A comparison of different methods of estimation for the flexible Weibull distribution

Year 2020, Volume: 69 Issue: 1, 794 - 814, 30.06.2020

Sajid Ali , Sanku Dey M H Tahir , Muhammad Mansoor

https://doi.org/10.31801/cfsuasmas.597680

Cited By: 7

Abstract

This article presents different parameter estimation methods for flexible Weibull distribution introduced by Bebbington et al. (Reliability Engineering and System Safety 92:719-726, 2007), which is a modified version of the Weibull distribution and is suitable to model different shapes of the hazard rate. We consider both frequentist and Bayesian estimation methods and present a comprehensive comparison of them. For frequentist estimation, we consider the maximum likelihood estimators, least squares estimators, weighted least squares estimators, percentile estimators, the maximum product spacing estimators, the minimum spacing absolute distance estimators, the minimum spacing absolute log-distance estimators, Cramér von Mises estimators, Anderson Darling estimators, and right tailed Anderson Darling estimators, and compare them using a comprehensive simulation study. We also consider Bayesian estimation by assuming gamma priors for both shape and scale parameters. We use a Markov Chain Monte Carlo algorithm to compute the posterior summaries. A real data example is also a part of this work.

Keywords

Weibull distribution; maximum likelihood estimators; least square estimators; weighted least square estimators; percentile estimators; maximum product spacing estimators; minimum spacing absolute distance estimators; minimum spacing absolute log-distance estimators; Cramer von Mises estimators; Anderson Darling estimators; right tailed Anderson Darling estimators; Bayesian estimation.

References

• Bebbington, M., Lai, C.-D., and Zitikis, R. (2007). A flexible weibull extension. Reliability Engineering &System Safety, 92(6):719-726.
• Devroye, L. (1986). Non-Uniform Random Variate Generation. Springer-Verlag.
• Kao, J. H. K. (1959). A graphical estimation of mixed Weibull parameters in life-testing of electron tubes.Technometrics, 1(4):389-407.
• Kenney, J. and Keeping, E. (1962). Mathematics of statistics. Number v. 2 in Mathematics of Statistics.Princeton: Van Nostrand.
• Mann, N., Schafer, R., and Singpurwalla, N. (1974). Methods for statistical analysis of reliability and lifedata. Wiley.
• Moors, J. J. A. (1988). A quantile alternative for kurtosis. Journal of the Royal Statistical Society. Series D(The Statistician), 37(1):25-32.
• Nadarajah, S. and Haghighi, F. (2011). An extension of the exponential distribution. Statistics, 45(6):543-558.
• Shaked, M. and Shanthikumar, J. (2007). Stochastic Orders. Springer New York.
• Anderson, T. W. and Darling, D. A. (1952). Asymptotic theory of certain goodness of fit criteria based on stochastic processes. The Annals of Mathematical Statistics, 23(2):193-212.
• Anderson, T. W. and Darling, D. A. (1954). A test of goodness of fit. Journal of the American Statistical Association, 49(268):765-769.
• Cheng, R. C. H. and Amin, N. A. K. (1979). Maximum product of spacings estimation with application to the lognormal distribution, math.
• Cheng, R. C. H. and Amin, N. A. K. (1983). Estimating parameters in continuous univariate distributions with a shifted origin. Journal of the Royal Statistical Society. Series B (Methodological), 45(3):394-403.
• Pettitt, A. N. (1976). A two-sample Anderson-Darling rank statistic. Biometrika, 63(1):161-168.
• Dey, S., Al-Zahrani, B., and Basloom, S. (2017a). Dagum distribution: Properties and different methods of estimation. International Journal of Statistics and Probability, 6(2):74-92.
• Dey, S., Ali, S., and Park, C. (2015). Weighted exponential distribution: properties and different methods of estimation. Journal of Statistical Computation and Simulation, 85(18):3641-3661.
• Dey, S., Dey, T., Ali, S., and Mulekar, M. S. (2016). Two-parameter Maxwell distribution: Properties and different methods of estimation. Journal of Statistical Theory and Practice, 10(2):291-310.
• Dey, S., Dey, T., and Kundu, D. (2014). Two-parameter Rayleigh distribution: Different methods of estimation. American Journal of Mathematical and Management Sciences, 33(1):55-74.
• Dey, S., Kumar, D., Ramos, P. L., and Louzada, F. (2017b). Exponentiated Chen distribution: Properties and estimation. Communications in Statistics - Simulation and Computation, 46(10):8118-8139.
• Dey, S., Raheem, E., and Mukherjee, S. (2017c). Statistical Properties and Different Methods of Estimation of Transmuted Rayleigh Distribution. Revista Colombiana de EstadAstica, 40:165-203.
• Dey, S., Raheem, E., Mukherjee, S., and Ng, H. K. T. (2017d). Two parameter exponentiated Gumbel distribution: properties and estimation with flood data example. Journal of Statistics and Management Systems, 20(2):197-233.
• Dey, S., Zhang, C., Asgharzadeh, A., and Ghorbannezhad, M. (2017e). Comparisons of methods of estimation for the NH distribution. Annals of Data Science, 4(4):441-455.
• Kao, J. H. K. (1958). Computer methods for estimating Weibull parameters in reliability studies. IRE Transactions on Reliability and Quality Control, PGRQC-13:15-22.
• MacDonald, P. D. M. (1971). Comment on "an estimation procedure for mixtures of distributions" by Choi and Bulgren. Journal of the Royal Statistical Society. Series B (Methodological), 33(2):326-329.
• Ranneby, B. (1984). The maximum spacing method. an estimation method related to the maximum likelihood method. Scandinavian Journal of Statistics, 11(2):93-112.
• Smith, R. L. and Naylor, J. C. (1987). A comparison of maximum likelihood and Bayesian estimators for the three- parameter weibull distribution. Journal of the Royal Statistical Society. Series C (Applied Statistics), 36(3):358-369.
• Stephens, M. A. (1974). EDF statistics for goodness of fit and some comparisons. Journal of the American Statistical Association, 69(347):730-737.
• Swain, J. J., Venkatraman, S., and Wilson, J. R. (1988). Least-squares estimation of distribution functions in Johnson's translation system. Journal of Statistical Computation and Simulation, 29(4):271-297.