Improved techniques for parametric and nonparametric evaluations of the first‐passage time for degradation processes

Abstract

For degradation data in reliability analysis, estimation of the first‐passage time (FPT) distribution to a threshold provides valuable information on reliability characteristics. Recently, Balakrishnan and Qin (2019; Applied Stochastic Models in Business and Industry, 35:571–590) studied a nonparametric method to approximate the FPT distribution of such degradation processes if the underlying process type is unknown. In this article, we propose some improved techniques based on saddlepoint approximation, which enhance those existing methods. Numerical examples and Monte Carlo simulation studies are used to illustrate the advantages of the proposed techniques. Limitations of the improved techniques are discussed and some possible solutions to such are proposed. Some concluding remarks and practical recommendations are provided based on the results.

Bootstrapping Through Discrete Convolutional Methods

“…When assuming the mean Monte Carlo estimates are the truth, the MAE for the convolutional method is 0.041 while the MAE for the saddlepoint method is 2.62. As in the previous comparison, the convolutional method appears to have an accuracy advantage over the saddlepoint method presented in Palayangoda et al (2020).…”

“…For example, Balakrishnan and Qin (2019) developed a nonparametric method to approximate FPT quantiles of a degradation process. This methodology was further improved in Palayangoda et al (2020) which increased the accuracy and proposed a method to handle unequally spaced data. Given the popularity of the saddlepoint method, it will be used as one of the main comparisons for the examples in this article and we will compare our proposed methods directly with results from both of these works as well as others.…”

“…In a more recent article, these laser data were analyzed again. The analysis was similar to that in Balakrishnan and Qin (2019), however in Palayangoda et al (2020), data from individual lasers were treated as indistinguishable. More specifically, rather than only sampling from one laser at a time, sampling could occur from any laser degradation measurement.…”

“…, 16}, define the discrete random measure for the random variable Y . Palayangoda et al (2020). Once again, (*) is an indicator that the saddlepoint approximation was not contained in the Monte Carlo bootstrap interval.…”

“…Once again, the computation time required for the convolutional method is substantially shorter than the time needed for the Monte Carlo method, clocking in at just 16 seconds for all 10 thresholds and for both the lower and upper bounds. Although no computation times were reported for the saddlepoint method in Palayangoda et al (2020), we assume they were much faster than the Monte Carlo method.…”