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A novel differential evolution algorithm approach for estimating the parameters of Gamma distribution: An application to the failure stresses of single carbon fibres

Year 2020, Volume: 49 Issue: 4, 1493 - 1514, 06.08.2020

Aynur Yonar Nimet Yapıcı Pehlivan

https://doi.org/10.15672/hujms.689381
Cited By: 2

Abstract

Three-parameter (3-p) Gamma distribution is widely used to model for skewed data in the reliability field. Thus, the problem of parameter estimation for the Gamma distribution has remained significant and interesting in all times. The maximum likelihood (ML) and the least square (LS) are the most popular methods in the parameter estimation. In this study, a novel Differential Evolution (DE) algorithm is proposed for the ML and LS estimation of the parameters of the 3-p Gamma distribution. This approach overcomes the problem of how to determine the search space of the DE by utilizing a new search space based on the confidence interval. The modified maximum likelihood and the profile likelihood methods are considered to constitute the confidence interval. In order to examine the performance of the proposed approach, an extensive Monte Carlo simulation study and a real data application are performed. The results show that this proposed approach is effective for estimating the parameters of the 3-p Gamma distribution with respect to mean square error and deficiency criteria.

Keywords

Differential evolution Gamma distribution Maximum Likelihood Least Square Monte Carlo simulation Parameter Estimation

Supporting Institution

Selçuk University

Project Number

2016-OYP-063.

References

  • [1] Ş. Acıtaş, Ç.H. Aladağ, and B. Şenoglu, A new approach for estimating the parameters of Weibull distribution via particle swarm optimization: An application to the strengths of glass fibre data, Reliability Engineering System Safety, 183, 116-127, 2019.
  • [2] M. Asim, W.M. Khan, Ö. Yeniay, M. A. Jan, N. Tairan, H. Hussian, and G.-G. Wang, Hybrid genetic algorithms for global optimization problems. Hacet. J. Math. Stat., 47 (3), 539-551, 2018.
  • [3] N. Balakrishnan and J., Wang, Simple efficient estimation for the three-parameter gamma distribution, J. Statist. Plann. Inference, 85 (1-2), 115-126, 2000.
  • [4] I. Başak and N. Balakrishnan, Estimation for the three-parameter gamma distribution based on progressively censored data, Stat. Methodol., 9 (3), 305-319, 2012.
  • [5] O.T. Bayrak and A.D. Akkaya, Autoregressive models with stochastic design variables and nonnormal innovations, Recent Researches in Applied Mathematics, Simulation and Modeling, Proceedings of the 5th International Conference on Applied Mathematics, Simulation, Modeling, 197-201, 2011.
  • [6] A.C. Cohen and B.J. Whitten, Modified moment and maximum likelihood estimators for parameters of the three-parameter gamma distribution, Comm. Statist. Simulation Comput., 11 (2), 197-216, 1982.
  • [7] A.C. Cohen and B.J. Whitten, Modified moment estimation for the three-parameter gamma distribution, Journal of Quality Technology, 18 (1), 53-62, 1986.
  • [8] M.J. Crowder, A.C. Kimber, R.L. Smith, and T.J. Sweating, The Statistical Analysis of Reliability Data, Chapman and Hall, London, 1991.
  • [9] S. Das, S.S. Mullick, and P.N. Suganthan, Recent advances in differential evolutionan updated surve, Swarm Evolutionary Computation, 27, 1-30, 2016.
  • [10] H. Hirose, Maximum likelihood parameter estimation in the three-parameter gamma distribution, Comput. Statist. Data Anal., 20,(4) 343-354, 1995.
  • [11] N.L. Johnson, S. Kotz, and N. Balakrishnan, Univariate continuous distributions: New York: John Wiley & Sons, 1994.
  • [12] V. Lakshmi and V. Vaidyanathan, Three-parameter gamma distribution: Estimation using likelihood, spacings and least squares approach, Journal of Statistics Management Systems, 19 (10), 37-53, 2016.
  • [13] W.K. Mashwani, Enhanced versions of differential evolution: state-of-the-art survey, Int. J. Comput. Sci. Math., 5 (2), 107-126, 2014.
  • [14] W.K. Mashwani, et al., Hybrid Constrained Evolutionary Algorithm for Numerical Optimization Problems. Int. J. Comput. Sci. Math., 48 (3), 931-950, 2018.
  • [15] E. Mezura-Montes, M.E. Miranda-Varela, and R. Carmen Gomez-Ramon, Differential evolution in constrained numerical optimization: an empirical study. Inform. Sci., 180 (22), , 4223-4262, 2010.
  • [16] A.W. Mohamed and H.Z. Sabry, Constrained optimization based on modified differential evolution algorithm, Inform. Sci., 194, 171-208, 2012.
  • [17] M.N. Omidvar, X. Li, Y. Mei, and X. Yao, Cooperative co-evolution with differential grouping for large scale optimization. IEEE Trans. Evol. Comput., 18(3), 378-393, 2013.
  • [18] E.O.J. Ouedraogo, B. Some, and S. Dossou-Gbete, On Maximum Likelihood Estimation for the Three Parameter Gamma Distribution Based on Left Censored Samples, Sci. J. Appl. Math. and Stat., 5(4), 147-163, 2017.
  • [19] H. Örkçü, E. Aksoy, and M.I. Doğan, Estimating the parameters of 3-p Weibull distribution through differential evolution, Appl. Math. Comput., 251, 211-224, 2015.
  • [20] V.S. Özsoy, M.G. Ünsal, and H.H. Örkçü, Use of the heuristic optimization in the parameter estimation of generalized gamma distribution: comparison of GA, DE, PSO and SA methods, Comput. Statist. Data Anal., 1-31, 2020.
  • [21] K. Price, R.M. Storn, and J.A. Lampinen, Differential evolution: a practical approach to global optimization: Springer Science and Business Media, 2006.
  • [22] K. Sindhya, S. Ruuska, T. Haanpaa, and K. Miettinen, A new hybrid mutation operator for multiobjective optimization with differential evolution, Soft Computing, 15 (10), 2041-2055, 2011.
  • [23] R. Storn, On the usage of differential evolution for function optimization. in: Fuzzy Information Processing Society, Biennial Conference of the North American, 519-523, IEEE, 1996.
  • [24] R. Storn and K. Price, Differential evolutiona simple and efficient heuristic for global optimization over continuous spaces, J. Global Optim., 11 (4), 341-359, 1997.
  • [25] E.-G. Talbi, Metaheuristics: from design to implementation: John Wiley & Sons, 2009.
  • [26] M.L. Tiku and A.D. Akkaya, Robust estimation and hypothesis testing: New Age International, 2004.
  • [27] G. Tzavelas,Maximum likelihood parameter estimation in the three-parameter gamma distribution with the use of Mathematica. J. Stat. Comput. Simul., 79 (12), 1457-1466, 2009.
  • [28] G. Tzavelas, Estimation in the Three-Parameter Gamma Distribution Based on the Profile Log-Likelihood Function, Comm. Statist. Theory Methods, 38 (5), 573-583, 2009.
  • [29] D. Vaughan and M. Tiku, Estimation and hypothesis testing for a nonnormal bivariate distribution with applications. Math. Comput. Model., 32 (4), 27, 2011. (1-2), 53-67, 2000.
  • [30] A. Yalçınkaya, B. Şenoglu, and U. Yolcu, Maximum likelihood estimation for the parameters of skew normal distribution using genetic algorithm, Swarm and Evolutionary Computation, 38, 127-138, 2018.
  • [31] X.-S. Yang, Engineering optimization: An introduction with metaheuristic applications: John Wiley & Sons, 2010.
  • [32] J.-H. Zhong and J. Zhang, SDE: A stochastic coding differential evolution for global optimization in: Proceedings of the 14th annual conference on Genetic and evolutionary computation, 975-982, ACM, 2012.

Year 2020, Volume: 49 Issue: 4, 1493 - 1514, 06.08.2020

Aynur Yonar Nimet Yapıcı Pehlivan

https://doi.org/10.15672/hujms.689381
Cited By: 2

Abstract

Project Number

2016-OYP-063.

References

  • [1] Ş. Acıtaş, Ç.H. Aladağ, and B. Şenoglu, A new approach for estimating the parameters of Weibull distribution via particle swarm optimization: An application to the strengths of glass fibre data, Reliability Engineering System Safety, 183, 116-127, 2019.
  • [2] M. Asim, W.M. Khan, Ö. Yeniay, M. A. Jan, N. Tairan, H. Hussian, and G.-G. Wang, Hybrid genetic algorithms for global optimization problems. Hacet. J. Math. Stat., 47 (3), 539-551, 2018.
  • [3] N. Balakrishnan and J., Wang, Simple efficient estimation for the three-parameter gamma distribution, J. Statist. Plann. Inference, 85 (1-2), 115-126, 2000.
  • [4] I. Başak and N. Balakrishnan, Estimation for the three-parameter gamma distribution based on progressively censored data, Stat. Methodol., 9 (3), 305-319, 2012.
  • [5] O.T. Bayrak and A.D. Akkaya, Autoregressive models with stochastic design variables and nonnormal innovations, Recent Researches in Applied Mathematics, Simulation and Modeling, Proceedings of the 5th International Conference on Applied Mathematics, Simulation, Modeling, 197-201, 2011.
  • [6] A.C. Cohen and B.J. Whitten, Modified moment and maximum likelihood estimators for parameters of the three-parameter gamma distribution, Comm. Statist. Simulation Comput., 11 (2), 197-216, 1982.
  • [7] A.C. Cohen and B.J. Whitten, Modified moment estimation for the three-parameter gamma distribution, Journal of Quality Technology, 18 (1), 53-62, 1986.
  • [8] M.J. Crowder, A.C. Kimber, R.L. Smith, and T.J. Sweating, The Statistical Analysis of Reliability Data, Chapman and Hall, London, 1991.
  • [9] S. Das, S.S. Mullick, and P.N. Suganthan, Recent advances in differential evolutionan updated surve, Swarm Evolutionary Computation, 27, 1-30, 2016.
  • [10] H. Hirose, Maximum likelihood parameter estimation in the three-parameter gamma distribution, Comput. Statist. Data Anal., 20,(4) 343-354, 1995.
  • [11] N.L. Johnson, S. Kotz, and N. Balakrishnan, Univariate continuous distributions: New York: John Wiley & Sons, 1994.
  • [12] V. Lakshmi and V. Vaidyanathan, Three-parameter gamma distribution: Estimation using likelihood, spacings and least squares approach, Journal of Statistics Management Systems, 19 (10), 37-53, 2016.
  • [13] W.K. Mashwani, Enhanced versions of differential evolution: state-of-the-art survey, Int. J. Comput. Sci. Math., 5 (2), 107-126, 2014.
  • [14] W.K. Mashwani, et al., Hybrid Constrained Evolutionary Algorithm for Numerical Optimization Problems. Int. J. Comput. Sci. Math., 48 (3), 931-950, 2018.
  • [15] E. Mezura-Montes, M.E. Miranda-Varela, and R. Carmen Gomez-Ramon, Differential evolution in constrained numerical optimization: an empirical study. Inform. Sci., 180 (22), , 4223-4262, 2010.
  • [16] A.W. Mohamed and H.Z. Sabry, Constrained optimization based on modified differential evolution algorithm, Inform. Sci., 194, 171-208, 2012.
  • [17] M.N. Omidvar, X. Li, Y. Mei, and X. Yao, Cooperative co-evolution with differential grouping for large scale optimization. IEEE Trans. Evol. Comput., 18(3), 378-393, 2013.
  • [18] E.O.J. Ouedraogo, B. Some, and S. Dossou-Gbete, On Maximum Likelihood Estimation for the Three Parameter Gamma Distribution Based on Left Censored Samples, Sci. J. Appl. Math. and Stat., 5(4), 147-163, 2017.
  • [19] H. Örkçü, E. Aksoy, and M.I. Doğan, Estimating the parameters of 3-p Weibull distribution through differential evolution, Appl. Math. Comput., 251, 211-224, 2015.
  • [20] V.S. Özsoy, M.G. Ünsal, and H.H. Örkçü, Use of the heuristic optimization in the parameter estimation of generalized gamma distribution: comparison of GA, DE, PSO and SA methods, Comput. Statist. Data Anal., 1-31, 2020.
  • [21] K. Price, R.M. Storn, and J.A. Lampinen, Differential evolution: a practical approach to global optimization: Springer Science and Business Media, 2006.
  • [22] K. Sindhya, S. Ruuska, T. Haanpaa, and K. Miettinen, A new hybrid mutation operator for multiobjective optimization with differential evolution, Soft Computing, 15 (10), 2041-2055, 2011.
  • [23] R. Storn, On the usage of differential evolution for function optimization. in: Fuzzy Information Processing Society, Biennial Conference of the North American, 519-523, IEEE, 1996.
  • [24] R. Storn and K. Price, Differential evolutiona simple and efficient heuristic for global optimization over continuous spaces, J. Global Optim., 11 (4), 341-359, 1997.
  • [25] E.-G. Talbi, Metaheuristics: from design to implementation: John Wiley & Sons, 2009.
  • [26] M.L. Tiku and A.D. Akkaya, Robust estimation and hypothesis testing: New Age International, 2004.
  • [27] G. Tzavelas,Maximum likelihood parameter estimation in the three-parameter gamma distribution with the use of Mathematica. J. Stat. Comput. Simul., 79 (12), 1457-1466, 2009.
  • [28] G. Tzavelas, Estimation in the Three-Parameter Gamma Distribution Based on the Profile Log-Likelihood Function, Comm. Statist. Theory Methods, 38 (5), 573-583, 2009.
  • [29] D. Vaughan and M. Tiku, Estimation and hypothesis testing for a nonnormal bivariate distribution with applications. Math. Comput. Model., 32 (4), 27, 2011. (1-2), 53-67, 2000.
  • [30] A. Yalçınkaya, B. Şenoglu, and U. Yolcu, Maximum likelihood estimation for the parameters of skew normal distribution using genetic algorithm, Swarm and Evolutionary Computation, 38, 127-138, 2018.
  • [31] X.-S. Yang, Engineering optimization: An introduction with metaheuristic applications: John Wiley & Sons, 2010.
  • [32] J.-H. Zhong and J. Zhang, SDE: A stochastic coding differential evolution for global optimization in: Proceedings of the 14th annual conference on Genetic and evolutionary computation, 975-982, ACM, 2012.
There are 32 citations in total.

Details

Primary Language English
Subjects Statistics
Journal Section Statistics
Authors

Aynur Yonar

Nimet Yapıcı Pehlivan 0000-0002-7094-8097

Project Number 2016-OYP-063.
Publication Date August 6, 2020
Published in Issue Year 2020 Volume: 49 Issue: 4

Cite

APA Yonar, A., & Yapıcı Pehlivan, N. (2020). A novel differential evolution algorithm approach for estimating the parameters of Gamma distribution: An application to the failure stresses of single carbon fibres. Hacettepe Journal of Mathematics and Statistics, 49(4), 1493-1514. https://doi.org/10.15672/hujms.689381
AMA Yonar A, Yapıcı Pehlivan N. A novel differential evolution algorithm approach for estimating the parameters of Gamma distribution: An application to the failure stresses of single carbon fibres. Hacettepe Journal of Mathematics and Statistics. August 2020;49(4):1493-1514. doi:10.15672/hujms.689381
Chicago Yonar, Aynur, and Nimet Yapıcı Pehlivan. “A Novel Differential Evolution Algorithm Approach for Estimating the Parameters of Gamma Distribution: An Application to the Failure Stresses of Single Carbon Fibres”. Hacettepe Journal of Mathematics and Statistics 49, no. 4 (August 2020): 1493-1514. https://doi.org/10.15672/hujms.689381.
EndNote Yonar A, Yapıcı Pehlivan N (August 1, 2020) A novel differential evolution algorithm approach for estimating the parameters of Gamma distribution: An application to the failure stresses of single carbon fibres. Hacettepe Journal of Mathematics and Statistics 49 4 1493–1514.
IEEE A. Yonar and N. Yapıcı Pehlivan, “A novel differential evolution algorithm approach for estimating the parameters of Gamma distribution: An application to the failure stresses of single carbon fibres”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 4, pp. 1493–1514, 2020, doi: 10.15672/hujms.689381.
ISNAD Yonar, Aynur - Yapıcı Pehlivan, Nimet. “A Novel Differential Evolution Algorithm Approach for Estimating the Parameters of Gamma Distribution: An Application to the Failure Stresses of Single Carbon Fibres”. Hacettepe Journal of Mathematics and Statistics 49/4 (August 2020), 1493-1514. https://doi.org/10.15672/hujms.689381.
JAMA Yonar A, Yapıcı Pehlivan N. A novel differential evolution algorithm approach for estimating the parameters of Gamma distribution: An application to the failure stresses of single carbon fibres. Hacettepe Journal of Mathematics and Statistics. 2020;49:1493–1514.
MLA Yonar, Aynur and Nimet Yapıcı Pehlivan. “A Novel Differential Evolution Algorithm Approach for Estimating the Parameters of Gamma Distribution: An Application to the Failure Stresses of Single Carbon Fibres”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 4, 2020, pp. 1493-14, doi:10.15672/hujms.689381.
Vancouver Yonar A, Yapıcı Pehlivan N. A novel differential evolution algorithm approach for estimating the parameters of Gamma distribution: An application to the failure stresses of single carbon fibres. Hacettepe Journal of Mathematics and Statistics. 2020;49(4):1493-514.

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