Some Robust Estimation Methods and Their Applications

Year 2015, , 73 - 82, 30.12.2015

Tolga Zaman Kamil Alakuş

https://doi.org/10.17093/aj.2015.3.2.5000152703

Abstract

This study examines robust regression methods which are used for the solution of problems caused by the situations in which the assumptions of LSM technique, which is commonly used for the prediction of linear regression models, cannot be used. Robust estimators are not influenced by small deviations and discrepancies. For this purpose, some robust regression techniques which are used in situations in which the assumptions cannot be made were introduced and parameter estimation algorithms of these techniques were analyzed. Regression models of the methods of Lad, Weighted –M regression, Theil regression and Least Median Squares, coefficients of determination and average absolute deviations were calculated and the results were discussed as to which of these methods gave better results. This study examines robust regression methods which are used for the solution of problems caused by the situations in which the assumptions of LSM technique, which is commonly used for the prediction of linear regression models, cannot be used. Robust estimators are not influenced by small deviations and discrepancies. For this purpose, some robust regression techniques which are used in situations in which the assumptions cannot be made were introduced and parameter estimation algorithms of these techniques were analyzed. Regression models of the methods of Lad, Weighted –M regression, Theil regression and Least Median Squares, coefficients of determination and average absolute deviations were calculated and the results were discussed as to which of these methods gave better results.

Bazı Robust Tahmin Yöntemleri Ve Uygulamaları

Abstract

Bu çalışmada doğrusal regresyon modellerinin tahmininde yaygın olarak kullanılan EKK tekniğinin varsayımlarının sağlanmamasından kaynaklanan problemlerin çözümü için kullanılan Robust regresyon yöntemleri incelenmiştir. Robust tahmin ediciler küçük sapmalardan, aykırılıklardan etkilenmezler. Bu amaçla, çalışmada varsayımların sağlanmadığı durumlarda kullanılan bazı robust regresyon teknikleri tanıtılmıştır ve bu tekniklere ait parametre tahmin algoritmaları incelenmiştir. Uygulamada Lad, Ağırlıklı –M regresyon, Theil regresyon ve En küçük Medyan Kareler yöntemlerine ait regresyon modeli, belirleme katsayıları ve ortalama mutlak sapmalar hesaplanmış olup, bu tahmin edicilerden hangisinin daha iyi sonuç verdiği tartışılmıştır.

References

• Birkes, D. and Dodge, Y. (1993), Alternative Methods of Regression. NY: Wiley.
• Clarke, G. M. and Cooke, D., (1992). A Basic Course in Statistics. 3rd Edition. P. 354-356, Exercises on Chapter 20, Excercis No: 9.
• Dietz, E. J. (1989), Teaching Regression in a Nonparametric Statistics Course. The American Statistician. 43, 35-40.
• Ergül, B. (2006), Robust Regresyon ve Uygulamaları. Eskişehir Osmangazi Üniversitesi Fen Bilimleri Enstitüsü İstatistik Anabilim Dalı Yüksek Lisans Tezi, Eskişehir.
• G. E., Granato. (2006), Kendall-Theil Robust Line (KTRLine-version 1.0) –A Visual Basic Program for Calculating and Graphing Robust Nonparametric Estimates of Linear-Regression Coefficients Between Two Continous Variables. Chapter 7, Section A, Statistical Anlaysis, Book 4, Hydrologic Analysis and Interpretation. U. S. Geological Survey Techniques and Methods 4-A7.
• Genceli, M. (2001), Ekonomide İstatistik İlkeler, İstanbul, Filiz Kitabevi.
• Huber, P. J. (1964), Robust Estimation of a Location Parameter. Ann. Math. Statist., 35, 73-101.
• Jabr, R. (2005), Power System Huber-M Estimation with Equilaty and Inequality Constraints, Electric Power System Research, 74, 239-246.
• Kıroğlu, G. (2001), Uygulamalı Parametrik Olmayan İstatistiksel Yöntemler. Mimar Sinan Üniversitesi Fen-Edebiyat Fakültesi, İstanbul.
• N. A., Erilli and K., Alakus. (2014), Non-Parametrıc Regression Estimation for Data With Equal Values. European Scientific Journal. February. Edition Vol. 10, No.4 ISSN:1857-7881 (Prınt) e-ISSN 1857-7434.
• Neter, J., Kutner, M. H., Nachtheim, C. J., Wasserman, W. (1993), Applied Linear Statistical Methods, Wiley.
• Rousseeuw, P. J. and Leroy, A. M. (1987), Robust Regression and Outlier Detection. New York: John Wiley& Sons, Inc.
• R. R., Wilcox. (2013), A Heteroscedastic Method for Comparing Regression Lines at Specified Design Points When Using a Robust Regression Estimator. Journal of Data Science 11,281-291.
• Sen, P. K. (1968), Estimates of the regression coefficient based on Kendall’s tau., J.Amer. Statist. Assoc., 63, 1379-1389.
• Sprent, R. (1993), Applied nonparametric statistical methods. 2 nd Ed. CRC Press, NY.
• Theil, H. (1950), A rank-invariant method of lineer and polynomial regression analysis, I. Proc. Kon. Ned. Akad. v . Wetensch. A53, 386-392.
• Yorulmaz, Ö. (2003), Robust Regresyon ve Mathematica Uygulamaları. Marmara Üniversitesi, Yüksek Lisans Tezi, Ankara.
• Wilcox, R. (1998), Simulations on the Theil-Sen regression estimator with right-censored data. Stat.& Prob. Letters 39, 43-47.