On Parameter Estimation with the Method of Moments

Yıl 2022, , 847 - 854, 31.12.2022

Yunus Bulut , Ahmet Demiralp

https://doi.org/10.35193/bseufbd.1091550

Öz

The method of moments uses to estimate the parameters of a statistical model. This method aims to find the values of the parameters which are solutions of equations given by relationship between the sample moments and the population moments. The Method of Moments, which is the first estimation method known in the literature, was first introduced by Pearson. This method is constantly used because of its applicability, simplicity and understanding. In this study, the estimators of the unknown parameters of the Binomial, Poisson, Continuous Uniform, and Gamma distributions are obtained by the Moments Method and the actual values and the estimation values are compared by simulating random data for the given distributions.

Anahtar Kelimeler

Parameter Estimation, Method of Moments, Binomial Distribution, Poisson Distribution, Gamma Distribution

Momentler Metodu ile Parametre Tahmini Üzerine

Öz

Momentler Metodu, bir istatistiksel modelin parametrelerini tahmin etmek için kullanılır. Bu yöntem örnek momentleri ile anakütle momentleri arasındaki ilişki ile verilen denklemlerin çözümü ile parametrelerin değerlerini bulmayı amaçlar. Literatürde bilinen ilk tahmin yöntemi olan Momentler Metodu ilk olarak Pearson tarafından ortaya atılmıştır. Uygulanabilirliği, basit ve anlaşılır olmasından dolayı sürekli başvurulan bir yöntemdir. Bu çalışmada, Binom, Poisson, Sürekli Düzgün ve Gamma dağılımlarının bilinmeyen parametrelerinin tahmincileri Momentler Metodu ile elde edilmiş ve verilen dağılımlar için tesadüfi veriler simüle edilerek gerçek değerleri ile tahmin değerleri karşılaştırılmıştır.

Anahtar Kelimeler

Parametre Tahmini, Momentler Metodu, Binom Dağılımı, Poisson Dağılımı, Gamma Dağılımı

Kaynakça

• Öztürk, F., Akdi, Y., Aydoğdu, H., & Karabulut, İ. (2015). Parametre Tahmini ve Hipotez Testi. Gazi Kitapevi, Ankara, 299.
• Accrachi, E. O., Dabye, A. S., & Gounoung, A. A. (2018). On the Parameter Estimation by Method of Moments and Wald Type Test for Poisson Processes. In A Collection of Papers in Mathematics and Related Sciences, 1, 57-74). Statistics and Probability African Society.
• Bickel, P. J., Chen, A., & Levina, E. (2011). The method of moments and degree distributions for network models. The Annals of Statistics, 39(5), 2280-2301.
• Bücher, A., & Jennessen, T. (2020). Method of moments estimators for the extremal index of a stationary time series. Electronic Journal of Statistics, 14(2), 3103-3156.
• Wang, X., & Peng, Z. (2014). Method of moments for estimating uncertainty distributions. Journal of Uncertainty Analysis and Applications, 2(1), 1-10.
• Wikström, D. (2015). Consistent method of moments estimation of the true fixed effects model. Economics Letters, 137, 62-69.
• Wang, L., & Hsiao, C. (2011). Method of moments estimation and identifiability of semiparametric nonlinear errors-in-variables models. Journal of Econometrics, 165(1), 30-44.
• Chen, R., Zhou, H., Yu, L., Jin, C., & Zhang, S. (2021). An efficient method for pricing foreign currency options. Journal of International Financial Markets, Institutions and Money, 74, 101295.
• Negri, I., & Nishiyama, Y. (2020). Change point detection based on method of moment estimators. arXiv preprint arXiv:2010.03334.
• Aoki, R., Bolfarine, H., & Singer, J. M. (2002). Asymptotic efficiency of method of moments estimators under null intercept measurement error regression models. Brazilian Journal of Probability and Statistics, 157-167.
• Carpenter, M., & Mishra, S. N. (2001). Bootstrap Bias Adjusted Estimators of Beta Distribution Parameters. Calcutta Statistical Association Bulletin, 51(1-2), 119-124.
• Abarin, T., Li, H., Wang, L., & Briollais, L. (2014). On method of moments estimation in linear mixed effects models with measurement error on covariates and response with application to a longitudinal study of gene-environment interaction. Statistics in Biosciences, 6(1), 1-18.
• Al Hayek, N. (2021). Parameter Estimation for Discrete Laplace Distribution. Lobachevskii Journal of Mathematics, 42(2), 368-373.
• Lee, R. S. (2019). Improving Standard Moment Estimators of Beta Random Variables. Academia Economic Papers, 47(4), 547-570.
• Jangphanish, K., & Budsaba, K. (2013). Parameter Estimation for Re-Parametrized Inverse Gaussian Distribution. Science & Technology Asia, 43-53.
• Dabye, A. S., Gounoung, A. A., & Kutoyants, Y. A. (2018). Method of Moments Estimators and Multi–step MLE for Poisson Processes. Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 53(4), 237-246.
• Gu, M., & Abraham, D. A. (1999, September). Parameter estimation for McDaniel's non-Rayleigh reverberation model. In Oceans' 99. MTS/IEEE. Riding the Crest into the 21st Century. Conference and Exhibition. Conference Proceedings (IEEE Cat. No. 99CH37008), 1, 279-283.