利用线性回归作预测的研究
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摘要
预测在许多领域如经济、生物、工业、农业、国防等方面都有广泛而重要的应用。预测离不开统计模型,在对某个变量进行预测之前,必须建立模型。线性回归模型是作预测的重要模型之一,本文对利用多元线性回归模型进行预测展开研究。首先针对一般的多元线性回归模型在度量误差标准为相对误差,即( ) ( )下,给出Y0的极小极大估计的定义,并找到Y0的极小极大估计,同时证明了所求的极小极大估计具有无偏性。
其次,在模型(A)的基础上,作如下的假设:
假设1: E[ V ec (ε)] = 0,
假设2: V [V ec (ε)]=σ2Δ?Σ,
假设3: E[ V ec (ε0)] = 0,
假设4: V [V ec (ε0 )]=σ2Δ?Σ0,
假设5: E[ V ec (ε)V ec′(ε0)]=σ2Δ? V,
假设6:∑0 ?V′∑?1V≠0,
其中σ2是未知参数,Δ和Σ分别为已知q阶和n阶正定矩阵,V为n×m阶已知矩阵,Σ0为已知m阶正定矩阵,我们将此模型称为模型(B).在模型(B)下我们得到Y0的最优预测矩阵。
Prediction is widely applied in economics、biology、industry、agriculture、medicine、national defence etc. Prediction must be based on models. Linear regression model is one of prediction models. This thesis studies on utilizing multivariate linear regression to prediction.
First, consider the multivariate linear regression model 0 0 0Y XBY X Bεε??? == ++, (A) Under the relative error criteria, we give the definition of minimax estimation of Y0 and find minimax estimation of Y0 .Meanwhile we prove the minimax estimation of Y0 is unbiased . Next ,according to model (3.1)、(3.2) we give the following assumptions: Assumption 1: E[ V ec (ε)] = 0, Assumption 2: V [V ec (ε)]=σ2Δ?Σ, Assumption 3: E[ V ec(ε0)] = 0, Assumption 4: V [V ec (ε0 )]=σ2Δ?Σ0, Assumption 5: E[ V ec (ε)V ec′(ε0)]=σ2Δ? V, Assumption 6:∑0 ?V′∑?1V≠0, whereσ2 is unknown parameter,Δis q×q known positive definite matrix,Σis n×n known positive definite matrix, V is n×m known matrix,Σ0 is m×m known positive definite matrix. The above model is called model(B).Under model(B) we obtain the optimal prediction matrix of Y0 .
其次,在模型(A)的基础上,作如下的假设:
假设1: E[ V ec (ε)] = 0,
假设2: V [V ec (ε)]=σ2Δ?Σ,
假设3: E[ V ec (ε0)] = 0,
假设4: V [V ec (ε0 )]=σ2Δ?Σ0,
假设5: E[ V ec (ε)V ec′(ε0)]=σ2Δ? V,
假设6:∑0 ?V′∑?1V≠0,
其中σ2是未知参数,Δ和Σ分别为已知q阶和n阶正定矩阵,V为n×m阶已知矩阵,Σ0为已知m阶正定矩阵,我们将此模型称为模型(B).在模型(B)下我们得到Y0的最优预测矩阵。
Prediction is widely applied in economics、biology、industry、agriculture、medicine、national defence etc. Prediction must be based on models. Linear regression model is one of prediction models. This thesis studies on utilizing multivariate linear regression to prediction.
First, consider the multivariate linear regression model 0 0 0Y XBY X Bεε??? == ++, (A) Under the relative error criteria, we give the definition of minimax estimation of Y0 and find minimax estimation of Y0 .Meanwhile we prove the minimax estimation of Y0 is unbiased . Next ,according to model (3.1)、(3.2) we give the following assumptions: Assumption 1: E[ V ec (ε)] = 0, Assumption 2: V [V ec (ε)]=σ2Δ?Σ, Assumption 3: E[ V ec(ε0)] = 0, Assumption 4: V [V ec (ε0 )]=σ2Δ?Σ0, Assumption 5: E[ V ec (ε)V ec′(ε0)]=σ2Δ? V, Assumption 6:∑0 ?V′∑?1V≠0, whereσ2 is unknown parameter,Δis q×q known positive definite matrix,Σis n×n known positive definite matrix, V is n×m known matrix,Σ0 is m×m known positive definite matrix. The above model is called model(B).Under model(B) we obtain the optimal prediction matrix of Y0 .
引文
[1] Henderson,C.R. Estimation of genetic parameters[J].Ann.Math.Statist.1950,21:309-310.
[2] Rao,C.R. Estimation of parameters in a linear model[J].Ann.Statist.1976,4:1023-1037.
[3] Kees Jan van Garderen. Optimal prediction in loglinear models[J]. Journal of Econometrics,2001,104 :119–140.
[4] 蒋春福,苏衡彦. 增长曲线模型中两个最优线性预测[J]. 数学研究,2003,36(3):322-330.
[5] Royall, R. M.. On Finite Population Sampling Theory under Certain Linear Regression Models[J]. Biometrika, 1970, 57: 377-387.
[6] Royall, R. M &Herson. J. H. Robust estimation in Finite Populations[J] .American Statistical Assoc, 1973(a), 68: 880-889.
[7] Pereira C A B, Rodrigues J, Robust Linear Prediction in Finite Populations[J]. International Statistical Review,1983,51: 293–300.
[8] Yoshikazu Takada. Relation of the best Invariant prediction And the best unbiased prediction Location And Scale Families[J]. The Annals of Statistics, 1981, 9(4): 917-921.
[9] Gauri Sankar Datta and Malay Ghoosh. Bayesian Prediction in Linear Models: Applications to Small Area Estimation[J]. The Annals of Statistics, 1991,19(4): 1748-1770.
[10] Richard M. Royall and Dany Pfeffermann.Balanced Samples and Robust Bayesian Inference in Finite Population Sampling[J]. Biometrika. 1982(69),2: 401-409.
[11] Arthur S. Best Linear Unbiased Prediction in the Generalized Linear Model[J].American Statistical Association Journal, 1962,6: 369-375.
[12] Richard M. Royall. The Linear Least-Squares Prediction Approach to Two-Stage Sampling[J].American Statistical Association Journal, 1976,71:657–664.
[13] Kenneth S. Kaminsky and Paul I. Nelson. Best Linear Unbiased Prediction of Order Statistics in Location and Scale Families[J]. American Statistical Association Journal, 1975,70:145–150.
[14] C. Radhakrishna Rao. Prediction of Future Observations in Growth Curve Models[J]. Statistical Science, 1987, 4: 434-471.
[15] Bolfarine H, Pereira C A B, Rodrigues J. Robust Linear Prediction in Finite Populations-A Bayesian Perspective[J]. Sankhya (Series B), 1987, 49: 23-35.
[16] Bolfarine H, Rodrigues J. On the Simple Projection Predictor in Finite Populations[J]. Aust.Jour. Statist. 1988, 30: 338–341.
[17] Bolfarine H, Zacks S. Bayes and Minimax Prediction in Finite Populations[J]. Journal Statistical Planning and Inference, 1991, 28: 139-151.
[18] Bolfarine H, Zacks S. Elian S N, Rodrigues J. Optimal Prediction of the Finite PopulationRegression Coeficient[J]. Sankhy a (Series B), 1994,56: 1–10.
[19] Rodrigues J, Bolfarine H, Rogakto A. A General Theory of Prediction in Finite Populations[J]. International Statistical Review, 1985, 53: 239-254.
[20] 喻胜华,何灿芝.有限总体中的最优预测[J].高校应用数学学报,2000, 15 (2): 199-205.
[21] 喻胜华,何灿芝.任意秩多元线性模型中的最优预测[J].应用数学报,2001,24(2):227-235.
[22] Bernhard F.Arnold,Peter Stahlecker. Linear regression analysis using the relative squared error[J]. Linear Algebra and its Applications,2002, 354:3-20.
[23] Bernhard F.Arnold,Peter Stahlecker.Relative squared error prediction in the generalized linear regression model[J] .statistical Papers,2003,44:107-115.
[24] Maciej Wilczyński . Minmax estimation in the linear model with a relative squared error[J]. Journal of statistical planning and Inference,2005,127:205-212.
[25] Klaus D.Schmidt. Prediction in the linear model: A direct approach[J]. Metrika,1998,48:141 -147.
[26] 倪国熙.常用的矩阵理论和方法[M] .上海:上海科学技术出版社,1984.
[27] Bernhard F.Arnold,Peter Stahlecker. Another view of the kuks-olman estimator[J]. Journal of statistical planning and Inference,2000, 89: 169-174.
[28] Maciej Wilczyński. Minimax estimation in linear regression with ellipsoidal constraints[J]. Journal of statistical planning and Inference,2007,137:79-86.
[29] 王松桂, 陈敏, 陈立萍.线性统计模型[M]. 北京:高等教育出版社,1999.
[30] 张尧庭,方开泰.多元统计分析引论[M]. 北京科学出版社,1982.
[31] Xu-Qing liu,,Jian-yingRong. Quadratic prediction populations[J] . Statistics&Probability Letters,2007,77:483-489.
[32] 袁伶俐. 多元线性模型中线性预测的可容许性[J].湖南第一师范学报,2006,6(1):112-114.
[33] 杨婷,杨虎,张洪阳.基于岭估计的最优预测与经典预测的最优性判别[J].重庆大学学报,2002,25(6): 56-59.
[34] Jiming Jiang. A derivation of BLUP-Best linear unbiased predictor[J]. Statistics & Probability Letters,1997,32:321-324.
[35] 茆诗松,王静龙,濮哓龙.高等数理统计[M].北京:高等教育出版社,1998.
[36] Halliwell LJ. Loss prediction by gengeralized least squares[J].Proc.Cassualty actuar. Soc,1996,83:436-489.
[37] Harry Haupt.,Walter Oberhofer. Best affine unbiased representations of the fully restricted general Gauss–Markov model[J]. Journal of Multivariate Analysis ,2006,97 : 759-764.
[38]刘英华.多元趋势面模型[J].湖北工业大学学报,2007,22:70-71.
[39] Anoop Chaturvedi, Alan T. K. Wan, Shri P. Singh.Improved Multivariate Prediction in a General Linear Model with an Unknown Error Covariance Matrix[J]. Journal ofMultivariate nalysis ,2002,83:166–182 .
[40] A.K. Srivastava. Improved predictors in survey sampling under a super population model[J]. Statistics & Probability Letters, 1995,25:249-252.
[41] Helge Blaker. Minimax estimation in linear regression under restrictions[J]. Journal of Statistical Planning and Inference,2000 ,90 :35-55.
[42] Klaus Th. Hess, Klaus D. Schmidt, Mathias Zocher. Multivariate loss prediction in the multivariate additive model[J]. Insurance: Mathematics and Economics ,2006,39 :185–191.
[43] 冯艳钦 , 蒋君 . 多元曲线模型的条件最好线性无偏估计 [J]. 武汉科技大学学报,2003,26(3):318-319.
[44] 刘严. 多元线性回归的数学模型[J]. 沈阳工程学院报,2005,1(2,3):128-129.
[45] 蒋春福,梁四安,戴永隆. 任意秩有限总体中无偏预测和估计的构造关系[J] .高校应用数学学报 A 辑, 2007,22(1):59-66.
[46] 苏衡彦,蒋春福.一般增长曲线模型中的最优线性无偏预测[J]. 湖南大学学报,2003,30(4):99-102.
[47] 袁 权 龙 , 王 浩 波 . 多 元 线 性 模 型 中 条 件 最 优 预 测 的 稳 健 性 [J]. 数 学 研究,2005,38(4):434-439.
[48] 徐 礼 文 , 喻 胜 华 . 矩 阵 损 失 下 线 性 预 测 的 可 容 许 性 [J]. 数 学 研 究 与 评论,2005,25(1):161-168.
[49] 喻胜华. 任意秩有限总体中的简单投影预测[J]. 经济数学,2001,18(4):49-52.
[50] Goldberger AS. Best linear unbiased prediction in the generalized linear regression model[J].Amer.Statist.Assoc,1962,57:369-375.
[51] Rao, C. R.,Toutenburg, H.. Linear models least squares and atternatives[M]. Heibelberg, 1995.
[52] Fuller. W. A.. Introduction to statistical time series[M]. Wiley, New York, 1976.
[53] Gotway. C. A., Cressie. N.. Improved multivariate prediction under a general linear model[J]. Journal of multivariate analysis,1993, 45: 56-72.
[54] Tapan K. Nayak . On best unbiased prediction and its relationships to unbiased estimation[J]. Journal of Statistical Planning and Inference ,2000,84 : 171-189.
[2] Rao,C.R. Estimation of parameters in a linear model[J].Ann.Statist.1976,4:1023-1037.
[3] Kees Jan van Garderen. Optimal prediction in loglinear models[J]. Journal of Econometrics,2001,104 :119–140.
[4] 蒋春福,苏衡彦. 增长曲线模型中两个最优线性预测[J]. 数学研究,2003,36(3):322-330.
[5] Royall, R. M.. On Finite Population Sampling Theory under Certain Linear Regression Models[J]. Biometrika, 1970, 57: 377-387.
[6] Royall, R. M &Herson. J. H. Robust estimation in Finite Populations[J] .American Statistical Assoc, 1973(a), 68: 880-889.
[7] Pereira C A B, Rodrigues J, Robust Linear Prediction in Finite Populations[J]. International Statistical Review,1983,51: 293–300.
[8] Yoshikazu Takada. Relation of the best Invariant prediction And the best unbiased prediction Location And Scale Families[J]. The Annals of Statistics, 1981, 9(4): 917-921.
[9] Gauri Sankar Datta and Malay Ghoosh. Bayesian Prediction in Linear Models: Applications to Small Area Estimation[J]. The Annals of Statistics, 1991,19(4): 1748-1770.
[10] Richard M. Royall and Dany Pfeffermann.Balanced Samples and Robust Bayesian Inference in Finite Population Sampling[J]. Biometrika. 1982(69),2: 401-409.
[11] Arthur S. Best Linear Unbiased Prediction in the Generalized Linear Model[J].American Statistical Association Journal, 1962,6: 369-375.
[12] Richard M. Royall. The Linear Least-Squares Prediction Approach to Two-Stage Sampling[J].American Statistical Association Journal, 1976,71:657–664.
[13] Kenneth S. Kaminsky and Paul I. Nelson. Best Linear Unbiased Prediction of Order Statistics in Location and Scale Families[J]. American Statistical Association Journal, 1975,70:145–150.
[14] C. Radhakrishna Rao. Prediction of Future Observations in Growth Curve Models[J]. Statistical Science, 1987, 4: 434-471.
[15] Bolfarine H, Pereira C A B, Rodrigues J. Robust Linear Prediction in Finite Populations-A Bayesian Perspective[J]. Sankhya (Series B), 1987, 49: 23-35.
[16] Bolfarine H, Rodrigues J. On the Simple Projection Predictor in Finite Populations[J]. Aust.Jour. Statist. 1988, 30: 338–341.
[17] Bolfarine H, Zacks S. Bayes and Minimax Prediction in Finite Populations[J]. Journal Statistical Planning and Inference, 1991, 28: 139-151.
[18] Bolfarine H, Zacks S. Elian S N, Rodrigues J. Optimal Prediction of the Finite PopulationRegression Coeficient[J]. Sankhy a (Series B), 1994,56: 1–10.
[19] Rodrigues J, Bolfarine H, Rogakto A. A General Theory of Prediction in Finite Populations[J]. International Statistical Review, 1985, 53: 239-254.
[20] 喻胜华,何灿芝.有限总体中的最优预测[J].高校应用数学学报,2000, 15 (2): 199-205.
[21] 喻胜华,何灿芝.任意秩多元线性模型中的最优预测[J].应用数学报,2001,24(2):227-235.
[22] Bernhard F.Arnold,Peter Stahlecker. Linear regression analysis using the relative squared error[J]. Linear Algebra and its Applications,2002, 354:3-20.
[23] Bernhard F.Arnold,Peter Stahlecker.Relative squared error prediction in the generalized linear regression model[J] .statistical Papers,2003,44:107-115.
[24] Maciej Wilczyński . Minmax estimation in the linear model with a relative squared error[J]. Journal of statistical planning and Inference,2005,127:205-212.
[25] Klaus D.Schmidt. Prediction in the linear model: A direct approach[J]. Metrika,1998,48:141 -147.
[26] 倪国熙.常用的矩阵理论和方法[M] .上海:上海科学技术出版社,1984.
[27] Bernhard F.Arnold,Peter Stahlecker. Another view of the kuks-olman estimator[J]. Journal of statistical planning and Inference,2000, 89: 169-174.
[28] Maciej Wilczyński. Minimax estimation in linear regression with ellipsoidal constraints[J]. Journal of statistical planning and Inference,2007,137:79-86.
[29] 王松桂, 陈敏, 陈立萍.线性统计模型[M]. 北京:高等教育出版社,1999.
[30] 张尧庭,方开泰.多元统计分析引论[M]. 北京科学出版社,1982.
[31] Xu-Qing liu,,Jian-yingRong. Quadratic prediction populations[J] . Statistics&Probability Letters,2007,77:483-489.
[32] 袁伶俐. 多元线性模型中线性预测的可容许性[J].湖南第一师范学报,2006,6(1):112-114.
[33] 杨婷,杨虎,张洪阳.基于岭估计的最优预测与经典预测的最优性判别[J].重庆大学学报,2002,25(6): 56-59.
[34] Jiming Jiang. A derivation of BLUP-Best linear unbiased predictor[J]. Statistics & Probability Letters,1997,32:321-324.
[35] 茆诗松,王静龙,濮哓龙.高等数理统计[M].北京:高等教育出版社,1998.
[36] Halliwell LJ. Loss prediction by gengeralized least squares[J].Proc.Cassualty actuar. Soc,1996,83:436-489.
[37] Harry Haupt.,Walter Oberhofer. Best affine unbiased representations of the fully restricted general Gauss–Markov model[J]. Journal of Multivariate Analysis ,2006,97 : 759-764.
[38]刘英华.多元趋势面模型[J].湖北工业大学学报,2007,22:70-71.
[39] Anoop Chaturvedi, Alan T. K. Wan, Shri P. Singh.Improved Multivariate Prediction in a General Linear Model with an Unknown Error Covariance Matrix[J]. Journal ofMultivariate nalysis ,2002,83:166–182 .
[40] A.K. Srivastava. Improved predictors in survey sampling under a super population model[J]. Statistics & Probability Letters, 1995,25:249-252.
[41] Helge Blaker. Minimax estimation in linear regression under restrictions[J]. Journal of Statistical Planning and Inference,2000 ,90 :35-55.
[42] Klaus Th. Hess, Klaus D. Schmidt, Mathias Zocher. Multivariate loss prediction in the multivariate additive model[J]. Insurance: Mathematics and Economics ,2006,39 :185–191.
[43] 冯艳钦 , 蒋君 . 多元曲线模型的条件最好线性无偏估计 [J]. 武汉科技大学学报,2003,26(3):318-319.
[44] 刘严. 多元线性回归的数学模型[J]. 沈阳工程学院报,2005,1(2,3):128-129.
[45] 蒋春福,梁四安,戴永隆. 任意秩有限总体中无偏预测和估计的构造关系[J] .高校应用数学学报 A 辑, 2007,22(1):59-66.
[46] 苏衡彦,蒋春福.一般增长曲线模型中的最优线性无偏预测[J]. 湖南大学学报,2003,30(4):99-102.
[47] 袁 权 龙 , 王 浩 波 . 多 元 线 性 模 型 中 条 件 最 优 预 测 的 稳 健 性 [J]. 数 学 研究,2005,38(4):434-439.
[48] 徐 礼 文 , 喻 胜 华 . 矩 阵 损 失 下 线 性 预 测 的 可 容 许 性 [J]. 数 学 研 究 与 评论,2005,25(1):161-168.
[49] 喻胜华. 任意秩有限总体中的简单投影预测[J]. 经济数学,2001,18(4):49-52.
[50] Goldberger AS. Best linear unbiased prediction in the generalized linear regression model[J].Amer.Statist.Assoc,1962,57:369-375.
[51] Rao, C. R.,Toutenburg, H.. Linear models least squares and atternatives[M]. Heibelberg, 1995.
[52] Fuller. W. A.. Introduction to statistical time series[M]. Wiley, New York, 1976.
[53] Gotway. C. A., Cressie. N.. Improved multivariate prediction under a general linear model[J]. Journal of multivariate analysis,1993, 45: 56-72.
[54] Tapan K. Nayak . On best unbiased prediction and its relationships to unbiased estimation[J]. Journal of Statistical Planning and Inference ,2000,84 : 171-189.