Chien-Yu Peng

Mis-Specification Analysis of Linear Degradation Models

Degradation models are widely used to assess the lifetime information of highly reliable products if there exists quality characteristics whose degradation over time can be related to reliability. The performance of a degradation model depends strongly on the appropriateness of the model describing a products degradation path. In this paper, motivated by laser data, we propose a general linear degradation path in which the unit-to-unit variation of all test units can be considered simultaneously with the time-dependent structure in degradation paths. Based on the proposed degradation model, we first derive an implicit expression of a products lifetime distribution, and its corresponding mean-time-to-failure (MTTF). By using the profile likelihood approach, maximum likelihood estimation of parameters, a products MTTF, and their confidence intervals can be obtained easily. In addition, laser degradation data are used to illustrate the proposed procedure. Furthermore, we also address the effects of model mis-specification on the prediction of the products MTTF. It shows that the effect of the model mis-specification on the predictions of a products MTTF is not critical under the case of large samples. However, when the sample size and the termination time are not large enough, a simulation study shows that these effects are not negligible.

Stochastic Diffusion Modeling of Degradation Data

Accelerated degradation tests (ADTs) can provide timely reliability information of product. Hence ADTs have been widely used to assess the lifetime distribution of highly reliable products. In order to properly predict the lifetime distribution, modeling the products degradation path plays a key role in a degradation analysis. In this paper, we use a stochastic diffusion process to describe the products degradation path and a recursive formula for the products lifetime distribution can be obtained by using the first passage time (FPT) of its degradation path. In addition, two approximate formulas for the products mean-time-to-failure (MTTF) and median life (B50) are given. Finally, we extend the proposed method to the case of ADT and a real LED data is used to illustrate the proposed procedure. The results demonstrate that the proposed method has a good performance for the LED lifetime prediction.

Progressive-Stress Accelerated Degradation Test for Highly-Reliable Products

For highly reliable products with very few test units, a progressive-stress accelerated degradation test (PSADT) has been proposed in the literature to obtain timely information of the products lifetime distribution. The results, however, are restricted to the case where the products degradation path follows a Wiener process (Brownian motion) with a linear drift rate. But in practical applications, the products mean degradation path may be non-linear. Hence, how to address the lifetime distribution in this situation is a worthy topic for reliability analysts. In this paper, a PSADT with a non-linear degradation path is constructed using the cumulative exposure model. Then the products lifetime distribution can be analytically obtained by the first passage time of its degradation path. Furthermore, we derive an exact relationship between the lifetime distributions of the PSADT, and the conventional constant-stress degradation test (CSDT), which allows us to extrapolate the products lifetime distribution under typical stress. Finally, the usage of the proposed model, and the efficiency of PSADT to reduce the products life testing time are demonstrated in the example.

Inverse Gaussian Processes With Random Effects and Explanatory Variables for Degradation Data

Degradation models are widely used to assess the lifetime information of highly reliable products. This study proposes a degradation model based on an inverse normal-gamma mixture of an inverse Gaussian process. This article presents the properties of the lifetime distribution and parameter estimation using the EM-type algorithm, in addition to providing a simple model-checking procedure to assess the validity of different stochastic processes. Several case applications are performed to demonstrate the advantages of the proposed model with random effects and explanatory variables. Technical details, data, and R code are available online as supplementary materials.

Statistical Lifetime Inference With Skew-Wiener Linear Degradation Models

Degradation models are widely used to assess the lifetime information of highly reliable products possessing quality characteristics that both degrade over time and can be related to reliability. The performance of a degradation model largely depends on an appropriate model description of the products degradation path. Conventionally, the random or mixed-effect model is one of the most well-known approaches presented in the literature in which the normal distribution is commonly adopted to represent unit-to-unit variability in the degradation model. However, this assumption may not appropriately signify accurate projections for practical applications. This paper is motivated by laser data wherein the normal distribution is relaxed with a skew-normal distribution that consequently provides greater flexibility as it can capture a broad range, non-normal, asymmetric behavior in unit-to-unit variability. Based on the proposed degradation model, we first derive analytical expressions for a products lifetime distribution along with its corresponding mean-time-to-failure (MTTF). We then utilize the laser data to illustrate advantages gained by the proposed model. Finally, we address effects from the skewness parameter with regard to the accuracy of both a products MTTF and its q th failure quantile; especially when the underlying skew-normal distribution is mis-specified as a normal distribution. The result demonstrates that effects from the skewness parameter on the tail probabilities of a products lifetime distribution are not negligible when the random effect of the true degradation model follows a skew-normal distribution.

On the Choice of Nugget in Kriging Modeling for Deterministic Computer Experiments

Kriging is commonly used for developing emulators as surrogates for computationally intensive simulations. One difficulty with kriging is the potential numerical instability in the computation of the inverse of the covariance matrix, which can lead to large variability and poor performance of the kriging predictor. First, we study some causes of ill-conditioning in kriging. We then study the use of nugget in kriging to overcome the numerical instability. Some asymptotic results on its interpolation bias and mean squared prediction errors are presented. Finally, we study the choice of the nugget parameter based on some algebraic lower bounds and use of a regularizing trace. A simulation study is performed to show the differences between kriging with and without nugget and to demonstrate the advantages of the former. This article has supplementary materials online.

A note on a Wiener process with measurement error

Abstract In this paper we give a closed form for the determinant and the inverse matrix of the covariance matrix of a Wiener process with measurement error. We will discuss its application in the analysis of degradation data for highly-reliable products.

Estimation for Weibull Distribution with Type II Highly Censored Data

Abstract Due to cost and time consideration, it is difficult to observe all of the product’s lifetime within a reasonable time period. Hence, censored lifetime data is usually collected in real applications. Even when accelerated life tests (ALT) are used, censoring is usually inevitable. Especially for highly reliable products nowadays, the censoring proportions are more likely greater than 0.5. Such data is called highly censored data. In such cases, it is not easy to obtain a precise estimation of reliability information that is of interest, even though the maximum likelihood (ML) method is utilized. With respect to the scenario that highly censored data occurs due to time restriction (i.e., cost is not of main concern), a remedy could be to put a great number of devices into testing. This is sometimes called quantity acceleration. The main purpose of the paper is to address this issue. For the whole censored data (including failure times and the running times of unfailed items), traditional methods (including the ML method) have been used to estimate the reliability information of interest. This paper provides an alternative approach based on the observed lifetime data. Specifically, with respect to a type II highly censored data from a Weibull distribution, we treat the failure data as the smallest extreme distribution and then model that extreme using the Peaks-over-Threshold (POT) model to estimate the lifetime quantile of interest. A comparison with the ML method is made to evaluate the effectiveness of the proposed method.

Dry heat forced degradation of buserelin peptide: Kinetics and degradant profiling

Buserelin is a GnRH agonist peptide drug, comprising a nine amino acid sequence (pGlu-His-Trp-Ser-Tyr-D-Ser(tBu)-Leu-Arg-Pro-NH-Et) and most commonly known for its application in hormone dependent cancer therapy, e.g. prostate cancer. In order to evaluate its hot-melt extrusion (HME) capabilities, buserelin powder in its solid state was exposed to elevated temperatures for prolonged time periods. A stability indicating UPLC-PDA method was used for quantification of buserelin and the formed degradants. Different solid state kinetic models were statistically evaluated of which the Ginstling-Brounshtein model fitted the data best. Extrapolation to and experimental verification of typical HME-related conditions, i.e. 5 min at 100°C and 125°C, showed no significant degradation, thus demonstrating the HME capabilities of buserelin. Mass spectrometric identification of the buserelin-related degradants formed under solid state heat stress was performed. Based upon the identity of these degradants, different degradation hypotheses were raised. First, direct β-elimination of the hydroxyl moiety at the serine residue, followed by fragmentation into an amide (pGlu-His-Trp-NH2) and pyruvoyl (pyruvoyl-Tyr-D-Ser(tBu)-Leu-Arg-Pro-NH-Et) peptide fragments, was postulated. Alternatively, internal esterification due to nucleophilic attack of the unprotected serine residue, followed by β-elimination or hydrolysis would yield pGlu-His-Trp, pGlu-His-Trp-NH2 and the pyruvoyl peptide fragment. Degradant pGlu-His-Trp-Ser-Tyr-NH2 is believed to be formed in a similar way. Secondly, direct backbone hydrolysis would yield pGlu-His-Trp and Tyr-D-Ser(tBu)-Leu-Arg-Pro-NH-Et peptide fragments. Moreover, the presence of Ala-Tyr-D-Ser(tBu)-Leu-Arg-Pro-NH-Et can be explained by hydrolysis of the Trp-Ser peptide bond and conversion of the serine moiety to an alanine moiety. Third and finally, isomerisation of aforementioned peptide fragments and buserelin itself was also observed.

Designing a Degradation Test with a Two-Parameter Exponential Lifetime Distribution

Degradation testing (DT) is a useful approach to assessing the reliability of highly reliable products which are not likely to fail under the traditional life tests or accelerated life tests. There have been a great number of excellent studies investigating the estimation of the failure time distribution and the optimal design (e.g., the optimal setting of the inspection frequency, the number of measurement, and the termination time) for DTs. However, the lifetime distributions considered in the studies mentioned above are all those without failure-free life. Here, failure-free life is characterized by a threshold parameter below which no failure is possible. The main purpose of this article is to deal with the optimal design of a DT with a two-parameter exponential lifetime distribution. More specifically, with respect to a DT where a linearized degradation model is used to model the degradation process and the lifetime is assumed to follow a two-parameter exponential distribution, under the constraint that the total experimental cost does not exceed a predetermined budget, the optimal combination of the inspection frequency, the sample size, and the termination time are determined by minimizing the mean squared error of the estimated 100p-th percentile of the lifetime distribution of the product. An example is provided to illustrate the proposed method and the corresponding sensitivity analysis is also discussed.