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It is assumed that the drift parameter is dependent on the acceleration variables and the diffusion coefficient remains the same across the whole accelerated degradation test (ADT) in most of the literature based on Wiener process. However, the diffusion coefficient variation would also become obvious in some applications with the stress increasing. Aiming at the phenomenon, the paper concludes that both the drift parameter and the diffusion parameter depend on stress variables based on the invariance principle of failure mechanism and Nelson assumption. Accordingly, constant stress accelerated degradation process (CSADP) and step stress accelerated degradation process (SSADP) with random effects are modeled. The unknown parameters in the established model are estimated based on the property of degradation and degradation increment, separately for CASDT and SSADT, by the maximum likelihood estimation approach with measurement error. In addition, the simulation steps of accelerated degradation data are provided and simulated step stress accelerated degradation data is designed to validate the proposed model compared to other models. Finally, a case study of CSADT is conducted to demonstrate the benefits of our model in the practical engineering.

For many highly reliable products, it is not an easy task to obtain their life information by using traditional life test because failures are not likely to occur in a certain period of time, even by censoring life test and accelerated life test. In such a case, degradation data which is related to life is used due to the following reasons: ease of obtaining, low cost, short test period, and informative data. And it has been widely used in classification [

Most of the degradation data mentioned above is degradation data under normal stress or field degradation data. However, the life information should be obtained in a shorter period of time for some products, especially for newly developed products and highly reliable components. Instead, it is a lengthy and drawn-out process to collect field degradation data. Under the circumstances, ADT is a suitable choice to gather the life information quickly and efficiently.

In general, with more accelerated degradation data and higher measuring precision, we can achieve higher accuracy for forecasting parameters, but the experiment cost would increase correspondingly. So we can deal with the optimal accelerated degradation plan (including the optimal settings for the sample size, accelerated stresses, measurement frequency, and termination time) for a Wiener degradation process by minimizing the approximate variance of the estimated mean time to failure under the constraint that the total experimental cost does not exceed a prespecified budget or minimizing the testing cost under the condition of a maximum acceptable approximate standard error. Some well-known references on the optimization of CSADT based on Wiener process are Lim and Yum [

The above literature all assumed that the drift parameter is dependent on the acceleration variables and the diffusion coefficient remains the same across the whole ADT. But when the degradation rate increases, the degradation variation would also become larger in some applications [

The paper is motivated by the latest paper of Wang et al. [

The remainder of this paper is organized as follows: Section

The time-transformed Wiener process is commonly used to model the nonlinear accelerated degradation data [

ADT is a method to accelerate the degradation of products by elevating stress, and the obtained degradation data are then used to extrapolate the information through accelerating model to obtain the estimates of life or performance of products at normal use condition. To ensure the accuracy of the extrapolation, the failure mechanism under the accelerated stress and the normal stress must keep the same which is also the premise of the ADT. One of the most common methods for consistency inspection of the failure mechanism is based on statistical method [

Specify

Then the acceleration factor

The expression

The expression of

The acceleration factor

Three accelerated models and their acceleration factor.

Accelerated models | Drift parameter | Diffusion parameter | Acceleration factor |
---|---|---|---|

Arrhenius model | | | |

Inverse power model | | | |

Eyring model | | | |

The observed degradation for products from the same population may be very different owing to unobservable factors [

Considering the random effects and the effects of accelerated stresses on the drift parameter

When

As the most-used ADT, CSADT and SSADT have been widely researched. But the models are quite different while the drift parameter

Let

Similarly, with CSADT, it was assumed that there are

The degradation process for CSADT is the same as SSADT under accelerated stress

Raise the accelerated stress up to

Similarly, the accelerated stress is turning up to

According to the analysis, the degradation process of SSADT can be formulated as

In real applications, it is inevitable that some measurement errors may be introduced during the observation process [

The unknown parameters in the models are

The increment

According to (P1), the degradation

Even though our models concern the unit-to-unite variability, the essence of the Wiener process remains the same. Owing to space constraints, this paper deals with the unknown parameter based on property (P2) in the case of CSADT and property (P1) in the case of SSADT.

Specify

The log-likelihood function of unknown parameters

Substituting (

The MLE of

The value of

The degradation process of SSADT shown as (

Specify

The foregoing transformation is equivalent to converting the degradation driven by stress

Define

Similarly, set the derivation of

It was assumed that the number of the measurements and the measurement points of each sample are the same for all of the samples under all of the accelerated stress. That is to say, the subscript of

Thus, the restricted MLE for

The first partial derivatives of the log-likelihood function to

For the special value of

The MLE of

The number of the measurements and the measurement points of each sample are different for all of the samples under all of the accelerated stress. In this case, the first partial derivatives of the log-likelihood function to

The MLE of

It is not to say that we can only use degradation for CSADT and increment for SSADT but just make an introduction to both of the two methods in the limited space. In addition, we could verify the results by comparing the estimation calculated by the two methods to avoid computation errors.

In order to validate the model described before and the parameter estimation methods, simulation test was conducted.

The parameters

Set

Generate

Let

Calculate

Set

Extract the degradation based on the predefined measurement time point

Simulate the measured degradation

There are some differences in the simulation process for SSADT compared with CSADT. The simulation process is shown as follows.

Generate

Let

Set

Calculate

Set

Extract the degradation based on the predefined measurement time point

Simulate the measured degradation

We just give the analysis of simulated SSADT data here because we would give a case study of the CSADT later. It was assumed that the accelerated stress is temperature and the simulation test contains 4 stresses which are 50°C, 60°C, 70°C, and 80°C. For simplicity, the transformed time function is set as

The simulation degradation paths of SSADT.

For simplicity, the degradation model for SSADT proposed in this paper is referred to as

The parameters of three degradation models with the SSADT simulated degradation data.

| | | | | | log-LF | AIC | RE | |
---|---|---|---|---|---|---|---|---|---|

Truth value | 5 | 1 | 0.25 | — | −3000 | 0.01 | — | — | — |

| 5.01 | 1.39 | 0.28 | — | −2975.61 | 0.0078 | 1422.28 | −2834.56 | 0.8402 |

| 5.12 | 1.75 | — | | −2980.81 | 0.0129 | 1122.89 | −2235.77 | 5.6271 |

| 6.20 | — | — | | −3051.75 | | 1093.61 | −2177.21 | 7.8049 |

It is assumed that the failure threshold

The comparison of the PDF and CDF of the three models for simulated SSADT data.

The CSADT model with covariates and random effects is verified by the accelerated degradation data of carbon-film resistors whose raw data set is explicitly given in Table C.3 of Meeker and Escobar [

The degradation paths of carbon-film resistors.

Similarly, the proposed degradation model for SSADT in this paper is referred to as

The parameters of three degradation models with the SSADT simulated degradation data.

| | | | | | | Log-LF | AIC | MTTF | |
---|---|---|---|---|---|---|---|---|---|---|

| 8.23 | 2.07 | | — | −4202.82 | | 0.50 | 518.48 | −1024.97 | |

| 14.71 | 4.33 | — | | −4586.56 | | 0.53 | 486.37 | −960.73 | |

| 11.11 | — | — | | −4479.82 | | 0.53 | 468.37 | −926.74 | |

The degradation paths of carbon-film resistors under transformed time scale.

Our model has the largest log-LF and smallest AIC compared with

The PDF and CDF under the standard operating temperature are as shown in Figure

The comparison of the PDF and CDF of the three models for real CSADT data.

In this paper, the degradation models based on nonlinear Wiener process are established for both constant stress accelerated degradation data and step stress accelerated degradation data. Before the establishment, the relationship between the drift parameter and stress variables is derived based on the invariance principle of failure mechanism and Nelson assumption, so is the relationship between the diffusion parameter and stress variables. It is concluded that the ratio of drift parameters under two stresses is a constant which is irrelevant to the testing time and depends only on the two stresses, as long as the ratio of diffusion parameters is equal to the ratio of drift parameters. And the ratio is defined as accelerated factor. Besides, the random effects are also taken into consideration where the drift parameter is assumed to be normally distributed and the diffusion parameter is same for all of the samples under a certain stress. Then the PDF and CDF of the FHT are deduced considering random effects.

Because of the dependency between the diffusion parameter and stress variables, the degradation process is quite different, either for CSADT or for SSADT. The CSADP and SSADP with random effects are modeled. Moreover, the unknown parameters are solved by MLE based on the two properties of Wiener process. At the end of the paper, the simulated data of SSADT and the CSADT data of carbon-film resistors are both analyzed to verify the proposed model. It is concluded that the model has the biggest log-LF and the smallest AIC compared with the two other models.

The innovation of this paper lies in the following: First, the random effects are considered under the new relationship between the diffusion parameter and accelerated stresses. Second, the degradation process was modeled for both CSADT and SSADT. Thirdly, the unknown parameters were estimated based on the two properties of Wiener process and the result of the MLE for

However, we have only considered the random effects of the drift parameter in this paper due to the complexity of the computation. A further research may consider the random effects of the diffusion parameter into the model. At the same time, the study of the paper may provide new ideas for the relativity analysis between the parameters of other stochastic process and stress variables.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was supported by Jiangsu Province Graduate Student Scientific Research Innovation Project of China (Project KYLX15_0330). The help is gratefully acknowledged.