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This paper studies the varying-coefficient heteroscedastic partially linear models where some covariates are measured with additive errors. To eliminate the bias of the usual profile least squares estimation when measurement errors are ignored, a modified profile least squares estimator of the regression parameter is suggested and the local polynomial smoother is applied to constructing estimators of the varying coefficient function and error variance function. Further, for the purpose of accounting for heteroscedasticity and the estimation accuracy, re-weighted estimations of the regression parameter and varying coefficient function are proposed. Asymptotic behaviors of the above estimators are established and the re-weighted estimator is shown to be more efficient than the modified profile least-squares estimator. Both simulated and real data examples are conducted to illustrate the applications of the proposed approaches.
varying-coefficient partially linear model, profile least squares, errors-in-variables, heteroscedasticity, re-weighted estimation
Carroll R. J., Ruppert D., Stefanski L. A. (1995). Measurement Error in Nonlinear Models, Chapman & Hall, London.
Chiou J. M., Mu¨ller H. G. (1999). Nonparametric quasi-likelihood. The Annals of Statistics, Vol. 27, No. 1, pp. 36-64. http://doi.org/10.1214/aos/1018031100
Fan G. L., Huang Z. S. (2013). Empirical likelihood for semiparametric varying-coefficient heteroscedastic partially linear errors-in-variables models. Communications in Statistics - Theory and Methods, Vol. 42, pp. 2437-2450. http://doi.org/10.3770/j.issn:2095-2651.2012.01.011
Fan G. L., Liang H. Y., Wang J. F. (2013). Statistical inference for partially time-varying coefficient errors-in-variables models. Journal of Statistical Planning and Inference, Vol. 143, pp. 505- 519. http://doi.org/10.1214/12-EJS701
Fan J., Gijbels I. (1992). Variable bandwidth and local linear regression smoothers. The Annals of Statistics, Vol. 20, pp. 2008-2036. http://doi.org/10.1214/aos/1176348900
Fan J., Gijbels I. (1996). Local Polynomial Modeling and Its Applications. London, Chap- man and Hall.
Fan J., Huang T. (2005). Profile likelihood inferences on semiparametric varying- coefficient partially linear models. Bernoulli, Vol. 11, pp. 1031-1057. http://doi.org/10.3150/bj/1137421639
Feng S. Y., Xue L. G. (2014). Bias-corrected statistical inference for partially linear varying coefficient errors-in-variables models with restricted condition. Annals of the Institute of Statistical Mathematics, Vol. 66, pp. 121-140. http://doi.org/10.1007/s10463-013-0407-z
Fuller W. A. (1987). Measurement Error Models. Wiley, New York.
Harrison D., Rubinfeld D. L. (1978). Hedonic housing prices and the demand for clean air. Journal of Environmental Economics and Management, Vol. 5, pp. 81-102. http://doi.org/10.1016/0095-0696(78)90006-2
Hastie T. J., Loader C. (1993). Local regression: automatic kernel carpentry (with Discussion). Statistical Science, Vol. 8, pp. 120-143. http://doi.org/10.1214/ss/1177011005
Hwang J. T. (1986). The multiplicative errors-in-variables models with applications to the recent data released by the U.S. department of energy. Journal of The American Statistical Association, Vol. 81, pp. 680-688. http://doi.org/10.1080/01621459.1986.10478321
Kai B., Li R., Zou H. (2011). New efficient estimation and variable selection methods for semiparametric varying-coefficient partially linear models. The Annals of Statistics, Vol. 39, pp. 305-332. http://doi.org/10.1214/10-AOS842
Liang H., H¨adle W., Carroll R. J. (1999). Estimation in a semiparametric partially linear errors-in-variables model. The Annals of Statistics, Vol. 27, pp. 1519-1535. http://doi.org/10.1214/aop/1022677563
Lu X. (2009). Empirical likelihood for heteroscedastic partially linear models. Journal of Multivariate Analysis, Vol. 100, pp. 387-396. http://doi.org/10.1016/j.jmva.2008.05.006
Ma Y., Chiou J. M., Wang N. (2006). Efficient semiparametric estimator for heteroscedastic partially linear models. Biometrika, Vol. 943, pp. 75-84. http://doi.org/10.1093/biomet/93.1.75
Ma Y., You J., Zhou Y. (2013). Generalized profile LSE in varying-coefficient partially linear models with measurement errors. Acta Mathematicae Applicatae Sinica, Vol. 29, pp. 477-490. http://doi.org/10.1007/s10255-013-0229-z
Mack Y. P., Silverman B. W. (1982). Weak and strong uniform consistency of kernel regression estimates. Probability Theory and Related Fields, Vol. 61, pp. 405-415. http://doi.org/10.1007/BF00539840
Menictas M., Wand M. P. (2015). Variational inference for heteroscedastic semiparametric regression. Australian & New Zealand Journal of Statistics, Vol. 57, pp. 119-138. http://doi.org/10.1111/anzs.12105
Ruppert D., Wand M. P. (1994). Multivariate weighted least squares regression. The Annals of Statistics, Vol. 22, pp. 1346-1370. http://doi.org/10.1214/aos/1176325632
Shen S. L., Cui J. L., Mei C. L., Wang C. W. (2014). Estimation and inference of semivarying coefficient models with heteroscedastic errors. Journal of Multivariate Analysis, Vol. 124, pp. 70-93. http://doi.org/10.1016/j.jmva.2013.10.010
Tosteson T., Stefanski L. A., Schafer D. W. (1989). A measurement error model for binary and ordinal regression. Statistical Medicine, Vol. 8, pp. 1139-1147. http://doi.org/10.1002/sim.4780080914
Wang N., Carroll R. J., Liang K. Y. (1996). Quasilikelihood and variance functions in measurement error models with replicates. Biometrics, Vol. 52, pp. 423-432. http://doi.org/10.2307/2532882
Wei C. H. (2012). Statistical inference for restricted partially linear varying coefficient errors- in-variables models. Journal of Statistical Planning and Inference, Vol. 142, pp. 2464-2472. http://doi.org/10.1016/j.jspi.2012.02.041
Xia Y. C., Li W. K. (1999). On the estimation and testing of functional-coefficient linear models. Statistica Sinica, Vol. 9, pp. 735-757.
You J. H., Chen G. M. (2006). Estimation of a semiparametric varying-coefficient partially linear errors-in-variables model. Journal of Multivariate Analysis, Vol. 97, pp. 324-341. http://doi.org/10.1016/j.jmva.2005.03.002
You J. H., Zhou Y., Chen G. M. (2006). Corrected local polynomial estimation in varying- coefficient models with measurement errors. The Canadian Journal of Statistics, Vol. 34, pp. 391-410. https://doi.org/10.1002/cjs.5550340303
Zhang W., Lee S. Y., Song X. (2002). Local polynomial fitting in semivarying coefficient models. Journal of Multivariate Analysis, Vol. 82, pp. 166-188. https://doi.org/10.1006/jmva.2001.2012
Zhou Y., Liang H. (2009). Statistical inference for semiparametric varying-coefficient partially linear models with error-prone linear covariates. The Annals of Statistics, Vol. 37, pp. 427- 458. http://doi.org/10.1214/07-AOS561
Zhu L., Cui H. (2003). A semiparametric regression model with errors in variables. Scandinavian Journal of Statistics, Vol. 30, pp. 429-442. http://doi.org/10.1111/1467-9469.00340