半参数空间多元回归模型两步估计及其性质研究
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摘要
半参数空间多元回归模型广泛的应用于地理,气象,经济环境,地质等领域的数据分析,是分析处理空间数据的有效工具.它以线性回归模型和非参数回归模型为基础,结合了二者的优点.参数分量用于分析确定性影响因素,非参数分量用于分析随机干扰影响因素.半参数空间多元回归模型不仅能够对现实世界建模,而且通过模型能够分析出内在规律.因此,进一步对半参数空间多元回归模型深入研究不仅有重要的学术意义,而且有广泛的应用价值.
本文的研究工作主要有以下三个方面的内容:(一)半参数空间变系数回归模型的两步估计.通过改进传统的半参数空间变系数回归模型,建立新模型,对其进行两步估计.采用加权最小二乘法对两步估计的变系数部分的初次估计量进行了稳健性分析.结果表明,偏差平方和的影响有所减小,有利于增强估计量的稳健性.(二)变系数部分两步估计的显著性检验.针对改进的半参数空间变系数回归模型,对其变系数部分两步估计值,采用一般性假设检验、准回归方程的显著性检验和系数的显著性检验,采用了t-分布检验了变系数初步估计值的显著性.从而有效地减少系统误差,提高了显著性检验的稳健性. (三)讨论变系数部分初次估计的可容许性.对于改进的半参数空间变系数回归模型的系数两步估计,本文对其变系数部分的初次估计进行了可容许性讨论,得到了变系数部分初次估计量满足可容许性的充要条件.
Semi-parametric spatially varying-coefficient regression model is widely used in geography, weather, economic conditions, geology and other fields of data analysis, and it is an effective tool for spatial data analysis and processing. It builds on base of the linear regression model and nonparametric regression model, combining the advantages of both. Parameter components are used in analyzing of uncertainty factors, and non-parametric components are used in analyzing of random interference factors. Semi-parametric spatially varying-coefficient regression model can not only model the real world, but can analyze the potential law by the model. Therefore, it is not only academic significance but also a wide range of application that we further study about semi-parametric spatially varying-coefficient regression model.
This research work mainly in the following three aspects: (a) Two-step procedure for semi-parametric spatially varying-coefficient regression model. By improving the traditional semi-parametric spatially varying-coefficient regression model, a new model is built and its two-step procedure are made in this paper. We have analyzed the robustness of initial estimator of variable coefficients in two-step procedure according to weighted least squares. The results show that the sum of squares of deviations has been reduced, are conducive to enhancing the robustness of estimators. (b) Test of significance to two-step procedure for variable coefficients. The improved semi-parametric spatially varying-coefficient regression model, for variable coefficients part of its two-step procedure, we adopt a general hypothesis testing, quasi-regression equation significance testing and significance testing of coefficients, and test the significance for first estimator of variable coefficients by t ? distribution. It can effectively reduce the system error and increase the robustness of the significance testing. (c) Discuss admissibility of the initial estimate of variable coefficients. For two-step procedure of the improved semi-parametric spatially varying-coefficient regression model, in this paper, we have discussed admissibility of the initial estimate of variable coefficients, and obtained the necessary and sufficient conditions of initial estimator of variable coefficients.
本文的研究工作主要有以下三个方面的内容:(一)半参数空间变系数回归模型的两步估计.通过改进传统的半参数空间变系数回归模型,建立新模型,对其进行两步估计.采用加权最小二乘法对两步估计的变系数部分的初次估计量进行了稳健性分析.结果表明,偏差平方和的影响有所减小,有利于增强估计量的稳健性.(二)变系数部分两步估计的显著性检验.针对改进的半参数空间变系数回归模型,对其变系数部分两步估计值,采用一般性假设检验、准回归方程的显著性检验和系数的显著性检验,采用了t-分布检验了变系数初步估计值的显著性.从而有效地减少系统误差,提高了显著性检验的稳健性. (三)讨论变系数部分初次估计的可容许性.对于改进的半参数空间变系数回归模型的系数两步估计,本文对其变系数部分的初次估计进行了可容许性讨论,得到了变系数部分初次估计量满足可容许性的充要条件.
Semi-parametric spatially varying-coefficient regression model is widely used in geography, weather, economic conditions, geology and other fields of data analysis, and it is an effective tool for spatial data analysis and processing. It builds on base of the linear regression model and nonparametric regression model, combining the advantages of both. Parameter components are used in analyzing of uncertainty factors, and non-parametric components are used in analyzing of random interference factors. Semi-parametric spatially varying-coefficient regression model can not only model the real world, but can analyze the potential law by the model. Therefore, it is not only academic significance but also a wide range of application that we further study about semi-parametric spatially varying-coefficient regression model.
This research work mainly in the following three aspects: (a) Two-step procedure for semi-parametric spatially varying-coefficient regression model. By improving the traditional semi-parametric spatially varying-coefficient regression model, a new model is built and its two-step procedure are made in this paper. We have analyzed the robustness of initial estimator of variable coefficients in two-step procedure according to weighted least squares. The results show that the sum of squares of deviations has been reduced, are conducive to enhancing the robustness of estimators. (b) Test of significance to two-step procedure for variable coefficients. The improved semi-parametric spatially varying-coefficient regression model, for variable coefficients part of its two-step procedure, we adopt a general hypothesis testing, quasi-regression equation significance testing and significance testing of coefficients, and test the significance for first estimator of variable coefficients by t ? distribution. It can effectively reduce the system error and increase the robustness of the significance testing. (c) Discuss admissibility of the initial estimate of variable coefficients. For two-step procedure of the improved semi-parametric spatially varying-coefficient regression model, in this paper, we have discussed admissibility of the initial estimate of variable coefficients, and obtained the necessary and sufficient conditions of initial estimator of variable coefficients.
引文
[1] Nuno Crato, Bonnie K Ray. Semi-parametric smoothing estimators for long memory processes with added noise[J]. Journal of Statistical Planning and Inference, 2002,105: 283-297
[2] Godambe. The foundation of finite sample estimation in stochastic processes [J].Biometrika 1985, 72:419-428
[3] T.M purairajan, Martin L.William. Optimal estimating function for estimation and prediction in semi-parametric model[J]. Journal of Statistical Planning and Inference, 2008,138:3283-3292
[4] Hyun Lim, Xu zhang. Semi-parametric additive risk models: Apllication to injury study[J]. Accident Analysis and Prevention ,2009,41:211-216
[5] Eric Wolsztynski ,Eric Thierry ,Luc Pronzato .Minimum-entropy estimation in semi-parametric models[J]. Signal Processing ,2005,85:937-949
[6] James B. McDonald, Yexiao J.Xu. A comparison of semi-parametric and partially adaptive estimators of censored regression model with possibly with possibly skewed and leptokurtic error distributions[J].Economic Letters,1996,51:153-159
[7] Erik Heitfield, Armando Levy. Parametric, semi-parametric and non-parametric models of telecommunications demand: An investigation of residential calling pattern[J].Information economics and Policy ,2001,13:311-329
[8] James B. McDnoald. An application and comparison of some flexible parametric and semi-parametric qualitative response models[J]. Economic Letters, 1996, 53: 145-152
[9] Okmyung Bin. A prediction comparison of housing sales prices by parametric versus semi-parametric regression[J]. Economics, 2004,13(1): 68-84
[10]R.H. Glendinning, A.J. Goode. Semi-parametric classification of noisy curves[J]. Pattern Recognition ,2003,36(1):35-44
[11]Gauthier Lanot,Lan Walke. The union /non-union wage differentical: An application of semi-parametric methods[J]. Journal of Econometrics ,1998,84(2): 327-349
[12]Myoung-jae Lee,Francis Vella. A semi-parametric estimator for censored selectionmodels with endogeneity[J]. Journal of Econometrics,2006,130: 3283-3292
[13]Oliver Lepez. Single-index regression models with right censored responses[J]. Journal of statistical Planning and Inference, 2009,139(1): 1082-1097
[14]Gregory M.Duncan. A semi-parametric censored regression estimator[J]. Journal of Econometrics,2005,131(2):209-229
[15]Bo E.Honore,James L Powell. Pairwise difference estimators of censored and truncated regression models[J]. Journal of Econometrics, 1991,64(1-2): 241-278
[16]Fotheringham A S, Chalton M, Brunsdon C. The geography of parameter space: An investigation into spatial non-stationarity[J]. International Journal of geographical Information System,1996,10:605-627
[17]Fothering A S.Trends in quantitative of Geographical Information Systems. Progress in Human Geography, 1997,(21):88-96
[18]Brunsdon C, Fotheringham A S,Charlton M. Geographically weighed regression: A method for exploring spatial nonsttionarity[J]. Geographically Analysis, 1996, 28: 281-298
[19]Fotheringham A S, Charlton M,Brunsdon C.Measureing spatial variation in relationhips with geographically weighted regression[J].In Recent Developments in Spatial Analysis, Editted by M M Fischer and A Getis, Spring. Verlag, London, 1997:60-82
[20]Jinhong You. Emprical likehood for semiparametric varying-coefficient partially linear regression models[J]. Statistics & Probxbility Letters,2006,76(4): 412-422
[21]Jinhong You,Gemai chen. Estimation of a semeparametric varying coefficient partially linear errors in variable model[J]. Jurnal of Multivariate Analysis, 2006, 97(1): 324-341
[22]Xian zhou, Jinhong You. Wavelet estimation in varying-coefficient partially linear regression models[J]. Statistics & Probability Letters,2004,68(1):91-104
[23]Yiqiang Liu. Generalized partially linear varying-coefficient models[J]. Journal of Statistical planning and Inference, 2008,138(4):901-914
[24]Yongzhou, Jinhong You,Xiaojing Wang: Strong convergence rates of several estimators in semiparametric varying-coefficient partially linear models[J]. Acta Mathematic Science , 2009,29(5):1113-1127
[25]Adrian Pagan. Some identification and estimation results for regression models with stochastically varying coefficients[J]. Journal of Econometrics,1980,13(3): 341-363
[26]Qihua Wang,Riqian zhang .Statistical estimation invarying-coefficient models with surrogate date and validation sampling[J].Journal of Multi variate Analysis,2009, 100(10): 2389-2405
[27]P.J. Huber. Robust Statistics(M), John Wiley & Sons,New York,1981
[28]Waid.A. Statistical Decision Functions(M), John Wiley,1950
[29]Cohen.A.AIladminisibile estimates of the mean vector,Ann[J].Math Statist, 1966,37: 458-463
[30]Zontek. S. Characterization of linear admissibile estimator in Gaura-Marker model under normality in linear statistical inference, proceeding&Rozon, Springer, Verlag,1984
[31]Stein. C. In admissibility of the usual estimator for the meal of a multivariate normal distribution,proc . Third Berkeley Symp[J]. Math Statist probab,1956, 1:197-206
[32]Yafeng Xia,Shaohui Li.Identification of Coefficients for Multivariate Regression Model.in ISAI2010,2010,Volume 1:Institute of Electrical and Electroneics Engine- er, Inc.27-30(EI待检索)
[33]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
[34]陈明华.污染数据回归分析中估计的强相合性[J].应用概率统计, 1998, 14: 73-78
[35]任哲,陈明华.污染数据回归分析中参数的最小一乘估计[J].应用概率统计, 2000, 16: 262-268
[36]刘丽萍.污染数据半参数回归模型中的强相合估计[J].同济大学学报:自然科学版, 2004, 32(6): 832-836
[37]黄四民,梁华.用半参数部分线性模型分析居民消费结构[J].数量经济技术经济研究, 1994, 10: 33-39
[38]梁华,熊健.再论半参数部分线性模型在居民消费结构分析中的应用[J].数理统计与管理, 1995(6): 9-17
[39]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
[40]魏传华,梅长林.半参数空间变系数回归模型的两步估计方法及其数值模拟[J],统计与信息论坛,2006(1):16-19
[41]魏传华,梅长林.半参数空间变系数回归模型的Back-Fitting估计[J],数学的实践与认识[J],2006(3):177-183
[42]高集体、洪圣岩、梁华等.半参数模型研究的若干进展[J],应用概率统计,1994(1):98-104
[43]陈希孺.数理统计引论[M].北京:科学出版社,1981
[44]杨文礼.线性模型引论[M].北京:北京师范东西出版社,1998
[45]王松桂.线性统计模型[M].北京:高等教育出版社,1999
[46]洪圣岩、成平.半参数回归模型参数估计的收敛速度[J],应用概率统计[J],1994(1):62-71
[47]柴根象,孙平,蒋泽云.半参数回归模型的二阶段估计[J].应用数学学报, 1995(3): 353-363
[48]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: 97-106
[49]崔恒建.半参数EV模型的参数估计理论[J].科学通报, 1995, 16: 1444-1448
[50]鹿长余,周光亚.正态均值参数的估计[J].吉林大学自然科学学报,1991,4:5-9
[51]吴启光.可容许线性估计的一个注记[J].应用数学学报,1982,5(1):19-24
[52]吴启光.二次损失下随机回归系数和参数的所有可容许性线性估计[J].系统科学和数学,1991,11(1) :306-312
[53]佟毅等.矩阵损失下的最优同变估计[J].抚顺石油学院学报,1996,3(1):72-76
[54]郑祖康.关于两类污染数据回归分析的参数估计[J].高校应用数学学报[J], 1996(1): 31-40
[2] Godambe. The foundation of finite sample estimation in stochastic processes [J].Biometrika 1985, 72:419-428
[3] T.M purairajan, Martin L.William. Optimal estimating function for estimation and prediction in semi-parametric model[J]. Journal of Statistical Planning and Inference, 2008,138:3283-3292
[4] Hyun Lim, Xu zhang. Semi-parametric additive risk models: Apllication to injury study[J]. Accident Analysis and Prevention ,2009,41:211-216
[5] Eric Wolsztynski ,Eric Thierry ,Luc Pronzato .Minimum-entropy estimation in semi-parametric models[J]. Signal Processing ,2005,85:937-949
[6] James B. McDonald, Yexiao J.Xu. A comparison of semi-parametric and partially adaptive estimators of censored regression model with possibly with possibly skewed and leptokurtic error distributions[J].Economic Letters,1996,51:153-159
[7] Erik Heitfield, Armando Levy. Parametric, semi-parametric and non-parametric models of telecommunications demand: An investigation of residential calling pattern[J].Information economics and Policy ,2001,13:311-329
[8] James B. McDnoald. An application and comparison of some flexible parametric and semi-parametric qualitative response models[J]. Economic Letters, 1996, 53: 145-152
[9] Okmyung Bin. A prediction comparison of housing sales prices by parametric versus semi-parametric regression[J]. Economics, 2004,13(1): 68-84
[10]R.H. Glendinning, A.J. Goode. Semi-parametric classification of noisy curves[J]. Pattern Recognition ,2003,36(1):35-44
[11]Gauthier Lanot,Lan Walke. The union /non-union wage differentical: An application of semi-parametric methods[J]. Journal of Econometrics ,1998,84(2): 327-349
[12]Myoung-jae Lee,Francis Vella. A semi-parametric estimator for censored selectionmodels with endogeneity[J]. Journal of Econometrics,2006,130: 3283-3292
[13]Oliver Lepez. Single-index regression models with right censored responses[J]. Journal of statistical Planning and Inference, 2009,139(1): 1082-1097
[14]Gregory M.Duncan. A semi-parametric censored regression estimator[J]. Journal of Econometrics,2005,131(2):209-229
[15]Bo E.Honore,James L Powell. Pairwise difference estimators of censored and truncated regression models[J]. Journal of Econometrics, 1991,64(1-2): 241-278
[16]Fotheringham A S, Chalton M, Brunsdon C. The geography of parameter space: An investigation into spatial non-stationarity[J]. International Journal of geographical Information System,1996,10:605-627
[17]Fothering A S.Trends in quantitative of Geographical Information Systems. Progress in Human Geography, 1997,(21):88-96
[18]Brunsdon C, Fotheringham A S,Charlton M. Geographically weighed regression: A method for exploring spatial nonsttionarity[J]. Geographically Analysis, 1996, 28: 281-298
[19]Fotheringham A S, Charlton M,Brunsdon C.Measureing spatial variation in relationhips with geographically weighted regression[J].In Recent Developments in Spatial Analysis, Editted by M M Fischer and A Getis, Spring. Verlag, London, 1997:60-82
[20]Jinhong You. Emprical likehood for semiparametric varying-coefficient partially linear regression models[J]. Statistics & Probxbility Letters,2006,76(4): 412-422
[21]Jinhong You,Gemai chen. Estimation of a semeparametric varying coefficient partially linear errors in variable model[J]. Jurnal of Multivariate Analysis, 2006, 97(1): 324-341
[22]Xian zhou, Jinhong You. Wavelet estimation in varying-coefficient partially linear regression models[J]. Statistics & Probability Letters,2004,68(1):91-104
[23]Yiqiang Liu. Generalized partially linear varying-coefficient models[J]. Journal of Statistical planning and Inference, 2008,138(4):901-914
[24]Yongzhou, Jinhong You,Xiaojing Wang: Strong convergence rates of several estimators in semiparametric varying-coefficient partially linear models[J]. Acta Mathematic Science , 2009,29(5):1113-1127
[25]Adrian Pagan. Some identification and estimation results for regression models with stochastically varying coefficients[J]. Journal of Econometrics,1980,13(3): 341-363
[26]Qihua Wang,Riqian zhang .Statistical estimation invarying-coefficient models with surrogate date and validation sampling[J].Journal of Multi variate Analysis,2009, 100(10): 2389-2405
[27]P.J. Huber. Robust Statistics(M), John Wiley & Sons,New York,1981
[28]Waid.A. Statistical Decision Functions(M), John Wiley,1950
[29]Cohen.A.AIladminisibile estimates of the mean vector,Ann[J].Math Statist, 1966,37: 458-463
[30]Zontek. S. Characterization of linear admissibile estimator in Gaura-Marker model under normality in linear statistical inference, proceeding&Rozon, Springer, Verlag,1984
[31]Stein. C. In admissibility of the usual estimator for the meal of a multivariate normal distribution,proc . Third Berkeley Symp[J]. Math Statist probab,1956, 1:197-206
[32]Yafeng Xia,Shaohui Li.Identification of Coefficients for Multivariate Regression Model.in ISAI2010,2010,Volume 1:Institute of Electrical and Electroneics Engine- er, Inc.27-30(EI待检索)
[33]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
[34]陈明华.污染数据回归分析中估计的强相合性[J].应用概率统计, 1998, 14: 73-78
[35]任哲,陈明华.污染数据回归分析中参数的最小一乘估计[J].应用概率统计, 2000, 16: 262-268
[36]刘丽萍.污染数据半参数回归模型中的强相合估计[J].同济大学学报:自然科学版, 2004, 32(6): 832-836
[37]黄四民,梁华.用半参数部分线性模型分析居民消费结构[J].数量经济技术经济研究, 1994, 10: 33-39
[38]梁华,熊健.再论半参数部分线性模型在居民消费结构分析中的应用[J].数理统计与管理, 1995(6): 9-17
[39]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
[40]魏传华,梅长林.半参数空间变系数回归模型的两步估计方法及其数值模拟[J],统计与信息论坛,2006(1):16-19
[41]魏传华,梅长林.半参数空间变系数回归模型的Back-Fitting估计[J],数学的实践与认识[J],2006(3):177-183
[42]高集体、洪圣岩、梁华等.半参数模型研究的若干进展[J],应用概率统计,1994(1):98-104
[43]陈希孺.数理统计引论[M].北京:科学出版社,1981
[44]杨文礼.线性模型引论[M].北京:北京师范东西出版社,1998
[45]王松桂.线性统计模型[M].北京:高等教育出版社,1999
[46]洪圣岩、成平.半参数回归模型参数估计的收敛速度[J],应用概率统计[J],1994(1):62-71
[47]柴根象,孙平,蒋泽云.半参数回归模型的二阶段估计[J].应用数学学报, 1995(3): 353-363
[48]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: 97-106
[49]崔恒建.半参数EV模型的参数估计理论[J].科学通报, 1995, 16: 1444-1448
[50]鹿长余,周光亚.正态均值参数的估计[J].吉林大学自然科学学报,1991,4:5-9
[51]吴启光.可容许线性估计的一个注记[J].应用数学学报,1982,5(1):19-24
[52]吴启光.二次损失下随机回归系数和参数的所有可容许性线性估计[J].系统科学和数学,1991,11(1) :306-312
[53]佟毅等.矩阵损失下的最优同变估计[J].抚顺石油学院学报,1996,3(1):72-76
[54]郑祖康.关于两类污染数据回归分析的参数估计[J].高校应用数学学报[J], 1996(1): 31-40