Year 2020, Volume 49 , Issue 4, Pages 1493 - 1514 2020-08-06

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

Aynur YONAR YONAR [1] , Nimet YAPICI PEHLİVAN YAPICI [2]


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.
Differential evolution, Gamma distribution, Maximum Likelihood, Least Square, Monte Carlo simulation, Parameter Estimation
  • [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.
Primary Language en
Subjects Statistics and Probability
Journal Section Statistics
Authors

Orcid: 0000-0003-1681-9298
Author: Aynur YONAR YONAR
Institution: SELÇUK UNIVERSITY, SCIENCE FACULTY
Country: Turkey


Orcid: 0000-0002-7094-8097
Author: Nimet YAPICI PEHLİVAN YAPICI (Primary Author)
Institution: SELÇUK UNIVERSITY, SCIENCE FACULTY
Country: Turkey


Supporting Institution Selçuk University
Project Number 2016-OYP-063.
Dates

Publication Date : August 6, 2020

Bibtex @research article { hujms689381, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {1493 - 1514}, doi = {10.15672/hujms.689381}, title = {A novel differential evolution algorithm approach for estimating the parameters of Gamma distribution: An application to the failure stresses of single carbon fibres}, key = {cite}, author = {Yonar, Aynur and Yapıcı Pehli̇van, Nimet} }
APA Yonar, A , Yapıcı Pehli̇van, 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 . DOI: 10.15672/hujms.689381
MLA Yonar, A , Yapıcı Pehli̇van, 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 49 (2020 ): 1493-1514 <https://dergipark.org.tr/en/pub/hujms/issue/56305/689381>
Chicago Yonar, A , Yapıcı Pehli̇van, 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 49 (2020 ): 1493-1514
RIS TY - JOUR T1 - A novel differential evolution algorithm approach for estimating the parameters of Gamma distribution: An application to the failure stresses of single carbon fibres AU - Aynur Yonar , Nimet Yapıcı Pehli̇van Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.689381 DO - 10.15672/hujms.689381 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1493 EP - 1514 VL - 49 IS - 4 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.689381 UR - https://doi.org/10.15672/hujms.689381 Y2 - 2020 ER -
EndNote %0 Hacettepe Journal of Mathematics and Statistics A novel differential evolution algorithm approach for estimating the parameters of Gamma distribution: An application to the failure stresses of single carbon fibres %A Aynur Yonar , Nimet Yapıcı Pehli̇van %T A novel differential evolution algorithm approach for estimating the parameters of Gamma distribution: An application to the failure stresses of single carbon fibres %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 4 %R doi: 10.15672/hujms.689381 %U 10.15672/hujms.689381
ISNAD Yonar, Aynur , Yapıcı Pehli̇van, 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
AMA Yonar A , Yapıcı Pehli̇van 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): 1493-1514.
Vancouver Yonar A , Yapıcı Pehli̇van 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): 1493-1514.