Estimation and Optimization for Step-Stress Accelerated Degradation Tests Under an Inverse Gaussian Process with Tampered Degradation Model

Abstract

This paper addresses the estimation and optimization problems for a step-stress accelerated degradation test (SSADT) when a product’s degradation follows an inverse Gaussian (IG) process. Developing a link model to connect the deterioration measurements at different stress levels is a primary concern for this design issue. In this paper, we propose a tampered degradation model and construct a new framework to model the effect of changing stress from a level to another on the degradation path. Parameter estimates are obtained by both maximum likelihood and Bayesian methods. Under the constraint that the total experimental cost does not exceed a pre-specified budget, the optimal settings such as sample size, measurement frequency, and the number of measurements at each stress level are obtained. Finally, a real-world example is analyzed to illustrate the application of the proposed method. A notable finding was that enlarging the difference between the two stress levels makes the test more efficient.

Appendix

We know that $t_j=jf$ for $j=1, \dots , n$. Taking this into account, the log-likelihood function with parameter set $\varTheta =(\beta _1, \beta _2,\gamma , q)$ can be rewritten as:

$$\begin{aligned}&l(\varTheta |{\underline{y}})\propto \frac{n\ell _1}{2}\ln \lambda _1 + \frac{n\ell _2}{2}\ln \lambda _2+n\sum _{j=1}^{\ell _1} \ln ( t_j^q-t_{j-1}^q)\\&\quad +n\sum _{j=\ell _1+1}^{\ell _1+\ell _2} \ln \left( (t_j-\tau _1)^q- (t_{j-1}-\tau _1)^q\right) \nonumber \\&\quad - \dfrac{\lambda _1}{2\mu _1^2}y_{1.}-\dfrac{\lambda _1}{2}y_{1r}+\dfrac{n\lambda _1}{\mu _1} \tau _1^q - \dfrac{\lambda _2}{2\mu _2^2}y_{2.}\\&\quad -\dfrac{\lambda _2}{2}y_{2r}+\dfrac{n\lambda _2}{\mu _2} (\tau _2-\tau _1)^q, , \end{aligned}$$

where $y_{1.}$, $y_{2.}$, $y_{1r}$, and $y_{2r}$ are the observed values of the statistics in (6) and $\mu _k, \lambda _k$ for $k=1,2$ are defined in (10) and (11), respectively.

It simply can be driven that:

$$\begin{aligned} \begin{array}{ll} E(Y_{1.})=n \mu _1\tau _1^q,&{}\quad E(Y_{1r})=\dfrac{n}{ \mu _1}\tau _1^q+\dfrac{n\ell _1}{\lambda _1}, \\ E(Y_{2.})=n\mu _2(\tau _2-\tau _1)^q,&{}\quad E(Y_{2r})=\dfrac{n}{ \mu _2}(\tau _2-\tau _1)^q+\dfrac{n\ell _2}{\lambda _2}. \end{array} \end{aligned}$$

Then, the expressions of the elements of the Fisher information $I(\varTheta )$ in (13) are:

$$\begin{aligned}&E\left( -\dfrac{\partial ^2 l}{\partial \beta _1^2}\right) =\dfrac{n\lambda _1}{\mu _1} \tau _1^q +\dfrac{n\lambda _2}{\mu _2}(\tau _2-\tau _1)^q, \\&E\left( - \dfrac{\partial ^2 l}{\partial \beta _2^2}\right) =\dfrac{n}{2}(\ell _1+\ell _2), \\&E\left( -\dfrac{\partial ^2 l}{\partial \gamma ^2}\right) =\dfrac{n}{2}(\ell _1 S_1^2+\ell _2 S_2^2) +\dfrac{nS_1^2\lambda _1}{\mu _1} \tau _1^q +\dfrac{nS_2^2\lambda _2}{\mu _2}(\tau _2-\tau _1)^q, \\&E\left( - \dfrac{\partial ^2 l}{\partial q^2}\right) =2n \sum _{j=1}^{\ell _1}\dfrac{ \left[ t_j^q\ln t_j -t_{j-1}^q\ln t_{j-1} \right] ^2}{\left[ t_j^q-t_{j-1}^q\right] ^2}\\&\qquad +n\dfrac{\lambda _1}{\mu _1} \sum _{j=1}^{\ell _1}\dfrac{ \left[ t_j^q\ln t_j -t_{j-1}^q\ln t_{j-1} \right] ^2}{ t_j^q-t_{j-1}^q}\\&\qquad +2n \sum _{j=\ell _1+1}^{\ell _1+\ell _2}\dfrac{ \left[ (t_j-\tau _1)^q\ln (t_j-\tau _1) -(t_{j-1}-\tau _1)^q\ln (t_{j-1}-\tau _1) \right] ^2}{\left[ (t_j-\tau _1)^q-(t_{j-1}-\tau _1)^q\right] ^2}\\&\qquad +n\dfrac{\lambda _2}{\mu _2} \sum _{j=\ell _1+1}^{\ell _1+\ell _2}\dfrac{ \left[ (t_j-\tau _1)^q\ln (t_j-\tau _1) -(t_{j-1}-\tau _1)^q\ln (t_{j-1}-\tau _1) \right] ^2}{(t_j-\tau _1)^q-(t_{j-1}-\tau _1)^q},\\&E\left( -\dfrac{\partial ^2 l}{\partial \beta _1\partial \beta _2}\right) =E\left( -\dfrac{\partial ^2 l}{\partial \beta _2\partial \beta _1}\right) =0,\\&E\left( -\dfrac{\partial ^2 l}{\partial \beta _1\partial \gamma }\right) =E\left( -\dfrac{\partial ^2 l}{\partial \gamma \partial \beta _1}\right) \\&\quad =\dfrac{nS_1\lambda _1}{\mu _1}\tau _1^q +\dfrac{nS_2\lambda _2}{\mu _2}(\tau _2-\tau _1)^q,\\&E\left( - \dfrac{\partial ^2 l}{\partial \beta _1\partial q}\right) =E\left( -\dfrac{\partial ^2 l}{\partial q\partial \beta _1}\right) \\&\quad =\dfrac{n\lambda _1}{\mu _1} \sum _{j=1}^{\ell _1} \left[ t_j^q \ln t_j -t_{j-1}^q \ln t_{j-1} \right] \\&\qquad +\dfrac{n\lambda _2}{\mu _2} \sum _{j=\ell _1+1}^{\ell _1+\ell _2} \left[ (t_j-\tau _1)^q \ln (t_j-\tau _1)- (t_{j-1}-\tau _1)^q \ln (t_{j-1}-\tau _1)\right] ,\\&E\left( -\dfrac{\partial ^2 l}{\partial \beta _2\partial \gamma }\right) =E\left( -\dfrac{\partial ^2 l}{\partial \gamma \partial \beta _2}\right) =\dfrac{n}{2}(\ell _1 S_1+\ell _2 S_2), \\&E\left( -\dfrac{\partial ^2 l}{\partial \beta _2\partial q}\right) =E\left( -\dfrac{\partial ^2 l}{\partial q\partial \beta _2}\right) = n \sum _{j=1}^{\ell _1}\dfrac{ t_j^q\ln t_j -t_{j-1}^q\ln t_{j-1} }{ t_j^q-t_{j-1}^q}\\&\qquad +n \sum _{j=\ell _1+1}^{\ell _1+\ell _2}\dfrac{ (t_j-\tau _1)^q\ln (t_j-\tau _1) -(t_{j-1}-\tau _1)^q\ln (t_{j-1}-\tau _1) }{ (t_j-\tau _1)^q-(t_{j-1}-\tau _1)^q}, \\&E\left( -\dfrac{\partial ^2 l}{\partial \gamma \partial q}\right) =E\left( -\dfrac{\partial ^2 l}{\partial q\partial \gamma }\right) \\&\quad = n S_1\sum _{j=1}^{\ell _1}\dfrac{ t_j^q\ln t_j -t_{j-1}^q\ln t_{j-1} }{ t_j^q-t_{j-1}^q}\\&\qquad +\dfrac{nS_1\lambda _1}{\mu _1}\sum _{j=1}^{\ell _1} \left[ t_j^q \ln t_j -t_{j-1}^q \ln t_{j-1} \right] \\&\qquad +n S_2 \sum _{j=\ell _1+1}^{\ell _1+\ell _2}\dfrac{ (t_j-\tau _1)^q\ln (t_j-\tau _1) -(t_{j-1}-\tau _1)^q\ln (t_{j-1}-\tau _1) }{ (t_j-\tau _1)^q-(t_{j-1}-\tau _1)^q} \\&\qquad +\dfrac{nS_2\lambda _2}{\mu _2} \sum _{j=\ell _1+1}^{\ell _1+\ell _2} \left[ (t_j-\tau _1)^q \ln (t_j-\tau _1)\right. \\&\qquad -\left. (t_{j-1}-\tau _1)^q \ln (t_{j-1}-\tau _1)\right] .\\ \end{aligned}$$

By considering $\varLambda (t)=t^q$, the p-quantile of the lifetime distribution can be rewritten as:

$$\begin{aligned} {\xi }_p=\left[ \dfrac{\mu _0}{4\lambda _0}\left( z_p+\sqrt{z_p^2+4\omega \lambda _0/\mu _0^2} \right) ^2\right] ^\frac{1}{q}, \end{aligned}$$

where $\mu _0$ and $\lambda _0$ are defined in (10) and (11). Hence, the elements of the vector $C(\varTheta )'$ can be expressed as:

$$\begin{aligned}&\dfrac{\partial \xi _p}{\partial \beta _1}=\dfrac{1}{q}\left( \dfrac{\mu _0}{4\lambda _0}\right) ^\frac{1}{q} \left( z_p+\sqrt{z_p^2+\dfrac{4\omega \lambda _0}{\mu _0^2}}\right) ^{\frac{2}{q}-1}\\&\quad \left[ z_p+\sqrt{z_p^2+\dfrac{4\omega \lambda _0}{\mu _0^2}}-\dfrac{8\omega \lambda _0}{\mu _0^2\sqrt{z_p^2+\dfrac{4\omega \lambda _0}{\mu _0^2}} } \right] ,\\&\dfrac{\partial \xi _p}{\partial \beta _2}=-\dfrac{1}{q}\left( \dfrac{\mu _0}{4\lambda _0}\right) ^\frac{1}{q} \left( z_p+\sqrt{z_p^2+\dfrac{4\omega \lambda _0}{\mu _0^2}}\right) ^{\frac{2}{q}-1} \\&\quad \left[ z_p+\sqrt{z_p^2+\dfrac{4\omega \lambda _0}{\mu _0^2}}-\dfrac{4\omega \lambda _0}{\mu _0^2\sqrt{z_p^2+\dfrac{4\omega \lambda _0}{\mu _0^2}} } \right] ,\\&\dfrac{\partial \xi _p}{\partial \gamma }=S_0 \dfrac{\partial \xi _p}{\partial \beta _1}+S_0 \dfrac{\partial \xi _p}{\partial \beta _2},\\&\dfrac{\partial \xi _p}{\partial q}=-\dfrac{1}{q^2}\left( \dfrac{\mu _0}{4\lambda _0}\right) ^\frac{1}{q} \left( z_p+\sqrt{z_p^2+\dfrac{4\omega \lambda _0}{\mu _0^2}}\right) ^{\frac{2}{q}} \\&\quad \ln \left[ \dfrac{\mu _0}{4\lambda _0}\left( z_p+\sqrt{z_p^2+\dfrac{4\omega \lambda _0}{\mu _0^2}}\right) ^2\right] . \end{aligned}$$