Estimating the Occurrence Rate for Alpha-Series Process in Rayleigh Distribution

Aya Mahmood Taha, Muthanna Subhi Sulaiman


The geometric process is sometimes appropriate for reliability and scheduling problems. Some previous studies suggested a possible alternative process that is alpha-series as to the geometric process when it decreases with time, as the decreasing geometric process shows that the expected number of events at an arbitrary time does not exist. In contrast, the expected number of events of the alpha-series process (ASP) exists at an arbitrary time under some conditions. In this paper, we assumed that the first arrival followed the Rayleigh distribution (RD). The modified moment estimator was proposed to estimate the alpha-series process parameters in the Rayleigh distribution and compare it with the maximum likelihood estimators. A simulation was conducted to compare the two estimators. The real-data application of intervals between successive failures of the Mosul Dam power station in Nineveh governorate in Iraq is provided to illustrate the results. When the initial occurrence time distribution is indicated to be RD, an estimate of the occurrence rate of an ASP is investigated in this study. Estimators are generated using modified moment (MM), and maximum likelihood (ML) approaches. According to the simulation study's findings, the MM estimator outperforms the ML estimator. In all cases, ASP with RD provides better data than the renewal process (RP) in real data sets. A test statistic has been devised to determine if the data conforms to an ASP.


Alpha-series process; Rayleigh distribution; maximum likelihood estimator; modified moment estimator; Monte Carlo simulation.

Full Text:



H. Aydogdu, B. Senoglu, and K. Mahmut, "Application of MML Methodology to an α–Series Process with Weibull Distribution," Hacettepe Journal of Mathematics and Statistics, vol. 39, pp. 449-460, 2010.

W. J. Braun, W. Li, and Y. Q. Zhao, "Properties of the geometric and related processes," Naval Research Logistics (NRL), vol. 52, pp. 607-616, 2005.

W. J. Braun, W. Li, and Y. Q. Zhao, "Some theoretical properties of the geometric and α-series processes," Communications in Statistics—Theory and Methods, vol. 37, pp. 1483-1496, 2008.

R. Hidayat, I. T. R. Yanto, A. A. Ramli, and M. F. M. Fudzee, "Similarity measure fuzzy soft set for phishing detection," International Journal of Advances in Intelligent Informatics, vol. 7, pp. 101-111, 2021.

J. Hillston, "Stochastic process algebras and their markovian semantics," ACM SIGLOG News, vol. 5, pp. 20-35, 2018.

P. Aguilar and R. Sommaruga, "The balance between deterministic and stochastic processes in structuring lake bacterioplankton community over time," Molecular ecology, vol. 29, pp. 3117-3130, 2020.

N. Masuda and L. E. Rocha, "A Gillespie algorithm for non-Markovian stochastic processes," Siam Review, vol. 60, pp. 95-115, 2018.

F. Feng, S. Teng, K. Liu, J. Xie, Y. Xie, B. Liu, et al., "Co-estimation of lithium-ion battery state of charge and state of temperature based on a hybrid electrochemical-thermal-neural-network model," Journal of Power Sources, vol. 455, p. 227935, 2020.

R. Arnold, S. Chukova, Y. Hayakawa, and S. Marshall, "Geometric-like processes: An overview and some reliability applications," Reliability Engineering & System Safety, vol. 201, p. 106990, 2020.

A. Gioia, M. F. Bruno, V. Totaro, and V. Iacobellis, "Parametric assessment of trend test power in a changing environment," Sustainability, vol. 12, p. 3889, 2020.

M. Kara, H. Aydoğdu, and B. Şenoğlu, "Statistical inference for α-series process with gamma distribution," Communications in Statistics-Theory and Methods, vol. 46, pp. 6727-6736, 2017.

M. Kara, Ö. Altındağ, M. H. Pekalp, and H. Aydoğdu, "Parameter estimation in α-series process with lognormal distribution," Communications in Statistics-Theory and Methods, vol. 48, pp. 4976-4998, 2019.

M. Kara, Ö. Türkşen, and H. Aydoğdu, "Statistical inference for α-series process with the inverse Gaussian distribution," Communications in Statistics-Simulation and Computation, vol. 46, pp. 4938-4950, 2017.

Y. Lam, The geometric process and its applications: World Scientific, 2007.

W.-X. Li, C. C.-S. Chen, and J. J. French, "The relationship between liquidity, corporate governance, and firm valuation: Evidence from Russia," Emerging Markets Review, vol. 13, pp. 465-477, 2012.

H. Demirci Biçer, "Statistical Inference for Alpha-Series Process with the Generalized Rayleigh Distribution," Entropy, vol. 21, p. 451, 2019.

X. Li, H. Zhang, R. Zhang, Y. Liu, and F. Nie, "Generalized uncorrelated regression with adaptive graph for unsupervised feature selection," IEEE transactions on neural networks and learning systems, vol. 30, pp. 1587-1595, 2018.

Y. L. Y. Lin, "Geometric processes and replacement problem," Acta Mathematicae Applicatae Sinica, vol. 4, pp. 366-377, 1988.

M. R. Vahid, B. Hanzon, and R. J. Ober, "Fisher information matrix for single molecules with stochastic trajectories," SIAM Journal on Imaging Sciences, vol. 13, pp. 234-264, 2020.

A. Eshragh, "Fisher Information, stochastic processes and generating functions," in Proceedings of the 21st International Congress on Modeling and Simulation, 2015.

I. Andrews and A. Mikusheva, "Maximum likelihood inference in weakly identified dynamic stochastic general equilibrium models," Quantitative Economics, vol. 6, pp. 123-152, 2015.

L. Meng and J. C. Spall, "Efficient computation of the fisher information matrix in the em algorithm," in 2017 51st Annual Conference on Information Sciences and Systems (CISS), 2017, pp. 1-6.

M.-K. Riviere, S. Ueckert, and F. Mentré, "An MCMC method for the evaluation of the Fisher information matrix for non-linear mixed effect models," Biostatistics, vol. 17, pp. 737-750, 2016.

E. Bulinskaya and A. Sokolova, "Limit theorems for generalized renewal processes," Theory of Probability & Its Applications, vol. 62, pp. 35-54, 2018.

M. P. Laurini and L. K. Hotta, "Generalized moment estimation of stochastic differential equations," Computational Statistics, vol. 31, pp. 1169-1202, 2016.

L. Yeh and S. K. Chan, "Statistical inference for geometric processes with lognormal distribution," Computational statistics & data analysis, vol. 27, pp. 99-112, 1998.

H. Demirci Biçer, "Statistical inference for geometric process with the Two-Parameter Lindley Distribution," Communications in Statistics-Simulation and Computation, vol. 49, pp. 2979-3000, 2020.

M. Kara, H. Aydoğdu, and Ö. Türkşen, "Statistical inference for geometric process with the inverse Gaussian distribution," Journal of Statistical Computation and Simulation, vol. 85, pp. 3206-3215, 2015.

C. Bicer, "Statistical inference for geometric process with the power Lindley distribution," Entropy, vol. 20, p. 723, 2018.

R. Diwakar, "An evaluation of normal versus lognormal distribution in data description and empirical analysis," Practical Assessment, Research, and Evaluation, vol. 22, p. 13, 2017.

C. Biçer, H. D. Biçer, K. Mahmut, and H. Aydoğdu, "Statistical inference for geometric process with the Rayleigh distribution," Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 68, pp. 149-160, 2019.

M. H. Pekalp, H. Aydoğdu, and K. F. Türkman, "Discriminating between some lifetime distributions in geometric counting processes," Communications in Statistics-Simulation and Computation, vol. 51, pp. 715-737, 2020.



  • There are currently no refbacks.

Published by INSIGHT - Indonesian Society for Knowledge and Human Development