APPLICATION OF MML METHODOLOGY TO AN α–SERIES PROCESS WITH WEIBULL DISTRIBUTION

Abstract

In an α-series process, explicit estimators of the parameters α, µ and σ 2 are obtained by using the methodology of modified maximum likelihood (MML) when the distribution of the first occurrence time of an event is assumed to be Weibull. Monte Carlo simulations are performed to compare the efficiencies of the MML estimators with the corresponding nonparametric (NP) estimators. We also apply the MML methodology to two real life data sets to show the performance of the MML estimators compared to the NP estimators.

Keywords

• α-series process,Modified likelihood,Nonparametric,Efficiency,Monte Carlo simulation,2000 AMS Classification: 60 G 55

APPLICATION OF MML METHODOLOGY TO AN α–SERIES PROCESS WITH WEIBULL DISTRIBUTION

Abstract

References

1. Andrews, D. F. and Herzberg, A. M. Data (Springer-Verlag, New York, 1985).
2. Ascher, H. and Feingold, H. Repairable systems reliability (Marcel Dekker, New York, 1984).
3. Aydogdu, H. and Kara, M. Nonparametric estimation in α-series processes, Computational Statistics and Data Analysis (Submitted, 2009).
4. Barnett, V. D. Evaluation of the maximum likelihood estimator where the likelihood equation has multiple roots, Biometrika 53, 151–165, 1966.
5. Bhattacharyya, G. K. The asymptotics of maximum likelihood and related estimators based on type II censored data, J. Amer. Stat. Assoc. 80, 398–404, 1985.
6. Braun, W. J., Li, W. and Zhao, Y. Q. Properties of the geometric and related processes, Naval Research Logistics 52, 607–616, 2005.
7. Braun, W. J., Li, W. and Zhao, Y. Q. Some theoretical properties of the geometric and α- series processes, Communication in Statistics: Theory and Methods 37, 1483–1496, 2008.
8. Chan, S. K., Lam, Y. and Leung, Y. P. Statistical inference for geometric processes with gamma distributions, Computational Statistics and Data Analysis 47, 565–581, 2004.

9. Cox, D. R. and Lewis, P. A. W. The statistical analysis of series of events (Methuen, London, 1966).
10. Islam, M. Q., Tiku, M. L. and Yildirim, F. Non-normal regression: Part I: Skew distribu- tions, Communications in Statistics: Theory and Methods 30, 993–1020, 2001.
11. Islam, M. Q. and Tiku, M. L. Multiple linear regression model under nonnormality, Com- munications in Statistics: Theory and Methods 33 (10), 2443–2467, 2004.
12. Jarrett, R. G. A note on the intervals between coal-mining disasters, Biometrika 66, 191– 193, 1979.
13. Lam, Y. A note on the optimal replacement problem, Adv. Appl. Probab. 20, 479–482, 1988.
14. Lam, Y. Geometric process and replacement problem, Acta Math. Appl. Sinica 4, 366–377, 1988.
15. Lam, Y. and Chan, S. K. Statistical inference for geometric processes with lognormal dis- tribution, Computational Statistics and Data Analysis 27, 99–112, 1998.
16. Proschan, F. Theoretical explanation of observed decreasing failure rate, Technometrics 5, 375–383, 1963.
17. Puthenpura, S. and Sinha, N. K. Modified maximum likelihood method for the robust esti- mation of system parameters from very noisy data,Automatica 22, 231–235, 1986.
18. Surucu, B. A power comparison and simulation study of goodness-of-fit tests, Computers and Mathematics with Applications 56, 1617–1625, 2008.
19. Tiku, M. L. Estimating the mean and standard deviation from censored normal samples, Biometrika 54, 155–165, 1967.
20. Tiku, M. L. Estimating the parameters of lognormal distribution from censored samples, J. Amer. Stat. Assoc. 63, 134–140, 1968.
21. Tiku, M. L. Goodness-of-fit statistics based on the spacings of complete or censored samples, Austral. J. Statist. 22, 260–275, 1980.
22. Tiku, M. L., Tan, W. Y. and Balakrishnan, N. Robust inference (Marcel Dekker, New York, 1986).
23. Tiku, M. L. and Akkaya, A. D. Robust estimation and hypothesis testing (New Age Interna- tional (P) Limited, Publishers, New Delhi, 2004).
24. Vaughan, D. C. On the Tiku-Suresh method of estimation, Communications in Statistics: Theory and Methods 21, 451–469, 1992.
25. Vaughan, D. C. and Tiku, M. L. Estimation and hypothesis testing for a nonnormal bivariate distribution with applications, Mathematical and Computer Modelling 32, 53–67, 2000.