Statistica Sinica

Ming-Yung Lee and Jen Tang

Abstract:Being the solution to the stochastic linear growth model, the Wiener process has recently been used to model the degradation (or cumulative decay) of certain characteristics of test units in lifetime data analyses. When the failure threshold is constant or linear in time, the failure time, which is defined as the first-passage time of the Wiener process over the failure threshold, will follow an inverse Gaussian (IG) distribution. In this paper we consider a time-censored degradation test where, in addition to the failure times of the failed units, we assume that the degradation values at the censor time of the censored units are also available. Then, based on these degradation values, we use a modified EM-algorithm to predict the failure times of the censored units. The resulting estimator of the mean failure time is shown to be a consistent estimator, and is also an estimator that maximizes the (modified) likelihood function of the available failure times and degradation values. For the scale parameter of the IG distribution, however, the algorithm produces an inconsistent estimator. We introduce two modified estimators to reduce bias. Analytical and numerical comparisons show that our proposed estimators perform well, as compared to the traditional MLEs and the modified MLEs, for both IG parameters. An example is used to illustrate the proposed methodology.

Key words and phrases:Bias, consistency, degradation test, EM-algorithm, inverse Gaussian distribution, maximum likelihood estimator, reliability, Wiener process.