The Performances of Spline and Kernel Regression Estimators in Models with Correlated Errors

Yıl 2012, Cilt: 9 Sayı: 2, 20 - 28, 15.08.2012

Serdar Demir , Dursun Aydın

Öz

It is well known that the performances of nonparametric regression estimators severely decrease in the models with correlated errors. In this paper, the performances of the cubic smoothing spline and the Nadaraya-Watson kernel estimators, which are often used for nonparametric estimation of regression function, are investigated in the models with correlated errors. For this purpose, a simulatian study is performed and the mean square errors of the estimators are compared. The simulatian results show that the Nadaraya-Watson kernel estimator performs well in small samples, while spline estimator performs well in large samples.

Anahtar Kelimeler

Kernel estimator, Correlated errors, Cubic spline, Nadaraya-Watson

İlişkili Hatalara Sahip Modellerde Splayn ve Çekirdek Regresyon Kestiricilerinin Performansları

Öz

Parametrik olmayan regresyon kestiricilerinin performanslarının, ilişkili hatalara sahip modellerde ciddi biçimde azaldığı iyi bilinmektedir. Bu çalışmada, regresyon fonksiyonunun parametrik olmayan kestirimi için sıkça kullanılan kübik düzleştirme splaynı ile Nadaraya-Watson çekirdek kestiricilerinin hatalı ilişkilere sahip modellerdeki performansları incelenmiştir. Bunun için bir benzetim çalışması gerçekleştirilerek elde edilen kestirimlerin hata kareler ortalamaları karşılaştırılmıştır. Benzetim sonuçlarına göre, Nadaraya-Watson çekirdek kestirisi küçük örneklemlerde iyi performans gösteriyorken, splayn kestirisi büyük örneklemlerde iyi performans göstermektedir.

Anahtar Kelimeler

Çekirdek kestiricisi, ilişkili hatalar, Kübik splayn, Nadaraya-Watson

Kaynakça

• Altman, N. S., 1990. Kernel Smoothing of Data with Correlated Errors, JASA, 85,749-759.
• Carroll, R. J., Linton, O., Mammen, E., Xiao, Z., 2002. More Efficient Kernel Estimation in Nonparametric Regression with Autocorrelated Errors, STlCERD Econometrics Paper Series, 2002/435.
• Chu, C. K., Marron, J. S. 1991. Comparison of Two Bandwidth Selectors with Dependent Errors, Ann. Statist., 19, 1906-1918.
• De Brabanter, K., De Brabanter, J., Suykens, J., De Moor, B., 2010. Kernel Regression with Correlated Errors. Proc. of the 11 th International Symposiurn on Computer Applications in Biotechnology (CAB), Leuven, Belgiurn, Jul. 2010, 13-18.
• De Gooijer, J. G., Gannoun, A., Larramendy, I., 2002. Nonparametric Regression with Serially Correlated Errors. Ann. Ins. Staı. Univ. Paris, 46, Fasc. 1 -2.
• Diggle, P. J , Hutchinson, M. F., 1989. On Spline Smoothing With Autocorrelated Errors. Australian Journal of Statistics. 31. 166-182.
• Eubank, R., 1999. Nonparametric Regression and Spline Smoothing, New York, Dekker.
• Green, D. J., Silverman, B. W., 1994. Nonparametric Regression and Generalized Linear Models, London, Chapmanı & Hall.
• Hardle, W., 1991. Applied Nonparametric Regression, Cambridge University Press, Cambridge.
• Hart, J. D., 1991. Kernel Regression Estimation with Time Series Errors, J. R. Statist. Soc. B, 53,173-187.
• Hermann, E., Gasser, T., Kneip, A., 1992. Choice of Bandwidth for Kernel Regression When Residuals are Correlated, Biometrika, 79, 783-795.
• Hurvich, C. M., Zeger, S. L., 1990. A Frequency Domain Selection Criterion for Regression with Autocorrelated Errors, J. Am. Statist. Ass., 85, 705-714.
• Kim, T. Y., Park, B. U., Moon, M. S., Kim, C., 2009. Using Bimodal Kernel Inference in Nonparametric Regression with Correlated Errors, J. Multivariate Analysis, 100, 1487-1497.
• Kohn, R., Ansley, C. G., Wong, C. M., 1992. Nonparametric Spline Regression with Autoregressive Moving Average Errors, Biometrika, 79, 335-346.
• Krivobokova, T., Kauermann, G., 2007. A Note on Penalized Spline Smoothing with Correlated Errors. Journal of the American Statistical Association, 102 (480), 1328-1337.
• Lee, Y. K., Mamnen, E., Park, B.U., 2010. Bandwidth Selection for Kernel Regression with Correlated Errors, 44, 4, 327-340.
• Liu Jun, M., 2009. Nonlinear Time Series Modeling Using Spline-Based Nonparametric Models, AMATH'09 Proceedings of the 15th American Conference on Applied Mathematics.
• Morton, R., Kang, E. L., Henderson, B., 2009. Smoothing Splines for Trend Estimation and Prediction in Time Series, Environmetrics, 20, 249-259.
• Nadaraya, E. A., 1964. On Estimating Regression, Theory Pb. Appl., Vol.10, 186-190.
• Opsomer, J., Wang, Y., Yang, Y., 2001. Nonparametric Regression with Correlated Errors, Statist. Sci. 16, 134-153.
• Pagan, A., Ullah A., 1999. Nonparametric Econometrics, Cambridge, Cambridge University Press.
• Park, B. U., Lee, Y. K., Kim, T. Y., Park, C., 2006. A Simple Estimator of Error Correlation in Nonparametric Regression Models, Scand. J. Statist, 33, 451-462.
• Ray, B. K., Tsay, R. S., 1997. Bandwidth Selection for Kernel Regression with Long-range Dependent Errors, Biometrika, 84(4),791-802.
• Rio, A. Q., 1996. Comparison of Bandwidth Selectors in Nonparametric Regression Under Dependence, Comput. Statist. DataAnal., 21, 563-580.
• Wang, Y., 1998. Smoothing Spline Models with Correlated Random Errors, JASA, 93, 34-348.
• Watson, G. S., 1964. Smooth Regression Analysis, Sankhya, Series A, 26, 359-372.