固定设计下一类半参数回归模型的渐近性质
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摘要
半参数回归模型是二十世纪八十年代发展起来的一种重要的统计模型。这种模型既有参数分量，又含有非参数分量，兼顾了参数回归模型和非参数回归模型的优点，较单纯的参数回归模型或非参数回归模型有更大的适应性，并具有更强的解释能力。
本论文主要研究一类重要的半参数回归模型
y_i=x′_iβ+g(x_i)+e_i，i=1，2，…，n，其中x_i=(x_i1，x_i2，…，x_id)＇，(d≥1)，g(·)是未知函数，β是未知待估参数，e_i是随机误差，且Ee_i=0，Ee_i~2=σ~2＜∞。
在许多实际问题中，我们遇到的多为x_i是非随机设计点列，即固定设计点列的情况。因而本论文主要是研究在x_i是固定设计情况下，此类半参数回归模型的大样本性质相合性。
与通常采用的两阶段估计方法即非参数权函数法结合最小二乘法不同，考虑到此模型本身的特性，本文采用最小二乘法结合一般非参数权函数估计方法，定义了未知待估参数β和未知函数g(·)及误差方差σ~2的估计量(?)_n，(?)_n(·)和(?)_n~2。其估计方法的基本思路是先基于线性模型y_i=x′_iβ+e_i，定义未知待估参数β的估计即此线性模型的最小二乘估计(?)_n；然后将所得估计(?)_n代入原半参数回归模型中，用一般的非参数权函数方法定义未知函数g(·)的估计(?)_n(·)；最后定义误差方差σ~2的估计(?)_n~2。
为讨论简单计，本文主要讨论模型在d=1时即一维的情况下的相合性。
北京工业大学理学硕士学位论文
同时指出多维情况可由一维情况平行的得到推广，从而得到该类模型在固定
设计下的相合性这一重要的大样本性质．
本论文在X；是固定设计的情况下，假定未知函数9（·）连续，对非参数权
函数的条件更为一般和基本，并对随机误差e；的矩的要求有限，讨论并证明
了在这些条件下，P；g（·）的估计量札lin（·）及误差方差a’的估计量枯相合
性和叭三2）阶平均相合性．
A semiparametric regression model has been an important statistics model since 1980s. This kind of model includes not only a parametric component but also a nonparametric component. So it has the advantages of the parametric regression model and the nonparametric regression model. It has the more implements and stronger explanations than the pure parametric or nonparametric regression model .
This paper considers an important semiparametric regression model
where is an unknown function, is an unkown parameter to be estimated. ei are iid. random errors with Bei = 0 and Eei2 = 2 < .
In many practical problems, Xi usually are some nonrandom design points, that is , fixed design points. So, the purpose of this paper concentrates on the semiparametric regression model's large sample property-consitency when xi are the fixed design points.
Unlike the normal two stages estimate method (the usual nonparametric weighted method combined with the least square estimate) , considering the characteristics of this model, this paper uses the least square estimate combining with the usual nonparametric weighted method and defines the estimators and n2 for the unknown parameter ,the unkown fuction g( ) and the unknown variance of errors 2 .
The basic idea of the estimate method is ,firstly ,based on the linear model yi = x'i + ei,defining the least square estimator n of the linear model for the unkown parametric ;secondly, using the estimator n we 've got to substitute for in the original semiparametric regression model yi = x'i + g(xi) + ei and using the usual nonparametric weighted function method to define the estimator gn(-) for the unknown function g( ); finally ,defining the estimator 2 for the unknown variance of errors 2.
For simplicity ,this paper focuses on the consistency of the model in one dimension case,that is d = 1. This paper also points out the consistency that can be generalized more than one dimension. So ,we achieve the large sample property -consistency of this class of model on the fixed design.
In this paper,for fixed design points xi; under the assumption that the unknown function g is continuous function and the moment of random error exists and is finity,we discuss and show that the estimators n,gn and n2 for ,g and 2 have strong consistency, p th-mean consistency for more general nonparametric weighted fuction.
本论文主要研究一类重要的半参数回归模型
y_i=x′_iβ+g(x_i)+e_i，i=1，2，…，n，其中x_i=(x_i1，x_i2，…，x_id)＇，(d≥1)，g(·)是未知函数，β是未知待估参数，e_i是随机误差，且Ee_i=0，Ee_i~2=σ~2＜∞。
在许多实际问题中，我们遇到的多为x_i是非随机设计点列，即固定设计点列的情况。因而本论文主要是研究在x_i是固定设计情况下，此类半参数回归模型的大样本性质相合性。
与通常采用的两阶段估计方法即非参数权函数法结合最小二乘法不同，考虑到此模型本身的特性，本文采用最小二乘法结合一般非参数权函数估计方法，定义了未知待估参数β和未知函数g(·)及误差方差σ~2的估计量(?)_n，(?)_n(·)和(?)_n~2。其估计方法的基本思路是先基于线性模型y_i=x′_iβ+e_i，定义未知待估参数β的估计即此线性模型的最小二乘估计(?)_n；然后将所得估计(?)_n代入原半参数回归模型中，用一般的非参数权函数方法定义未知函数g(·)的估计(?)_n(·)；最后定义误差方差σ~2的估计(?)_n~2。
为讨论简单计，本文主要讨论模型在d=1时即一维的情况下的相合性。
北京工业大学理学硕士学位论文
同时指出多维情况可由一维情况平行的得到推广，从而得到该类模型在固定
设计下的相合性这一重要的大样本性质．
本论文在X；是固定设计的情况下，假定未知函数9（·）连续，对非参数权
函数的条件更为一般和基本，并对随机误差e；的矩的要求有限，讨论并证明
了在这些条件下，P；g（·）的估计量札lin（·）及误差方差a’的估计量枯相合
性和叭三2）阶平均相合性．
A semiparametric regression model has been an important statistics model since 1980s. This kind of model includes not only a parametric component but also a nonparametric component. So it has the advantages of the parametric regression model and the nonparametric regression model. It has the more implements and stronger explanations than the pure parametric or nonparametric regression model .
This paper considers an important semiparametric regression model
where is an unknown function, is an unkown parameter to be estimated. ei are iid. random errors with Bei = 0 and Eei2 = 2 < .
In many practical problems, Xi usually are some nonrandom design points, that is , fixed design points. So, the purpose of this paper concentrates on the semiparametric regression model's large sample property-consitency when xi are the fixed design points.
Unlike the normal two stages estimate method (the usual nonparametric weighted method combined with the least square estimate) , considering the characteristics of this model, this paper uses the least square estimate combining with the usual nonparametric weighted method and defines the estimators and n2 for the unknown parameter ,the unkown fuction g( ) and the unknown variance of errors 2 .
The basic idea of the estimate method is ,firstly ,based on the linear model yi = x'i + ei,defining the least square estimator n of the linear model for the unkown parametric ;secondly, using the estimator n we 've got to substitute for in the original semiparametric regression model yi = x'i + g(xi) + ei and using the usual nonparametric weighted function method to define the estimator gn(-) for the unknown function g( ); finally ,defining the estimator 2 for the unknown variance of errors 2.
For simplicity ,this paper focuses on the consistency of the model in one dimension case,that is d = 1. This paper also points out the consistency that can be generalized more than one dimension. So ,we achieve the large sample property -consistency of this class of model on the fixed design.
In this paper,for fixed design points xi; under the assumption that the unknown function g is continuous function and the moment of random error exists and is finity,we discuss and show that the estimators n,gn and n2 for ,g and 2 have strong consistency, p th-mean consistency for more general nonparametric weighted fuction.
引文
1 Stone,C.J. Consitent Nonparametric Regression[J]. The Annuals of Statistics.5:4(1977), 595-645
2 Engel R,Granger C,Rice J,etal. Nonparametric Estimation of the Ratetion Between Weather and Electricity Sales. J.amer.Statist.Assoc.81(1988), 310-320
3 Heckman,N. Spline smoothing in a partly linear model. J.Roy.Statist.Soc.Set.B48(1986),244-248
4 Rice,J. Covergence rates for partial linear models. Statist. and Probab. Letters. 4(1986),203-208
5 Chen,H. Covergence rates for parameteic Components in a partly linear. Ann. Statist. 16{1988),136-146
6 Gao Jiti. Covergence Rate of some Estemators for a Semiparametric Regression Model. Journal of China University of Science and Technology.22:2(1992)
7 Liang,H. Asymtotic efficiency in semiparametric regression models. Ph.D.thesis. Institute of Systems Science Academia Sinica.(1992)
8 柴根象，徐克军．半参数回归模型的小波光滑[C]．第五届中国统计讨论会议论文集．(1997)
9 钱伟民，柴根象．半参数回归模型小波估计的强逼近[J]．中国科学．29：3(1999)，
233-240
10 钱伟民，柴根象，蒋凤瑛．半参数回归模型的误差方差的小波估计．数学年刊．21A：3(2000)，341-350
11 Shi Peide,Li Guoying. A Note on the Convergence Rates of M-Estimates for Partly Linear Model. Statistics. 26(1995),27-47
12 薛留根．半参数回归模型中随机加权 M 估计的强逼．应用数学学报．25：4(2002)，591-603
13 王启华，郑忠国．随机删失下半参数回归模型中估计的渐近性质．中国科学．A辑，27：7(1997)，583-594
14 薛留根．随机删失下半参数回归模型的估计理论．数学年刊．20A：6(1999)，745-754
15 Gao,J.T.and Zhao,L.C.Adaptive estimation in partly linear regression models. Scientia Sinica. Ser. A,1(1993),14-27
16 高集体，洪圣岩，梁华．部分线性模型中估计的收敛速度．数学学报．38：5(1995)，658-669
17 王启华．随机截断下半参数回归模型的相合估计．中国数学．25：8(1995)，819-832
18 陈明华．固定设计下半参数回归模型估计的收敛速度．应用概率统计．14：2(1998)，149-158
19 陈明华．固定设计下半参数回归模型估计的相合性．高校应用数学学报．
A辑，13：3(1998)，301-310
20 柴根象，孙平．半参数回归模型的二阶段估计的渐近正态性[J]．应用数学学报．18：3(1995)，354-363
21 薛留根，韩建国，．半参数回归模型中二阶段估计的渐近性质．高校应用数学学报．A辑，16：1(2001)，87-94
22 Stone,C.J.Additive regression and other nonparametric models. Ann. Statist. 13(1985),689-705
23 Hastie,T.and R.Tibshirani. Statist.Sci. 1(1986),197-318
24 Hastie,T.and R.Tibshirani. J.Amer.Statist.Assoc. 82(1987),371-386
25 Buja.A..T.Hastie and R.Tibshirani. Ann.Statist. 17(1989),453-555
26 赵江．半参数回归函数混合型估计均方收敛性．应用数学学报．8：2(1985)
27 张辉霞，薛留根．用混合回归模型评价高等数学教学效果．数理统计与管理．O212．7(2001)
28 Gao,J.T. A large sample theory in semiparametric regression models,Ph.D. thesis. Graduate school at the University of Science & Technology of China. Hefei
29 欧阳资生，鄢茵．一种包含递归的核回归估计的回归预测模型．数理统计与管理．5：20(2001)
30 高集体，洪圣岩，梁华，施沛德．半参数回归研究德若干进展．应用概率统计．10：1(1994)，96-103
31 Fraiman R,Iribarren P. Nonparametric regression estimation in models with weak error's structure.J.Mult.Anal,37:2(1991),180-196
32 胡舒合．一类新的半参数回归模型中的相合估计．数学学报．40：4(1997)
33 薛留根．半参数回归模型中小波估计的随机加权逼近速度．应用数学学报．26：1(2003)
34 Gao,J.T. Asymptotically efficient estimatiaon in partial linear models(1992)
35 Bennett,G. Probability inequalities for the sums of independent random variables.JASA. 57(1962),33-45
36 Whittle,P. Bounds for the moments of linear and quadradic forms in independent random variables. Theory Probab.Appl.5(1960),302-305
37 严士健等．概率论基础第一版．科学出版社．北京，(1982)
38 柴根象，洪圣岩编著．半参数回归模型安徽教育出版社．(1995)
39 洪圣岩，赵忠柏．偏线性模型的核-最小二乘估计法的渐进性质．数学年刊．14A：6(1993)，717-731
40 林正炎，陆传荣，苏中根．概率极限理论基础．高等教育出版社．(1998)
41 陈希孺，柴根象．非参数统计教程．华东师范大学出版社．(1989)
42 高惠璇．统计计算．北京大学出版社．(1995)
43 肖云茹．概率统计计算方法．南开大学出版社．(1994)
44 刘宏友．Matlab 6基础及应用．重庆大学出版社．(2002)
2 Engel R,Granger C,Rice J,etal. Nonparametric Estimation of the Ratetion Between Weather and Electricity Sales. J.amer.Statist.Assoc.81(1988), 310-320
3 Heckman,N. Spline smoothing in a partly linear model. J.Roy.Statist.Soc.Set.B48(1986),244-248
4 Rice,J. Covergence rates for partial linear models. Statist. and Probab. Letters. 4(1986),203-208
5 Chen,H. Covergence rates for parameteic Components in a partly linear. Ann. Statist. 16{1988),136-146
6 Gao Jiti. Covergence Rate of some Estemators for a Semiparametric Regression Model. Journal of China University of Science and Technology.22:2(1992)
7 Liang,H. Asymtotic efficiency in semiparametric regression models. Ph.D.thesis. Institute of Systems Science Academia Sinica.(1992)
8 柴根象，徐克军．半参数回归模型的小波光滑[C]．第五届中国统计讨论会议论文集．(1997)
9 钱伟民，柴根象．半参数回归模型小波估计的强逼近[J]．中国科学．29：3(1999)，
233-240
10 钱伟民，柴根象，蒋凤瑛．半参数回归模型的误差方差的小波估计．数学年刊．21A：3(2000)，341-350
11 Shi Peide,Li Guoying. A Note on the Convergence Rates of M-Estimates for Partly Linear Model. Statistics. 26(1995),27-47
12 薛留根．半参数回归模型中随机加权 M 估计的强逼．应用数学学报．25：4(2002)，591-603
13 王启华，郑忠国．随机删失下半参数回归模型中估计的渐近性质．中国科学．A辑，27：7(1997)，583-594
14 薛留根．随机删失下半参数回归模型的估计理论．数学年刊．20A：6(1999)，745-754
15 Gao,J.T.and Zhao,L.C.Adaptive estimation in partly linear regression models. Scientia Sinica. Ser. A,1(1993),14-27
16 高集体，洪圣岩，梁华．部分线性模型中估计的收敛速度．数学学报．38：5(1995)，658-669
17 王启华．随机截断下半参数回归模型的相合估计．中国数学．25：8(1995)，819-832
18 陈明华．固定设计下半参数回归模型估计的收敛速度．应用概率统计．14：2(1998)，149-158
19 陈明华．固定设计下半参数回归模型估计的相合性．高校应用数学学报．
A辑，13：3(1998)，301-310
20 柴根象，孙平．半参数回归模型的二阶段估计的渐近正态性[J]．应用数学学报．18：3(1995)，354-363
21 薛留根，韩建国，．半参数回归模型中二阶段估计的渐近性质．高校应用数学学报．A辑，16：1(2001)，87-94
22 Stone,C.J.Additive regression and other nonparametric models. Ann. Statist. 13(1985),689-705
23 Hastie,T.and R.Tibshirani. Statist.Sci. 1(1986),197-318
24 Hastie,T.and R.Tibshirani. J.Amer.Statist.Assoc. 82(1987),371-386
25 Buja.A..T.Hastie and R.Tibshirani. Ann.Statist. 17(1989),453-555
26 赵江．半参数回归函数混合型估计均方收敛性．应用数学学报．8：2(1985)
27 张辉霞，薛留根．用混合回归模型评价高等数学教学效果．数理统计与管理．O212．7(2001)
28 Gao,J.T. A large sample theory in semiparametric regression models,Ph.D. thesis. Graduate school at the University of Science & Technology of China. Hefei
29 欧阳资生，鄢茵．一种包含递归的核回归估计的回归预测模型．数理统计与管理．5：20(2001)
30 高集体，洪圣岩，梁华，施沛德．半参数回归研究德若干进展．应用概率统计．10：1(1994)，96-103
31 Fraiman R,Iribarren P. Nonparametric regression estimation in models with weak error's structure.J.Mult.Anal,37:2(1991),180-196
32 胡舒合．一类新的半参数回归模型中的相合估计．数学学报．40：4(1997)
33 薛留根．半参数回归模型中小波估计的随机加权逼近速度．应用数学学报．26：1(2003)
34 Gao,J.T. Asymptotically efficient estimatiaon in partial linear models(1992)
35 Bennett,G. Probability inequalities for the sums of independent random variables.JASA. 57(1962),33-45
36 Whittle,P. Bounds for the moments of linear and quadradic forms in independent random variables. Theory Probab.Appl.5(1960),302-305
37 严士健等．概率论基础第一版．科学出版社．北京，(1982)
38 柴根象，洪圣岩编著．半参数回归模型安徽教育出版社．(1995)
39 洪圣岩，赵忠柏．偏线性模型的核-最小二乘估计法的渐进性质．数学年刊．14A：6(1993)，717-731
40 林正炎，陆传荣，苏中根．概率极限理论基础．高等教育出版社．(1998)
41 陈希孺，柴根象．非参数统计教程．华东师范大学出版社．(1989)
42 高惠璇．统计计算．北京大学出版社．(1995)
43 肖云茹．概率统计计算方法．南开大学出版社．(1994)
44 刘宏友．Matlab 6基础及应用．重庆大学出版社．(2002)