## 3 ADDT Analysis
### ADDT Model  
Suppose that the ADDT of a return-spring is conducted at two higher stress levels: \(x_{0}\leq x_{1}\leq x_{2}\), where \(x_{0}\) is the normal use stress, that is, \(230\ ^{\circ}\)C in a bi-functional DC motor system. The observed sample degradation path at measurement time \(t\) on the \(j\)th testing unit under stress level \(x_{i}\) is given by  
\[y_{ij}(t)=\mathcal{D}(t;x_{i},\mathbf{\theta})+\epsilon_{ij}(t)\qquad i=1,\ 2,\quad j=1,\dots,n_{i},\quad t=1,\dots,l_{ij}, \tag{1}\]  
where \(\mathcal{D}(t;x_{i},\mathbf{\theta})\) is the mean degradation path at time \(t\) under stress level \(x_{i}\), and \(n_{i}\) is the number of samples measured at the \(i\)th stress level. \(\mathcal{D}(\cdot)\) is assumed to be a monotonic decreasing function of time \(t\) and the parameter vector \(\mathbf{\theta}\); it is also a twice differentiable continuous function. \(\epsilon_{ij}(t)\) is a normal error term describing unit-to-unit variability as \(\mathcal{N}(0,\sigma^{2})\). For temperature-acceleration processes, the Arrhenius model is often used to represent the relationship between the degradation process and temperature as  
\[x_{i}=h(Temp_{i})=-\frac{11,605}{Temp_{i}+273.15}, \tag{2}\]  
Here, \(Temp_{i}\) is in degrees Celsius, and the value 11,605 is the reciprocal of [PERSON]'s constant in units of electron volts per Kelvin (eV/K). The value of 273.15 in the denominator is used to convert to the absolute Kelvin temperature scale. Note that the transformed function \(h(\cdot)\) is of the exponential form for the Arrhenius model, that is, \(h(\cdot)\equiv\exp(\cdot)\). With respect to the ADDT model in Equation (1), the regression based general path models have been widely applied in many applications, e.g., (transformed) linear path model [14] and nonlinear path model [15]. Recently, a semiparametric model has been suggested by [PERSON] et al. [4]. For a given time and accelerated variable level, the cumulative distribution function (CDF) for the degradation \(y\) is  
\[F_{Y}(y;t,x)=\Pr(Y\leq y;t,x)=\Phi\bigg{[}\frac{y-\mathcal{D}(t;x,\mathbf{\theta}) }{\sigma}\bigg{]}, \tag{3}\]  
where \(\Phi(\cdot)\) is the CDF of a standard normal distribution. For the mean degradation path, four popular degradation models are employed; exponential, power, logarithmic, and logistic models. Table 1 lists the formulas of the four models and plots the outlined featuresunder specified values of their parameters with \(\beta_{1}<0\). For example, the exponential ADDT model with the Arrhenius relationship at temperature level \(x_{i}\) is given by  
\[y_{ij}(t)=\beta_{0}\cdot\exp\bigg{\{}\ \beta_{1}\cdot\exp\bigg{(}E_{a}\times \bigg{(}\frac{11.605}{Temp_{u}+273.16}-\frac{11.605}{Temp_{i}+273.16}\bigg{)} \bigg{)}\cdot t\bigg{\}}+e_{ij}(t), \tag{4}\]  
for \(i=1\), \(2\), \(j=1,\ldots,n_{i}\), \(t=1,\ldots,l_{ij}\), and \(\theta\equiv(\beta_{0},\ \beta_{1},\ E_{a})\). Here, \(Temp_{u}\) denotes the normal use temperature level and \(E_{a}\) is the activation energy of the return-spring. Under the parametric specification, the parameters can be estimated by using the maximum likelihood (ML) method or least-squares (LS) approach. In this work, we use the ML method since the ML method is more efficient in the nonlinear degradation model and we can easily quantify uncertainty through asymptotic normality for the ML estimator or its function. To derive the failure-time distribution from the ADDT model in Equation (1), the failure-time \(T\) is defined as the first crossing time that the actual degradation path \(\mathcal{D}(t;x,\theta)\) reaches the pre-determined threshold level \(\mathcal{D}_{f}\).