Statistical inference for geometric process with the Rayleigh distribution

Abstract

The aim of this study is to investigate the solution of the statistical inference problem for the geometric process (GP) when the distribution of first occurrence time is assumed to be Rayleigh. Maximum likelihood (ML) estimators for the parameters of GP, where a and λ are the ratio parameter of GP and scale parameter of Rayleigh distribution, respectively, are obtained. In addition, we derive some important asymptotic properties of these estimators such as normality and consistency. Then we run some simulation studies by different parameter values to compare the estimation performances of the obtained ML estimators with the non-parametric modified moment (MM) estimators. The results of the simulation studies show that the obtained estimators are more efficient than the MM estimators.

Keywords

• Parameter estimation,geometric process,maximum likelihood estimators,asymptotic distribution

References

1. Andrews D. F., Herzberg. A. M., Data. New York: Springer, 1985.
2. Ascher H, Feingold H., Repairable systems reliability, New York: Marcel Dekker; 1984.
3. Aydoğdu H., ¸Senoğlu B., Kara M., Parameter estimation in geometric process with Weibull distribution, Appl Math Comput. (2010), 217, 2657--2665.
4. Barndorff-Nielsen O. E., Cox D. R., Inference and asymptotics, London: Chapman & Hall, 1994.
5. Braun W. J. , Li W., Zhao Y. P., Properties of the geometric and related processes, Nav Res Log., (2005), 52, 607--616.
6. Chan S. K., Lam Y., Leung Y.P., Statistical inference for geometric processes with gamma distribution, Comput. Stat. Data Anal. (2004), 47, 565--581..
7. Cox D.R., Lewis P.A.W., The statistical analysis of series of events, London: Mathuen, 1966.
8. Forbes C., Evans M., Hastings N., Peacock B., Statistical distributions, New Jersey: John Wiley & Sons; 2011.

9. Kara, M., Aydoğdu, H., & Türkşen, Ö. Statistical inference for geometric process with the inverse Gaussian distribution, Journal of Statistical Computation and Simulation, (2015), 85(16), 3206-3215.
10. Kececioglu, Dimitri. Reliability engineering handbook, Prentice-Hall Inc., 1991.
11. Lam Y., A note on the optimal replacement problem, Adv Appl Probab. (1988), 20, 479--482.
12. Lam Y., Geometric process and replacement problem, Acta Math Appl Sin. (1988), 366--377.
13. Lam Y., Nonparametric inference for geometric processes. Commun Stat Theor M. (1992), 21, 2083--2105.
14. Lam Y, Chan S. K., Statistical inference for geometric processes with lognormal distribution, Comput Stat Data Anal. (1998), 27, 99--112.
15. Lam Y, Zheng Y. H, Zhang Y. L., Some limit theorems in geometric process, Acta Math Appl Sin. (2003),19(3), 405-- 416.
16. Lam Y, Zhu L.X., Chan JSK, Liu Q. Analysis of data from a series of events by a geometric process model, Acta Math Appl Sin. (2004),20(2), 263--282.
17. Lam Y., The geometric process and its applications, Singapore: World Scientific; 2007.
18. Tiku M. L., Goodness-of-fit statistics based on the spacings of complete or censored samples, Austral. J. Statist. (1980) 22, 260--275.