Bayesian Parameter Estimation for Geometric Process with Rayleigh Distribution

Abstract

The main purpose of this study is to deal with the parameter estimation problem for the geometric process (GP) when the distribution of the first occurrence time of an event is assumed to be Rayleigh. For this purpose, maximum likelihood and Bayesian parameter estimation methods are discussed. Lindley and MCMC approximation methods are used in Bayesian calculations. Additionally, a novel method called the Modified-Lindley approximation has been proposed as an alternative to the Lindley approximation. An extensive simulation study was conducted to compare the performances of the prediction methods. Finally, a real data set is analyzed for illustrative purposes.

Keywords

• Rayleigh distribution
• Geometric process
• Maximum likelihood estimator
• Bayesian parameter estimation
• Simulation study.

References

1. [1] Y. Lam, “A note on the optimal replacement problem,”Advances in Applied Probability, vol. 20, pp. 479-482, 1988
2. [2]Y. Lam, “Geometric processes and replacement problem,” Acta Math Appl Sin., vol. 4, pp. 366-377, 1988.
3. [3] W.J. Braun, W. Li, and Y.P. Zhao, “Properties of the geometric and related processes,”.Nav Res Log., vol. 52, pp.607–616, 2005.
4. [4] Y. Lam, Y.H. Zheng, and Y.L. Zhang, “Some limit theorems in geometric process,” Acta Math Appl Sin., vol. 19, pp. 405– 416, 2003.
5. [5] L. Yeh, and S. K. Chan, “Statistical inference for geometric processes with lognormal distribution,” Computational statistics & data analysis, vol. 27, no. 1, pp. 99-112, 1998.
6. [6] M. Kara, G. Güven, B. Şenoğlu, and H. Aydoğdu, “Estimation of the parameters of the gamma geometric process,” Journal of Statistical Computation and Simulation, vol. 92, no. 12, pp. 2525-2535, 2022.
7. [7] H. Aydoğdu, B. Şenoğlu, and M. Kara, “Parameter estimation in geometric process with Weibull distribution.” Applied Mathematics and Computation, vol. 217, no.6, pp. 2657-2665,2010.
8. [8] I.Usta, “Statistical inference for geometric process with the inverse Rayleigh distribution,” Sigma Journal of Engineering and Natural Sciences, vol. 37, no. 3, pp. 871-882,2019.

9. [9] C. Biçer, H.D. Biçer, M. Kara, and A. Yılmaz, “Statistical inference for geometric process with the generalized rayleigh distribution,” Facta Universitatis, Series: Mathematics and Informatics, 1107-1125, 2021.
10. [10] C. Biçer, H.D. Biçer, M.Kara, 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, no. 1, pp. 149-160,2019.
11. [11] M. Kara, H. Aydoğdu, and Ö. Türkşen, 2015. “Statistical inference for geometric process with the inverse Gaussian distribution,” Journal of Statistical Computation and Simulation, vol. 85, no. 16, pp. 3206-3215, 2015.
12. [12] A. Yılmaz, M. Kara, and H. Kara, “Bayesian inference for geometric process with lindley distribution and its applications,” Fluctuation and Noise Letters, vol. 21, no. 05, 2250048,2022.
13. [13] I. Usta, “Bayesian estimation for geometric process with the Weibull distribution,”Communications in Statistics-Simulation and Computation, vol. 53, no. pp. 1-27, 2022.
14. [14] S. Ali, M. Aslam and S.M.A. Kazmi, 2013. “A study of the effect of the loss function on Bayes Estimate, posterior risk and hazard function for Lindley distribution”. Applied Mathematical Modelling 37, 6068-6078,2013.
15. [15] M.Y. Danish, and M. Aslam, 2013. “Bayesian estimation for randomly censored generalized exponential distribution under asymmetric loss functions,” Journal of Applied Statistics, vol. 40, no. 5, pp. 1106-1119, 2013.
16. [16] A.Helu, and H. Samawi, H. 2015. “The inverse Weibull distribution as a failure model under various loss functions and based on progressive first-failure censored data,” Quality Technology & Quantitative Management, vol. 12, no. 4, pp. 517-535, 2015.
17. [17] D.V. Lindley, “Approximate Bayesian methods,” Trabajos Estadistica Investig Oper., vol. 31, pp. 223- 245, 1980.
18. [18] A.F. Smith, and G. O. Roberts, “Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods,” Journal of the Royal Statistical Society: Series B (Methodological).vol. 55, pp. 3-23, 1993.
19. [19] N. Metropolis, A.W. Rosenbluth, and M.N. Rosenbluth, “Equation of state calculations by fast computing machines,” J Chem Phys., vol. 21, pp. 1087-1092, 1953.
20. [20] D. Andrews, and A. Herzberg, Data. New York: Springer,1985.
21. [21] Y.Lam, “Nonparametric inference for geometric process,” Communications in Statistics - Theory and Methods. vol 21, pp. 2083–105,1992.
22. [22]M.L. Tiku, “Goodness of fit statistics based on the spacings of complete or censored samples,” Australian Journal of Statistics, vol. 22, no. 3, pp. 260-275,1980