污染线性模型的参数和非参数估计的研究
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摘要
线性模型是数理统计学中发展较早，理论丰富且应用性很强的一个重要分支，过去的百余年，线性模型不仅在理论研究方面甚为活跃，获得了长足的发展，而且在工农业、气象地质、经济管理、医药卫生、教育心理学等领域的应用也日渐广泛，作为线性模型前沿科学研究的一部分，污染线性模型由于在实际生活中的广泛存在，越来越受到关注，具有很高的应用价值。
本文研究了污染数据下线性回归模型的参数与非参数估计。论文第一部分是绪论，在这部分首先介绍了有关污染线性模型的发展历史，并对整篇论文做了简要的概括。论文的第二部分与第三部分是这篇论文的主体部分。在这两个部分，从三个不同的角度给出了在绪论中提出的三种不同污染线性模型的讨论结果。第二部分，利用矩估计和极大似然估计两种不同的统计方法，给出了模型Ⅰ和模型Ⅱ的参数估计。第三部分，对于污染线性模型Ⅲ：y_i=βx_i+e_i，i=1，2，…，n，设误差序列{e_i}是平稳的α~-混合序列，f(x)为其公共的未知密度函数。讨论了基于残差的f(x)核估计的相合性及其收敛速度。并构造了污染系数ε及回归参数β的非参数估计，证明了估计量的强相合性和强收敛速度。
论文的第四部分是论文的主要应用。在这部分中，通过三个不同的实例分析了污染线性模型的实际应用价值。这几个实例分别讨论了污染线性模型的最小二乘估计、污染线性模型的非参数拟合和污染系数的模拟结果。
Linear model is one of the most important branches of the statistics, which develops earlier and concludes plentiful theories in mathematical statistics. During the past several hundred years, linear model is not only active in theoretic research areas but also applies widely in some fields such as industry and agriculture meteorology and geology, economical management, medical affairs, educational psychology gradually. As a most popular part in the scientific research in the linear model, contaminated linear model is the focus because of the widespread existence in actual life and has the very high value in the application.
The thesis is a research on non-parameter estimate of contamination data for simple linear regression model. In the first part of this paper, we suggest the history about the progress of the contaminated linear model and give a brief summary about the whole paper. The second part and third part of this paper is the main body. We suggest three different discussion results about the three different contaminated models which have been brought up in the preface. In the second part, we give the estimation about the parameters of Model I and Model II with both the moment method and the maximum likelihood method. In the third part, we consider the contaminated linear model: y_i =βx_i+e_i, i = l,2,…,n.Let the errors sequence {e_i} is a stationary with unknown density f(x). We obtain consistency the kernel estimation of f(x) based on the residuals. Then we establish nonparametric estimation in contaminated coefficientεand regression parameterβ. We prove the strong consistency and convergence rate almost surely of the estimators.
The fourth part of this paper is the applications. In this part we analyze the actual of the contaminated model by three different examples. These examples discuss the MLE of the contaminated model, the non-parametric fitting of the contaminated model and the simulation on coefficient of contamination.
本文研究了污染数据下线性回归模型的参数与非参数估计。论文第一部分是绪论，在这部分首先介绍了有关污染线性模型的发展历史，并对整篇论文做了简要的概括。论文的第二部分与第三部分是这篇论文的主体部分。在这两个部分，从三个不同的角度给出了在绪论中提出的三种不同污染线性模型的讨论结果。第二部分，利用矩估计和极大似然估计两种不同的统计方法，给出了模型Ⅰ和模型Ⅱ的参数估计。第三部分，对于污染线性模型Ⅲ：y_i=βx_i+e_i，i=1，2，…，n，设误差序列{e_i}是平稳的α~-混合序列，f(x)为其公共的未知密度函数。讨论了基于残差的f(x)核估计的相合性及其收敛速度。并构造了污染系数ε及回归参数β的非参数估计，证明了估计量的强相合性和强收敛速度。
论文的第四部分是论文的主要应用。在这部分中，通过三个不同的实例分析了污染线性模型的实际应用价值。这几个实例分别讨论了污染线性模型的最小二乘估计、污染线性模型的非参数拟合和污染系数的模拟结果。
Linear model is one of the most important branches of the statistics, which develops earlier and concludes plentiful theories in mathematical statistics. During the past several hundred years, linear model is not only active in theoretic research areas but also applies widely in some fields such as industry and agriculture meteorology and geology, economical management, medical affairs, educational psychology gradually. As a most popular part in the scientific research in the linear model, contaminated linear model is the focus because of the widespread existence in actual life and has the very high value in the application.
The thesis is a research on non-parameter estimate of contamination data for simple linear regression model. In the first part of this paper, we suggest the history about the progress of the contaminated linear model and give a brief summary about the whole paper. The second part and third part of this paper is the main body. We suggest three different discussion results about the three different contaminated models which have been brought up in the preface. In the second part, we give the estimation about the parameters of Model I and Model II with both the moment method and the maximum likelihood method. In the third part, we consider the contaminated linear model: y_i =βx_i+e_i, i = l,2,…,n.Let the errors sequence {e_i} is a stationary with unknown density f(x). We obtain consistency the kernel estimation of f(x) based on the residuals. Then we establish nonparametric estimation in contaminated coefficientεand regression parameterβ. We prove the strong consistency and convergence rate almost surely of the estimators.
The fourth part of this paper is the applications. In this part we analyze the actual of the contaminated model by three different examples. These examples discuss the MLE of the contaminated model, the non-parametric fitting of the contaminated model and the simulation on coefficient of contamination.
引文
[1] McKendrick AG. Applications of mathematics to medical problems. Proceedings Edinburgh Mathematics Society, 1926; 44: 98-130.
[2] Davis D. J. An analysis of some failure. JASA 1952; 47: 113-159.
[3] Huber PJ. Robust estimation of a location parameter. Ann Math Statics, 1964, 35: 73-102.
[4] 郑祖康，丁邦俊。关于两类污染数据回归分析的数学的估计高校应用数学报。1996，11(1)：31-40．
[5] 陈明华。污染数据回归分析中估计的强相合性．应用概率统计．1998，14(1)：73-80．
[6] Chai Gen Xiang, Tian Hong, Li ZhuYu, Consistent non-parametric estimation of error distributions in linear model, Acta. Math. Appl. Sinica, 1991, 7: 245-256
[7] 高集体等．部分线性模型中估计的渐进正态性．数学学报，1994，37(2)：256-268．
[8] 缪柏其．概率论教程．安徽合肥：中国科学技术大学出版社，1998．
[9] 陈希孺 数理统计引论．科学出版社，北京，1981．
[10] Liebscher E. Strong convergence of suns of α~-mixing random variables with applications to density estimation. Stochastic Processes and Applications, 1996, 65: 69-80.
[11] 胡舒合，陈贵景。相依误差下回归LS估计的强相合性。数学物理学报，1996，6A(1)：27-34．
[12] 杨筱菡，柴根象。相依样本下线性模型误差分布相合估计。数学物理学报，2002，22A(4)：536-547．
[13] 陈希孺，陈桂景，吴启光等．线性模型参数的估计理论．北京：科学出版社，1985．92．
[14] 柴根象，李竹渝，田红．Consistent nonparametric estimation of error distributions in linear model. Acta Math Applicate Sinica, 1991(7): 245-256.
[15] 成平．回归函数改量核估计的强相合性及收敛速度。系统科学与数学，1983，3(4)：304-315．
[16] Wu Shanshan, MiaoBaiqi, Statistical Inference for Contamination Distribution.系统科学与复杂性(2001).
[17] 王巍，污染分布的非参数推断，复旦博士论文(1999)．
[18] 郑祖康，吴雪明，饶刚．污染数据的处理．应用概率统计，1998(8)：307-312。
[19] 柴根象，花虹．污染的可识别条件与判别分析．生物数学学报，2004，19(3)：364-372．
[20] 查道庆，污染线性模型的讨论，安徽大学硕士学位论文(2006)．
[21] 张文扬．线性模型误差分布的相合核估计．四川大学学报(自然科学版)，1990，27(2)．
[22] 秦永松．相依误差下回归函数导数估计的强收敛速度．应用数学，1994，7(1)．
[23] 杨善朝．线性模型中φ混合误差下回归系数最小二乘估计的相合性．数理统计与应用概率，1994，9(1)．
[24] Billings ley, P. Convergence of Probability Measure, John, Welleg and Sons, Inc, 1968.
[25] 凌能祥．相依样本下线性模型误差分布的相合核估计．安徽工学院学报，1997，16(3)；61-67．
[26] Dhomponfsa S. A note on the almost sure approximation of the empirical process of weakly dependent r. v. s. YokohamaMath J, 1984, 32: 113-121.
[27] Pall H. Martingale Limit Theory and its Application. New York: Academic Press, 1980
[28] Devroye L. The equivalence of weak strong and complete convergence in for kernel density estimates. Ann Statics, 1983, 11: 896-904.
[29] Athreya K B, Pantula S G. Mixing properties of Harris chains and autoregressive processes. J Appl Probab, 1986, 23: 880-892.
[30] Ahmad I A. A note on non-parametric density estimation for dependent variables using a delta sequence. Ann Inst Statist Math, 1981, 33: 247-254.
[31] Folders A. Density estimation for dependent sample. Studia Scientiarum Mathematicarum Hungarica, 1974, 9: 443-452.
[30] Roussas G G. Non-parametric estimation in mixing sequence of random variables. Math Methods Stastist, 1995, 4: 216-229.
[32] Roussas G G. Asymptotic normality of the kernel estimation under dependent conditions: Application to hazard rate. J Statist Plan Inference, 1990, (25): 81-104.
[33] Tran L T. Kernel estimation under dependent conditions. Statist Probab Lett, 1991, 10: 193-201.
[34] Vieu P. Quadratic errors for non-parametric density estimation under dependence. Multivaiate Anal, 1991, 39: 324-347.
[35] Yu K F. A note on the estimation of the mixing parameter in a mixture of two exponential distributions. Commun Stat(Theory Method), 1991, 20: 595-609.
[36] Miao Baiqi, Hui Jun. Nonparametric Estimation for Contamination Distribution. Working Paper.
[2] Davis D. J. An analysis of some failure. JASA 1952; 47: 113-159.
[3] Huber PJ. Robust estimation of a location parameter. Ann Math Statics, 1964, 35: 73-102.
[4] 郑祖康，丁邦俊。关于两类污染数据回归分析的数学的估计高校应用数学报。1996，11(1)：31-40．
[5] 陈明华。污染数据回归分析中估计的强相合性．应用概率统计．1998，14(1)：73-80．
[6] Chai Gen Xiang, Tian Hong, Li ZhuYu, Consistent non-parametric estimation of error distributions in linear model, Acta. Math. Appl. Sinica, 1991, 7: 245-256
[7] 高集体等．部分线性模型中估计的渐进正态性．数学学报，1994，37(2)：256-268．
[8] 缪柏其．概率论教程．安徽合肥：中国科学技术大学出版社，1998．
[9] 陈希孺 数理统计引论．科学出版社，北京，1981．
[10] Liebscher E. Strong convergence of suns of α~-mixing random variables with applications to density estimation. Stochastic Processes and Applications, 1996, 65: 69-80.
[11] 胡舒合，陈贵景。相依误差下回归LS估计的强相合性。数学物理学报，1996，6A(1)：27-34．
[12] 杨筱菡，柴根象。相依样本下线性模型误差分布相合估计。数学物理学报，2002，22A(4)：536-547．
[13] 陈希孺，陈桂景，吴启光等．线性模型参数的估计理论．北京：科学出版社，1985．92．
[14] 柴根象，李竹渝，田红．Consistent nonparametric estimation of error distributions in linear model. Acta Math Applicate Sinica, 1991(7): 245-256.
[15] 成平．回归函数改量核估计的强相合性及收敛速度。系统科学与数学，1983，3(4)：304-315．
[16] Wu Shanshan, MiaoBaiqi, Statistical Inference for Contamination Distribution.系统科学与复杂性(2001).
[17] 王巍，污染分布的非参数推断，复旦博士论文(1999)．
[18] 郑祖康，吴雪明，饶刚．污染数据的处理．应用概率统计，1998(8)：307-312。
[19] 柴根象，花虹．污染的可识别条件与判别分析．生物数学学报，2004，19(3)：364-372．
[20] 查道庆，污染线性模型的讨论，安徽大学硕士学位论文(2006)．
[21] 张文扬．线性模型误差分布的相合核估计．四川大学学报(自然科学版)，1990，27(2)．
[22] 秦永松．相依误差下回归函数导数估计的强收敛速度．应用数学，1994，7(1)．
[23] 杨善朝．线性模型中φ混合误差下回归系数最小二乘估计的相合性．数理统计与应用概率，1994，9(1)．
[24] Billings ley, P. Convergence of Probability Measure, John, Welleg and Sons, Inc, 1968.
[25] 凌能祥．相依样本下线性模型误差分布的相合核估计．安徽工学院学报，1997，16(3)；61-67．
[26] Dhomponfsa S. A note on the almost sure approximation of the empirical process of weakly dependent r. v. s. YokohamaMath J, 1984, 32: 113-121.
[27] Pall H. Martingale Limit Theory and its Application. New York: Academic Press, 1980
[28] Devroye L. The equivalence of weak strong and complete convergence in for kernel density estimates. Ann Statics, 1983, 11: 896-904.
[29] Athreya K B, Pantula S G. Mixing properties of Harris chains and autoregressive processes. J Appl Probab, 1986, 23: 880-892.
[30] Ahmad I A. A note on non-parametric density estimation for dependent variables using a delta sequence. Ann Inst Statist Math, 1981, 33: 247-254.
[31] Folders A. Density estimation for dependent sample. Studia Scientiarum Mathematicarum Hungarica, 1974, 9: 443-452.
[30] Roussas G G. Non-parametric estimation in mixing sequence of random variables. Math Methods Stastist, 1995, 4: 216-229.
[32] Roussas G G. Asymptotic normality of the kernel estimation under dependent conditions: Application to hazard rate. J Statist Plan Inference, 1990, (25): 81-104.
[33] Tran L T. Kernel estimation under dependent conditions. Statist Probab Lett, 1991, 10: 193-201.
[34] Vieu P. Quadratic errors for non-parametric density estimation under dependence. Multivaiate Anal, 1991, 39: 324-347.
[35] Yu K F. A note on the estimation of the mixing parameter in a mixture of two exponential distributions. Commun Stat(Theory Method), 1991, 20: 595-609.
[36] Miao Baiqi, Hui Jun. Nonparametric Estimation for Contamination Distribution. Working Paper.