Journal of the Korean Data and Information Science Society

The Korean Data and Information Science Society (한국데이터정보과학회)
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Varying coefficient model with errors in variables

가변계수 측정오차 회귀모형

  • Sohn, Insuk (Statistics and Data Center, Samsung Medical Center) ;
  • Shim, Jooyong (Department of Statistics, Inje University)
  • 손인석 (삼성서울병원 통계자료센터) ;
  • 심주용 (인제대학교 통계학과)
  • Received : 2017.08.29
  • Accepted : 2017.09.13
  • Published : 2017.09.30
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Abstract

The varying coefficient regression model has gained lots of attention since it is capable to model dynamic changes of regression coefficients in many regression problems of science. In this paper we propose a varying coefficient regression model that effectively considers the errors on both input and response variables, which utilizes the kernel method in estimating the varying coefficient which is the unknown nonlinear function of smoothing variables. We provide a generalized cross validation method for choosing the hyper-parameters which affect the performance of the proposed model. The proposed method is evaluated through numerical studies.

가변계수 회귀모형은 회귀계수의 동적변화를 모형화함으로써 종속변수와 입력변수의 관계에 대한 쉬운 해석이 가능하고 회귀계수의 변동성도 추정할 수 있는 장점을 지니고 있으므로, 여러 과학 분야에서 많은 주목을 받고 있다. 본 논문에서 입력변수와 출력변수의 오차를 효과적으로 고려한 가변계수 오차모형을 제안한다. 가변계수가 평활변수의 알려지지 않은 형태의 비선형함수이므로 이를 추정하기 위하여 커널 방법을 사용한다. 제안된 모형의 성능에 영향을 미치는 초모수의 최적값을 구하기 위하여 일반화 교차타당성 방법 또한 제안한다. 제안된 방법은 모의자료와 실제자료를 이용한 수치적 연구를 통하여 평가된다.

Keywords

References

  1. Boggs, P. T. and Rogers, J. E. (1990). Orthogonal distance regression. Contemporary Mathematics, 112, 183-194.
  2. Carroll, R. J., Ruppert, D. and Stefanski, L. A. (1997). Measurement error in nonlinear models, Monographs on Statistics and Applied Probability, Chapman & Hall, New York.
  3. Craven, P. and Wahba, G. (1979). Smoothing noisy data with spline functions: Estimating the correct degree of smoothing by the method of generalized cross validation. Numerical Mathematics, 31, 377-403.
  4. Fan, J. and Zhang, W. (2008). Statistical methods with varying coefficient models. Statistics and Its Interface, 1, 179-195. https://doi.org/10.4310/SII.2008.v1.n1.a15
  5. Fuller, W. A. (1987). Measurement error models, Wiley, New York.
  6. Hastie, T. and Tibshirani, R. (1993). Varying-coefficient models. Journal of the Royal Statistical Society: B, 55, 757-796.
  7. Hoover, D. R., Rice, J. A., Wu, C. O. and Yang, L. P. (1998). Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data. Biometrika, 85, 809-822. https://doi.org/10.1093/biomet/85.4.809
  8. Hu, Y. and Schennach, S. M. (2008). Identification and estimation of nonclassical nonlinear errors-invariables models with continuous distributions using instruments. Econometrica, 76, 195-216. https://doi.org/10.1111/j.0012-9682.2008.00823.x
  9. Lee, Y. K., Mammen, E. and Park, B. U. (2012). Projection-type estimation for varying coefficient regression models. Bernoulli, 18, 177-205. https://doi.org/10.3150/10-BEJ331
  10. Li, Q. and Racine, J. S. (2010). Smooth varying-coefficient estimation and inference for qualitative and quantitative data. Econometric Theory, 26, 1607-1637. https://doi.org/10.1017/S0266466609990739
  11. Madansky, A. (1959). The fitting of straight lines when both variables are subject to error. Journal of the American Statistical Association, 54, 173-205. https://doi.org/10.1080/01621459.1959.10501505
  12. Mercer, J. (1909). Function of positive and negative type and their connection with theory of integral equations. Philosophical Transactions of Royal Society A, 415-416.
  13. Shim, J. (2014). Quantile regression with errors in variables. Journal of the Korean Data & Information Science Society, 25, 439-446. https://doi.org/10.7465/jkdi.2014.25.2.439
  14. Shim, J and Hwang, C. (2015). Varying coefficient modeling via least squares support vector regression. Neurocomputing, 161, 254-259. https://doi.org/10.1016/j.neucom.2015.02.036
  15. Van Gorp, J., Schoukens, J. and Pintelon, R. (2000). Learning neural networks with noisy inputs using the errors-in-variables approach. IEEE Transactions on Neural Networks, 11, 402-414. https://doi.org/10.1109/72.839010
  16. Wooldridge, J. M. (2003). Introductory econometrics: A modern approach, South-Western Cengage Learning, Mason.
  17. Xue, L. and Qu, A. (2012). Variable selection in high-dimensional varying-coefficient models with global optimality. Journal of Machine Learning Research, 13, 1973-1998.
  18. Zhang, W., Lee, S. Y. and Song, X. (2002). Local polynomial fitting in semivarying coefficient models. Journal of Multivariate Analysis, 82, 166-188. https://doi.org/10.1006/jmva.2001.2012