响应变量随机缺失下的半参数变系数部分线性变量含误差模型的统计推断
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摘要
半参数变系数部分线性模型是近年来统计研究的热点之一,它涵盖了很多的参数、非参数回归模型,其中线性回归模型和部分线性回归模型以及变系数模型都是该模型的退化情形.由于该模型结合了参数线性模型和非参数模型,吸收了各自的优点,而比一般的参数线性模型或半参数部分线性模型具有更强的解释能力,同时避免了许多“维数祸根”问题.因此,它在经济、金融以及医学等领域得到了广泛的应用.
在许多实际问题中,由于人为或系统的原因,度量误差总是存在的.因此,研究度量误差模型具有很大的实际意义.当数据可以完全观测到时,You & Chen[27]对半参数变系数部分线性变量含误差模型进行了广泛深入的研究.
而在现实中由于仪器精度不高、实验费用昂贵、或被调查者拒绝回答等因素,在数据的收集与分析过程中不仅会出现度量误差还会容易导致大量缺失数据的产生,所以缺失数据问题在实际应用中越来越引起人们的普遍重视.在有数据缺失的情况下,通常的统计方法通常不能直接应用,需要对数据进行适当的处理,处理方法往往是先对缺失值进行填补,得到“完全样本”,再按统计方法进行推断.
在文献中,对缺失数据的处理往往采用删除有缺失的单元(C-C方法)、加权或填补方法.通常使用的填补方法有线性回归填补法,非参数核回归填补法和半参数回归填补法.本文利用C-C方法研究了响应变量随机缺失下的半参数变系数部分线性变量含误差模型的参数估计并研究了估计的渐近性质.
本文分成四章。第一章为引言和相关文献综述.
第二章在响应变量随机缺失下利用C-C方法处理缺失数据,构造半参数变系数部分线性变量含误差模型参数的估计,并证明了估计的渐近正态性,利用此结果分别构造了模型参数基于正态逼近的渐近置信区间(域).
第三章在响应变量随机缺失下利用C-C方法处理缺失数据,构造半参数变系数部分线性变量含误差模型参数的经验似然比统计量,证明了经验似然比统计量的极限分布是卡方分布,并利用这个结果构造了模型参数的经验似然置信区间(域).
第四章通过数值模拟发现用经验似然方法构造响应变量随机缺失下的半参数变系数部分线性变量含误差模型中参数分量的置信区间比用最小二乘法构造的置信区间有更精确的覆盖率,并且所得到的置信区间的平均长度也比最小二乘法得到的置信区间的平均长度短.
Semiparametric variable coefficient partly linear model in recent years is one of the hot spot of statistical studies, it covers A lot of parameters, nonparametric regression model, including linear regression model and the partial linear regression model And variable coefficient model is the model of degradation situation. Because the model combining the parameters of linear model and the parameter model, absorbs the respective advantages, Rather than the average linear-in-parameter model or semi-parametric partly linear model has stronger explain capabilities, Avoiding many "dimension curse" problem. Therefore, It in economic, financial and medical fields been widely used.
In many practical problems, due to human or system, the reason of the measure error is always exist. Therefore, the study measurement error model of great practical significance. When data can be completely observation arrives, You & Chen parameters variable coefficient partly linear variable contains error model an extensive and in-depth research.
But in reality the instrumentation precision, experimental expensive, or respondents refused to answer and other factors, In data collect and analysis process not only can appear measurement error will also easy, resulting in a large number of missing data of generation, So the missing data problems in practical application more aroused people's universal attention. In the background of lack of data, usually a statistical method usually cannot be applied directly, Need for data proper treatment, the method is often first to fill, get on missing value "Completely sample", then press the statistical methods of inference.
In the literature, For the missing data processing is usually adopted delete defects unit (C-C methods), weighted or packing method. Commonly used to fill method has linear regression fill method, Non-parameter kernel regression fill method and semi-parametric regression fill method. This paper using C-C method was used to study the response variables random missing semi-parametric variable coefficient partly linear Variable contains error model parameter estimation and study estimated the asymptotic properties.
This paper is divided into three chapters. The first chapter for preface and related literature review. The second chapter in response variables are missing at random by using C-C under method to deal with the missing data, Tectonic semi-parametric variable coefficient partly linear variable contains error model parameter estimation, And prove the estimation of asymptotic normality, and using these results are constructed respectively Model parameters based on normal approximation of the asymptotic confidence interval (regions);
The third chapter in response variables are missing at random by using C-C under method to deal with the missing data, Tectonic semi-parametric variable coefficient partly linear variable con-tains error model parameters empirical likelihood ratio statistics, Proved empirical likelihood ratio statistics limit distributions was chi-square distribution, And by using this result was constructed model parameters empirical likelihood confidence interval (regions).
This article features embodied in the following two aspects. First, in the response variables are missing at random study the semi-parametric variable coefficient Partly linear variable contains error model parameters estimation and asymptotic normality.
Second, the empirical likelihood applied to response variables random missing semi-parametric variable coefficient Partly linear variable contains error model, the tectonic model pa-rameters of the empirical likelihood confidence interval (regions).
在许多实际问题中,由于人为或系统的原因,度量误差总是存在的.因此,研究度量误差模型具有很大的实际意义.当数据可以完全观测到时,You & Chen[27]对半参数变系数部分线性变量含误差模型进行了广泛深入的研究.
而在现实中由于仪器精度不高、实验费用昂贵、或被调查者拒绝回答等因素,在数据的收集与分析过程中不仅会出现度量误差还会容易导致大量缺失数据的产生,所以缺失数据问题在实际应用中越来越引起人们的普遍重视.在有数据缺失的情况下,通常的统计方法通常不能直接应用,需要对数据进行适当的处理,处理方法往往是先对缺失值进行填补,得到“完全样本”,再按统计方法进行推断.
在文献中,对缺失数据的处理往往采用删除有缺失的单元(C-C方法)、加权或填补方法.通常使用的填补方法有线性回归填补法,非参数核回归填补法和半参数回归填补法.本文利用C-C方法研究了响应变量随机缺失下的半参数变系数部分线性变量含误差模型的参数估计并研究了估计的渐近性质.
本文分成四章。第一章为引言和相关文献综述.
第二章在响应变量随机缺失下利用C-C方法处理缺失数据,构造半参数变系数部分线性变量含误差模型参数的估计,并证明了估计的渐近正态性,利用此结果分别构造了模型参数基于正态逼近的渐近置信区间(域).
第三章在响应变量随机缺失下利用C-C方法处理缺失数据,构造半参数变系数部分线性变量含误差模型参数的经验似然比统计量,证明了经验似然比统计量的极限分布是卡方分布,并利用这个结果构造了模型参数的经验似然置信区间(域).
第四章通过数值模拟发现用经验似然方法构造响应变量随机缺失下的半参数变系数部分线性变量含误差模型中参数分量的置信区间比用最小二乘法构造的置信区间有更精确的覆盖率,并且所得到的置信区间的平均长度也比最小二乘法得到的置信区间的平均长度短.
Semiparametric variable coefficient partly linear model in recent years is one of the hot spot of statistical studies, it covers A lot of parameters, nonparametric regression model, including linear regression model and the partial linear regression model And variable coefficient model is the model of degradation situation. Because the model combining the parameters of linear model and the parameter model, absorbs the respective advantages, Rather than the average linear-in-parameter model or semi-parametric partly linear model has stronger explain capabilities, Avoiding many "dimension curse" problem. Therefore, It in economic, financial and medical fields been widely used.
In many practical problems, due to human or system, the reason of the measure error is always exist. Therefore, the study measurement error model of great practical significance. When data can be completely observation arrives, You & Chen parameters variable coefficient partly linear variable contains error model an extensive and in-depth research.
But in reality the instrumentation precision, experimental expensive, or respondents refused to answer and other factors, In data collect and analysis process not only can appear measurement error will also easy, resulting in a large number of missing data of generation, So the missing data problems in practical application more aroused people's universal attention. In the background of lack of data, usually a statistical method usually cannot be applied directly, Need for data proper treatment, the method is often first to fill, get on missing value "Completely sample", then press the statistical methods of inference.
In the literature, For the missing data processing is usually adopted delete defects unit (C-C methods), weighted or packing method. Commonly used to fill method has linear regression fill method, Non-parameter kernel regression fill method and semi-parametric regression fill method. This paper using C-C method was used to study the response variables random missing semi-parametric variable coefficient partly linear Variable contains error model parameter estimation and study estimated the asymptotic properties.
This paper is divided into three chapters. The first chapter for preface and related literature review. The second chapter in response variables are missing at random by using C-C under method to deal with the missing data, Tectonic semi-parametric variable coefficient partly linear variable contains error model parameter estimation, And prove the estimation of asymptotic normality, and using these results are constructed respectively Model parameters based on normal approximation of the asymptotic confidence interval (regions);
The third chapter in response variables are missing at random by using C-C under method to deal with the missing data, Tectonic semi-parametric variable coefficient partly linear variable con-tains error model parameters empirical likelihood ratio statistics, Proved empirical likelihood ratio statistics limit distributions was chi-square distribution, And by using this result was constructed model parameters empirical likelihood confidence interval (regions).
This article features embodied in the following two aspects. First, in the response variables are missing at random study the semi-parametric variable coefficient Partly linear variable contains error model parameters estimation and asymptotic normality.
Second, the empirical likelihood applied to response variables random missing semi-parametric variable coefficient Partly linear variable contains error model, the tectonic model pa-rameters of the empirical likelihood confidence interval (regions).
引文
[1]Owen A B. Empirical likelihood ratio confidence intervals for a single functional [J]. Biometrika,1988,75:237-249.
[2]Owen A B. Empirical likelihood ratio confidence regions [J]. Ann Statist,1990,18:90-120.
[3]Owen A B. Empirical likelihood for linear models[J]. Ann Statist,1991,19(4):1725-1747.
[4]Qin J, Lawless J. Empirical likelihood and general estimating equations[J]. Ann Statist, 1994,22:300-325.
[5]Chen S X. Empirical likelihood confidence intervals for nonparametric density estimation [J]. Biometrika,1996,83(2):329-341.
[7]Chen S X, Qin Y S. Empirical likelihood confidence intervals for local linear smoothers [J]. Biometrika,2000,87:946-953.
[8]Zhong B, Rao J N K. Empirical likelihood inference under stratified random sampling using auxiliary population information[J]. Biometrika,2000,87:929-938.
[9]Kitamura Y. Asymptotic optimality of empirical likelihood for testing moment restrictions[ J]. Econometrica,2001,69:1661-1672.
[10]Wang Q, Rao J N K. Empirical likelihood-based inference in linear models with missing data[J]. Scandinavian Journal of Statistics,2002,29(2):563-576.
[11]Wang Q, Rao J N K. Empirical likelihood-based inference under imputation for missing response data[J]. Ann Statist,2002,30(3):896-924.
[12]Cui H J, Kong E.Empirical likelihood Confidence Region for Parameters in Semi-linear Errors-in-Variables Models[J]. The Scandinavian Journal of Statistics,2006,33(1):153-168.
[13]Wang Q, Rao J N K. Empirical likelihood for linear regression models under imputation for missing responses [J]. Canad J Statist,2001,29(4):597-608.
[14]Wang Q, Oliver L, Wolfgang H. Semiparametric regression analysis with response missing at random [J]. J Amer Statist Assoc,2004,99:334-345.
[15]Anderson T W. Estimating linear statistieal relationshiPs [J].The Annals of Statistics,1984,12:1-45.
[16]Carroll R J, Spiegeman C H, Lan K G. On errors-in-variables for binary regression mod-els[J]. Biometrika,1984,71:19-25.
[17]Stefanski L A. The effects of measurement error on parameter estimation [J]. Biometrika, 1985,72:583-592.
[18]Fuller W A. Meastirement Error Models [M]. New York:Wiley,1987.
[19]Wang Q H. Estimation of linear error-in-response models with validation data under random censorship. Journal of Multivariate Analysis,2000,74:245,266.
[20]Cui H J, Chen 5 X.Empirieal likelihood confidence regions for parameter in the errors-in-variables models.Journal of Multivariate Analysis,2003,84:101-1-5.
[21]Fan J Q, Truong Y K. Nonparametric regression with errors in variables.The Annals of Statistics,21:1900-1925.
[22]Carroll R J, Ruppert D Stefanski L A.Measurement Error in nonlinear models[M]. London: Chapman, Hall,1995.
[23]Liang H. Asymptotic normality of parametric part in partially linear models with measure-ment erorr in the nonparametric. Journal of Statistical Plannig and Inference.2000,86:51-62.
[24]Cui H J, Li R C. On Parameter estimation for Semi-linear Errors-in-Variables Models. Jour-nal of Multivariate Analysis,1998,64:1-24.
[25]Cui H J. Estimation in Partial linear EV models with replieated observations. Science in China, Series A Mathematies,2004,47(1):144-159.
[26]Liang W, Wang N. Partially linear single-index measurement erorr models. Statistica Sinica, 2005,15:99-116.
[27]You J H, Chen G M. Estimation of a semiparametric varying-coefficient Partially linear errors-in-variables model. Journal of Multivariate Analysis,2006,97:324-341.
[28]李志强,薛留根.缺失数据下的半参数变系数模型的借补估计.于稿.
[29]Carroll R J, Ruppert D, Stefanski L A and Crainiceanu C M Measurement Error in Nonlinear Models,2nd ed.Chapman and Hall,New York.
[30]Fan J, Huang T. Profile likelihood inferences on semiparametric vary-coefficient paratially linear regression models. Bernoulli,2005,11:1031-1057.
[31]Liang H, Wang S, Robins J M, Carroll R J. Estimation in partially linear models with missing covariates[J]. J Amer Statist Assoc,2004,99:357-367.
[32]Mack, Y.P., Silverman, B.W. Weak and strong uniform consistency of kernel regression estimates. Z. Wahrsch. Verw. Gebiete 1982,61:405-415.
[33]Xia Y, Li W K. On the estimation and testing of functional coefficient linear models. Statist Sinica,1999,9:737-757.
[34]李志强,薛留根.响应变量随机缺失下的半参数变系数模型的均值估计.数学的实践与认识,2006,36(10)：191-196.
[35]You J, Zhou Y. Empirical likelihood for semiparametric vary-coefficient paratially linear regression models. Statistical and Probablity letters,2006,76:412-422.
[36]Zhou Y, Liang H. Statistical inference for semiparametric vary-coefficient paratially linear models with error-prone linear covariates.Annals of Statistics.2009,37(1):427-458.
[37]Zhou Y, Wan A.T.K., Wang X J. Estimating equations inference with missing data. J Amer Statist Assoc,2008,103(483):1187-1199.
[38]Little R J A, Rubin D B. Statistical Analysis with Missing Data[M]. New York:John Wiley, 1987.
[39]Lehmann E L. Theory of Point Estimation[M]. New York:Wiley,1983.
[40]Lehmann E L. Testing Statistical Hypotheses (2nd edition)[M]. New York:Wiley,1986.
[2]Owen A B. Empirical likelihood ratio confidence regions [J]. Ann Statist,1990,18:90-120.
[3]Owen A B. Empirical likelihood for linear models[J]. Ann Statist,1991,19(4):1725-1747.
[4]Qin J, Lawless J. Empirical likelihood and general estimating equations[J]. Ann Statist, 1994,22:300-325.
[5]Chen S X. Empirical likelihood confidence intervals for nonparametric density estimation [J]. Biometrika,1996,83(2):329-341.
[7]Chen S X, Qin Y S. Empirical likelihood confidence intervals for local linear smoothers [J]. Biometrika,2000,87:946-953.
[8]Zhong B, Rao J N K. Empirical likelihood inference under stratified random sampling using auxiliary population information[J]. Biometrika,2000,87:929-938.
[9]Kitamura Y. Asymptotic optimality of empirical likelihood for testing moment restrictions[ J]. Econometrica,2001,69:1661-1672.
[10]Wang Q, Rao J N K. Empirical likelihood-based inference in linear models with missing data[J]. Scandinavian Journal of Statistics,2002,29(2):563-576.
[11]Wang Q, Rao J N K. Empirical likelihood-based inference under imputation for missing response data[J]. Ann Statist,2002,30(3):896-924.
[12]Cui H J, Kong E.Empirical likelihood Confidence Region for Parameters in Semi-linear Errors-in-Variables Models[J]. The Scandinavian Journal of Statistics,2006,33(1):153-168.
[13]Wang Q, Rao J N K. Empirical likelihood for linear regression models under imputation for missing responses [J]. Canad J Statist,2001,29(4):597-608.
[14]Wang Q, Oliver L, Wolfgang H. Semiparametric regression analysis with response missing at random [J]. J Amer Statist Assoc,2004,99:334-345.
[15]Anderson T W. Estimating linear statistieal relationshiPs [J].The Annals of Statistics,1984,12:1-45.
[16]Carroll R J, Spiegeman C H, Lan K G. On errors-in-variables for binary regression mod-els[J]. Biometrika,1984,71:19-25.
[17]Stefanski L A. The effects of measurement error on parameter estimation [J]. Biometrika, 1985,72:583-592.
[18]Fuller W A. Meastirement Error Models [M]. New York:Wiley,1987.
[19]Wang Q H. Estimation of linear error-in-response models with validation data under random censorship. Journal of Multivariate Analysis,2000,74:245,266.
[20]Cui H J, Chen 5 X.Empirieal likelihood confidence regions for parameter in the errors-in-variables models.Journal of Multivariate Analysis,2003,84:101-1-5.
[21]Fan J Q, Truong Y K. Nonparametric regression with errors in variables.The Annals of Statistics,21:1900-1925.
[22]Carroll R J, Ruppert D Stefanski L A.Measurement Error in nonlinear models[M]. London: Chapman, Hall,1995.
[23]Liang H. Asymptotic normality of parametric part in partially linear models with measure-ment erorr in the nonparametric. Journal of Statistical Plannig and Inference.2000,86:51-62.
[24]Cui H J, Li R C. On Parameter estimation for Semi-linear Errors-in-Variables Models. Jour-nal of Multivariate Analysis,1998,64:1-24.
[25]Cui H J. Estimation in Partial linear EV models with replieated observations. Science in China, Series A Mathematies,2004,47(1):144-159.
[26]Liang W, Wang N. Partially linear single-index measurement erorr models. Statistica Sinica, 2005,15:99-116.
[27]You J H, Chen G M. Estimation of a semiparametric varying-coefficient Partially linear errors-in-variables model. Journal of Multivariate Analysis,2006,97:324-341.
[28]李志强,薛留根.缺失数据下的半参数变系数模型的借补估计.于稿.
[29]Carroll R J, Ruppert D, Stefanski L A and Crainiceanu C M Measurement Error in Nonlinear Models,2nd ed.Chapman and Hall,New York.
[30]Fan J, Huang T. Profile likelihood inferences on semiparametric vary-coefficient paratially linear regression models. Bernoulli,2005,11:1031-1057.
[31]Liang H, Wang S, Robins J M, Carroll R J. Estimation in partially linear models with missing covariates[J]. J Amer Statist Assoc,2004,99:357-367.
[32]Mack, Y.P., Silverman, B.W. Weak and strong uniform consistency of kernel regression estimates. Z. Wahrsch. Verw. Gebiete 1982,61:405-415.
[33]Xia Y, Li W K. On the estimation and testing of functional coefficient linear models. Statist Sinica,1999,9:737-757.
[34]李志强,薛留根.响应变量随机缺失下的半参数变系数模型的均值估计.数学的实践与认识,2006,36(10)：191-196.
[35]You J, Zhou Y. Empirical likelihood for semiparametric vary-coefficient paratially linear regression models. Statistical and Probablity letters,2006,76:412-422.
[36]Zhou Y, Liang H. Statistical inference for semiparametric vary-coefficient paratially linear models with error-prone linear covariates.Annals of Statistics.2009,37(1):427-458.
[37]Zhou Y, Wan A.T.K., Wang X J. Estimating equations inference with missing data. J Amer Statist Assoc,2008,103(483):1187-1199.
[38]Little R J A, Rubin D B. Statistical Analysis with Missing Data[M]. New York:John Wiley, 1987.
[39]Lehmann E L. Theory of Point Estimation[M]. New York:Wiley,1983.
[40]Lehmann E L. Testing Statistical Hypotheses (2nd edition)[M]. New York:Wiley,1986.