Estimating the Parameters Geographically Weighted Regression (GWR) with Measurement Error
Affiliation(s)
Departement of Mathematics, Cenderawasih University, Jayapura, Indonesia;Departement of Statistics, Bogor Agricultural University, Bogor, Indonesia. Departement of Mathematics, Bogor Agricultural University, Bogor, Indonesia.
Departement of Mathematics, Cenderawasih University, Jayapura, Indonesia;Departement of Statistics, Bogor Agricultural University, Bogor, Indonesia. Departement of Mathematics, Bogor Agricultural University, Bogor, Indonesia.
ABSTRACT
Geographically weighted regression models with the measurement error are a modeling method that combines the global regression models with the measurement error and the weighted regression model. The assumptions used in this model are a normally distributed error with that the expectation value is zero and the variance is constant. The purpose of this study is to estimate the parameters of the model and find the properties of these estimators. Estimation is done by using the Weighted Least Squares (WLS) which gives different weighting to each location. The variance of the measurement error is known. Estimators obtained are 
. The properties of the estimator are unbiased and have a minimum variance.
KEYWORDS
Geographical Weighted Regression; Measurement Error; Instrumental Variable; Weighted Least Squares
Geographical Weighted Regression; Measurement Error; Instrumental Variable; Weighted Least Squares
Cite this paper
I. Hutabarat, A. Saefuddin, A. Djuraidah and I. Mangku, "Estimating the Parameters Geographically Weighted Regression (GWR) with Measurement Error," Open Journal of Statistics, Vol. 3 No. 6, 2013, pp. 417-421. doi: 10.4236/ojs.2013.36049.
I. Hutabarat, A. Saefuddin, A. Djuraidah and I. Mangku, "Estimating the Parameters Geographically Weighted Regression (GWR) with Measurement Error," Open Journal of Statistics, Vol. 3 No. 6, 2013, pp. 417-421. doi: 10.4236/ojs.2013.36049.
References
[1] W. A. Fuller “Measurement Error Models,” John Wiley and Sons Inc., New York, 1987.
http://dx.doi.org/10.1002/9780470316665 [2] R. J. Carroll, D. Ruppert and L. A. Stefanski “Measurement Error in Nonlinear Models,” Chapman and Hall, New York, 1995.
http://dx.doi.org/10.1007/978-1-4899-4477-1 [3] X. H. Chen, H. Hong and D. Nekipelov, “Nonlinear Models of Measurement Errors,” Journal of Economic Literature, Vol. 49, No. 4, 2011, pp. 901-937.
http://dx.doi.org/10.1257/jel.49.4.901 [4] W. A. Fuller and M. A. Hidiroglou, “Regression Estimation after Correction for Attenuation,” Journal of the American Statistical Association, Vol. 73, No. 361, 1978, pp. 99-104.
http://dx.doi.org/10.1080/01621459.1978.10480011 [5] J. Fan and Y. K. Truong, “Nonparametric Regression with Errors in Variables,” Annals of Statistics, Vol. 21, No. 4, 1993, pp. 1900-1925.
http://dx.doi.org/10.1214/aos/1176349402 [6] R. J. Carroll, J. D. Maca and D. Ruppert, “Nonparametric Regression in the Presence of Measurement Error,” Biometrika, Vol. 86, No. 3, 1999, pp. 541-554.
http://dx.doi.org/10.1093/biomet/86.3.541 [7] L. A. Stefanski and R. J. Carrol, “Covariate Measurement Error in Logistic Regression,” The Annals of Statistics, Vol. 13, No. 4, 1985, pp. 1335-135.
http://dx.doi.org/10.1214/aos/1176349741 [8] L. A. Stefanski, “Measurement Errors in Generalized Linear Model Explanatory Variables,” 3rd International Workshop on Statistical Modelling, Vienna, 4-8 July 1988, pp. 1-40. [9] T. Nakamura, “Proportional Hazards Model with Covariates Subject to Measurement Error,” Biometrics, Vol. 48, No. 3, 1992, pp. 829-839.
http://dx.doi.org/10.2307/2532348 [10] Y. Li, H. C. Tang and X. H. Lin, “Spatial Linear Mixed Models with Covariate Measurement Errors,” Statistica Sinica, Vol. 19, No. 3, 2009, pp. 1077-1093. [11] A. S. Fotheringham, C. Brunsdon and M. E. Charlton, “Geographically Weighted Regression, the Analysis of Spatially Varying Relationships,” John Wiley and Sons, Ltd., Hoboken, 2002. [12] C. L. Mei, “Geographically Weighted Regression Technique for Spatial Data Analysis,” School of Science Xi’an Jiaotong University, Xi’an, 2005. [13] A. Madansky, “The Fitting of Straight Lines When Both Variables Are Subject to Error,” Journal of the American Statistician Association, Vol. 54, No. 285, 1959, pp. 173205.
http://dx.doi.org/10.1080/01621459.1959.10501505 [14] L. M. Bartels, “Instrumental and Quasi-Instrumental Variables,” American Journal of Political Science, Vol. 35, No. 3, 1991, pp. 777-800.
http://dx.doi.org/10.2307/2111566 [15] R. J. Carroll and L. A. Stefanski, “Measurement Error, Instrumental Variables and Corrections for Attenuation with Application to Meta-Analyses,” Statistics in Medicine, Vol. 13, No. 12, 1994, pp. 1265-1282.
http://dx.doi.org/10.1002/sim.4780131208
[1] W. A. Fuller “Measurement Error Models,” John Wiley and Sons Inc., New York, 1987.
http://dx.doi.org/10.1002/9780470316665 [2] R. J. Carroll, D. Ruppert and L. A. Stefanski “Measurement Error in Nonlinear Models,” Chapman and Hall, New York, 1995.
http://dx.doi.org/10.1007/978-1-4899-4477-1 [3] X. H. Chen, H. Hong and D. Nekipelov, “Nonlinear Models of Measurement Errors,” Journal of Economic Literature, Vol. 49, No. 4, 2011, pp. 901-937.
http://dx.doi.org/10.1257/jel.49.4.901 [4] W. A. Fuller and M. A. Hidiroglou, “Regression Estimation after Correction for Attenuation,” Journal of the American Statistical Association, Vol. 73, No. 361, 1978, pp. 99-104.
http://dx.doi.org/10.1080/01621459.1978.10480011 [5] J. Fan and Y. K. Truong, “Nonparametric Regression with Errors in Variables,” Annals of Statistics, Vol. 21, No. 4, 1993, pp. 1900-1925.
http://dx.doi.org/10.1214/aos/1176349402 [6] R. J. Carroll, J. D. Maca and D. Ruppert, “Nonparametric Regression in the Presence of Measurement Error,” Biometrika, Vol. 86, No. 3, 1999, pp. 541-554.
http://dx.doi.org/10.1093/biomet/86.3.541 [7] L. A. Stefanski and R. J. Carrol, “Covariate Measurement Error in Logistic Regression,” The Annals of Statistics, Vol. 13, No. 4, 1985, pp. 1335-135.
http://dx.doi.org/10.1214/aos/1176349741 [8] L. A. Stefanski, “Measurement Errors in Generalized Linear Model Explanatory Variables,” 3rd International Workshop on Statistical Modelling, Vienna, 4-8 July 1988, pp. 1-40. [9] T. Nakamura, “Proportional Hazards Model with Covariates Subject to Measurement Error,” Biometrics, Vol. 48, No. 3, 1992, pp. 829-839.
http://dx.doi.org/10.2307/2532348 [10] Y. Li, H. C. Tang and X. H. Lin, “Spatial Linear Mixed Models with Covariate Measurement Errors,” Statistica Sinica, Vol. 19, No. 3, 2009, pp. 1077-1093. [11] A. S. Fotheringham, C. Brunsdon and M. E. Charlton, “Geographically Weighted Regression, the Analysis of Spatially Varying Relationships,” John Wiley and Sons, Ltd., Hoboken, 2002. [12] C. L. Mei, “Geographically Weighted Regression Technique for Spatial Data Analysis,” School of Science Xi’an Jiaotong University, Xi’an, 2005. [13] A. Madansky, “The Fitting of Straight Lines When Both Variables Are Subject to Error,” Journal of the American Statistician Association, Vol. 54, No. 285, 1959, pp. 173205.
http://dx.doi.org/10.1080/01621459.1959.10501505 [14] L. M. Bartels, “Instrumental and Quasi-Instrumental Variables,” American Journal of Political Science, Vol. 35, No. 3, 1991, pp. 777-800.
http://dx.doi.org/10.2307/2111566 [15] R. J. Carroll and L. A. Stefanski, “Measurement Error, Instrumental Variables and Corrections for Attenuation with Application to Meta-Analyses,” Statistics in Medicine, Vol. 13, No. 12, 1994, pp. 1265-1282.
http://dx.doi.org/10.1002/sim.4780131208