Nonparametric evaluation of the first passage time of degradation processes

Abstract

This paper discusses a nonparametric method to approximate the first passage time (FPT) distribution of the degradation processes incorporating random effects if the process type is unknown. The FPT of a degradation process is unnecessarily observed since its density function can be approximated by inverting the empirical Laplace transform using the empirical saddlepoint method.The empirical Laplace transform is composed of the measured increments of the degradation processes. To evaluate the performance of the proposed method, the approximated FPT is compared with the theoretical FPT assuming a true underlying process. The nonparametric method discussed in this paper is shown to possess the comparatively small relative errors in the simulation study and performs well to capture the heterogeneity in the practical data analysis. To justify the fitting results, the goodness-of-fit tests including Kolmogorov-Smirnov test and Cramér-von Mises test are conducted, and subsequently, a bootstrap confidence interval is constructed in terms of the 90th percentile of the FPT distribution.

Bootstrapping Through Discrete Convolutional Methods

“…From that point on, the basic theory of using multiplication in the Fourier domain to compute a convolution was utilized as in the previous example. Balakrishnan and Qin (2019) Balakrishnan and Qin (2019) provide an application using laser degradation data; this analysis lends itself well to comparison. Their application looks at cumulative laser device degradation using several different values as failure thresholds.…”

“…More recently, the saddlepoint method was used to approximate first passage quantiles of a stochastic process. 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.…”

“…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.…”

“…A.3 Laser Data Analysis from Balakrishnan and Qin (2019) #Find the data in Meeker and Escobar. A.5 Asthma Application from Warr and Woodfield (2020) #The data is contained in the SemiMarkov library library(SemiMarkov); data(asthma) #Initializing vectors and constants t12 <-numeric(); t13 <-numeric(); t21 <-numeric(); t23 <-numeric(); j <-0; k <-0; m <-0; n <-0 #Creating separate vectors for each of the transitions for(i in 1:nrow(asthma…”

Bootstrapping through discrete convolutional methods

“…More recently, the saddlepoint method was used to approximate first passage quantiles of a stochastic process. For example, Reference 6 developed a nonparametric method to approximate FPT quantiles of a degradation process. This methodology was further improved in Reference 7 which increased the accuracy and proposed a method to handle unequally spaced data.…”