Cenker Biçer, Hayrinisa D. Biçer, Mahmut Kara, Asuman Yılmaz

DOI Number
First page
Last page


In the present paper, statistical inference problem is considered for the geometric process (GP) by assuming the distribution of the first arrival time is generalized Rayleigh with the parameters $\alpha$ and $\lambda$. We use the maximum likelihood method for obtaining the ratio parameter of the GP and distributional parameters of the generalized Rayleigh distribution. By a series of Monte-Carlo simulations evaluated through the different samples of sizes small, moderate and large, we also compare the estimation performances of the maximum likelihood estimators with the other estimators available in the literature such as modified moment, modified L-moment, and modified least squares. Furthermore, we present two real-life dataset analyzes to show the modeling behavior of GP with generalized Rayleigh distribution.


Monotone processes; non-parametric estimation; parametric estimation; stochastic process; data with trend.

Full Text:



M. Abramowitz and I. A. Irene: Handbook of mathematical functions: with formulas, graphs, and mathematical tables. Courier Corporation, 1964

H. Ascher and H. Feingold: Repairable Systems Reliability. Marcel Dekker, New York, 1984

H. Aydoğdu and B. Şenoğlu and M. Kara: Parameter estimation in geometric process with Weibull distribution. Applied Mathematics and Computation. 217(6) (2010), 2657–2665.

C. Biçer: Statistical Inference for Geometric Process with the Power Lindley Distribution. Entropy, 20(10), 2018, 723.

C. Biçer: Statistical inference for geometric process with the Two-parameter Rayleigh Distribution. The Most Recent Studies in Science and Art, 1, (2018), 576–583.

H. D. Biçer: Statistical inference for geometric process with the Two-Parameter Lindley Distribution. Communications in Statistics-Simulation and Computation, (2019), 1–22.

C. Biçer and H. D. Biçer: Statistical Inference for Geometric Process with the Lindley Distribution. Researches on Science and Art in 21st Century Turkey, 2, (2017), 2821–2829.

C. Biçer, and H. D. Biçer and M. Kara and H. Aydoğdu, Halil: Statistical inference for geometric process with the Rayleigh distribution. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(1), 2019, 149–160.

W. J. Braun, and W. Li and Y. Q. Zhao: Properties of the geometric and related processes. Naval Research Logistics. 52(7), (2005), 607–616.

W. J. Braun, and W. Li and Y. Q. Zhao: Some theoretical properties of the geometric and alpha-series processes. Communications in Statistics Theory and Methods. 37(9), (2008), 1483-1496.

M. J. Crowder and A. C. Kimber and R. L. Smith and T. J. Sweeting: Statistical concepts in reliability. Chapman and Hall, London, 1991.

J. S. K. Chan and, Y. Lam and D. Y. P. Leung: Statistical inference for geometric processes with gamma distributions. Computational statistics & data analysis. 47(3), (2004), 565–581.

M. Kara and H. Aydoğdu and Ö. Türkşen: Statistical inference for geometric process with the inverse Gaussian distribution. Journal of Statistical Computation and Simulation. 85(16), (2015), 3206–3215.

D. Kundu and M. Z. Raqab: Generalized Rayleigh distribution: different methods of estimations. Computational statistics & data analysis. 49(1), (2005), 187–200.

Y. Lam: A note on the optimal replacement problem. Advances in Applied Probability. 20(2), (1988), 479–482.

Y. Lam: Nonparametric inference for geometric processes. Communications in statistics-theory and methods. 21(7), 1992, 2083–2105.

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

Y. Lam and S. K. Chan: Statistical inference for geometric processes with lognormal distribution. Computational statistics & data analysis. 27(1), (1998), 99–112.

Y. Lam and Y. Zheng and Y. Zhang: Some limit theorems in geometric processes. Acta Mathematicae Applicatae Sinica, English Series. 19(3), (2003), 405-416.

Y. Lam and L. Zhu and J. S. K. Chan and Q. Liu: Analysis of data from a series of events by a geometric process model. Acta Mathematicae Applicatae Sinica, English Series. 20(2), (2004), 263–282.

M. Z. Raqab and D. Kundu: Burr type X distribution: revisited. Journal of probability and statistical sciences. 4(2), (2006), 179–193.

J.G. Surles and W. J. Padgett: Inference for reliability and stress-strength for a scaled Burr type X distribution. Lifetime Data Analysis, 7(2), (2001), 187–200.



  • There are currently no refbacks.

© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online)