }=\sum _{j=0}^{i-1}\left[ j(j-1)\gamma ^{j-2} \xi _{ij}^p\varDelta t_{i-j}\right] ,\nonumber \\&\varepsilon _{pp}=\sum _{j=0}^{i-1}\left[ \gamma ^{j} \xi _{ij}^p(\log \xi _{ij})^2\varDelta t_{i-j}\right] ,\quad \varepsilon _{kk}=\sum _{j=0}^{i-1}\left[ \gamma ^{j} p(p-1)k^{p-2}\xi _{ij}^p\varDelta t_{i-j}\right] ,\nonumber \\&\varepsilon _{\gamma p}=\sum _{j=0}^{i-1}\left[ j\gamma ^{j-1} \xi _{ij}^p(\log \xi _{ij})\varDelta t_{i-j}\right] ,\quad \varepsilon _{\gamma k}=\sum _{j=0}^{i-1}\left[ j\gamma ^{j-1} pk^{p-1}\xi _{ij}^p\varDelta t_{i-j}\right] ,\nonumber \\&\varepsilon _{pk}=\sum _{j=0}^{i-1}\left[ \gamma ^{j} pk^{n-1}\xi _{ij}^p(\log \xi _{ij})\varDelta t_{i-j} +\gamma ^{j} k^{-1}\xi _{ij}^p \varDelta t_{i-j}\right] . \end{aligned}$$

(25.34)

We can obtain the Jacobian and Hessian metrices from Eqs. (25.7)–(25.34).