Chanseok Park

Stochastic degradation models with several accelerating variables

Many products & systems age, wear, or degrade over time before they fail or break down. Thus, in many engineering reliability experiments, measures of degradation or wear toward failure can often be observed over a period of time before failure occurs. Because the degradation values provide additional information beyond that provided by the failure observations, both sets of observations need to be considered when doing inference on the statistical parameters of the product or system lifetime distributions. For highly-reliable modern products, it often takes much more time to obtain lifetime & degradation data under usual use conditions, and this requires one to use accelerated tests. Accelerated tests expose the products to greater environmental stress levels so that we can obtain lifetime & degradation measurements in a more timely fashion. In addition, many products are exposed to several environmental variables in some manufacturing processes, or under some operating conditions. This motivates the need for developing general accelerated test models with several accelerating variables for inference based on both observed failure values, and degradation measurements. In this paper, new accelerated test models are developed based on a generalized cumulative damage approach with a stochastic process characterizing a degradation phenomenon.

New cumulative damage models for failure using stochastic processes as initial damage

Based on a generalized cumulative damage approach with a stochastic process describing initial damage for a material specimen, a broad class of statistical models for material strength is developed. Plausible choices of stochastic processes for the initial damage include Brownian motion, geometric Brownian motion, and the gamma process; and additive & multiplicative cumulative damage functions are considered. The resulting general statistical model gives an accelerated test form of the inverse Gaussian distribution, special cases of which include some existing models in addition to several new models. Model parameterizations & estimation by maximum likelihood from accelerated test data are discussed, and the applicability of the general model is illustrated for three sets of strength data. The proposed models are compared with the power-law Weibull model, and the inverse Gaussian generalized linear models.

Robust design modeling and optimization with unbalanced data

The usual assumption behind robust design is that the number of replicates at each design point during an experimental stage is equal. In practice, however, it is often the case that this assumption is not met due to physical limitations and/or cost constraints. In this situation, using the usual method of ordinary least squares (OLS) to obtain fitted response functions for the mean and variance of the quality characteristic of interest may not be an effective tool. In this paper, we first show simulation results, indicating that an alternative method, called the method of weighted least squares (WLS), outperforms the OLS method in terms of mean squared error. We then lay out the WLS-based robust design modeling and optimization. A case study is presented for numerical purposes.

Parameter estimation of incomplete data in competing risks using the EM algorithm

Consider a system which is made up of multiple components connected in a series. In this case, the failure of the whole system is caused by the earliest failure of any of the components, which is commonly referred to as competing risks. In certain situations, it is observed that the determination of the cause of failure may be expensive, or may be very difficult to observe due to the lack of appropriate diagnostics. Therefore, it might be the case that the failure time is observed, but its corresponding cause of failure is not fully investigated. This is known as masking. Moreover, this competing risks problem is further complicated due to possible censoring. In practice, censoring is very common because of time and cost considerations on experiments. In this paper, we deal with parameter estimation of the incomplete lifetime data in competing risks using the EM algorithm, where incompleteness arises due to censoring and masking. Several studies have been carried out, but parameter estimation for incomplete data has mainly focused on exponential models. We provide the general likelihood method, and the parameter estimation of a variety of models including exponential, s-normal, and lognormal models. This method can be easily implemented to find the MLE of other models. Exponential and lognormal examples are illustrated with parameter estimation, and a graphical technique for checking model validity.

A bootstrap control chart for Birnbaum–Saunders percentiles

The problem of detecting a shift in the percentile of a Birnbaum–Saunders population in a process monitoring situation is considered. For example, such problems may arise when the quality characteristic of interest is tensile strength or breaking stress. The parametric bootstrap method is used to develop a quality control chart for monitoring percentiles when process measurements have a Birnbaum–Saunders distribution. Through extensive Monte Carlo simulations, we investigate the behavior and performance of the proposed bootstrap percentile charts. Average run lengths of the proposed percentile chart are also investigated. Illustrative examples with the data concerning the tensile strength of the aluminum sheeting are presented. Copyright

Parameter estimation for the reliability of load-sharing systems

Consider a multi-component system connected in parallel. In this system, as components fail one by one, the total load or traffic applied to the system is redistributed among the remaining surviving components, which is commonly referred to as load-sharing. This develops parameter estimation methods for these type of systems. A closed-form Maximum Likelihood Estimator (MLE) and Best Unbiased Estimator (BUE) are provided under a general load-sharing rule when the underlying lifetime distribution of the components in the system is exponential. As an extension, it is assumed that the underlying lifetime distribution of the components is Weibull and it is shown that, after the shape parameter is estimated by solving the one-dimensional log-likelihood estimating equation, the closed-form MLE and conditional BUE of the rate parameter are easily obtained. The asymptotic distribution of the proposed MLE is also provided. Illustrative examples and Monte Carlo simulation results are also presented and these substantiate the proposed methods.

Development of Robust Design Under Contaminated and Non-normal Data

The usual assumptions behind robust design are that the distribution of experimental data is approximately normal and that there is no major contamination due to outliers in the data. Under these assumptions, sample mean and variance are often used to estimate process mean and variance. In this article, we first show simulation results indicating that sample mean and variance may not be the best choice when one or both assumptions are not met. The results further show that sample median and median absolute deviation or sample median and interquartile range are indeed more resistant to departures from normality and to contaminated data. We then show how to incorporate this observation into robust design modeling and optimization. A case study is presented.

Development of a highly efficient and resistant robust design

Robust design uses the ordinary least squares method to obtain adequate response functions for the process mean and variance by assuming that experimental data are normally distributed and that there is no major contamination in the data set. Under these assumptions, the sample mean and variance are often used to estimate the process mean and variance. In practice, the above assumptions are not always satisfied. When these assumptions are violated, one can alternatively use the sample median and median absolute deviation to estimate the process mean and variance. However, the median and median absolute deviation both suffer from a lack of efficiency under the normal distribution, although they are fairly outlier-resistant. To remedy this problem, we propose new robust design methods based on a highly efficient and outlier-resistant estimator. Numerical studies substantiate the new methods developed and compare the performance of the proposed methods with the ordinary dual-response robust design.

Parameter estimation from load-sharing system data using the expectation–maximization algorithm

This article considers a system of multiple components connected in parallel. As components fail one by one, the remaining working components share the total load applied to the system. This is commonly referred to as load sharing in the reliability engineering literature. This article considers the traditional approach to the modeling of a load-sharing system under the assumption of the existence of underlying hypothetical latent random variables. Using the Expectation–Maximization (EM) algorithm, a methodology is proposed to obtain the maximum likelihood estimates in such a model in the case where the underlying lifetime distribution of the components is lognormal or normal. The proposed EM method is also illustrated and substantiated using numerical examples. The estimates obtained using the EM algorithm are compared with those obtained using the Broyden–Fletcher–Goldfarb–Shanno algorithm, which falls under the class of numerical methods known as Newton or quasi-Newton methods. The results show that the estimates obtained using the proposed EM method always converge to a unique global maximizer, whereas the estimates obtained using the Newton-type method are highly sensitive to the choice of starting values and thus often fail to converge.

Parametric inference of incomplete data with competing risks among several groups

We develop parametric inferential methods for the competing risks problem where data arise due to multiple causes of failure in several groups with censoring and possibly missing causes. We provide the general likelihood method and the closed-form maximum-likelihood estimators for the exponential model. Parametric tests are given for comparing different causes and groups. An extensive numerical and graphical investigation is presented to substantiate the proposed methods. A real-data example is illustrated.