几类数据下部分性单调回归模型的统计推断
|
摘要
部分线性回归模型是参数回归模型和非参数回归模型的结合,参数部分可避免维数灾难和提高非参数回归的解释,非参数部分保持了模型的灵活性,因此在描述实际问题时更具有灵活性和解释力。在解决实际问题时,经常会碰到模型非参数部分与解释变量具有明显单调关系的情形。在此基础上,统计学者提出了部分线性单调回归模型。实际问题中,我们经常遇到以下几种类型数据,如测量误差数据、缺失数据、删失数据等。因此,研究这几类数据下的部分线性单调回归模型的统计推断方法具有一定的理论意义和实际价值。
本文主要研究几类数据下部分线性单调回归模型的估计问题,考虑了度量误差数据、随机右删失数据和缺失数据等数据类型。
首先,我们引入了部分线性单调回归度量误差模型。在该模型下,我们研究了模型的参数部分和非参数部分的估计问题。利用局部线性方法来估计条件期望从而得到了参数部分的相合估计。在此基础上,利用分组Brunk B-样条方法得到了单调的非参数函数的估计。在一定的正则条件下,给出了参数估计的渐近正态性以及非参数函数的收敛速度。通过模拟实验研究了估计的有限样本性质,并且比较了参数部分用核方法估计条件期望和局部线性方法估计条件期望,非参数部分的Brunk估计方法和分组Brunk B-样条估计方法的有限样本性质。
其次,我们研究了响应变量随机右删失、回归模型线性部分协变量带有度量误差情况下部分线性单调回归模型的估计问题。采用了把完全观测数据垫高的思想,定义一个与响应变量同均值的合成变量,来处理删失问题,进而转化为完全数据的情况下处理相关问题。利用局部线性方法来估计条件期望得到了参数部分的(?)相合估计。进一步利用分组Brunk B-样条方法得到了单调的非参数部分的估计。在一定的正则条件下,给出了参数估计的渐近正态性和非参数部分估计的渐近分布。通过随机模拟研究了不同删失概率下参数部分用核方法和局部线性方法估计条件期望,非参数部分用Brunk方法和分组Brunk B-样条估计方法的有限样本性质。
最后,我们研究了响应变量随机缺失(MAR),回归模型线性部分协变量带有度量误差情况下部分线性单调回归模型的参数和非参数部分的估计问题。我们对参数部分和非参数部分均采用了两部估计。首先,根据完全数据采用局部线性光滑方法得到了参数和非参数部分的初始估计,接下来利用逆边际加权借补的方法得到了参数部分的借补估计,在此基础上利用分组Brunk B-样条方法给出了非参数部分的两步估计。我们证明了在一定的正则条件下,β的借补估计的渐近分布为正态分布,且收敛速度为Op(n-1/2),f(w)的两步估计收敛到真实函数的速度为Op(n-1/3),并且在非参数函数的定义域边界点是相合的。此外将我们提出的方法与参数部分采用核光滑方法来估计,非参数部分利用Brunk估计来估计的方法下得到的参数,非参数估计做了模拟比较,发现我们提出的方法在估计效果上是有所改进的。
Partial linear regression model is the combination of parametric regression model and non-parametric regression model. The parametric part can avoid the curse of di-mensionality and improve the explanation of non-parametric part. The non-parametric part can maintain the flexibility of the model. Therefore, in the description of the prac-tical problems, this model has more flexible and explanatory power. In solving practi-cal problems, we often encounter the case that the non-parametric part has obviously monotonic relationship with explanatory variables. On this basis, statisticians raise partially linear monotonic regression model. In real problems, we often encounter the following types of data, such as measurement error data, missing data, Censored data and so on. Therefore, the study of statistical inference method with these types of data under partially linear monotonic regression model has some theoretical significance and Real value.
In this thesis, we are mainly concerned with the estimation procedures of the par-tially linear monotonic regression model with several datas including mea-surement error data, missing data, randomly censored data and so on.
First, we introduce partially linear monotonic errors-in-variables model.Under this model, we study the estimation of parametric and non-parametric part. We ob-tain the consistent estimation of the Parametric part by using local linear method to estimate the conditional expectation. On this basis, using grouped Brunk B-spline method to obtain the estimation nonparametric part. Under certain regularity condi-tions, we give the asymptotic normality of parametric part and the convergence rate of non-parametric function. We give the finite sample properties of the estimation through simulation experiments. Also, we compare the finite sample properties of the proposed estimators with original estimators through simulation experiments.
Second, the estimation of the partially linear monotonic regression model with randomly right censored response and errors in variables is considered.We elevate complete data through making a variable whose mean is as same as response vari-able. The censoring problem is solved. By using local linear method to estimate the conditional expectation, we obtain the (?) consistent estimators of parametric part. By using grouped Brunk B-spline method, we obtain the estimation of nonparametric part.Under certain regularity conditions, we give the asymptotic normality of paramet-ric part and the asymptotic distribution of non-parametric function. We compare the finite sample properties of the proposed estimators with original estimators through simulation experiments under different censoring probabilities.
Finally, the estimation of the partially linear monotonic regression model with missing response and errors in variables is considered.We using two-step method in estimating parametric and non-parametric part. Based on the full data, we get the initial estimators of the parametric and nonparametric part by using Local Linear S-moothing method. After that, inverse marginal probability weighted imputation ap-proach is developed to estimate the regression parameter and a least-square approach under monotone constraint is employed to estimate the functional component. It is shown that the proposed estimator of the regression parameter is root-n consistent and asymptotically normal and the monotonic estimator of the functional component, at a fixed point,regardless of whether is the boundary point, is cubic root-n consisten-t. We compare the finite sample properties of the proposed estimators with original estimators through simulation experiments under different missing probabilities.
本文主要研究几类数据下部分线性单调回归模型的估计问题,考虑了度量误差数据、随机右删失数据和缺失数据等数据类型。
首先,我们引入了部分线性单调回归度量误差模型。在该模型下,我们研究了模型的参数部分和非参数部分的估计问题。利用局部线性方法来估计条件期望从而得到了参数部分的相合估计。在此基础上,利用分组Brunk B-样条方法得到了单调的非参数函数的估计。在一定的正则条件下,给出了参数估计的渐近正态性以及非参数函数的收敛速度。通过模拟实验研究了估计的有限样本性质,并且比较了参数部分用核方法估计条件期望和局部线性方法估计条件期望,非参数部分的Brunk估计方法和分组Brunk B-样条估计方法的有限样本性质。
其次,我们研究了响应变量随机右删失、回归模型线性部分协变量带有度量误差情况下部分线性单调回归模型的估计问题。采用了把完全观测数据垫高的思想,定义一个与响应变量同均值的合成变量,来处理删失问题,进而转化为完全数据的情况下处理相关问题。利用局部线性方法来估计条件期望得到了参数部分的(?)相合估计。进一步利用分组Brunk B-样条方法得到了单调的非参数部分的估计。在一定的正则条件下,给出了参数估计的渐近正态性和非参数部分估计的渐近分布。通过随机模拟研究了不同删失概率下参数部分用核方法和局部线性方法估计条件期望,非参数部分用Brunk方法和分组Brunk B-样条估计方法的有限样本性质。
最后,我们研究了响应变量随机缺失(MAR),回归模型线性部分协变量带有度量误差情况下部分线性单调回归模型的参数和非参数部分的估计问题。我们对参数部分和非参数部分均采用了两部估计。首先,根据完全数据采用局部线性光滑方法得到了参数和非参数部分的初始估计,接下来利用逆边际加权借补的方法得到了参数部分的借补估计,在此基础上利用分组Brunk B-样条方法给出了非参数部分的两步估计。我们证明了在一定的正则条件下,β的借补估计的渐近分布为正态分布,且收敛速度为Op(n-1/2),f(w)的两步估计收敛到真实函数的速度为Op(n-1/3),并且在非参数函数的定义域边界点是相合的。此外将我们提出的方法与参数部分采用核光滑方法来估计,非参数部分利用Brunk估计来估计的方法下得到的参数,非参数估计做了模拟比较,发现我们提出的方法在估计效果上是有所改进的。
Partial linear regression model is the combination of parametric regression model and non-parametric regression model. The parametric part can avoid the curse of di-mensionality and improve the explanation of non-parametric part. The non-parametric part can maintain the flexibility of the model. Therefore, in the description of the prac-tical problems, this model has more flexible and explanatory power. In solving practi-cal problems, we often encounter the case that the non-parametric part has obviously monotonic relationship with explanatory variables. On this basis, statisticians raise partially linear monotonic regression model. In real problems, we often encounter the following types of data, such as measurement error data, missing data, Censored data and so on. Therefore, the study of statistical inference method with these types of data under partially linear monotonic regression model has some theoretical significance and Real value.
In this thesis, we are mainly concerned with the estimation procedures of the par-tially linear monotonic regression model with several datas including mea-surement error data, missing data, randomly censored data and so on.
First, we introduce partially linear monotonic errors-in-variables model.Under this model, we study the estimation of parametric and non-parametric part. We ob-tain the consistent estimation of the Parametric part by using local linear method to estimate the conditional expectation. On this basis, using grouped Brunk B-spline method to obtain the estimation nonparametric part. Under certain regularity condi-tions, we give the asymptotic normality of parametric part and the convergence rate of non-parametric function. We give the finite sample properties of the estimation through simulation experiments. Also, we compare the finite sample properties of the proposed estimators with original estimators through simulation experiments.
Second, the estimation of the partially linear monotonic regression model with randomly right censored response and errors in variables is considered.We elevate complete data through making a variable whose mean is as same as response vari-able. The censoring problem is solved. By using local linear method to estimate the conditional expectation, we obtain the (?) consistent estimators of parametric part. By using grouped Brunk B-spline method, we obtain the estimation of nonparametric part.Under certain regularity conditions, we give the asymptotic normality of paramet-ric part and the asymptotic distribution of non-parametric function. We compare the finite sample properties of the proposed estimators with original estimators through simulation experiments under different censoring probabilities.
Finally, the estimation of the partially linear monotonic regression model with missing response and errors in variables is considered.We using two-step method in estimating parametric and non-parametric part. Based on the full data, we get the initial estimators of the parametric and nonparametric part by using Local Linear S-moothing method. After that, inverse marginal probability weighted imputation ap-proach is developed to estimate the regression parameter and a least-square approach under monotone constraint is employed to estimate the functional component. It is shown that the proposed estimator of the regression parameter is root-n consistent and asymptotically normal and the monotonic estimator of the functional component, at a fixed point,regardless of whether is the boundary point, is cubic root-n consisten-t. We compare the finite sample properties of the proposed estimators with original estimators through simulation experiments under different missing probabilities.
引文
[1]A.Tsiatis. Semiparametric Theory and Missing Data. Springer.2006
[2]Brunk, H. D. (1958) On the estimation of parameters restricted by inequalities. Ann. Math. Statist.29,437-454.
[3]Caroll,R.J.,Knickerbocker,R.K. and Wang.C.Y,1995.Dimension Reduction in a Semipara-metric Regression Model with Errors in Covariates. The Annals of Statistics.23,161 181.
[4]C. Srinivasan, M. Zhou. Linear Regression with Censoring.J. Multivariate Anal.1994, 49:179 201
[5]Chatterjee, S.& Hadi, A.S. Regression Analysis by Example. John Wiley & Sons.2006.
[6]D. L. Dohoho. Nonlinear Solution of Linear Inverse Problems by Wavelet-vaguelette De-composition. Appl. Comput. Harm. Anal.1995,2:101 126.1999
[7]D. M. Dabrowska. Nonparametric Regression with Censored Survival Time Data. Scand. J. Statist.1987,14:181 197
[8]Engle, R., Granger, C., Rice, J. and Weiss, A. (1986) Semiparametric estimates of the rela-tion between weather and electricity sales. Journal of the American Statistical Association. 81,310-320.
[9]Eubank, R. L. Spline Smoothing and Nonparametric Regression. New York.1988
[10]Fan, J. Local linear regression smoothers and their minimax efficiency. The Annals of Statis-tics..21 (1993),196-216.
[11]Fan, J.& Gijbels, I. Local Polynomial Modelling and Its Application. Chapman & Hal-l,London.1996.
[12]Fuller, W. A. (1987) Measurement Error Models. New York, Wiley.
[13]F.T. Wright.(1981) The Asymptotic Behavior of Monotone Regression Estimates. The An-nals of Statistics.9(2):443448.
[14]G. Cheng. Semiparametric Additive Isotonic Regression. Journal of Statistical Planning and Inference.2009,139:1980 1991
[15]G. Duncan, D. Hill. An Investigations of the Extent and Consequences of Measurement Error in Labor-economics Survey data. Journal of Labor Economics.1985,3:508 532
[16]Geoeneboom, P. and Weller, J. (2001) Computing Chernoff's Distribution. J. Comp. Graph. Statist.10,388-400.
[17]G. Qin, B.Y. Jing. Empirical Likelihood for Censored Linear Regression. Scand. J. Statist. 2001,28:661673
[18]G. Wahba. Partial Spline Models for the Semiparametric Estimation of Functions of Several Variables in:Statistical Analysis of Time Series. Proceedings of the Japan US Joint Seminar, Inst. Statist. Math, Tokyo.1984
[19]H. B. Zhou, J. H. You. Statistical Inference for a Semiparametric Measurement Error Re-gression Models with Heteroscedastic Errors. Journal of Statistical Planning and Inference. 2007,137:2263 2276
[20]Horvitz D G.Thompson D J.1952.A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association,47:663 685
[21]H. Ichimura. Semiparametric Least Squares(SLS) and Weighted SLS Estimation of Single-index Models. J. Econometrics.1993,58:71 120
[22]Huang, J. (2002) A note on estimating a partly linear model under monotonicity constraints. Journal of statistical planning and inference.107,345-351
[23]H. J.Cui, E.F.Kong. Empirical Likelihood Confidence Region for Parameters in Semi-linear Errors-in-variables Models. Scand. J. Statist.2006,33(1):153 168.
[24]H. J. Cui, R. C. Li. On Parameter Estimation for Semi-linear Errors-in-variables Models. J. Multi.Anal.1998,64:1 24
[25]H. J. Cui. Estimation in Partial Linear EV Models with Replicated Observations. Science in China, Ser.A.2004,34(4):467482
[26]H. Liang, W. HAardle, R. J Carroll. Estimation in a Semiparametric Partially Linear Errors-in-Variables Model. The Annals of Statistics.1999,27:1519 1535
[27]H.Liang, S. Wang, R.J.Carroll. Partially Linear Models with Missing Response Variables and Error-prone Covariates. Biometrika.2007,94:185 198
[28]H. Schneeweiss. Consistent Estimation of a Regression with Errors in the Variables. Metrika. 1976,23:101 115
[29]H. T. Kim, Y. K. Truong. Nonparametric Regression Estimates with Censored Data:Local Linear Smoothers and Their Applications. Biometrics.1998,54:1434 1444
[30]J.Buckley, J.James. Linear Regression with Censored Data.Biometrika.1979,66:429 436
[31]J. G. Wang. A Note on the Uniform Consistency of the Kaplan-Meier Estimator. The Annals of Statistics.1987,15(3):1313 1316
[32]J. Huang. Efficient Estimation for the Proportional Hazards Model with Interval Censoring. The Annals of Statistics.1996,24:540568
[33]J. L. Horowitz. Semiparametric Methods in Econometrics. London:Springer-Verlag.1998
[34]J. L. Powell. Censored Regression Quantiles. Journal of Econometrics.1986,32:143 145
[35]J. L. Powell. Least Absolute Deviations Estimation for the Censored Regression Model. Journal of Econometrics.1984,25:303325
[36]J. Q.Fan, W. Y.Zhang. Statistical Estimation in Varying Coe±cient Models.The Annals of Statistics.1999,27:1491 1518
[37]J. Q.Fan, I. Gijbels. Censored Regression:Local Linear Approximations and Their Applica-tions.J.Amer. Statist. Assoc.1994,89:560 570
[38]J. T. Gao, L. C. Zhao. Adaptive Estimation in Partly Linear Regression Models. Sci. China, Ser.A.1993,23(1):1427
[39]K. Chen, S.H. Lo. On the Rate of Uniform Convergence of the Product-limit Estimators: Strong and Weak Laws. Ann. Statist.1997,25:1050 1087
[40]K.C. Li, J.L. Wang, C.H. Chen. Dimension Reduction for Censored Regression Data.Ann. Statist.1999,27:1 23
[41]K. Chen, Z. Jin, Z. Ying. Semiparametric Analysis of Transformation Models with Censored Data. Biometrika.2002,89:659 668
[42]Little R J A.Rubin D B.Statistical Analysis with Missing Data [M].New York:Wiley,1987.
[43]Liang, H.(1999) Estimation in a Semiparametric Partially Linear Errors-in-variables Model. The Annals of Statistics.27,1519-1535.
[44]Liang, H., Hardle, W. (2000) Partially linear models.
[45]Liang, H., Wang, S., Carroll, R. J. (2007) Partially Linear Models with Missing Response Variables and Error-prone Covariates. Biometrika.94,185-198.
[46]L. X. Zhu, H. J. Cui. A Semi-parametric Regression Model with Errors in Variables. Scan-dinavian Journal of Statistics.2003,30:429442
[47]M.Fekri, A. R. Gazen. Robust Weighted Orthogonal Regression in the Errors-in-variables Model. J. Multivar. Anal.2004,88:89108
[48]M. Gu, T.L. Lai. Functional Laws of the Iterated Logarithm for the Product-limit Estimator of a Distribution Function under Random Censorship or Truncation. Ann. Probab.1990, 18:160 189
[49]Miller R G.1976. Least squares regression with censored data.Biometrika,63:449 464.
[50]M. Y. Wong. Likelihood Estimation of a Simple Linear Regression Model when Both Vari-ables Have Error. Biometrika.1989,76(1):141148
[51]M. Zhou. Asymptotic Normality of the Synthetic Estimator for Censored Survival Data. The Annals of Statistics.1992,20:1002 1021
[52]M. Zhou. Empirical Likelihood Ratio with Arbitrarily Censored/truncated Data by a Modi-fied EM Algorithm.J. Comput.Graph. Statist.2005,14:643 656
[53]Nadaraya, E.A. On estimating regression. Theory of Probability ans Its Applications.9 (1964),141-142
[54]P. D. Shi. M Estimation for Partly Linear Models. Ph.D Thesis. Institute of Systems Science, Chinese Academy of Sciences. Beijing, P.R. China.1992
[55]P. J. Bickel, C. A. J.Klaassen, Y. Ritov, J. A.Wellner. E±cient and Adaptive Estimation for Semiparametric Models. Johns Hopkins University Press, Baltimore.1993
[56]Q. Li, C. J. Huang, D. Li, T. T. Fu. Semiparametric Smooth Coefficient Models. Journal of Business & Economic Statistics.2002,20:412 422
[57]R. D. Gill. Large Sample Behavior of the Product-limit Estimator on the Whole Line. Ann. Statist.1983,11:49 58
[58]R. F. Engle, C. W. J. Granger, J. Rice, A. Weiss. Semiparametric Estimates of the Relation between Weather and Electricity Scales. J. Amer. Statist. Assoc.1986,81:310320
[59]R.H.Zamar. Robust Estimation in the Errors-in-variables Model. Biometrika.1989,76(1): 149160.
[60]R. J.Carroll, R. K.Knickerbocker, C. Y.Wang. Dimension Reduction in a Semiparametric Regression Model with Errors in Covariates.Ann.Starist.1995,23:161 181
[61]R. J.Carroll, M. P. Wand. Semiparametric Estimation in Logistic Measure Error Models, J. Roy. Statist. Soc.Ser.B.1991,53:652 663
[62]R.J.Carroll, D.Ruppert, L. A.Stefansk. Measurement Error in Nonlinear Models. New York: Chapman and Hall.1995
[63]R. J.Carroll, R. K.Knickerbocker, C. Y.Wang. Dimension Reduction in a Semiparametric Regression Model with Errors in Covariates. Ann.Starist.1995,23:161 181
[64]Robins J M,Rotnitzky A,Zhao L P.1994.Estiamtion of regression coefficients when some regressors are not always observed. Journal of the American Statistical Association,89:846 866.
[65]R. J. A.Little, D. B.Rubin. Statistical Analysis with Missing Data. NewYork:Wiley.1987
[66]Robertson, T., Wright, F. T. and Dykstar, R. (1988) Order Restricted Statistical Inference. John Wiley & Sons.
[67]S. A. Hamilton, J. M. G. Taylor. A Comparison of the Box-Cox Transformation Method and Nonparametric Methods for Estimating Quantiles in Clinical Data with Repeated Measures. J. Statist. Comput. Simulation.1993,45:185201
[68]Scharfstein D O,Rotnizky A.Robins J.1999.Adjusting for nonignorable drop out in semi-parametric nonresponse models(with discussion). Journal of the American Statistical Asso-ciation,94:1096 1146.
[69]Schumaker, L. (1981) Spline Functions:Basic Theory, Wiley, New York.
[70]S. C. Cheng, L. J. Wei, Z. Ying. Prediction of Survival Probabilities with Semi-parametric Rransformation Models. J.Am.Statist.Assoc.1997,92:227 235
[71]S. Ma, R. Michael, Kosorok. Penalized Log-likelihood Estimation for Partly Linear Trans-formation Models with Current Status Data. The Annals of Statistics.2005,33:2256 2290
[72]S. V. Huffel, P. Lemmerling. Total Least Squares and Errors-in-variables Modeling:Analy-sis, Algorithms and Applications, Kluwer Academic Publishers, Dordrecht.2002
[73]Sun, Z.,2011. Semi-parametric Monotone Regression Model Estimated in the Complex Da-ta(doctor thesis).
[74]T. A. Severini, J. G. Staniswalis. Quasilikelihood Estimation in Semiparametric Models. J. Amer.Statist.Assoc.1994,89:501511
[75]T.Hastie, R. Tibshirani. Varying-Coefficient Models. J. Roy. Stat. Soc.Ser. B.1993,55:757 796
[76]T. Lai, Z. L. Ying, Z. K. Zheng. Asymptotic Normality of a Class of Adaptive Statistics with Applications to Synthetic Data Method for Censored Regression. Journal of multivariate analysis.1995,52:259 279
[77]T. Robertson, F. T. Wright, R.Dykstar. Order Restricted Statistical Inference. John Wiley & Sons.1988
[78]Wright,F.T.,1981. The Asymptotic Behavior of Monotone Regression Estimates. The Annals of Statistics.9(2),443-448.
[79]W. Hardle, H. Liang, J. T. Gao. Partially Linear Models. Physica Verlag.Heidelberg.2000
[80]Wang Q H,Linton O,Hardle W.2004.Semiparametric regression analysis with missing re-sponse at random. Journal of the American Statistical Association,99:334 345.
[81]Wang, X.& Shen, J. (2010) A Class of Grouped Brunk Estimators and Penalized Spline Estimators for Monotone Regression.
[82]W. Y. Zhang, S. Y. Lee, X. Y. Song. Local Polynomial Fitting in Semivarying Coefficient Model. Journal of Multivariate Analysis.2002,82:166 188
[83]X. He, Shi P. Bivariate Tensor-product B-splines in a Partly Linear Model. J.Multivariate Statist.1996,58:162 181
[84]X. Zhou, J. H. You. Wavelet Estimation in Varying-Coefficient Partially Linear Regression Models. Statistics & Probability Letters.2004,68:91 104
[85]X. He, H. Liang. Quantile Regression Estimates for a Class of Linear and Partially Linear Errors-in-variables Models. Statist. Sinica.2000,10:129 140
[86]Y. Ritov. Estimation in a Linear Regression Model with Censored Data. Ann. Statist.1990, 18:303 328
[87]Z. W.Cai. Two-step likelihood estimation procedure for varying-coefficient models. Journal of Multivariate Analysis.2002,82:189 209
[88]Z. Jin, D.Y. Lin, Z.L. Ying. On Least-squares Regression with Censored Data. Biometrika. 2006,93:147161
[89]Z. Ying. A Note on Asymptotic Properties of the Product-Limit Estimator on the Whole Line. Statistics and Probability Letters.1989,7:311314
[90]Zhou, M.,1992. Asymptotic Normality of the Synthetic Estimator for Censored Survival Data. The Annals of Statistics.20,1002 1021.
[2]Brunk, H. D. (1958) On the estimation of parameters restricted by inequalities. Ann. Math. Statist.29,437-454.
[3]Caroll,R.J.,Knickerbocker,R.K. and Wang.C.Y,1995.Dimension Reduction in a Semipara-metric Regression Model with Errors in Covariates. The Annals of Statistics.23,161 181.
[4]C. Srinivasan, M. Zhou. Linear Regression with Censoring.J. Multivariate Anal.1994, 49:179 201
[5]Chatterjee, S.& Hadi, A.S. Regression Analysis by Example. John Wiley & Sons.2006.
[6]D. L. Dohoho. Nonlinear Solution of Linear Inverse Problems by Wavelet-vaguelette De-composition. Appl. Comput. Harm. Anal.1995,2:101 126.1999
[7]D. M. Dabrowska. Nonparametric Regression with Censored Survival Time Data. Scand. J. Statist.1987,14:181 197
[8]Engle, R., Granger, C., Rice, J. and Weiss, A. (1986) Semiparametric estimates of the rela-tion between weather and electricity sales. Journal of the American Statistical Association. 81,310-320.
[9]Eubank, R. L. Spline Smoothing and Nonparametric Regression. New York.1988
[10]Fan, J. Local linear regression smoothers and their minimax efficiency. The Annals of Statis-tics..21 (1993),196-216.
[11]Fan, J.& Gijbels, I. Local Polynomial Modelling and Its Application. Chapman & Hal-l,London.1996.
[12]Fuller, W. A. (1987) Measurement Error Models. New York, Wiley.
[13]F.T. Wright.(1981) The Asymptotic Behavior of Monotone Regression Estimates. The An-nals of Statistics.9(2):443448.
[14]G. Cheng. Semiparametric Additive Isotonic Regression. Journal of Statistical Planning and Inference.2009,139:1980 1991
[15]G. Duncan, D. Hill. An Investigations of the Extent and Consequences of Measurement Error in Labor-economics Survey data. Journal of Labor Economics.1985,3:508 532
[16]Geoeneboom, P. and Weller, J. (2001) Computing Chernoff's Distribution. J. Comp. Graph. Statist.10,388-400.
[17]G. Qin, B.Y. Jing. Empirical Likelihood for Censored Linear Regression. Scand. J. Statist. 2001,28:661673
[18]G. Wahba. Partial Spline Models for the Semiparametric Estimation of Functions of Several Variables in:Statistical Analysis of Time Series. Proceedings of the Japan US Joint Seminar, Inst. Statist. Math, Tokyo.1984
[19]H. B. Zhou, J. H. You. Statistical Inference for a Semiparametric Measurement Error Re-gression Models with Heteroscedastic Errors. Journal of Statistical Planning and Inference. 2007,137:2263 2276
[20]Horvitz D G.Thompson D J.1952.A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association,47:663 685
[21]H. Ichimura. Semiparametric Least Squares(SLS) and Weighted SLS Estimation of Single-index Models. J. Econometrics.1993,58:71 120
[22]Huang, J. (2002) A note on estimating a partly linear model under monotonicity constraints. Journal of statistical planning and inference.107,345-351
[23]H. J.Cui, E.F.Kong. Empirical Likelihood Confidence Region for Parameters in Semi-linear Errors-in-variables Models. Scand. J. Statist.2006,33(1):153 168.
[24]H. J. Cui, R. C. Li. On Parameter Estimation for Semi-linear Errors-in-variables Models. J. Multi.Anal.1998,64:1 24
[25]H. J. Cui. Estimation in Partial Linear EV Models with Replicated Observations. Science in China, Ser.A.2004,34(4):467482
[26]H. Liang, W. HAardle, R. J Carroll. Estimation in a Semiparametric Partially Linear Errors-in-Variables Model. The Annals of Statistics.1999,27:1519 1535
[27]H.Liang, S. Wang, R.J.Carroll. Partially Linear Models with Missing Response Variables and Error-prone Covariates. Biometrika.2007,94:185 198
[28]H. Schneeweiss. Consistent Estimation of a Regression with Errors in the Variables. Metrika. 1976,23:101 115
[29]H. T. Kim, Y. K. Truong. Nonparametric Regression Estimates with Censored Data:Local Linear Smoothers and Their Applications. Biometrics.1998,54:1434 1444
[30]J.Buckley, J.James. Linear Regression with Censored Data.Biometrika.1979,66:429 436
[31]J. G. Wang. A Note on the Uniform Consistency of the Kaplan-Meier Estimator. The Annals of Statistics.1987,15(3):1313 1316
[32]J. Huang. Efficient Estimation for the Proportional Hazards Model with Interval Censoring. The Annals of Statistics.1996,24:540568
[33]J. L. Horowitz. Semiparametric Methods in Econometrics. London:Springer-Verlag.1998
[34]J. L. Powell. Censored Regression Quantiles. Journal of Econometrics.1986,32:143 145
[35]J. L. Powell. Least Absolute Deviations Estimation for the Censored Regression Model. Journal of Econometrics.1984,25:303325
[36]J. Q.Fan, W. Y.Zhang. Statistical Estimation in Varying Coe±cient Models.The Annals of Statistics.1999,27:1491 1518
[37]J. Q.Fan, I. Gijbels. Censored Regression:Local Linear Approximations and Their Applica-tions.J.Amer. Statist. Assoc.1994,89:560 570
[38]J. T. Gao, L. C. Zhao. Adaptive Estimation in Partly Linear Regression Models. Sci. China, Ser.A.1993,23(1):1427
[39]K. Chen, S.H. Lo. On the Rate of Uniform Convergence of the Product-limit Estimators: Strong and Weak Laws. Ann. Statist.1997,25:1050 1087
[40]K.C. Li, J.L. Wang, C.H. Chen. Dimension Reduction for Censored Regression Data.Ann. Statist.1999,27:1 23
[41]K. Chen, Z. Jin, Z. Ying. Semiparametric Analysis of Transformation Models with Censored Data. Biometrika.2002,89:659 668
[42]Little R J A.Rubin D B.Statistical Analysis with Missing Data [M].New York:Wiley,1987.
[43]Liang, H.(1999) Estimation in a Semiparametric Partially Linear Errors-in-variables Model. The Annals of Statistics.27,1519-1535.
[44]Liang, H., Hardle, W. (2000) Partially linear models.
[45]Liang, H., Wang, S., Carroll, R. J. (2007) Partially Linear Models with Missing Response Variables and Error-prone Covariates. Biometrika.94,185-198.
[46]L. X. Zhu, H. J. Cui. A Semi-parametric Regression Model with Errors in Variables. Scan-dinavian Journal of Statistics.2003,30:429442
[47]M.Fekri, A. R. Gazen. Robust Weighted Orthogonal Regression in the Errors-in-variables Model. J. Multivar. Anal.2004,88:89108
[48]M. Gu, T.L. Lai. Functional Laws of the Iterated Logarithm for the Product-limit Estimator of a Distribution Function under Random Censorship or Truncation. Ann. Probab.1990, 18:160 189
[49]Miller R G.1976. Least squares regression with censored data.Biometrika,63:449 464.
[50]M. Y. Wong. Likelihood Estimation of a Simple Linear Regression Model when Both Vari-ables Have Error. Biometrika.1989,76(1):141148
[51]M. Zhou. Asymptotic Normality of the Synthetic Estimator for Censored Survival Data. The Annals of Statistics.1992,20:1002 1021
[52]M. Zhou. Empirical Likelihood Ratio with Arbitrarily Censored/truncated Data by a Modi-fied EM Algorithm.J. Comput.Graph. Statist.2005,14:643 656
[53]Nadaraya, E.A. On estimating regression. Theory of Probability ans Its Applications.9 (1964),141-142
[54]P. D. Shi. M Estimation for Partly Linear Models. Ph.D Thesis. Institute of Systems Science, Chinese Academy of Sciences. Beijing, P.R. China.1992
[55]P. J. Bickel, C. A. J.Klaassen, Y. Ritov, J. A.Wellner. E±cient and Adaptive Estimation for Semiparametric Models. Johns Hopkins University Press, Baltimore.1993
[56]Q. Li, C. J. Huang, D. Li, T. T. Fu. Semiparametric Smooth Coefficient Models. Journal of Business & Economic Statistics.2002,20:412 422
[57]R. D. Gill. Large Sample Behavior of the Product-limit Estimator on the Whole Line. Ann. Statist.1983,11:49 58
[58]R. F. Engle, C. W. J. Granger, J. Rice, A. Weiss. Semiparametric Estimates of the Relation between Weather and Electricity Scales. J. Amer. Statist. Assoc.1986,81:310320
[59]R.H.Zamar. Robust Estimation in the Errors-in-variables Model. Biometrika.1989,76(1): 149160.
[60]R. J.Carroll, R. K.Knickerbocker, C. Y.Wang. Dimension Reduction in a Semiparametric Regression Model with Errors in Covariates.Ann.Starist.1995,23:161 181
[61]R. J.Carroll, M. P. Wand. Semiparametric Estimation in Logistic Measure Error Models, J. Roy. Statist. Soc.Ser.B.1991,53:652 663
[62]R.J.Carroll, D.Ruppert, L. A.Stefansk. Measurement Error in Nonlinear Models. New York: Chapman and Hall.1995
[63]R. J.Carroll, R. K.Knickerbocker, C. Y.Wang. Dimension Reduction in a Semiparametric Regression Model with Errors in Covariates. Ann.Starist.1995,23:161 181
[64]Robins J M,Rotnitzky A,Zhao L P.1994.Estiamtion of regression coefficients when some regressors are not always observed. Journal of the American Statistical Association,89:846 866.
[65]R. J. A.Little, D. B.Rubin. Statistical Analysis with Missing Data. NewYork:Wiley.1987
[66]Robertson, T., Wright, F. T. and Dykstar, R. (1988) Order Restricted Statistical Inference. John Wiley & Sons.
[67]S. A. Hamilton, J. M. G. Taylor. A Comparison of the Box-Cox Transformation Method and Nonparametric Methods for Estimating Quantiles in Clinical Data with Repeated Measures. J. Statist. Comput. Simulation.1993,45:185201
[68]Scharfstein D O,Rotnizky A.Robins J.1999.Adjusting for nonignorable drop out in semi-parametric nonresponse models(with discussion). Journal of the American Statistical Asso-ciation,94:1096 1146.
[69]Schumaker, L. (1981) Spline Functions:Basic Theory, Wiley, New York.
[70]S. C. Cheng, L. J. Wei, Z. Ying. Prediction of Survival Probabilities with Semi-parametric Rransformation Models. J.Am.Statist.Assoc.1997,92:227 235
[71]S. Ma, R. Michael, Kosorok. Penalized Log-likelihood Estimation for Partly Linear Trans-formation Models with Current Status Data. The Annals of Statistics.2005,33:2256 2290
[72]S. V. Huffel, P. Lemmerling. Total Least Squares and Errors-in-variables Modeling:Analy-sis, Algorithms and Applications, Kluwer Academic Publishers, Dordrecht.2002
[73]Sun, Z.,2011. Semi-parametric Monotone Regression Model Estimated in the Complex Da-ta(doctor thesis).
[74]T. A. Severini, J. G. Staniswalis. Quasilikelihood Estimation in Semiparametric Models. J. Amer.Statist.Assoc.1994,89:501511
[75]T.Hastie, R. Tibshirani. Varying-Coefficient Models. J. Roy. Stat. Soc.Ser. B.1993,55:757 796
[76]T. Lai, Z. L. Ying, Z. K. Zheng. Asymptotic Normality of a Class of Adaptive Statistics with Applications to Synthetic Data Method for Censored Regression. Journal of multivariate analysis.1995,52:259 279
[77]T. Robertson, F. T. Wright, R.Dykstar. Order Restricted Statistical Inference. John Wiley & Sons.1988
[78]Wright,F.T.,1981. The Asymptotic Behavior of Monotone Regression Estimates. The Annals of Statistics.9(2),443-448.
[79]W. Hardle, H. Liang, J. T. Gao. Partially Linear Models. Physica Verlag.Heidelberg.2000
[80]Wang Q H,Linton O,Hardle W.2004.Semiparametric regression analysis with missing re-sponse at random. Journal of the American Statistical Association,99:334 345.
[81]Wang, X.& Shen, J. (2010) A Class of Grouped Brunk Estimators and Penalized Spline Estimators for Monotone Regression.
[82]W. Y. Zhang, S. Y. Lee, X. Y. Song. Local Polynomial Fitting in Semivarying Coefficient Model. Journal of Multivariate Analysis.2002,82:166 188
[83]X. He, Shi P. Bivariate Tensor-product B-splines in a Partly Linear Model. J.Multivariate Statist.1996,58:162 181
[84]X. Zhou, J. H. You. Wavelet Estimation in Varying-Coefficient Partially Linear Regression Models. Statistics & Probability Letters.2004,68:91 104
[85]X. He, H. Liang. Quantile Regression Estimates for a Class of Linear and Partially Linear Errors-in-variables Models. Statist. Sinica.2000,10:129 140
[86]Y. Ritov. Estimation in a Linear Regression Model with Censored Data. Ann. Statist.1990, 18:303 328
[87]Z. W.Cai. Two-step likelihood estimation procedure for varying-coefficient models. Journal of Multivariate Analysis.2002,82:189 209
[88]Z. Jin, D.Y. Lin, Z.L. Ying. On Least-squares Regression with Censored Data. Biometrika. 2006,93:147161
[89]Z. Ying. A Note on Asymptotic Properties of the Product-Limit Estimator on the Whole Line. Statistics and Probability Letters.1989,7:311314
[90]Zhou, M.,1992. Asymptotic Normality of the Synthetic Estimator for Censored Survival Data. The Annals of Statistics.20,1002 1021.