具有AR（1）误差的变系数模型的统计分析
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摘要
近年来，变系数模型的研究引起了众多学者的广泛关注，并成为当今回归分析中研究的热点课题。变系数模型是经典的线性模型的一种有用推广，在处理许多实际问题，尤其是在经济学、生物医学、流行病学中有着广泛的应用。
本文研究了误差项具有一阶自回归的变系数模型，并着重于模型统计诊断方面的研究。首先给出模型函数系数的B样条估计、方差估计以及自回归系数的估计。在自回归系数已知且样本容量充分大时，我们证明了这样得到的函数系数的B样条估计要优于不考虑误差自回归的情况。然后基于数据删除模型给出诊断统计量，并证明了数据删除模型与均值漂移模型的等价性。接着对模型误差项进行相关性和异方差性检验，分别得到D-W检验统计量和Score检验统计量。最后给出识别模型异常点和强影响点的诊断统计量，得到了似然比统计量、残差、广义Cook距离等诊断统计量的简洁计算公式。并用实例验证了这些方法的有效性。
In recent years, the research of varying-coefficient model attracts considerable attenti-on and becomes an important field in the regression analysis. Varying-coefficient model,useful in many problems especially for econometrics, biomedicine and epidemiology, is anexcellent generalization from linear regression model.
This paper discusses the varying-coefficient model with first-order autoregressiveerrors, especially for the method and application of the diagnostics. At first, we obtain theestimation of functional coefficients, the deviation and autoregression coefficients. Whenthe number of case is sufficient, we proved that the functional coefficients' B-splineestimation on thinking over autoregression is better than omitting the autoregression.Then, we get the diagnostic expressions based on case deletion, establish an equivalencebetween the case deletion model and mean shift outlier model. The D-W test and score testare proposed to test the autocorrelation and heteroscedasticity of the random errors invarying-coefficient model. At last, the diagnostic statistic for test outliers and influentialpoint is given. Numerical examples are given to illustrate our results.
本文研究了误差项具有一阶自回归的变系数模型，并着重于模型统计诊断方面的研究。首先给出模型函数系数的B样条估计、方差估计以及自回归系数的估计。在自回归系数已知且样本容量充分大时，我们证明了这样得到的函数系数的B样条估计要优于不考虑误差自回归的情况。然后基于数据删除模型给出诊断统计量，并证明了数据删除模型与均值漂移模型的等价性。接着对模型误差项进行相关性和异方差性检验，分别得到D-W检验统计量和Score检验统计量。最后给出识别模型异常点和强影响点的诊断统计量，得到了似然比统计量、残差、广义Cook距离等诊断统计量的简洁计算公式。并用实例验证了这些方法的有效性。
In recent years, the research of varying-coefficient model attracts considerable attenti-on and becomes an important field in the regression analysis. Varying-coefficient model,useful in many problems especially for econometrics, biomedicine and epidemiology, is anexcellent generalization from linear regression model.
This paper discusses the varying-coefficient model with first-order autoregressiveerrors, especially for the method and application of the diagnostics. At first, we obtain theestimation of functional coefficients, the deviation and autoregression coefficients. Whenthe number of case is sufficient, we proved that the functional coefficients' B-splineestimation on thinking over autoregression is better than omitting the autoregression.Then, we get the diagnostic expressions based on case deletion, establish an equivalencebetween the case deletion model and mean shift outlier model. The D-W test and score testare proposed to test the autocorrelation and heteroscedasticity of the random errors invarying-coefficient model. At last, the diagnostic statistic for test outliers and influentialpoint is given. Numerical examples are given to illustrate our results.
引文
[1] T. Hastie,R. Tibshirani. Varying-Coefficient Models(with discussion).Journal of the Royal Statistical Society(Series B). 1993(55):757-796.
[2] Chin-Tsang Chiang, Rice, J. A .and Wu ,C .O. Smoothing spline estimation for varying-coefficient models with repeatedly measure dependent variables. Journal of the Americ-an statistical Association. 2001 (96):605-619.
[3] Fan, J. and Zhang J.T. Two-step estimation of functional linear models with applications to longitudinal data. Journal of the Royal Statistical Society(Series B)2000(62): 303-322.
[4] Cai.Z, Fan J, Li R. Efficient estimation and inferences for varying-coefficient models. Journal of the American statistical Association. 2000(95): 888-902.
[5] Hoover D R, Rice J A, Wu C O, et al. Nonparametric smoothing estimates of time varying coefficient models with longitudinal data. Biometrika. 1998(85):809-822.
[6] Cleveland W.S, Grosse E, ShyuW.M. Local regression models.In: Chambers J M, Hastie T J,eds.Statistical Models in S.Pacific Grove,CA: Wadsworth/Brooks-Cole, 1992:309-376
[7] Fan, J, Zhang, W. Statistical estimation in varying-coefficient model Ann Statist 1999 (27): 1491-1518
[8] 唐庆国，王金德．变系数模型中的一步估计法．中国科学A辑．2005(35(1))：23-38．
[9] Mei Changlin, Zhang Wenxiu, Leung Yee. Statistical Inferences for Varying-coefficient models based on locally weighted regression technique. Acta Mathematicae Application sinica. 2001(17(3)): 407-417.
[10] 卢一强．变系数模型的研究与分析．华东师范大学博士学位论文．2003．
[11] 张曰权，卢一强．变系数模型．科学出版社．2004．
[12] Wai Cheung Ip Heung Wong Riquan ZhangGeneralized likelihood ratio test for varyiny-coeficient models with different smoothing variables Computational statistics & data analysis. 2006(16): 1-19.
[13] 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.
[14] T.Hastie , R .Tibshirani. Generalized Additive Models. Chapman and Hall. 1990.
[15] Epps, T.M and Epps, M.I, The robustness of some standard test for autocorrelation and heteroscedasticity when both problems are present Econometrika, 45(1977), 745-753.
[16] Cai, Z, Tiwari, R.C. Application of a local linear autoregressive model to B OD time series.Environmetrics. 2000(11):341-350
[17] Kim, W. Nonparametric kernel estimation of evolutionary autoregressive processes. Working Paper, Institute of Statistics and Economics. Humboldt Unversity, Germany.
[18] Robinson, P.M. Nonparametric estimation of time-varying parameters. In Statistical Analysis and Forecasting of Economic Structural Change. Springer-Verlag, Berlin 1989: 164-253.
[19] Robinson, P.M. Time-varying nonlinear regression. In Economic Structure Change, Analysis and Forecasting. Springer-Verlag, Berlin. 1991:179-190
[20] G.A.F.Seber and C.J.Wild. Nonlinear Regression .John Wilcy&Sons, NewYork, 1989.
[21] 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．
[22] Puterman, M.L. Leverage and influence in autocorrelated regression models. Appl Statist.1988(37):76-86.
[23] Hossain A. Detection of influential observation in regression model with autocorrelated errors. Comm Statist Theory Methods 1990(19): 1047-1060
[24] 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.
[25] Zongwu Cai. Trending time-varying coefficient models with serially correlated errors. Journal of Econometrics, 2007(136(1)): 163-188.
[26] Cook,R.D.andWeisberg,S. Diagnostics for heteroscedasticity in regression. Biometrika.1983(70):1-10.
[27] 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．
[28] Tsai,C.L. Score test for the First-order autoregressive model with heteroscedasticity. Biometrika. 1986(73):455-460.
[29] 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．
[30] 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.
[31] 韦博成，鲁国斌，史建清．统计诊断引论．东南大学出版社．1991．
[32] 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．
[33] 林金官，韦博成．非线性随机效应模型的异方差性检验．系统科学与数学．2002(22(2))：245-256．
[34] 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．
[35] 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．
[36] McCullagh, P.and Nelder, J.A.Generalized linear models.London:Chapman&Hall, 1989.
[37] Wei, B, C. Exponential family nonlinear models.Sinapore:Springer-Verlag, 1998.
[38] 宗序平．系统工程中指数族非线性模型和测量误差模型的统计分析 东南大学博士学位论文．1999．
[39] 朱仲义，韦博成．半参数非线性模型的统计诊断和影响分析．应用数学学报．2001(24)：568-581．
[40] Wing-Kam Fung, Zhong_Yi Zhu. Influence diagnostics and outlier tests for semi-para-metric mixed models .J .R. Statist. Soc.B. 2002(64):565-579.
[41] 赵为华，冯予．于旨数族半参数非线性模型的统计诊断和影响分析．应用数学学报．2006(29(4))：734-746．
[42] 关治，陆金甫．数值分析基础．高等教育出版社．1998．
[43] Schumaker L L. Spline Functions NewYork: Wiley 1981.
[44] 王松桂，陈敏，陈立萍．线性统计模型．高等教育出版社．1999．
[45] 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．
[46] 常学将，陈敏，王明生．时间序列分析．高等教育出版社．1993．
[2] Chin-Tsang Chiang, Rice, J. A .and Wu ,C .O. Smoothing spline estimation for varying-coefficient models with repeatedly measure dependent variables. Journal of the Americ-an statistical Association. 2001 (96):605-619.
[3] Fan, J. and Zhang J.T. Two-step estimation of functional linear models with applications to longitudinal data. Journal of the Royal Statistical Society(Series B)2000(62): 303-322.
[4] Cai.Z, Fan J, Li R. Efficient estimation and inferences for varying-coefficient models. Journal of the American statistical Association. 2000(95): 888-902.
[5] Hoover D R, Rice J A, Wu C O, et al. Nonparametric smoothing estimates of time varying coefficient models with longitudinal data. Biometrika. 1998(85):809-822.
[6] Cleveland W.S, Grosse E, ShyuW.M. Local regression models.In: Chambers J M, Hastie T J,eds.Statistical Models in S.Pacific Grove,CA: Wadsworth/Brooks-Cole, 1992:309-376
[7] Fan, J, Zhang, W. Statistical estimation in varying-coefficient model Ann Statist 1999 (27): 1491-1518
[8] 唐庆国，王金德．变系数模型中的一步估计法．中国科学A辑．2005(35(1))：23-38．
[9] Mei Changlin, Zhang Wenxiu, Leung Yee. Statistical Inferences for Varying-coefficient models based on locally weighted regression technique. Acta Mathematicae Application sinica. 2001(17(3)): 407-417.
[10] 卢一强．变系数模型的研究与分析．华东师范大学博士学位论文．2003．
[11] 张曰权，卢一强．变系数模型．科学出版社．2004．
[12] Wai Cheung Ip Heung Wong Riquan ZhangGeneralized likelihood ratio test for varyiny-coeficient models with different smoothing variables Computational statistics & data analysis. 2006(16): 1-19.
[13] 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.
[14] T.Hastie , R .Tibshirani. Generalized Additive Models. Chapman and Hall. 1990.
[15] Epps, T.M and Epps, M.I, The robustness of some standard test for autocorrelation and heteroscedasticity when both problems are present Econometrika, 45(1977), 745-753.
[16] Cai, Z, Tiwari, R.C. Application of a local linear autoregressive model to B OD time series.Environmetrics. 2000(11):341-350
[17] Kim, W. Nonparametric kernel estimation of evolutionary autoregressive processes. Working Paper, Institute of Statistics and Economics. Humboldt Unversity, Germany.
[18] Robinson, P.M. Nonparametric estimation of time-varying parameters. In Statistical Analysis and Forecasting of Economic Structural Change. Springer-Verlag, Berlin 1989: 164-253.
[19] Robinson, P.M. Time-varying nonlinear regression. In Economic Structure Change, Analysis and Forecasting. Springer-Verlag, Berlin. 1991:179-190
[20] G.A.F.Seber and C.J.Wild. Nonlinear Regression .John Wilcy&Sons, NewYork, 1989.
[21] 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．
[22] Puterman, M.L. Leverage and influence in autocorrelated regression models. Appl Statist.1988(37):76-86.
[23] Hossain A. Detection of influential observation in regression model with autocorrelated errors. Comm Statist Theory Methods 1990(19): 1047-1060
[24] 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.
[25] Zongwu Cai. Trending time-varying coefficient models with serially correlated errors. Journal of Econometrics, 2007(136(1)): 163-188.
[26] Cook,R.D.andWeisberg,S. Diagnostics for heteroscedasticity in regression. Biometrika.1983(70):1-10.
[27] 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．
[28] Tsai,C.L. Score test for the First-order autoregressive model with heteroscedasticity. Biometrika. 1986(73):455-460.
[29] 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．
[30] 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.
[31] 韦博成，鲁国斌，史建清．统计诊断引论．东南大学出版社．1991．
[32] 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．
[33] 林金官，韦博成．非线性随机效应模型的异方差性检验．系统科学与数学．2002(22(2))：245-256．
[34] 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．
[35] 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．
[36] McCullagh, P.and Nelder, J.A.Generalized linear models.London:Chapman&Hall, 1989.
[37] Wei, B, C. Exponential family nonlinear models.Sinapore:Springer-Verlag, 1998.
[38] 宗序平．系统工程中指数族非线性模型和测量误差模型的统计分析 东南大学博士学位论文．1999．
[39] 朱仲义，韦博成．半参数非线性模型的统计诊断和影响分析．应用数学学报．2001(24)：568-581．
[40] Wing-Kam Fung, Zhong_Yi Zhu. Influence diagnostics and outlier tests for semi-para-metric mixed models .J .R. Statist. Soc.B. 2002(64):565-579.
[41] 赵为华，冯予．于旨数族半参数非线性模型的统计诊断和影响分析．应用数学学报．2006(29(4))：734-746．
[42] 关治，陆金甫．数值分析基础．高等教育出版社．1998．
[43] Schumaker L L. Spline Functions NewYork: Wiley 1981.
[44] 王松桂，陈敏，陈立萍．线性统计模型．高等教育出版社．1999．
[45] 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．
[46] 常学将，陈敏，王明生．时间序列分析．高等教育出版社．1993．