The performance of units in the same batch can exhibit considerable heterogeneity due to the variation in the raw materials and fluctuation in the manufacturing process. For products suffering performance degradation in their use, such heterogeneity often results in an increase in the dispersion of the degradation paths of units in a population. The degradation rate of products can be unit-specific and often treated as random effects. This paper develops a novel random-effects Wiener process model to account for the unit-to-unit heterogeneity in the degradation, where the generalized inverse Gaussian (GIG) distribution is used to model the unit-specific degradation rate. The GIG distribution is a very general distribution with broad applications, which includes the inverse Gaussian (IG) distribution and the Gamma distribution as special cases. We investigate the model properties and develop an expectation maximization (EM) algorithm for parameter estimation. By comparing the proposed model with existing models on two real degradation datasets of the infrared LEDs and the GaAs lasers, we show that the proposed model is quite effective for degradation modeling with heterogeneous rates.

Many large organizations have developed ambitious programs to build reliability databases by collecting field failure data from a large variety of components. To make the database concise, only the number of component replacements in a component position during an operation time interval is reported in these databases. This leads to time-censoring in the aggregate failure data. Statistical inference for the time-censored aggregate data is challenging, because the likelihood function based on some common lifetime distributions can be intractable. In this study, we propose a general parametric estimation framework for the aggregate data. We first use the gamma distribution and the Inverse Gaussian (IG) distribution to model the aggregate data. Bayesian inference for the two models is discussed. Unlike the gamma/IG distribution, other lifetime distributions involve multiple integrals in the likelihood function, making the standard Bayesian inference difficult. To address the estimation problem, an approximate Bayesian computation algorithm that does not require evaluating the likelihood function is proposed, and its performance is assessed by simulation. As there are several candidate distributions, we further propose a model selection procedure to identify an appropriate distribution for the time-censored aggregate data. A real aggregate dataset extracted from a reliability database is used for illustration.

Due to the variability of raw materials and the fluctuation in the manufacturing process, degradation of products may exhibit unit-to-unit variability in a population. The heterogeneous degradation rates can be viewed as random effects, which are often modeled by a normal distribution. Despite of its mathematical convenience, the normal distribution has certain limitations in modeling the random effects. In this study, we propose a novel random-effects Wiener process model based on ideas from accelerated failure time principle. An inverse Gaussian (IG) distribution can be used to characterize the unit-specific heterogeneity in degradation paths, which overcomes the disadvantages of the traditional models and provides more flexibility in the degradation modeling using Wiener processes. Properties of the model are investigated, and statistical inference based on the maximum likelihood estimation and the EM algorithm is established. An extension of the model to the constant-stress accelerated degradation test (ADT) is developed. The effectiveness and applicability of the proposed model are validated using a laser degradation dataset and an LED ADT dataset.