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Comparison Of Maximum Likelihood And Bayes Estimators Under Symmetric And Asymmetric Loss Functions By Means Of Tierney Kadane’s Approximation For Weibull Distribution

Year 2019, Volume: 14 Issue: 2, 69 - 78, 27.09.2019

Abstract



In this study, it is considered the problem
of comparing the performances of the Maximum Likelihood (ML) and Bayes
estimators under symmetric and asymmetric loss function  for the unknown parameters of Weibull
distribution. ML estimators are computed by using the Newton Raphson method. Bayesian
estimations under Squared, Linex and General Entropy loss functions by using
Jeffrey’s extension prior are introduced with Tierney Kadane approximation for
Weibull distribution. For different sample sizes, estimators are compared to
obtain the best estimator in terms of mean squared errors using a Monte Carlo
simulation study.




References

  • [1] Abernethy R. B. The New Weibull Handbook, 5th edition. 2006.
  • [2] Nadarajah, S., Cordeiro, G. M. and Ortega E. M. M. The exponentiated Weibull distribution: a survey.Statistical Papers 2012; 54(3), 839-877.
  • [3] Zhang,L. F.,Xie M. and Tang, L.C. On weighted least square estimation for the parameters of Weibull distribution.Recent Advances in Reliability and Quality Design 2008; P. Hoang, Ed., pp. 57–84, Springer, London, UK.
  • [4] Guure, C.B., İbrahim, N.A. and Ahmed, A.O.M. Bayesian Estimation of Two-Parameter Weibull Distribution Using Extension of Jeffreys' Prior Information with Three Loss Functions, Mathematical Problems in Engineering 2012;, 1-13.
  • [5] Al-Kutubi, H. S. and Ibrahim, N. A. Bayes estimator for exponential distribution with extension of Jeffery prior information. Malaysian Journal of Mathematical Sciences 2009; 3(2), 297–313.
  • [6] Jeffreys, H. Theory of Probability. Oxford at the Clarendon Press. 1948.
  • [7] Legendre, A. Nouvelles M´ethodes pour la D´etermination des Orbites des Com`etes. Courcier, Paris. 1805.
  • [8] Gauss, C.F. M´ethode des Moindres Carr´es. M´emoire sur la Combination des Observations. Transl. J. Bertrand (1955). Mallet-Bachelier,Paris. 1810.
  • [9] Varian, H. R. Variants in Economic Theory. Norhampton-USA, Edward Elgar. 2000.
  • [10] Zellner, A. Bayesian estimation and prediction using asymmetric loss functions.Journal of the American Statistical Association 1986; 81(394), 446-451.
  • [11] Calabria, R. and Pulcinia, G. Point estimation under asymmetric loss functions for left-truncated exponential samples. Communications in Statistics - Theory and Methods 1996;, 25(3), 585-600.
  • [12] Dey, D. K. and Pei-San Liao, L. On comparison of estimators in a generalized life model.Microelectronics Reliability 1992; 32(1–2), 207-221.
  • [13] Tierney, L. and Kadane, J. B. Accurate Approximations for Posterior Moments and Marginal Densities.Journal of the American Statistical Association 1986; 81(393), 82-86.
  • [14] Danish, M. Y. ve Aslam, M. Bayesian estimation for randomly censored generalized exponential distribution under asymmetric loss functions, Journal of Applied Statistics 2013; 40 (5), 1106-1119.
Year 2019, Volume: 14 Issue: 2, 69 - 78, 27.09.2019

Abstract

References

  • [1] Abernethy R. B. The New Weibull Handbook, 5th edition. 2006.
  • [2] Nadarajah, S., Cordeiro, G. M. and Ortega E. M. M. The exponentiated Weibull distribution: a survey.Statistical Papers 2012; 54(3), 839-877.
  • [3] Zhang,L. F.,Xie M. and Tang, L.C. On weighted least square estimation for the parameters of Weibull distribution.Recent Advances in Reliability and Quality Design 2008; P. Hoang, Ed., pp. 57–84, Springer, London, UK.
  • [4] Guure, C.B., İbrahim, N.A. and Ahmed, A.O.M. Bayesian Estimation of Two-Parameter Weibull Distribution Using Extension of Jeffreys' Prior Information with Three Loss Functions, Mathematical Problems in Engineering 2012;, 1-13.
  • [5] Al-Kutubi, H. S. and Ibrahim, N. A. Bayes estimator for exponential distribution with extension of Jeffery prior information. Malaysian Journal of Mathematical Sciences 2009; 3(2), 297–313.
  • [6] Jeffreys, H. Theory of Probability. Oxford at the Clarendon Press. 1948.
  • [7] Legendre, A. Nouvelles M´ethodes pour la D´etermination des Orbites des Com`etes. Courcier, Paris. 1805.
  • [8] Gauss, C.F. M´ethode des Moindres Carr´es. M´emoire sur la Combination des Observations. Transl. J. Bertrand (1955). Mallet-Bachelier,Paris. 1810.
  • [9] Varian, H. R. Variants in Economic Theory. Norhampton-USA, Edward Elgar. 2000.
  • [10] Zellner, A. Bayesian estimation and prediction using asymmetric loss functions.Journal of the American Statistical Association 1986; 81(394), 446-451.
  • [11] Calabria, R. and Pulcinia, G. Point estimation under asymmetric loss functions for left-truncated exponential samples. Communications in Statistics - Theory and Methods 1996;, 25(3), 585-600.
  • [12] Dey, D. K. and Pei-San Liao, L. On comparison of estimators in a generalized life model.Microelectronics Reliability 1992; 32(1–2), 207-221.
  • [13] Tierney, L. and Kadane, J. B. Accurate Approximations for Posterior Moments and Marginal Densities.Journal of the American Statistical Association 1986; 81(393), 82-86.
  • [14] Danish, M. Y. ve Aslam, M. Bayesian estimation for randomly censored generalized exponential distribution under asymmetric loss functions, Journal of Applied Statistics 2013; 40 (5), 1106-1119.
There are 14 citations in total.

Details

Primary Language English
Journal Section TJST
Authors

Gülcan Gencer 0000-0002-3543-041X

Kerem Gencer 0000-0002-2914-1056

Publication Date September 27, 2019
Submission Date December 24, 2018
Published in Issue Year 2019 Volume: 14 Issue: 2

Cite

APA Gencer, G., & Gencer, K. (2019). Comparison Of Maximum Likelihood And Bayes Estimators Under Symmetric And Asymmetric Loss Functions By Means Of Tierney Kadane’s Approximation For Weibull Distribution. Turkish Journal of Science and Technology, 14(2), 69-78.
AMA Gencer G, Gencer K. Comparison Of Maximum Likelihood And Bayes Estimators Under Symmetric And Asymmetric Loss Functions By Means Of Tierney Kadane’s Approximation For Weibull Distribution. TJST. September 2019;14(2):69-78.
Chicago Gencer, Gülcan, and Kerem Gencer. “Comparison Of Maximum Likelihood And Bayes Estimators Under Symmetric And Asymmetric Loss Functions By Means Of Tierney Kadane’s Approximation For Weibull Distribution”. Turkish Journal of Science and Technology 14, no. 2 (September 2019): 69-78.
EndNote Gencer G, Gencer K (September 1, 2019) Comparison Of Maximum Likelihood And Bayes Estimators Under Symmetric And Asymmetric Loss Functions By Means Of Tierney Kadane’s Approximation For Weibull Distribution. Turkish Journal of Science and Technology 14 2 69–78.
IEEE G. Gencer and K. Gencer, “Comparison Of Maximum Likelihood And Bayes Estimators Under Symmetric And Asymmetric Loss Functions By Means Of Tierney Kadane’s Approximation For Weibull Distribution”, TJST, vol. 14, no. 2, pp. 69–78, 2019.
ISNAD Gencer, Gülcan - Gencer, Kerem. “Comparison Of Maximum Likelihood And Bayes Estimators Under Symmetric And Asymmetric Loss Functions By Means Of Tierney Kadane’s Approximation For Weibull Distribution”. Turkish Journal of Science and Technology 14/2 (September 2019), 69-78.
JAMA Gencer G, Gencer K. Comparison Of Maximum Likelihood And Bayes Estimators Under Symmetric And Asymmetric Loss Functions By Means Of Tierney Kadane’s Approximation For Weibull Distribution. TJST. 2019;14:69–78.
MLA Gencer, Gülcan and Kerem Gencer. “Comparison Of Maximum Likelihood And Bayes Estimators Under Symmetric And Asymmetric Loss Functions By Means Of Tierney Kadane’s Approximation For Weibull Distribution”. Turkish Journal of Science and Technology, vol. 14, no. 2, 2019, pp. 69-78.
Vancouver Gencer G, Gencer K. Comparison Of Maximum Likelihood And Bayes Estimators Under Symmetric And Asymmetric Loss Functions By Means Of Tierney Kadane’s Approximation For Weibull Distribution. TJST. 2019;14(2):69-78.