Parameter Estimation of Probability Distributions Using Bernstein and Rational Bernstein Polynomial-Based Approaches

Yıl 2025, Cilt: 25 Sayı: 6, 1316 - 1322

Mahmut Sami Erdoğan

Öz

This study examines two different approaches based on Bernstein polynomials and rational Bernstein polynomials for parameter estimation of probability distributions. It is discussed how both methods can be used in the parameter estimation process, and it is aimed to determine the optimal parameters with the least squares method. Monte Carlo simulations are performed to evaluate the effectiveness of the methods, and their estimation performances are analyzed for various distributions. Simulation results demonstrate that rational Bernstein polynomials achieve lower mean squared error values, which consequently raise parameter estimation accuracy through enhanced flexibility.

Anahtar Kelimeler

Bernstein polynomials , Rational Bernstein Polynomials , Parameter Estimation , Nonparametric Estimation , Mean Squared Error

Bernstein ve Rasyonel Bernstein Polinom Tabanlı Yaklaşımlar Kullanılarak Olasılık Dağılımlarının Parametre Tahmini

Öz

Bu çalışmada olasılık dağılımlarının parametre kestirimi için Bernstein polinomları ve rasyonel Bernstein polinomlarına dayalı iki farklı yaklaşım incelenmektedir. Her iki yöntemin parametre kestirim sürecinde nasıl kullanılabileceği tartışılmakta ve en küçük kareler yöntemi ile optimum parametrelerin belirlenmesi amaçlanmaktadır. Yöntemlerin etkinliğini değerlendirmek için Monte Carlo simülasyonları gerçekleştirilmekte ve çeşitli dağılımlar için kestirim performansları analiz edilmektedir. Simülasyon sonuçları rasyonel Bernstein polinomlarının daha düşük ortalama karesel hata değerlerine ulaştığını ve bunun sonucunda artan esneklik yoluyla parametre kestirim doğruluğunu artırdığını göstermektedir.

Anahtar Kelimeler

Bernstein polinomları , Rasyonel Bernstein polinomları , Parametre kestirimi , Parametrik olmayan kestirim , Ortalama karesel hata

Kaynakça

• Babu, G.J., Canty, A.J., Chaubey, Y.P., 2002. Application of Bernstein polynomials for smooth estimation of a distribution and density function. Journal of Statistical Planning and Inference, 105(2), 377–392. https://doi.org/10.1016/S0378-3758(01)00265-8
• Babu, G.J., Chaubey, Y.P., 2006. Smooth estimation of a distribution and density function on a hypercube using Bernstein polynomials for dependent random vectors. Statistics & Probability Letters, 76(9), 959–969. https://doi.org/10.1016/j.spl.2005.10.031
• Barry, P.J., Beatty, J.C., Goldman, R.N., 1992. Unimodal properties of B-spline and Bernstein-basis functions. Computer-Aided Design, 24(12), 627–636. https://doi.org/10.1016/00104485(92)90017-5
• Bernšteín, S., 1912. Démonstration du théoreme de Weierstrass fondée sur le calcul des probabilités. Communication of the Kharkov Mathematical Society, 13, 1–2.
• Budakçı, G., Oruç, H., 2012. Bernstein–Schoenberg operator with knots at the q-integers. Mathematics and Computer Modelling, 56(3–4), 56–59. https://doi.org/10.1016/j.mcm.2011.12.049
• Carnicero, J.A., Wiper, M.P. and Ausín, M.C., 2018. Density estimation of circular data with Bernstein polynomials. Hacettepe Journal of Mathematics and Statistics, 47(2), 273–286. https://doi.org/10.15672/HJMS.2014437525
• Cordeiro, G.M. and Brito, R.S., 2012. The beta power distribution. Brazilian Journal of Probability and Statistics, 26(1), 88–112. https://doi.org/10.1214/10-BJPS124
• Erdoğan, M.S., Dişibüyük, Ç., Oruç, Ö.E., 2019. An alternative distribution function estimation method using rational Bernstein polynomials. Journal of Computational and Applied Mathematics, 353, 232–242. https://doi.org/10.1016/j.cam.2018.12.033
• Ghosal, S., 2001. Convergence rates for density estimation with Bernstein polynomials. The Annals of Statistics, 29(5), 1264–1280. https://doi.org/10.1214/aos/1013203453
• Kakizawa, Y., 2004. Bernstein polynomial probability density estimation. Journal of Nonparametric Statistics, 16(5), 709–729. https://doi.org/10.1080/1048525042000191486
• Korkmaz, M.Ç., Altun, E., Alizadeh, M. and El-Morshedy, M., 2021. The Log Exponential-Power Distribution: Properties, estimations and quantile regression model. Mathematics, 9(21), 2634. https://doi.org/10.3390/math9212634
• Korkmaz, M.Ç., Karakaya, K. and Akdoğan, Y., 2022. Parameter estimation procedures for log exponential-power distribution with real data applications. Adıyaman University Journal of Science, 12(2), 193–202. https://doi.org/10.37094/adyujsci.1073616
• Kutner, M.H., Nachtsheim, C.J., Neter, J. and Li, W., 2005. Applied linear statistical models. McGraw-Hill.
• Lorentz, G. G., 1986. Bernstein polynomials. American Mathematical Society.
• Oruç, H., Phillips, G.M., Davis, P.J., 1999. A generalization of the Bernstein polynomials. Proceedings of the Edinburgh Mathematical Society, 42(2), 403–413. https://doi.org/10.1017/S0013091500020332
• Petrone, S., 1999. Bayesian density estimation using Bernstein polynomials. Canadian Journal of Statistics, 27(1), 105–126. https://doi.org/10.2307/3315494
• Petrone, S. and Wasserman, L., 2002. Consistency of Bernstein polynomial posteriors. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64(1), 79–100. https://doi.org/10.1111/1467-9868.00326
• Turnbull, B.C., Ghosh, S.K., 2014. Unimodal density estimation using Bernstein polynomials. Computational Statistics & Data Analysis, 72, 13–29. https://doi.org/10.1016/j.csda.2013.10.021
• Vitale, R.A., 1975. A Bernstein polynomial approach to density function estimation. Statistical Inference and Related Topics, Elsevier, pp. 87–99. https://doi.org/10.1016/B978-0-12-568002-8.50011-2