Efficient estimation of Pareto model using modified maximum likelihood estimators

Document Type : Research Note

Authors

Department of Statistics, Government College University, Faisalabad, Pakistan

Abstract

In this article, we have proposed some modifications in the maximum likelihood estimation for estimating the parameters of the Pareto distribution and evaluated performance of these modified estimators in comparison to the existing maximum likelihood estimators. Total Relative Deviation (TRD) and Mean Square Error (MSE) have been used as performance indicators for goodness of fit analysis. The modified and traditional estimators have been compared for different sample sizes and different parameter combinations using a Monto Carlo simulation in R-language. We have concluded that the modified maximum likelihood estimator based on expectation of empirical Cumulative Distribution Function (CDF) of first-order statistic performs much better than the traditional ML estimator and other modified estimators based on median and coefficient of variation. The superiority of the said estimator is independent of sample size and choice of true parameter values. The simulation results were further corroborated by employing the proposed estimation strategies on two real-life data sets.

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References

References:
1. Pareto, V. "The new theories of economics", J. Polit. Econ., 5(4), pp. 485-502 (1897).
2. Munir, R., Saleem, M., Aslam, M., and Ali, S. "Comparison of different methods of parameters estimation for Pareto model", Casp. J. Appl. Sci. Res., 2(1), pp. 45-56 (2013).
3. Arnold, B.C. Encyclopaedia of Statistical Sciences, John Wiley (2008).
4. Abdel-All, N.H., Mahmoud, M.A.W., and Abd-Ellah, H.N. "Geometrical properties of Pareto distribution", Appl. Math. Comput., 145(2), pp. 321-339 (2003).
5. Sankaran, P.G. and Nair, M.T. "On finite mixture of Pareto distributions", Calcutta Stat. Assoc. Bull., 57(1-2), pp. 225-226 (2005).
6. Burroughs, S.M. and Tebbens, S.F. "Upper-truncated power law distributions", Fractals, 9(1), pp. 209-222 (2001).
7. Castillo, E. and Hadi, A.S. "Fitting the generalized Pareto distribution to data", J. Am. Stat. Assoc., 92(440), pp. 1609-1620 (1997).
8. Bourguignon, M., Ghosh, I., and Cordeiro, G.M. "General results for the transmuted family of distributions and new models", J. Probab. Stat., 2016, pp. 1-12 (2016).
9. Quandt, R.E. "Old and new methods of estimation and the Pareto distribution", Metrika, 10(1), pp. 55- 82 (1966).
10. Afify, E.E. "Order statistics from Pareto distribution", J. Appl. Sci., 6(10), pp. 2151-2157 (2006).
11. Lu, H.-L. and Tao, S.H. "The estimation of Pareto distribution by a weighted least square method", Qual. Quant., 41(6), pp. 913-926 (2007).
12. Pobockova, I. and Sedliackova, Z. "Comparison of four methods for estimating the Weibull distribution parameters", Appl. Math. Sci., 8(83), pp. 4137-4149 (2014).
13. Shawky, A.I. and Abu-Zinadah, H.H. "Exponentiated Pareto distribution: different method of estimations", Int. J. Contemp. Mathematical Sci., 4(14), pp. 677-693 (2009).
14. Grimshaw, S.D. "Computing maximum likelihood estimates for the generalized Pareto distribution", Technometrics, 35(2), pp. 185-191 (1993).
15. Cohen, A.C. and Whitten, B. "Modified maximum likelihood and modified moment estimators for the three-parameterWeibull distribution", Commun. Stat. Methods, 11(23), pp. 2631-2656 (1982).
16. Iwase, K. and Kanefuji, K. "Estimation for 3- parameter lognormal distribution with unknown shifted origin", Stat. Pap., 35(1), pp. 81-90 (1994).
17. Lalitha, S. and Mishra, A. "Modified maximum likelihood estimation for Rayleigh distribution", Commun. Stat. - Theory Methods, 25(2), pp. 389-401 (1996).
18. Rashid, M.Z. and Akhter, A.S. "Estimation accuracy of exponential distribution parameters", Pakistan J. Stat. Oper. Res., 7(2), pp. 217-232 (2011).
19. Zaka, A. and Akhter, A.S. "Modified moment, maximum likelihood and percentile estimators for the parameters of the power function distribution", Pakistan J. Stat. Oper. Res., 10(4), pp. 361-368 (2014).
20. Fisher, R.A. "On the mathematical foundations of theoretical statistics", Philos. Trans. R. Soc. London. Ser. A, Contain. Pap. a Math. or Phys. Character, 222, pp. 309-368 (1922).
21. Cohen, C.A. and Whitten, B.J. "Modified moment and maximum likelihood estimators for parameters of the three-parameter Gamma distribution", Commun. Stat. Comput., 11(2), pp. 197-216 (1982).
22. Al-Fawzan, M.A., Methods for Estimating the Parameters of the Weibull Distribution, King Abdulaziz City Sci. Technol. (2000).
23. Khalaf, G., Mansson, K., and Shukur, G. "Modified ridge regression estimators", Commun. Stat. - Theory Methods, 42(8), pp. 1476-1487 (2013).
24. Aydin, D. and Senoglu, B. "Monte Carlo comparison of the parameter estimation methods for the twoparameter Gumbel distribution", J. Mod. Appl. Stat. Methods, 14(2), pp. 123-140 (2015).
25. Shakeel, M., Haq, M.A. ul, Hussain, I., Abdulhamid, A.M., and Faisal, M. "Comparison of two new robust parameter estimation methods for the power function distribution", PLoS ONE, 11(8), p. e016069 (2016).
26. James, W. and Stein, C. "Estimation with quadratic loss", Proc. Fourth Berkeley Symp. Math. Stat. Probab., 1(1961), pp. 361-379 (1961).
27. Team, R.C.R, A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria (2016).
28. Clark, D.R., A Note on the Upper-truncated Pareto Distribution, Casualty Actuarial Society E-Forum, Winter, pp. 1-22 (2013).
29. Kantar, Y.M. "Generalized least squares and weighted least squares estimation methods for distributional parameters", REVSTAT - Stat. J., 13(3), pp. 263-282 (2015).
30. Beirliant, J., Teugels, J.L., and Vynckier, P., Practical Analysis of Extreme Values, Leuven University Press, Leuven, Belgium (1996).
31. Obradovic, M. "On Asymptotic efficiency of goodness of fit tests for Pareto distribution based on characterizations", Filomat, 29(10), pp. 2311-2324 (2015).
32. Kolmogorov, A. "On the determination of empirical fit of a distribution" [Sulla determinazione empirica diuna legge di distribuzione], G. dell'Istituto Ital. degli Attuari, 4, pp. 83-91 (1933).
33. Smirnov, N.V. "Estimate of deviation between empirical distribution functions in two independent samples", Bull. Moscow Univ., 2, pp. 3-16 (1939).
Scientia Iranica
Volume 26, Issue 1
Transactions on Industrial Engineering (E)
January and February 2019
Pages 605-614

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History

  • Receive Date: 09 June 2017
  • Revise Date: 17 September 2017
  • Accept Date: 06 June 2018

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