具有AR（1）误差的半参数回归模型的统计分析
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摘要
近年来,半参数回归模型的研究引起了众多学者的关注,并成为当今回归分析的热点。半参数回归模型可以看作是参数线性模型和非参数回归模型的混合模型,是线性模型的推广。它放宽了线性模型中对某些解释变量的假定,使模型适应数据变化的能力更强,在处理许多实际问题,尤其是在经济学、医学、试验设计、GPS定位等学科中有着广泛的应用。
本文研究了误差为一阶自回归(AR(1))的半参数回归模型,着重在于统计诊断方面的研究。首先给出模型中未知参数的惩罚最小二乘估计,并对模型的误差项分别进行相关性和异方差性检验,得到了D-W检验统计量和Score检验统计量;其次基于数据删除模型得到估计的诊断公式,证明了数据删除模型与均值漂移模型的等价性,定义了残差、杠杆值、Cook距离和惩罚似然距离等诊断统计量;最后,对该模型进行局部影响分析,分别对加权扰动模型,响应变量扰动模型得到了影响矩阵的计算公式,数值实例分析验证了诊断方法的有效性。
In recent years, the research of semi-parametric regression model has attracted considerable attention and becomes an important field in the regression analysis. Semi-parametric regression model can be seen as mixed model of the parametric linear model and the nonparametric regression model, it is extensions of linear model. It has relaxed the assumption of certain explanatory variable in linear model, causes the model adaptation data change ability to be stronger. Semi-parametric regression model, useful in many problems especially for econometrics, medical, experimental design, GPS localization, is an excellent generalization from linear regression model.
This paper discusses the semi-parametric regression model with first-order auto-regressive errors, especially for the method and application of the diagnostics. The penalized least square estimation of the model is given first. The D-W test and Score test are proposed to test the autocorrelation and heteroscedasticity of the random errors in semi-parametric model. Then, we get the concise expressions based on case deletion model, establish an equivalence between the case deletion model and mean shift outlier model from which we derive tests for outliers. The basic diagnostic statistics such as residuals, generalized leverage, Cook distance, penalized likelihood displacement et al. are introduced. Finally, we also discuss the local influence analysis and get concise influence matrix for case weight perturbation and perturbation of response. Numerical examples are given to illustrate our diagnostics methods.
本文研究了误差为一阶自回归(AR(1))的半参数回归模型,着重在于统计诊断方面的研究。首先给出模型中未知参数的惩罚最小二乘估计,并对模型的误差项分别进行相关性和异方差性检验,得到了D-W检验统计量和Score检验统计量;其次基于数据删除模型得到估计的诊断公式,证明了数据删除模型与均值漂移模型的等价性,定义了残差、杠杆值、Cook距离和惩罚似然距离等诊断统计量;最后,对该模型进行局部影响分析,分别对加权扰动模型,响应变量扰动模型得到了影响矩阵的计算公式,数值实例分析验证了诊断方法的有效性。
In recent years, the research of semi-parametric regression model has attracted considerable attention and becomes an important field in the regression analysis. Semi-parametric regression model can be seen as mixed model of the parametric linear model and the nonparametric regression model, it is extensions of linear model. It has relaxed the assumption of certain explanatory variable in linear model, causes the model adaptation data change ability to be stronger. Semi-parametric regression model, useful in many problems especially for econometrics, medical, experimental design, GPS localization, is an excellent generalization from linear regression model.
This paper discusses the semi-parametric regression model with first-order auto-regressive errors, especially for the method and application of the diagnostics. The penalized least square estimation of the model is given first. The D-W test and Score test are proposed to test the autocorrelation and heteroscedasticity of the random errors in semi-parametric model. Then, we get the concise expressions based on case deletion model, establish an equivalence between the case deletion model and mean shift outlier model from which we derive tests for outliers. The basic diagnostic statistics such as residuals, generalized leverage, Cook distance, penalized likelihood displacement et al. are introduced. Finally, we also discuss the local influence analysis and get concise influence matrix for case weight perturbation and perturbation of response. Numerical examples are given to illustrate our diagnostics methods.
引文
[1]Engle,R.E.,Granger,C.W.J.,Rice,J.A.and Wiess,A.Semiparametric estimates of the relation between weather and electricity sales.JASA.1986(81):310-320
[2]Heckman,N.E.Spline Smoothing in a partly linear models.J.R.Statist.Soc.B.1986(48):244-249
[3]Rice,J.Convergence rates for partly linear models.Statist.&Probab.Lett.1986(4),203-208
[4]Chen,H.Convergence rates for parametric component in partly linear model.Ann.Statist.1988(16):136-146
[5]Speckman,P.Kernel smothing in partial linear models.J.R.Statist.Soc.B.1988(50):413-436
[6]Robinson,P.Root-n-consistent semiparametric regression.Econometric.1988(56):931-954
[7]Cuizick,J.Semiparametric additive regression.J.R.Statist Soc.B.1992(54):831-843
[8]Donald,S.G.and Neweg,W.K.Series estimation of semilinear models.J.Multi.Anal.1994(50):30-40
[9]高集体.半参数回归模型的大样本理论.博士论文.合肥:中国科技大学,1992
[10]柴根象,洪圣岩.半参数回归模型.合肥:安徽教育出版社,1995
[11]施沛得,李国英.部分线性模型中参数分量的M估计的渐进正态性.科学通报.1994(39):1-2
[12]Green,P.J.Penealized likelihood for general semiparametric regression models.Int,Statist.Rev.1987(55):245-259
[13]Severini,T.A.and Wong,W.H.Profile likelihood and conditionally parametric models.Ann.Statist.1992(20):1768-1802
[14]Severini,T.A.and Staniswalis,J.G.Quasilikelihood estimation in semiparametric models.JASA.1994(89):501-511
[15]Chen,H.Asymptotic efficient estimation in semiparametric generalized linear models.Ann.Statist.1995(23):1102-1129
[16]Cui,H.J.and Qin,H.Z.The estimation for nonlinear semiparametric EV models.Manuscript,1998
[17]Liang,H.and Hardle,W.Large sample theory in a semiparametric partially linear errors-in-variables models.Manuscript,1998
[18]王启华,郑忠国.随机删失半参数回归模型中估计的渐进性质.中国科学,1997(27(A)):583-594
[19]高集体,洪圣岩等.半参数回归模型研究的若干进展.数学学报.1995(38):685-669
[20]Gao,J.T.and Liang,H.Statistical inference in single-index partial nonlinear models.Ann.Inst.Statist.Math.1997(49):493-517
[21]韦博成,鲁国斌,史建清.统计诊断引论.南京:东南大学出版社,1991
[22]王松桂,陈敏,陈立萍.线性统计模型.高等教育出版社,1999
[23]G.A.F.Seber and C.J.Wild.Nonlinear Recession.John Wilcy&Sons.NewYork,1989
[24]胡舒合,潘光明,高启兵.误差为线性过程时回归模型的估计问题.高校应用数学学报 A辑.2003(18(1)):81-90
[25]林金官,韦博成.具有ARMA(0,1,0)误差的非线性模型的异方差检验.数学杂志.2005(25):71-76.
[26]Puterman,M.L.Leverage and influence in autocorrelated regression models.Appl Statist.1988(37):76-86
[27]Hossain A.Detection of influential observation in regression model with autocorrelated errors.Comm Statist Theory Methods.1990(19):1047-1060
[28]Richard F,Rob J H,Alan V.Residual diagnostic plots for model mis-speciflcation in time series regression.Aust N Z J Stat.2000(42):463-477
[29]杨爱军,林金官,韦博成.具有AR(1)误差的非线性随机效应模型中自相关系数的扰动诊断.应用数学学报,2006(19(4)):818-822
[30]Chi,E.M,Reinsel,G..C.Asymptotic properties of the score test for autocorrelation in effects with AR(1)errors models.Statistics & Probability Letter.1991(11):453-457
[31]Epps,T.M and Epps,M.I.The robustness of some standard test for autocorrelation and Heteroscedasticity when both problems are present.Econometrika.1977(45):745-753
[32]Tsai,C.L.Score test for the First-order autoregressive model with heteroscedasticity.Biometrika.1986(73):455-460
[33]韦博成,胡跃清.非线性回归模型相关性和异方差性的检验.工程数学学报.1994(11(4)):1-12
[34]刘应安,韦博成,林金官.AR(1)误差的线性随机效应模型中方差齐性和自相关的检验.应用概率统计.2004(20(1)):27-36
[35]Lin J.G.,Wei,B.C.,Testing for heteroscedasticity and correlation in nonlinear models with correlated errors.Comm Statist Theory Methods.2004(32(2)):251-275
[36]刘应安,韦博成,林金官.误差为ARMA(1,1)的非线性回归模型相关性和异方差的检验.东南大学学报(自然科学版),2001(31(6)):98-102
[37]Cook,R.D.Deletion of influential observations in linear regression.Technometric.1977(19):15-18
[38]Cook,R.D.and Weisberg,S.Residual and influence in regression.New York:Chapman and Hall,1982
[39]Pregibon,D.Logistic regression diagnostics.Ann.Statist.1981(9):705-724
[40]Wei,B,C.Exponential family nonlinear models.Sinapore:Spring-Verlag,1998
[41]Eubank,R.L.The hat matrix for smoothing spline.Statist.& Probab.1984(Lett.2):9-16
[42]Eubank,R.L.Diagnostics for smoothing splines.J.R.Statist.,Soc.B.1985(47):322-341
[43]Kim,C.Cook's distance in spline smoothing.Statist.& Probab.1996(Lett.,32):139-144
[44]Kim,C.,Park,B.U.and Kim,W.Influence diagnostics in semiparametic regression models.Statist&Probab.2002(Lett.60):49-58
[45]朱仲义,韦博成.半参数非线性模型的统计诊断与影响分析.应用数学学报,2001(24(4)):568-581
[46]曾林蕊,朱仲义.半参数广义线性模型的局部影响分析.华东师范大学学报(自然科学版).2005(4):18-25
[47]Green,P.J.and Silverman,B.W.Nonparametric Regression and Generalized Linear models.London:Chapman and Hall,1994
[48]胡宏昌.误差为AR(1)情形的半参数回归模型拟极大似然估计的存在性.湖北师范学院学报(自然科学版),2006(26(3)):12-16
[49]Eubank,R.L.Spline smoothing and nonparametric regression.New York:Narcel Dekker,1988
[50]安鸿志,陈兆国,杜金观,潘一民.时间序列的分析与应用.科学出版社,1983.
[51]王燕.应用时间序列分析.高等教育出版社,2006
[52]Cook,R.D.Assessment of local influence(with discussion).J.R.Statist.Soc.B.1986(48):133-169
[53]Thomas,W.and Cook,D.R.Aseessing influence on regression coefficients ingeneralized linear models.Biometrika.1989(76):741-749
[54]Escobar,L.A.and Meeker,W.Q.Aseessing influence in regression analysis with censored data.Biometrics.1992(48):507-508
[55]Poon,W.Y.and Poon,Y.S.Conformal normal curvature and assessment of local influence.J.R.Statist.Soc.B.1999(61):51-61
[2]Heckman,N.E.Spline Smoothing in a partly linear models.J.R.Statist.Soc.B.1986(48):244-249
[3]Rice,J.Convergence rates for partly linear models.Statist.&Probab.Lett.1986(4),203-208
[4]Chen,H.Convergence rates for parametric component in partly linear model.Ann.Statist.1988(16):136-146
[5]Speckman,P.Kernel smothing in partial linear models.J.R.Statist.Soc.B.1988(50):413-436
[6]Robinson,P.Root-n-consistent semiparametric regression.Econometric.1988(56):931-954
[7]Cuizick,J.Semiparametric additive regression.J.R.Statist Soc.B.1992(54):831-843
[8]Donald,S.G.and Neweg,W.K.Series estimation of semilinear models.J.Multi.Anal.1994(50):30-40
[9]高集体.半参数回归模型的大样本理论.博士论文.合肥:中国科技大学,1992
[10]柴根象,洪圣岩.半参数回归模型.合肥:安徽教育出版社,1995
[11]施沛得,李国英.部分线性模型中参数分量的M估计的渐进正态性.科学通报.1994(39):1-2
[12]Green,P.J.Penealized likelihood for general semiparametric regression models.Int,Statist.Rev.1987(55):245-259
[13]Severini,T.A.and Wong,W.H.Profile likelihood and conditionally parametric models.Ann.Statist.1992(20):1768-1802
[14]Severini,T.A.and Staniswalis,J.G.Quasilikelihood estimation in semiparametric models.JASA.1994(89):501-511
[15]Chen,H.Asymptotic efficient estimation in semiparametric generalized linear models.Ann.Statist.1995(23):1102-1129
[16]Cui,H.J.and Qin,H.Z.The estimation for nonlinear semiparametric EV models.Manuscript,1998
[17]Liang,H.and Hardle,W.Large sample theory in a semiparametric partially linear errors-in-variables models.Manuscript,1998
[18]王启华,郑忠国.随机删失半参数回归模型中估计的渐进性质.中国科学,1997(27(A)):583-594
[19]高集体,洪圣岩等.半参数回归模型研究的若干进展.数学学报.1995(38):685-669
[20]Gao,J.T.and Liang,H.Statistical inference in single-index partial nonlinear models.Ann.Inst.Statist.Math.1997(49):493-517
[21]韦博成,鲁国斌,史建清.统计诊断引论.南京:东南大学出版社,1991
[22]王松桂,陈敏,陈立萍.线性统计模型.高等教育出版社,1999
[23]G.A.F.Seber and C.J.Wild.Nonlinear Recession.John Wilcy&Sons.NewYork,1989
[24]胡舒合,潘光明,高启兵.误差为线性过程时回归模型的估计问题.高校应用数学学报 A辑.2003(18(1)):81-90
[25]林金官,韦博成.具有ARMA(0,1,0)误差的非线性模型的异方差检验.数学杂志.2005(25):71-76.
[26]Puterman,M.L.Leverage and influence in autocorrelated regression models.Appl Statist.1988(37):76-86
[27]Hossain A.Detection of influential observation in regression model with autocorrelated errors.Comm Statist Theory Methods.1990(19):1047-1060
[28]Richard F,Rob J H,Alan V.Residual diagnostic plots for model mis-speciflcation in time series regression.Aust N Z J Stat.2000(42):463-477
[29]杨爱军,林金官,韦博成.具有AR(1)误差的非线性随机效应模型中自相关系数的扰动诊断.应用数学学报,2006(19(4)):818-822
[30]Chi,E.M,Reinsel,G..C.Asymptotic properties of the score test for autocorrelation in effects with AR(1)errors models.Statistics & Probability Letter.1991(11):453-457
[31]Epps,T.M and Epps,M.I.The robustness of some standard test for autocorrelation and Heteroscedasticity when both problems are present.Econometrika.1977(45):745-753
[32]Tsai,C.L.Score test for the First-order autoregressive model with heteroscedasticity.Biometrika.1986(73):455-460
[33]韦博成,胡跃清.非线性回归模型相关性和异方差性的检验.工程数学学报.1994(11(4)):1-12
[34]刘应安,韦博成,林金官.AR(1)误差的线性随机效应模型中方差齐性和自相关的检验.应用概率统计.2004(20(1)):27-36
[35]Lin J.G.,Wei,B.C.,Testing for heteroscedasticity and correlation in nonlinear models with correlated errors.Comm Statist Theory Methods.2004(32(2)):251-275
[36]刘应安,韦博成,林金官.误差为ARMA(1,1)的非线性回归模型相关性和异方差的检验.东南大学学报(自然科学版),2001(31(6)):98-102
[37]Cook,R.D.Deletion of influential observations in linear regression.Technometric.1977(19):15-18
[38]Cook,R.D.and Weisberg,S.Residual and influence in regression.New York:Chapman and Hall,1982
[39]Pregibon,D.Logistic regression diagnostics.Ann.Statist.1981(9):705-724
[40]Wei,B,C.Exponential family nonlinear models.Sinapore:Spring-Verlag,1998
[41]Eubank,R.L.The hat matrix for smoothing spline.Statist.& Probab.1984(Lett.2):9-16
[42]Eubank,R.L.Diagnostics for smoothing splines.J.R.Statist.,Soc.B.1985(47):322-341
[43]Kim,C.Cook's distance in spline smoothing.Statist.& Probab.1996(Lett.,32):139-144
[44]Kim,C.,Park,B.U.and Kim,W.Influence diagnostics in semiparametic regression models.Statist&Probab.2002(Lett.60):49-58
[45]朱仲义,韦博成.半参数非线性模型的统计诊断与影响分析.应用数学学报,2001(24(4)):568-581
[46]曾林蕊,朱仲义.半参数广义线性模型的局部影响分析.华东师范大学学报(自然科学版).2005(4):18-25
[47]Green,P.J.and Silverman,B.W.Nonparametric Regression and Generalized Linear models.London:Chapman and Hall,1994
[48]胡宏昌.误差为AR(1)情形的半参数回归模型拟极大似然估计的存在性.湖北师范学院学报(自然科学版),2006(26(3)):12-16
[49]Eubank,R.L.Spline smoothing and nonparametric regression.New York:Narcel Dekker,1988
[50]安鸿志,陈兆国,杜金观,潘一民.时间序列的分析与应用.科学出版社,1983.
[51]王燕.应用时间序列分析.高等教育出版社,2006
[52]Cook,R.D.Assessment of local influence(with discussion).J.R.Statist.Soc.B.1986(48):133-169
[53]Thomas,W.and Cook,D.R.Aseessing influence on regression coefficients ingeneralized linear models.Biometrika.1989(76):741-749
[54]Escobar,L.A.and Meeker,W.Q.Aseessing influence in regression analysis with censored data.Biometrics.1992(48):507-508
[55]Poon,W.Y.and Poon,Y.S.Conformal normal curvature and assessment of local influence.J.R.Statist.Soc.B.1999(61):51-61