Estimation in α-Series Processes with Exponential Inter-Arrival Times under Censored Data

Abstract

The α-series process is an important counting process commonly used to model data sets having monotonic trend. It is especially utilized in reliability analysis of deteriorating systems and warranty analysis of repairable systems. When a data set is compatible with the α-series process, it is important to make inference for model parameters of the process. All the studies in the literature only consider single realization of the process which only has complete samples. However, multi-sample of the process may be observed. In this situation, the data set includes both complete and censored samples. In this study, estimation problem for an α-series process under censored data is studied by assuming inter-arrival times of the process have exponential distribution and all samples are homogeneous. Maximum likelihood estimators of the model parameters are obtained and their asymptotic properties such as asymptotic normality and consistency are proved. Also, their small sample performances have been investigated by a simulation study.

Keywords

• α-series process
• Maximum likelihood method
• Exponential distribution
• Multi-sample
• Censored data

References

1. Altındağ, Ö., & Aydoğdu, H. (2021). Estimation of renewal function under progressively censored data and its applications. Reliability Engineering & System Safety, 216, 107988.
2. Aydoğdu, H., & Altındağ, Ö. (2016). Computation of the mean value and variance functions in geometric process. Journal of Statistical Computation and Simulation, 86(5), 986-995.
3. Aydoğdu, H., & Kara, M. (2012). Nonparametric estimation in α-series processes. Computational statistics & data analysis, 56(1), 190-201.
4. Barlow, R. E., & Proschan, F. (1996). Mathematical theory of reliability. Society for Industrial and Applied Mathematics.
5. Blischke, W. R., & Murthy, D. P. (2011). Reliability: modeling, prediction, and optimization. John Wiley & Sons.
6. Braun, W. J., Li, W., & Zhao, Y. Q. (2005). Properties of the geometric and related processes. Naval Research Logistics (NRL), 52(7), 607-616.
7. Braun, W. J., Li, W., & Zhao, Y. Q. (2008). Some theoretical properties of the geometric and α-series processes. Communications in Statistics—Theory and Methods, 37(9), 1483-1496.
8. Chukova, S., & Hayakawa, Y. (2004). Warranty cost analysis: Non‐zero repair time. Applied Stochastic Models in Business and Industry, 20(1), 59-71.

9. Fleming, T. R., & Harrington, D. P. (2013). Counting processes and survival analysis (Vol. 625). John Wiley & Sons.
10. Jiang, R. (2020). A novel two-fold sectional approximation of renewal function and its applications. Reliability Engineering & System Safety, 193, 106624
11. Kara, M., Aydoğdu, H., & Şenoğlu, B. (2017a). Statistical inference for α-series process with gamma distribution. Communications in Statistics-Theory and Methods, 46(13), 6727-6736.
12. Kara, M., Türkşen, Ö., & Aydoğdu, H. (2017b). Statistical inference for α-series process with the inverse Gaussian distribution. Communications in Statistics-Simulation and Computation, 46(6), 4938-4950.
13. Kara, M., Altındağ, Ö., Pekalp, M. H., & Aydoğdu, H. (2019). Parameter estimation in α-series process with lognormal distribution. Communications in Statistics-Theory and Methods, 48(20), 4976-4998.
14. Lam, Y. (1988). Geometric processes and replacement problem. Acta Mathematicae Applicatae Sinica, 4, 366-377.
15. Lam, Y. (2007). The geometric process and its applications. World Scientific.
16. Park, S., Balakrishnan, N., & Zheng, G. (2008). Fisher information in hybrid censored data. Statistics & probability letters, 78(16), 2781-2786.
17. Pekalp, M. H., & Aydoğdu, H. (2021). Power series expansions for the probability distribution, mean value and variance functions of a geometric process with gamma interarrival times. Journal of Computational and Applied Mathematics, 388, 113287.
18. Zheng, G., & Gastwirth, J. L. (2001). On the Fisher information in randomly censored data. Statistics & probability letters, 52(4), 421-426.