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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
The HCF is the largest real number, which when used to divide the time increments leads to all the ratios as positive integers.
References
Arnold, B. C., & Seshadri, V. (2009). Some new independence properties of the inverse Gaussian law. Sankhyā: The Indian Journal of Statistics, 71, 94–108.
Balakrishnan, N., & Kundu, D. (2019). Birnbaum-Saunders distribution: A review of models, analysis, and applications. Applied Stochastics Models in Business and Industry, 35, 4–49.
Balakrishnan, N., & Qin, C. (2019). Nonparametric evaluation of the first passage time of degradation processes. Applied Stochastic Models in Business and Industry, 35, 571–590.
Balakrishnan, N., Qin, C. (2020). Nonparametric optimal designs for degradation tests. Journal of Applied Statistics, 47, 624–641.
Bertoin, J. (1996). Lévy processes (Vol. 121). Cambridge, UK: Cambridge University Press.
Birnbaum, Z. W., & Saunders, S. C. (1969). A new family of life distributions. Journal of Applied Probability, 6, 319–327.
Chhikara, R. S., & Folks, J. L. (1989). The inverse Gaussian distribution: Theory, methodology, and applications. New York, NY: Marcel Dekker.
Donsker, M. D. (1951). An invariance principle for certain probability limit theorems. Memoirs of the American Mathematical Society, 6, 12.
Lugannani, R., & Rice, S. (1980). Saddle point approximation for the distribution of the sum of independent random variables. Advances in Applied Probability, 12, 475–490.
Meeker, W. Q., & Escobar, L. A. (1998). Statistical Methods for Reliability Data. New York, NY: Wiley.
Palayangoda, L. (2020). Statistical models and analysis of univariate and multivariate degradation data. Ph.D. thesis, Southern Methodist University, Dallas, Texas, USA.
Palayangoda, L. K., Ng, H. K. T., & Butler, R. W. (2020). Improved techniques for parametric and nonparametric evaluations of the first-passage time for degradation processes. Applied Stochastic Models in Business and Industry, 36(4), 730–753.
Park, C., & Padgett, W. (2005). Accelerated degradation models for failure based on geometric Brownian motion and Gamma processes. Lifetime Data Analysis, 11, 511–527.
Peng, C.-Y. (2015). Inverse Gaussian processes with random effects and explanatory variables for degradation data. Technometrics, 57, 100–111.
Tseng, S. T., & Lee, I.-C. (2016). Optimum allocation rule for accelerated degradation tests with a class of exponential-dispersion degradation models. Technometrics, 58, 244–254.
Wang, X., & Xu, D. (2010). An inverse Gaussian process model for degradation data. Technometrics, 52, 188–197.
Ye, Z., & Chen, N. (2014). The inverse Gaussian process as a degradation model. Technometrics, 56, 302–311.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Palayangoda, L.K., Ng, H.K.T. (2021). Nonparametric Approximation Methods for the First-Passage Time Distribution for Degradation Data Measured with Unequal Time Intervals. In: Ghosh, I., Balakrishnan, N., Ng, H.K.T. (eds) Advances in Statistics - Theory and Applications. Emerging Topics in Statistics and Biostatistics . Springer, Cham. https://doi.org/10.1007/978-3-030-62900-7_19
Download citation
DOI: https://doi.org/10.1007/978-3-030-62900-7_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-62899-4
Online ISBN: 978-3-030-62900-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)