部分线性回归模型的估计
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摘要
部分线性回归模型由于兼具参数部分和非参数部分,比较容易解释各个变量的影响情况,从而在实际应用中有着更强的灵活性和适用性。从1986年Engle等在研究电力需求和气候变化之间的关系时建立该模型,至今,国内外学者对部分线性回归模型的研究已具有相当的规模。在现有文献资料的基础上,本文就部分线性回归模型在误差为相依的情形下,运用一般权函数法并综合最小二乘法得到了未知参数和未知函数的估计量,接着讨论了估计量的相合性及渐近正态性。
首先,本文对部分线性回归模型的历史背景及国内外的研究发展状况作以简介。主要是在观察值为固定设计时,对部分线性回归模型的误差项作各种不同假设,以便寻求对参数部分及非参数部分合理的估计方法及对所得估计量的大样本性质的讨论途径。
其次,对部分线性回归模型在误差为线性过程时,利用两阶段估计法并综合一般权函数法及最小二乘法,得到了β和g(·)的估计量,然后利用鞅差序列加权和的中心极限定理,讨论了估计量的渐近正态性,进而证明了估计量的相合性。相比现有文献中关于估计量的相关性质的证明,本文的证明方法假设条件较弱,主要是去掉了一些跟对数运算有关的限制条件,引进了关于权函数的部分极限性质,使得条件更具有普遍性,也很容易验证,从而更便于操作和运用。
最后,对上述模型在误差为ρ混合序列情形下,利用小波方法得到β和g(·)的估计量,进而证明了估计量的r-阶相合性。
Including parametric component and non-paramertic component, partially linear regression models allow easier interpretation of the effect on each variable, so they are more flexible and universal than the classic linear models or non-paramertic regression models. Engle, Grabger, Rice and Weiss were among the first to consider the partially linear regression models. In 1986, they analyzed the ralationship between temperature and electricity usage. From then on, there have been maken great progress in the studies on partially linear models. In this paper, the consistency and asymptotic normality of the estimators are studied on the base of domestic and overseas scholar researches.
Firstly, the partially linear regression models are introduced from the origin to the development at home and abroad, mainly including all kinds of estimation methods to parametric component and non-parametric component and studies to the properties of large sample. The studies are based on the different supposes to the errors in the models with fixed designed points.
Secondly, considering the partially linear models with linear process errors Bying least squares and usual weighted function method combining two-stage estimation, we define the estimatorsβand g forβand g(·), then we obtain their r-order mean consistency, complete consistency and asymptotic normality under suitable conditions. Since given conditions become weaker, the means in this paper are more universal and flexible than other methods, there we remove some restrictions on logrithm calculation, and introduce some Iimite properties on power function.
Finally, to the above model withρ-mixing sequence, we use Wavelet methods and obtain the estimators of/βand g(·), then we study the r-order consistency of the estimators.
首先,本文对部分线性回归模型的历史背景及国内外的研究发展状况作以简介。主要是在观察值为固定设计时,对部分线性回归模型的误差项作各种不同假设,以便寻求对参数部分及非参数部分合理的估计方法及对所得估计量的大样本性质的讨论途径。
其次,对部分线性回归模型在误差为线性过程时,利用两阶段估计法并综合一般权函数法及最小二乘法,得到了β和g(·)的估计量,然后利用鞅差序列加权和的中心极限定理,讨论了估计量的渐近正态性,进而证明了估计量的相合性。相比现有文献中关于估计量的相关性质的证明,本文的证明方法假设条件较弱,主要是去掉了一些跟对数运算有关的限制条件,引进了关于权函数的部分极限性质,使得条件更具有普遍性,也很容易验证,从而更便于操作和运用。
最后,对上述模型在误差为ρ混合序列情形下,利用小波方法得到β和g(·)的估计量,进而证明了估计量的r-阶相合性。
Including parametric component and non-paramertic component, partially linear regression models allow easier interpretation of the effect on each variable, so they are more flexible and universal than the classic linear models or non-paramertic regression models. Engle, Grabger, Rice and Weiss were among the first to consider the partially linear regression models. In 1986, they analyzed the ralationship between temperature and electricity usage. From then on, there have been maken great progress in the studies on partially linear models. In this paper, the consistency and asymptotic normality of the estimators are studied on the base of domestic and overseas scholar researches.
Firstly, the partially linear regression models are introduced from the origin to the development at home and abroad, mainly including all kinds of estimation methods to parametric component and non-parametric component and studies to the properties of large sample. The studies are based on the different supposes to the errors in the models with fixed designed points.
Secondly, considering the partially linear models with linear process errors Bying least squares and usual weighted function method combining two-stage estimation, we define the estimatorsβand g forβand g(·), then we obtain their r-order mean consistency, complete consistency and asymptotic normality under suitable conditions. Since given conditions become weaker, the means in this paper are more universal and flexible than other methods, there we remove some restrictions on logrithm calculation, and introduce some Iimite properties on power function.
Finally, to the above model withρ-mixing sequence, we use Wavelet methods and obtain the estimators of/βand g(·), then we study the r-order consistency of the estimators.
引文
[1]Green P. J.. Penalized likelihood for general semiparametric regression models [J]. Int. Statist. Rev.,1985,55:245-259
[2]Engle R. E., Rice J.. Semiparametric estimates of the relation between weather and electricity sales [J]. JASA,1986,81:310-320
[3]Heckman N. E.. Spline smoothing in partly linear models:Journal of the Royal Statistical Society [J]. Series B,1986,48:244-248
[4]Hamilton S. A., Troung Y. K.. Local linear estimation in partly linear models [J]. Jour-nal of Multivariate Analysis,1997,60:1-19
[5]高集体,洪圣岩,梁华.部分线性模型中估计的收敛速度[J].数学学报,1995,38(5)：658-669
[6]薛留根.随机删失下半参数回归模型的估计理论[J].数学年刊,1999,620A：745-754
[7]陈明华.完全与截尾样本时半参数回归模型中估计的强相合性[J].数学物理学报,1999,19(5)：501-506
[8]陈明华,任哲,胡舒合.部分线性模型模型中估计的强相合性[J].数学学报,1999,19(5)： 501-506
[9]柴根象,徐克军.半参数回归的线性小波光滑[J].应用概率统计,1992,15(1)：97-105
[10]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,1993,29(3)：233-240
[11]薛留根.半参数回归模型中小波估计的随机加权逼近速度[J].应用数学学报,2003,26(1)：11-25
[12]刘强,姜玉英,吴可法.半参数变量含误差函数关系模型的小波估计[J].应用数学学报,2005,28(2)：296-307
[13]Hamilton S. A., Truong Y. K.. Loacal linear estimation in partly models [J]. Multivariate Anal,1997,60:1-19
[14]金丽红.半参数回归模型误差为鞅差序列的r阶矩相合性[J].武汉工业学院学报,2004,23(1)：82-84
[15]胡舒合.误差为线过程时回归模型的估计问题[J].高校应用数学学报,2003,18(1)：81-90 大学硕士学位论文
[15]胡舒合.误差为线过程时回归模型的估计问题[J].高校应用数学学报,2003,18(1)： 81-90
[16]Hall P, Heyde C C.. Martinggale limit theory and its application [M]. Academic Prees, New York,1980
[17]Liptser R S, Shiryayev A N. Theorem of martingales [M]. Nethelands:Kluwer Academic Pulishers,1989
[18]凌能祥.线性过程误差下的半参数模型[J].安徽大学学报,2004,28(4)：15-18
[19]胡舒合.鞅差序列加权和的中心极限定理[J].高校应用数学学报,2001,24(4)：539-546
[20]吴群英.ρ混合序列的若干收敛性质[J].工程数学学报,2002,18(3)：58-64
[21]Bradley R C. Equivalent mixing conditions for random fields [R]. Technical Report No 336:Center for Stochastic Processes, Chaple Hill:Univ of North Carolina, 1990
[22]Bradley R C. On the spectral density and asymptotic normality of weakly random fields [J]. Theor Probab,1992,5:355-374
[23]Bryc W, Smolenski W. Moment conditions for almost sure convergence of wealy correlated random variables [J]. Proc Amer Math Soc,1993,119(2):629-635
[24]杨善朝.一类随机变量部分和的矩不等式及应用[J].科学通报,1998,43(17)：1823-1827
[25]吴群英.不同分布ρ混合序列的强收敛速度[J].数学研究与评论,2004,24(1)：173-179
[26]黄海牛,吴群英,贾贞.不同分布ρ混合序列的完全收敛性[J].高校应用数学学报,2009,24(3)：253-258
[27]冯凤香.不同分布ρ混合序列加权和的完全收敛性和强收敛性[J].桂林工学院学报,2008,28(2)：282-284
[28]吴群英.混合序列加权和的完全收敛性和强收敛性[J].应用数学,2002,15(1)：1-4
[29]冯凤香.不同分布φ混合序列加权和的完全收敛性和强收敛性[J].广西科 1025-1034
[32]Lin Z. Y, Lu C. R.. The limit theory for mixing dependent random variables [M]. Beijing, Science Press,1997
[33]Kim T.Y.. Monment bounds for non-stationary dependent sequences [J]. Appl Prob,1994, 31:731-742
[34]Zhou Y.. Convergence of oscillation module of empirical processes under dependent sample [J]. Chin. Ann. Math,1996,17:1-8
[35]Yang S. C.. Moment inequality for mixing sequences and non-parametric estimation [J]. Acta Math Sinica,1997,40:271-279
[36]Dsukhan D.. Mixing properties and examples [M]. Lecture Notes in Statistics, Berlin: Springer,1994
[37]Prakasa R.. Non-parametric Functional Estimation [M]. Newyork: Academic Press,1983
[38]吴群英,林亮.φ混合序列的完全收敛性和强收敛性[J].工程数学学报,2004,21(1)：75-80
[39]Baum L. E, Katz M.. Convergence rates in the lay of large numbers [J]. Trar Amer Math Soc,1965,120:108-123
[40]Herrndorf N.. The invariance principle for φ-mixing sequence [J]. Wahr Verw Gebiete, 1983,63:97-108
[41]潘雄,孙海燕.半参数模型误差为NA序列时的r阶矩相合性[J].武汉工业学院学报,2004,23(4)：104-108
[42]Su Chun, Zhao Linchen. Mean Inequality and Weak Convergence for NA φ Sequence [J]. Science in China (series A),1996,26(12):1091-1099
[43]Ren Zhe, Chen Minhua. Strong Consistency of A class of Estimators in Semiparametric Regression Model When the Sample is A Sequence of NA [J]. Appl Math J China Univ Ser A, 2000,15(4):467-474
[44]Antoniads A, Gregoire G, Mckeague I W.. Wavelet Methods for Curve Estimation [J]. Amer Statist Assoc,1994,89:1340-1353
[45]Qian Weimin, Chai Genxiang. Strong Approximation of Wavelet Estimate in Semiparametric Regression Model [J]. Science in China(series A),1999,29(3):233-240
[46]Chai Genxiang, Xu Kejun. Wavelet smoothing in semiparametric regression model [J]. China J of AAPPL Prob and Stati,1999,15(1):97-105
[47]何宝珠.混合序列的几乎处处收敛速度[J].桂林工学院学报,2006,25(2)：256-258
[48]唐国强,伍艳春.混合序列加权和的完全收敛性和强收敛性[J].桂林工学院学报,2004,24(1)：100-102
[49]刘萍.不同分布φ混合序列的完全收敛性[J].桂林工学院学报,2006,26(2)：298-301
[50]陈希孺.数理统计引论[M].北京：科学出版社,1981
[51]柴根象,洪圣岩.半参数回归模型[M].合肥：安徽教育出版社,1995
[52]茆诗松,王静龙,濮晓龙.高等数理统计[M].北京：高等教育出版社,1998
[53]王松桂.线性模型的理论及其应用[M].合肥：安徽教育出版社,1996
[54]林正炎,陆传荣,苏中根.概率极限理论基础[M].北京：高等教育出版社,1999
[2]Engle R. E., Rice J.. Semiparametric estimates of the relation between weather and electricity sales [J]. JASA,1986,81:310-320
[3]Heckman N. E.. Spline smoothing in partly linear models:Journal of the Royal Statistical Society [J]. Series B,1986,48:244-248
[4]Hamilton S. A., Troung Y. K.. Local linear estimation in partly linear models [J]. Jour-nal of Multivariate Analysis,1997,60:1-19
[5]高集体,洪圣岩,梁华.部分线性模型中估计的收敛速度[J].数学学报,1995,38(5)：658-669
[6]薛留根.随机删失下半参数回归模型的估计理论[J].数学年刊,1999,620A：745-754
[7]陈明华.完全与截尾样本时半参数回归模型中估计的强相合性[J].数学物理学报,1999,19(5)：501-506
[8]陈明华,任哲,胡舒合.部分线性模型模型中估计的强相合性[J].数学学报,1999,19(5)： 501-506
[9]柴根象,徐克军.半参数回归的线性小波光滑[J].应用概率统计,1992,15(1)：97-105
[10]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,1993,29(3)：233-240
[11]薛留根.半参数回归模型中小波估计的随机加权逼近速度[J].应用数学学报,2003,26(1)：11-25
[12]刘强,姜玉英,吴可法.半参数变量含误差函数关系模型的小波估计[J].应用数学学报,2005,28(2)：296-307
[13]Hamilton S. A., Truong Y. K.. Loacal linear estimation in partly models [J]. Multivariate Anal,1997,60:1-19
[14]金丽红.半参数回归模型误差为鞅差序列的r阶矩相合性[J].武汉工业学院学报,2004,23(1)：82-84
[15]胡舒合.误差为线过程时回归模型的估计问题[J].高校应用数学学报,2003,18(1)：81-90 大学硕士学位论文
[15]胡舒合.误差为线过程时回归模型的估计问题[J].高校应用数学学报,2003,18(1)： 81-90
[16]Hall P, Heyde C C.. Martinggale limit theory and its application [M]. Academic Prees, New York,1980
[17]Liptser R S, Shiryayev A N. Theorem of martingales [M]. Nethelands:Kluwer Academic Pulishers,1989
[18]凌能祥.线性过程误差下的半参数模型[J].安徽大学学报,2004,28(4)：15-18
[19]胡舒合.鞅差序列加权和的中心极限定理[J].高校应用数学学报,2001,24(4)：539-546
[20]吴群英.ρ混合序列的若干收敛性质[J].工程数学学报,2002,18(3)：58-64
[21]Bradley R C. Equivalent mixing conditions for random fields [R]. Technical Report No 336:Center for Stochastic Processes, Chaple Hill:Univ of North Carolina, 1990
[22]Bradley R C. On the spectral density and asymptotic normality of weakly random fields [J]. Theor Probab,1992,5:355-374
[23]Bryc W, Smolenski W. Moment conditions for almost sure convergence of wealy correlated random variables [J]. Proc Amer Math Soc,1993,119(2):629-635
[24]杨善朝.一类随机变量部分和的矩不等式及应用[J].科学通报,1998,43(17)：1823-1827
[25]吴群英.不同分布ρ混合序列的强收敛速度[J].数学研究与评论,2004,24(1)：173-179
[26]黄海牛,吴群英,贾贞.不同分布ρ混合序列的完全收敛性[J].高校应用数学学报,2009,24(3)：253-258
[27]冯凤香.不同分布ρ混合序列加权和的完全收敛性和强收敛性[J].桂林工学院学报,2008,28(2)：282-284
[28]吴群英.混合序列加权和的完全收敛性和强收敛性[J].应用数学,2002,15(1)：1-4
[29]冯凤香.不同分布φ混合序列加权和的完全收敛性和强收敛性[J].广西科 1025-1034
[32]Lin Z. Y, Lu C. R.. The limit theory for mixing dependent random variables [M]. Beijing, Science Press,1997
[33]Kim T.Y.. Monment bounds for non-stationary dependent sequences [J]. Appl Prob,1994, 31:731-742
[34]Zhou Y.. Convergence of oscillation module of empirical processes under dependent sample [J]. Chin. Ann. Math,1996,17:1-8
[35]Yang S. C.. Moment inequality for mixing sequences and non-parametric estimation [J]. Acta Math Sinica,1997,40:271-279
[36]Dsukhan D.. Mixing properties and examples [M]. Lecture Notes in Statistics, Berlin: Springer,1994
[37]Prakasa R.. Non-parametric Functional Estimation [M]. Newyork: Academic Press,1983
[38]吴群英,林亮.φ混合序列的完全收敛性和强收敛性[J].工程数学学报,2004,21(1)：75-80
[39]Baum L. E, Katz M.. Convergence rates in the lay of large numbers [J]. Trar Amer Math Soc,1965,120:108-123
[40]Herrndorf N.. The invariance principle for φ-mixing sequence [J]. Wahr Verw Gebiete, 1983,63:97-108
[41]潘雄,孙海燕.半参数模型误差为NA序列时的r阶矩相合性[J].武汉工业学院学报,2004,23(4)：104-108
[42]Su Chun, Zhao Linchen. Mean Inequality and Weak Convergence for NA φ Sequence [J]. Science in China (series A),1996,26(12):1091-1099
[43]Ren Zhe, Chen Minhua. Strong Consistency of A class of Estimators in Semiparametric Regression Model When the Sample is A Sequence of NA [J]. Appl Math J China Univ Ser A, 2000,15(4):467-474
[44]Antoniads A, Gregoire G, Mckeague I W.. Wavelet Methods for Curve Estimation [J]. Amer Statist Assoc,1994,89:1340-1353
[45]Qian Weimin, Chai Genxiang. Strong Approximation of Wavelet Estimate in Semiparametric Regression Model [J]. Science in China(series A),1999,29(3):233-240
[46]Chai Genxiang, Xu Kejun. Wavelet smoothing in semiparametric regression model [J]. China J of AAPPL Prob and Stati,1999,15(1):97-105
[47]何宝珠.混合序列的几乎处处收敛速度[J].桂林工学院学报,2006,25(2)：256-258
[48]唐国强,伍艳春.混合序列加权和的完全收敛性和强收敛性[J].桂林工学院学报,2004,24(1)：100-102
[49]刘萍.不同分布φ混合序列的完全收敛性[J].桂林工学院学报,2006,26(2)：298-301
[50]陈希孺.数理统计引论[M].北京：科学出版社,1981
[51]柴根象,洪圣岩.半参数回归模型[M].合肥：安徽教育出版社,1995
[52]茆诗松,王静龙,濮晓龙.高等数理统计[M].北京：高等教育出版社,1998
[53]王松桂.线性模型的理论及其应用[M].合肥：安徽教育出版社,1996
[54]林正炎,陆传荣,苏中根.概率极限理论基础[M].北京：高等教育出版社,1999