On Estimating Parameters of the Kumaraswamy Distribution with Ranked Set Sampling Using Genetic Algorithms

Öz

In this paper, genetic algorithm approach is used to estimate parameters of the Kumaraswamy distribution with maximum likelihood method. In addition ranked set sampling is used since it is expected to give better results in comparison to simple random sampling. Genetic algorithm approach is chosen because it is relatively more convenient in terms of satisfying positivity constraints for the parameters of the Kumaraswamy distribution. Also there is no need to use derivatives in the genetic algorithm approach. Bias, MSE and efficiency is calculated to compare performaces of maximum likelihood estimators for ranked set sampling and simple random sampling obtained by using genetic algorithms. The R software and related packages are preferred for calculations in the simulation study.

Anahtar Kelimeler

• Kumaraswamy distribution,Ranked set sampling,Maximum likelihood estimation,Genetic algorithm,R software

Sıralı Küme Örneklemesi ile Kumaraswamy Dağılımı Parametrelerinin Tahmin Edilmesinde Genetik Algoritma Kullanılması

Öz

Bu çalışmada, Kumaraswamy dağılımının parametrelerinin en çok olabilirlik yöntemi ile tahmin edilmesi genetik algoritma yaklaşımı kullanılarak araştırılmıştır. Ayrıca basit rasgele örneklemeye göre daha iyi sonuç verebileceği düşünülerek parametrelerin tahmin edilmesinde sıralı küme örneklemesi de incelenmiştir. Genetik algoritma yaklaşımı, Kumaraswamy dağılımı parametrelerinin pozitif olma koşulunun hesaba katılması nedeniyle tercih edilmiştir. Ek olarak genetik algoritma yaklaşımında en çok olabilirlik fonksiyonunun türev bilgisine ihtiyaç duyulmaması da hesaplamalarda kolaylık sağlamaktadır. Genetik algoritma kullanılarak elde edilen her iki örnekleme yöntemine ait olabilirlik tahmin edicilerinin performanslarının karşılaştırılması için yan, hata kareler ortalaması ve etkinlikleri hesaplanmıştır. Simülasyon çalışmasındaki hesaplamalar için R yazılımı ve ilgili paketler kullanılmıştır.

Anahtar Kelimeler

• Kumaraswamy dağılımı,Sıralı küme örneklemesi,En çok olabilirlik tahmini,Genetik algoritma

Kaynakça

1. [1] Kumaraswamy, P. 1980. A generalized Probability Density Function for Double-Bounded Random Processes. Journal of Hydrology, 46(1980), 79-88.
2. [2] Jones, M. C. 2009. Kumaraswamy’s distribution: A beta-type distribution with some tractability advantages. Statistical Methodology, 6(2009), 70-81.
3. [3] Hussian, M. A. 2014. Bayesian and Maximum Likelihood Estimation for Kumaraswamy Distribution based on Ranked Set Sampling. American Journal of Mathematics and Statistics, 4(2014), 30-37.
4. [4] McIntyre, G. A. 1952. A Method for Unbiased Selective Sampling, using Ranked Sets. Australian Journal of Agricultural Research, 1952 385-390.
5. [5] Patil, G. P., Surucu, B. and Egemen D. 2002. Ranked set sampling. Wiley StatsRef: Statistics Reference Online, 2002.
6. [6] Takahasi, K., and Wakimoto, K. 1968. On unbiased estimates of the population mean based on the sample stratifed by means of ordering. Annals of the Institute of Statistical Mathematics, 1968, 1-31.
7. [7] Dell, T. R., and Clutter, J. L. 1972. Ranked set sampling theory with order statistics background. Biometrics, 1972, 545-555.
8. [8] Stokes, S. L. 1977. Ranked Set Sampling with Concomitant Variables. Communications in Statistics-Theory and Methods, 1977, 1207-1211.

9. [9] Samawi, H. M. 1996. Stratified Ranked Set Sample. Pakistan Journal of Statıstıcs-All Serıes, 12(1996), 9-16.
10. [10] Samawi, H. M., Ahmed, M. S., and Abu-Dayyeh, W. 1996. Estimating the Population Mean using Extreme Ranked Set Sampling. Biometrical Journal, 38(1996), 577-586.
11. [11] Al-Saleh, M. F., and Al-Kadiri M. A. 2000. Double-Ranked Set Sampling. Statistics & Probability Letters, 48(2000), 205-212.
12. [12] Al- Saleh, M. F., and Al-Omari A. I. 2002. Multistage Ranked Set Sampling. Journal of Statistical planning and Inference, 102(2002), 273-286.
13. [13] Muttlak, H. A. 2003. Investigating the Use of Quartile Ranked Set Samples for Estimating the Population Mean. Applied Mathematics and Computation, 146(2003), 437-443.
14. [14] Holland, J.H. 1975. Adaptation in Natural and Artifcial Systems. MIT Press.
15. [15] Goldberg, D. 1989. Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley.
16. [16] Chong, E. KP., Zak, S. H. 2013. An Introduction to Optimization. 2nd, John Wiley & Sons.
17. [17] R Core Team (2017). R: A language and environment for statistical computing., R Foundation for Statistical Computing, Vienna, Austria., [Çevrimiçi]. Available: https://www.R-project.org/.
18. [18] C. D. Marie Laure Delignette-Muller, fitdistrplus: An R Package for Fitting Distributions, Journal of Statistical Software 64(4), 1-34, 2015. [Çevrimiçi]. Available: http://www.jstatsoft.org/v64/i04/.
19. [19] R. V. John C. Nash, Unifying Optimization Algorithms to Aid Software System Users: optimx for R, Journal of Statistical Software, 43(9), 1-14, 2011. [Çevrimiçi]. Available: http://www.jstatsoft.org/v43/i09/.
20. [20] Scrucca, L., GA: A Package for Genetic Algorithms in R., Journal of Statistical Software, 53(4), 1-37., 2013. [Çevrimiçi]. Available: http://www.jstatsoft.org/v53/i04/.
21. [21] Yee, T. W., The VGAM Package for Categorical Data Analysis., Journal of Statistical Software, 32(10), 1-34., 2010. [Çevrimiçi]. Available: http://www.jstatsoft.org/v32/i10/.