半参数回归模型的渐进性质及其应用
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摘要
半参数回归模型是一类重要的统计模型,有着广泛的实际应用背景.该模型综合了参数与非参数的信息,比一般的参数模型或非参数模型有更强的解释能力.在理论上,处理这种模型的方法既融合了两类单一性质回归模型的处理方法又不是它们的简单叠加.
考虑半参数回归模型y i = xi'β+g(ti)+eii=1,2,,n (1)其中( xi ,ti)是已知的固定设计点列,β∈Rd为未知参数向量,g (?)为定义于[0,1]上的未知可测函数, ei是随机误差. {e i ,i= 1,,n}的均值为零,方差为σ2的独立同分布的随机误差向量.
文献中一般用最小二乘法估计参数分量β,而非参数分量g (?)估计问题的大多采用样条估计、核估计、L. S.估计和M-估计等.本文采用了新的推断方法近邻估计,这种方法为半参数回归模型的研究提供了新的途径,使得参数统计量β和g (?)估计应用起来更方便,更切合实际,并在给定的基本假设条件下证明了统计量β和g (?)的渐进性质及其在实际问题中应用,这就为进行大样本参数假设检验奠定了基础,也为以后研究统计量g (?)的估计的收敛速度提供了参考价值.研究结果表明,参数统计量β和g (?)的估计方法比其他估计方法更简洁,参数g (?)的渐进性质得到了证实.
半参数回归模型的研究已成为一个重要的研究方向.将极具优势的近邻权函数应用于具有强解释能力的半参数回归模型,这在实际中有着更为广阔的应用背景.深信随着二者在理论和方法上的不断完善和发展,对经济、通信、生物等各个领域都将起着积极的促进作用.
Semi-parametric regression model is an important kind of statistical model which has a wide range of practical applications. This model synthesizes the parametric and non-parametric information and possesses more strong explanatory ability than normal parametric and non-parametric model. Theoretically, the way to deal with this model which integrated the methods of the two kinds of regression models of the single property.
Consider the semi-parametric regression model as follows: y i = xi'β+g(ti)+eii=1,2,,n (1)
In the above equation, ( xi ,ti)is the known fixed design lattice,β∈Rdis the unknown parameter vector, g (?)is the unknown measurable function which define in the range of [0, 1], ei is the stochastic error. {e i ,i= 1,,n}is the independent identically distributed(i. i. d. ) stochastic error vector which the mean value is zero and the variance isσ2.
In the references, the normal method to estimate parametric componentβis least square method, but the methods to estimate non-parametric component g (?)are spline approximation, kernel estimation, L. S. estimation, M-estimation et al. This paper presents a new neighbor estimation of deduction method. This method offers new research way for the semi-parametric regression model. After the new strategy used, to estimate the parametric statisticβand g (?)will be more convenient and practical. We have proven the asymptotic property of the statisticβand g (?)and the practical applications of the method, it will lay the foundation for hypothesis testing of the large sample parameters and provide referential value of the further study of convergence rate of the statistic g (?). The results show that the estimation method of the parametric statisticβand g (?)is more brief and compact than other methods and the asymptotic property of parametric statisticβis also has proven.
The research of semi-parametric regression model is an important research direction at present. The model which chooses dominant neighborhood function as weight function will have extensive application background. Coupled with the development of the two factors, they will play an active role in the area of communications, economy, biology et al. .
考虑半参数回归模型y i = xi'β+g(ti)+eii=1,2,,n (1)其中( xi ,ti)是已知的固定设计点列,β∈Rd为未知参数向量,g (?)为定义于[0,1]上的未知可测函数, ei是随机误差. {e i ,i= 1,,n}的均值为零,方差为σ2的独立同分布的随机误差向量.
文献中一般用最小二乘法估计参数分量β,而非参数分量g (?)估计问题的大多采用样条估计、核估计、L. S.估计和M-估计等.本文采用了新的推断方法近邻估计,这种方法为半参数回归模型的研究提供了新的途径,使得参数统计量β和g (?)估计应用起来更方便,更切合实际,并在给定的基本假设条件下证明了统计量β和g (?)的渐进性质及其在实际问题中应用,这就为进行大样本参数假设检验奠定了基础,也为以后研究统计量g (?)的估计的收敛速度提供了参考价值.研究结果表明,参数统计量β和g (?)的估计方法比其他估计方法更简洁,参数g (?)的渐进性质得到了证实.
半参数回归模型的研究已成为一个重要的研究方向.将极具优势的近邻权函数应用于具有强解释能力的半参数回归模型,这在实际中有着更为广阔的应用背景.深信随着二者在理论和方法上的不断完善和发展,对经济、通信、生物等各个领域都将起着积极的促进作用.
Semi-parametric regression model is an important kind of statistical model which has a wide range of practical applications. This model synthesizes the parametric and non-parametric information and possesses more strong explanatory ability than normal parametric and non-parametric model. Theoretically, the way to deal with this model which integrated the methods of the two kinds of regression models of the single property.
Consider the semi-parametric regression model as follows: y i = xi'β+g(ti)+eii=1,2,,n (1)
In the above equation, ( xi ,ti)is the known fixed design lattice,β∈Rdis the unknown parameter vector, g (?)is the unknown measurable function which define in the range of [0, 1], ei is the stochastic error. {e i ,i= 1,,n}is the independent identically distributed(i. i. d. ) stochastic error vector which the mean value is zero and the variance isσ2.
In the references, the normal method to estimate parametric componentβis least square method, but the methods to estimate non-parametric component g (?)are spline approximation, kernel estimation, L. S. estimation, M-estimation et al. This paper presents a new neighbor estimation of deduction method. This method offers new research way for the semi-parametric regression model. After the new strategy used, to estimate the parametric statisticβand g (?)will be more convenient and practical. We have proven the asymptotic property of the statisticβand g (?)and the practical applications of the method, it will lay the foundation for hypothesis testing of the large sample parameters and provide referential value of the further study of convergence rate of the statistic g (?). The results show that the estimation method of the parametric statisticβand g (?)is more brief and compact than other methods and the asymptotic property of parametric statisticβis also has proven.
The research of semi-parametric regression model is an important research direction at present. The model which chooses dominant neighborhood function as weight function will have extensive application background. Coupled with the development of the two factors, they will play an active role in the area of communications, economy, biology et al. .
引文
[1] Engle R.F, Granger CWJ , Rice J, et.al. Semiparametric estimates of the relation between weather and elect ricity sales[J]. Journal of the American Statistical Association , 1986 , 81 : 310-320.
[2] Gao J T , Chen X R , Zhao L C. Asymptotic normalityof a class estimates in partly linear models [ J ] . Acta Mathematica Sinica ,1994 , 37 : 256-268.
[3]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, 1995, 18 (3) : 354～363.
[4] U spensky, J. V. , Int roduct ion to Mathematical Probability[M ], M cGurw-Hill, New York, 1937.
[5]洪圣岩,一类半参数回归模型的估计理论[J ],中国科学A辑, 1991, 21 (12) : 1258-1272.
[6]洪圣岩,最近邻回归估计的渐近正态性,应用概率统计[J ], 1991, 7 (2) : 187-191.
[7]赵林城,白志东,非参数回归函数最近邻估计的强相合性[J ],中国科学A辑, 1984, 14 (5) : 387-393.
[8]吴友平,彭军还,鲁铁定.半参数估计与LS估计的有效性比较.测绘科学,34(1):98-99,2009
[9]潘雄.半参数模型的估计理论及应用研究[D]武汉,武汉大学, 2005
[10]洪圣岩,Asymptotic properties of kernel smoothing in partial linear model,第四届中日统计讨论会, 1991.
[11]潘建敏.污染数据半参数回归模型的估计方法[J].工程数学学报, 1997(3): 81-85.
[12]陈明华.固定设计下半参数回归模型参数估计的收敛速度[J].应用概率统计, 1998(2):1 49-158.
[13]金丽红.半参数回归模型两步估计的渐进性[J]. 23(2). 102-108. 2004.
[14]钱伟民,柴根象.半参数回归模型的估计的渐进性质. 14(2). A辑161-168 1997.
[15]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:A辑, 1999(6): 745-754.
[16]田萍,薛留根.纵向数据半参数回归模型估计的强相合性[J].工程数学学报, 2006, 23(2): 369-373
[17]陈长生,徐勇勇,夏结来.光滑样条非参数回归方法及医学应用.中国卫生统计,1999 ,16 (6) :342.
[18]陈长生,徐勇勇,尚磊.儿童生长曲线的光滑样条和核估计拟合.中国卫生统计,1997 ;14 (4) :1.
[19] Bates DM ,Lindstrom MJ ,Wahba G,Yandell BS. The GCVPACK- Routines for generalized cross validation. Communication in statistics - simuilation and computa-tion, 1987 ,16 :263.
[20] SAS Institute Inc. SAS INSIGHT user's Guide. Version 6 Second Edition. Cary ,NC ,USA. 1993.
[21] CHANG Xiao2wen ,QU Le2ming. Wavelet estimation ofpartially linear models [J ] . Computational Statistics &Data Analysis , 2004 , 47 : 31248.
[22]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,1999 ,15 (1) :972105.
[23]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) , 11225.
[24]钱伟民,柴根象,将凤瑛.半参数回归模型的误差方差的小波估计[J ] .数学年刊,2000 ,21A(3) :3412350.
[25]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(A辑) ,1999 , 29 (3) :2332240.
[26] Ana Bianco, Graciela Boente. Robust estimators insemiparametric partly linear regression models[J]. Journal of Statistical Planning and Inference, 2004, 122 : 2292252.
[27] SHI Pei-de , TENG Xing-dong. Asymptotic distriburitions of M-estimators of the partly linear models with fixed carriers [J ] . Advancesin Mathematics , 1999 ,28 : 4472461.
[28]薛留根.半参数回归模型中随机加权M估计的强逼近[J ] .应用数学学报, 2002 ,25 (4) :5912603.
[29]陈希孺.线性模型参数M估计的线性表示[J ] .中国科学(A辑) ,1993 , 23 (12) :126421275.
[30] CHEN H. Convergence rates for parametric components in a partly linear model[J]. Ann. Statist. 1988, 16: 136-146.
[31]中国国家统计局.中国统计年鉴2006 [M ].北京:中国统计出版社, 2006.
[32]钱伟民,柴根象.一类半参数回归模型二阶段估计的渐近理论[J].同济大学学报:自然科学版, 1998(1): 77-82.
[33]李玉梅,钱伟民.半参数混合效应模型中的估计及其强相合性[J]. . 28(1): 107-120,系统科学与数学. 2008.
[34]田萍,薛留根.纵向数据下半参数回归模型估计的渐近性质[J]. . 23(4):369-376.应用概率统计. 2007.
[35]胡宏昌.半参数回归模型M估计的线性强表示[J]. 25(5): 5-7,周口师范学院学报. 2008.
[36] Ana Bianco, Graciela Boente. Robust estimators in semiparametric partly linear regression models [J]. . Journal of Statistical Planning and Inference, 2004 ,122 : 229-252.
[37]刘志杰,季令,叶玉玲,耿志民.基于径向基神经网络的铁路货运量预测[J]. . 28(5): 1-5铁道学报2006.
[38]郝佳,李澜.铁路货运量组合预测模型的研究[ J ].铁道运输与经济, 2004, 26 (11) : 73-75. HAO Jia, LILan. The study of comp rehensive forecastmodel for railway freight transportvolume[ J ]. Railway Transport and Economy, 2004, 26 (11) : 73-75.
[39] FRAIMAN R, INBARREN P. Nonparametric regression estimation in models with weak error's structure[J]. J Mult Anal, 1991, 37(2): 180-196
[40]王一兵.半参数回归模型在商品房价格指数中的应用研究[J].统计研究, 2005(4): 25-30.
[41] Loh, W. L. : Fixed-domain asymptotics for a subclass of matern-type Gaussian random fields. Ann. Stat. 33(5), 2344–2394 (2005)
[42] Marinucci, D. : Testing for non-Gaussianity on cosmic microwave background radiation: a review. Stat. Sci. 19(2), 294–307 (2004)
[43] Marinucci, D. , Piccioni, M. : The empirical process on Gaussian spherical harmonics. Ann. Stat. 32(3), 1261–1288 (2004)
[44] Marinucci, D. : High resolution asymptotics for the angular bispectrum. Ann. Stat. 34(1), 1–41 (2006)
[2] Gao J T , Chen X R , Zhao L C. Asymptotic normalityof a class estimates in partly linear models [ J ] . Acta Mathematica Sinica ,1994 , 37 : 256-268.
[3]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, 1995, 18 (3) : 354～363.
[4] U spensky, J. V. , Int roduct ion to Mathematical Probability[M ], M cGurw-Hill, New York, 1937.
[5]洪圣岩,一类半参数回归模型的估计理论[J ],中国科学A辑, 1991, 21 (12) : 1258-1272.
[6]洪圣岩,最近邻回归估计的渐近正态性,应用概率统计[J ], 1991, 7 (2) : 187-191.
[7]赵林城,白志东,非参数回归函数最近邻估计的强相合性[J ],中国科学A辑, 1984, 14 (5) : 387-393.
[8]吴友平,彭军还,鲁铁定.半参数估计与LS估计的有效性比较.测绘科学,34(1):98-99,2009
[9]潘雄.半参数模型的估计理论及应用研究[D]武汉,武汉大学, 2005
[10]洪圣岩,Asymptotic properties of kernel smoothing in partial linear model,第四届中日统计讨论会, 1991.
[11]潘建敏.污染数据半参数回归模型的估计方法[J].工程数学学报, 1997(3): 81-85.
[12]陈明华.固定设计下半参数回归模型参数估计的收敛速度[J].应用概率统计, 1998(2):1 49-158.
[13]金丽红.半参数回归模型两步估计的渐进性[J]. 23(2). 102-108. 2004.
[14]钱伟民,柴根象.半参数回归模型的估计的渐进性质. 14(2). A辑161-168 1997.
[15]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:A辑, 1999(6): 745-754.
[16]田萍,薛留根.纵向数据半参数回归模型估计的强相合性[J].工程数学学报, 2006, 23(2): 369-373
[17]陈长生,徐勇勇,夏结来.光滑样条非参数回归方法及医学应用.中国卫生统计,1999 ,16 (6) :342.
[18]陈长生,徐勇勇,尚磊.儿童生长曲线的光滑样条和核估计拟合.中国卫生统计,1997 ;14 (4) :1.
[19] Bates DM ,Lindstrom MJ ,Wahba G,Yandell BS. The GCVPACK- Routines for generalized cross validation. Communication in statistics - simuilation and computa-tion, 1987 ,16 :263.
[20] SAS Institute Inc. SAS INSIGHT user's Guide. Version 6 Second Edition. Cary ,NC ,USA. 1993.
[21] CHANG Xiao2wen ,QU Le2ming. Wavelet estimation ofpartially linear models [J ] . Computational Statistics &Data Analysis , 2004 , 47 : 31248.
[22]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,1999 ,15 (1) :972105.
[23]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) , 11225.
[24]钱伟民,柴根象,将凤瑛.半参数回归模型的误差方差的小波估计[J ] .数学年刊,2000 ,21A(3) :3412350.
[25]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(A辑) ,1999 , 29 (3) :2332240.
[26] Ana Bianco, Graciela Boente. Robust estimators insemiparametric partly linear regression models[J]. Journal of Statistical Planning and Inference, 2004, 122 : 2292252.
[27] SHI Pei-de , TENG Xing-dong. Asymptotic distriburitions of M-estimators of the partly linear models with fixed carriers [J ] . Advancesin Mathematics , 1999 ,28 : 4472461.
[28]薛留根.半参数回归模型中随机加权M估计的强逼近[J ] .应用数学学报, 2002 ,25 (4) :5912603.
[29]陈希孺.线性模型参数M估计的线性表示[J ] .中国科学(A辑) ,1993 , 23 (12) :126421275.
[30] CHEN H. Convergence rates for parametric components in a partly linear model[J]. Ann. Statist. 1988, 16: 136-146.
[31]中国国家统计局.中国统计年鉴2006 [M ].北京:中国统计出版社, 2006.
[32]钱伟民,柴根象.一类半参数回归模型二阶段估计的渐近理论[J].同济大学学报:自然科学版, 1998(1): 77-82.
[33]李玉梅,钱伟民.半参数混合效应模型中的估计及其强相合性[J]. . 28(1): 107-120,系统科学与数学. 2008.
[34]田萍,薛留根.纵向数据下半参数回归模型估计的渐近性质[J]. . 23(4):369-376.应用概率统计. 2007.
[35]胡宏昌.半参数回归模型M估计的线性强表示[J]. 25(5): 5-7,周口师范学院学报. 2008.
[36] Ana Bianco, Graciela Boente. Robust estimators in semiparametric partly linear regression models [J]. . Journal of Statistical Planning and Inference, 2004 ,122 : 229-252.
[37]刘志杰,季令,叶玉玲,耿志民.基于径向基神经网络的铁路货运量预测[J]. . 28(5): 1-5铁道学报2006.
[38]郝佳,李澜.铁路货运量组合预测模型的研究[ J ].铁道运输与经济, 2004, 26 (11) : 73-75. HAO Jia, LILan. The study of comp rehensive forecastmodel for railway freight transportvolume[ J ]. Railway Transport and Economy, 2004, 26 (11) : 73-75.
[39] FRAIMAN R, INBARREN P. Nonparametric regression estimation in models with weak error's structure[J]. J Mult Anal, 1991, 37(2): 180-196
[40]王一兵.半参数回归模型在商品房价格指数中的应用研究[J].统计研究, 2005(4): 25-30.
[41] Loh, W. L. : Fixed-domain asymptotics for a subclass of matern-type Gaussian random fields. Ann. Stat. 33(5), 2344–2394 (2005)
[42] Marinucci, D. : Testing for non-Gaussianity on cosmic microwave background radiation: a review. Stat. Sci. 19(2), 294–307 (2004)
[43] Marinucci, D. , Piccioni, M. : The empirical process on Gaussian spherical harmonics. Ann. Stat. 32(3), 1261–1288 (2004)
[44] Marinucci, D. : High resolution asymptotics for the angular bispectrum. Ann. Stat. 34(1), 1–41 (2006)