Nonparametric Approximation Methods for the First-Passage Time Distribution for Degradation Data Measured with Unequal Time Intervals

Abstract

Evaluating the first-passage time (FPT) distribution of a stochastic process is a prominent statistical problem, which has long been studied. This problem has important applications in reliability and degradation data analysis since the time that a degradation process of a product passes a critical level is considered as the failure time of the product. While most of the studies for FPT distribution focus on parametric stochastic process in the past, nonparametric approaches based on empirical saddlepoint approximation (ESA) are proposed recently. An advantage of those nonparametric approaches for evaluating the FPT distribution is that it does not require the specification of the parametric form of the underlying stochastic process. However, those nonparametric approaches based on ESA require the degradation measurements to be measured at equally distanced time points. For this reason, several random imputation methods are proposed by Palayangoda et al. (Appl Stoch Model Bus Ind 36(4):730–753, 2020). To facilitate the ESA method when the degradation data are measured at unequal time intervals, in this chapter, a least-squares modeling approach is proposed as an alternative approach to the random imputation methods. Monte Carlo simulation studies are used to evaluate the performance of the proposed methods for estimating the quantiles of the FPT distribution and for estimating the standard deviation of the FPT. Finally, degradation data analysis of a laser data is used to illustrate the methodologies presented in this chapter.