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## entrMaximum Likelihood Estimation of the Parameters of Progressively Type-2 Censored Samples From Weibull Distribution Using Genetic AlgorithmGenetik Algoritma ile İlerleyen Tür Tip 2 Sansürlü Örneklemlerde Weibull Dağılımının Parametrelerinin En Çok Olabilirlik Tahmini

#### Aydın Karakoca [1] , Ahmet Pekgör [2]

##### 66 148

In this study we suggested an alternative solution to the parameter estimation problem of the Weibull distribution based on progressively Type-II censored samples with Newton method. Newton is one of the widely used methods for solving the system of equations especially in maximum likelihood estimation. Even though it is popular, the biggest disadvantage of the Newton method is that it is a valid method for only functions that derivativable at least two times. Since the likelihood functions are in more complex form for censored samples than in full samples, calculations of derivatives and related processes are more complicated. We proposed to use the Genetic Algorithm an alternative to the limitations of the Newton method in solution of system of equations in maximum likelihood estimation. Performance of these methods are evaluated by the simulated bias and mean square error criteria by an intensive simulation study. Simulation results of the study showed that the suggested method give better results than Newton method for scale parameter for all conditions. Also shape parameter results for simulated biases are similar for GA and Newton method but Newton has better mean squared error values for some censoring schemes.

Bu çalışmada Weibull dağılımına sahip ilerleyen tür tip 2 sansürlü örneklemlerde parametre tahmini probleminde Newton yöntemine alternatif bir çözüm önerilmiştir. Newton yöntemi en çok olabilirlik tahmininde sıklıkla kullanılmaktadır. Newton yöntemi popüler olmasına rağmen en büyük dezavantajı en az iki kez türevlenebilir fonksiyonlar için kullanılabilmesidir. Olabilirlik fonksiyonu sansürlü örneklemlerde tam örneklemlere göre fonksiyonel olarak daha kompleks bir yapıda olduğundan, türev ve diğer hesaplamalar nispeten daha karışıktır. Bu çalışmada en çok olabilirlik yönteminde elde edilen denklem sisteminin çözümü için Newton metodunun kullanımındaki kısıtlamalara bir alternatif olarak Genetik Algoritma önerilmiştir. Detaylı bir simülasyon çalışması yardımıyla yan ve hata kareler ortalaması ile iki yöntemin performansları değerlendirilmiştir. Simülasyon sonuçlarına göre önerilen yöntemin karşılaştırılan tüm durumlar için ölçek parametresi için daha iyi sonuçlar verdiği, şekil parametresi için ise yanlar açısından sonuçların benzer olduğu ancak hata kareler ortalamasına göre bazı sansür şemaları için Newton yönteminin iyi sonuç verdiği bulunmuştur.

• [1] A. C. Cohen, "Maximum likelihood estimation in the Weibull distribution based on complete and on censored samples," Technometrics, vol. 7, no. 4, pp. 579-588, 1965.
• [2] D. R. Thomas and W. M. Wilson, "Linear order statistic estimation for the two-parameter Weibull and extreme-value distributions from type II progressively censored samples," Technometrics, vol. 14, no. 3, pp. 679-691, 1972.
• [3] N. R. Mann, "Best linear invariant estimation for Weibull parameters under progressive censoring," Technometrics, vol. 13, no. 3, pp. 521-533, 1971.
• [4] R. Viveros and N. Balakrishnan, "Interval estimation of parameters of life from progressively censored data," Technometrics, vol. 36, no. 1, pp. 84-91, 1994.
• [5] N. Balakrishnan, N. Kannan and C. Lin, "Point and interval estimation for Gaussian distribution, based on progressively," IEEE Transactions on Reliability, vol. 52, no. 1, pp. 90-95, 2003.
• [6] R. R. Abu Awwad, M. Z. Raqap and M. A.-M. Intesar, "Statistical inference based on progressively type II censored data from Weibull model," Communications in Statistics-Simulation and Computation, vol. 44, no. 10, pp. 2654-2670, 2015.
• [7] H. K. T. Ng, "Parameter estimation for a modified Weibull distribution, for progressively type-II censored samples," IEEE Transactions on Reliability, vol. 54, no. 3, pp. 374-380, 2005.
• [8] Y. Hak-Keung and T. Siu-Keung, "Parameters estimation for weibull distributed lifetimes under progressive censoring with random removeals," Journal of Statistical Computation and Simulation, vol. 55, no. 1-2, pp. 57-71, 1996.
• [9] S.-J. Wu, "Estimations of the parameters of the Weibull distribution with progressively censored data," Journal of the Japan Statistical Society, vol. 32, no. 2, pp. 155-163, 2002.
• [10] H. Ng, P. Chan and N. Balakrishnan, "Estimation of parameters from progressively censored data using EM algorithm," Computational Statistics & Data Analysis, vol. 39, no. 4, pp. 371-386, 2002.
• [11] W. Weibull, "A Statistical distribution function of Wide Applicability," ASME Journal of applied Mechanics, pp. 293-297, 1951.
• [12] C. G. Broyden, "Quasi-Newton methods and their application to function minimisation," Mathematics of Computation, vol. 21, no. 99, pp. 368-381, 1967.
• [13] R. Fletcher and M. Powell, "A rapidly convergent descent method for minimization," The computer journal, vol. 6, no. 2, pp. 163-168, 1963. [14] D. F. Shanno, "Conditioning of quasi-Newton methods for function minimization," Mathematics of computation, vol. 24, no. 111, pp. 647-656, 1970.
• [15] D. E. Goldberg and J. H. Holland, "Genetic algorithms and machine learning," Machine learning, vol. 3, no. 2, pp. 95-99, 1988.
• [16] N. Balakrishnan and R.A.Sandhu, "A Simple Simulation algorithm for Generating Progressive Type-II Censored Samples," The American Statistician, vol. 2, no. 49, pp. 229-230, 1995.
• [17] M. Burkschat, E. Cramer and U. Kamps, "Optimality criteria and optimal schemes in progressive censoring," Communications in Statistics—Theory and Methods, vol. 36, no. 7, pp. 1419-1431, 2007.
Primary Language en Engineering Articles Orcid: 0000-0001-6503-3872Author: Aydın Karakoca (Primary Author)Institution: NECMETTİN ERBAKAN ÜNİVERSİTESİCountry: Turkey Orcid: 0000-0001-9446-7960Author: Ahmet PekgörInstitution: NECMETTİN ERBAKAN ÜNİVERSİTESİCountry: Turkey Publication Date: May 25, 2019
 Bibtex @research article { apjes452564, journal = {Akademik Platform Mühendislik ve Fen Bilimleri Dergisi}, issn = {}, eissn = {2147-4575}, address = {Academic Platform}, year = {2019}, volume = {7}, pages = {189 - 199}, doi = {10.21541/apjes.452564}, title = {Maximum Likelihood Estimation of the Parameters of Progressively Type-2 Censored Samples From Weibull Distribution Using Genetic Algorithm}, key = {cite}, author = {Karakoca, Aydın and Pekgör, Ahmet} } APA Karakoca, A , Pekgör, A . (2019). Maximum Likelihood Estimation of the Parameters of Progressively Type-2 Censored Samples From Weibull Distribution Using Genetic Algorithm. Akademik Platform Mühendislik ve Fen Bilimleri Dergisi, 7 (2), 189-199. DOI: 10.21541/apjes.452564 MLA Karakoca, A , Pekgör, A . "Maximum Likelihood Estimation of the Parameters of Progressively Type-2 Censored Samples From Weibull Distribution Using Genetic Algorithm". Akademik Platform Mühendislik ve Fen Bilimleri Dergisi 7 (2019): 189-199 Chicago Karakoca, A , Pekgör, A . "Maximum Likelihood Estimation of the Parameters of Progressively Type-2 Censored Samples From Weibull Distribution Using Genetic Algorithm". Akademik Platform Mühendislik ve Fen Bilimleri Dergisi 7 (2019): 189-199 RIS TY - JOUR T1 - Maximum Likelihood Estimation of the Parameters of Progressively Type-2 Censored Samples From Weibull Distribution Using Genetic Algorithm AU - Aydın Karakoca , Ahmet Pekgör Y1 - 2019 PY - 2019 N1 - doi: 10.21541/apjes.452564 DO - 10.21541/apjes.452564 T2 - Akademik Platform Mühendislik ve Fen Bilimleri Dergisi JF - Journal JO - JOR SP - 189 EP - 199 VL - 7 IS - 2 SN - -2147-4575 M3 - doi: 10.21541/apjes.452564 UR - https://doi.org/10.21541/apjes.452564 Y2 - 2018 ER - EndNote %0 Academic Platform Journal of Engineering and Science Maximum Likelihood Estimation of the Parameters of Progressively Type-2 Censored Samples From Weibull Distribution Using Genetic Algorithm %A Aydın Karakoca , Ahmet Pekgör %T Maximum Likelihood Estimation of the Parameters of Progressively Type-2 Censored Samples From Weibull Distribution Using Genetic Algorithm %D 2019 %J Akademik Platform Mühendislik ve Fen Bilimleri Dergisi %P -2147-4575 %V 7 %N 2 %R doi: 10.21541/apjes.452564 %U 10.21541/apjes.452564 ISNAD Karakoca, Aydın , Pekgör, Ahmet . "Maximum Likelihood Estimation of the Parameters of Progressively Type-2 Censored Samples From Weibull Distribution Using Genetic Algorithm". Akademik Platform Mühendislik ve Fen Bilimleri Dergisi 7 / 2 (May 2019): 189-199. https://doi.org/10.21541/apjes.452564 AMA Karakoca A , Pekgör A . Maximum Likelihood Estimation of the Parameters of Progressively Type-2 Censored Samples From Weibull Distribution Using Genetic Algorithm. APJES. 2019; 7(2): 189-199. Vancouver Karakoca A , Pekgör A . Maximum Likelihood Estimation of the Parameters of Progressively Type-2 Censored Samples From Weibull Distribution Using Genetic Algorithm. Akademik Platform Mühendislik ve Fen Bilimleri Dergisi. 2019; 7(2): 199-189.