带污染数据的回归模型参数估计
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摘要
污染数据是生物统计和金融统计中常见的一类数据,它也是一类不完全数据.由于试验设计、设备误差、条件限制以及观测者主观因素等原因,我们得到的是不完全数据.不完全数据并不是完全不能利用的数据,虽然有时可以再做一次数据的统计工作,但大多数时候是不可重复、费时太长或是代价太高的.而且,在固定的污染源未查明或被消除的情况下,只可能得到被污染的数据.因此,关于污染数据的统计分析已发展成为统计推断一个重要专题.本文重点研究了带污染数据的回归模型参数估计问题.
第一章简要介绍了选题的背景,包括“回归”的特点、发展,污染数据的产生背景以及国内外研究状况.
第二章介绍了截断情形下带污染数据的模型参数估计,包括截断模型下的污染系数的估计,有两种方法:一种是基于截断数据的均值估计的想法,得到了污染系数的估计,且估计具有渐近正态性,另一种是利用研究截断数据回归分析的想法来做,同样得到了具有良好性质的估计;接着使用矩估计法得到了截断的污染数据的参数估计;最后利用最小二乘法及矩估计方法得到了截断情形下污染数据半参数回归模型的参数估计.
第三章介绍了污染数据线性回归模型的估计问题.首先介绍了一些基础知识,然后重点介绍了我所做的工作:讨论简单回归模型中响应变量受到另一随机变量序列污染时,模型参数和污染系数的估计方法.利用贝叶斯统计原理,给出了污染系数的后验置信区间及模型参数估计.由于引入先验信息,增加了被估参数的信息,它对于提高估计的性能是有益的.在实际中,有广泛的应用价值.
第四章介绍了污染数据半参数回归模型的估计方法.首先利用矩方法给出了两种污染方式下污染参数及污染系数的估计.然后介绍了我所做的另一工作:利用(线性)小波估计方法,给出了未知待估参数β,未知函数g (? )以及污染参数υ的估计,并证明了它们的弱相合性.小波方法用于回归模型,在对待估函数要求较低的情况下,得到了比较优良的性质.
第五章是总结和展望.
Contamination data is a common statistical data in Biological Statistics and Financial Statistics, it is also a incomplete data .There are much factors such as experimental design ,equipment error, restrictions and observers subjective factors,so we are given incomplete data. However, the use of incomplete data is not entirely.Although sometimes we can do a statistical data, but most of the time it is cannot be duplicated, time-consuming too long or the price too is high.Moreover, in the situation which the stationary source has not verified or eliminated,we only obtain the contaminated data.Therefore, the statistical analysis of contamination data have become an important topic. This paper importantly studies the estimation of parameter in regression model which contains contamination data.
In chapter one,the problem’s background is introduced,including the characteristics and development of regression,the background of contamination data and research at home and abroad.
In chapter two,it introduces the estimation of parameter in regression model which contains contamination data under the situation of censord data,including the estimation of contamination coefficients of censord data:one is based on the mean estimate of censord data ,the other is to use the idea to do research on the analysis of regression with censord data;the use of moment estimation method in parametric estimation of broken contamination data;estimation method of semiparametric regression model with contaminated and censord data,and this paper presents the estimations of model parametric and contamination parametric respectively by using least-squares method and moment estimation method.
In chaper three,it introduces the estimation method of linear regression model with contaminated data.First,much basic knowledge is introduced.Then,it importantly introduces my perspective : in this paper the estimations of the parameters and coefficient of contamination for the simple regression model and studied when its response variables are contaminated by another random variable sequence it discusses that the interval estimation of posterior confidence probability of parameters on coefficient of contamination by using the method of Bayes inference. And it gives the point estimation of the parameters in the model.
As a result of the introduction of priori,it increased the information of estimated parameter, which is useful for improving the performance of the estimates. Application of this method have been a lot of attention to other area.
In chaper four, it introduces the estimation method of semiparametric regression model with contaminated data.First , it presents the estimations of model parametric and contamination parametric respectively by using moment estimation method. Then, it importantly introduces my another perspective:we apply (linear) wavelet estimation method to model and give the definition of the unknown parameters to be estimatedβ, unknown function g (? ) and contaminated parameterυ, then prove their weak consistency.When wavelet method is used in regression model, the treatment of conditions that require lower , good nature will be succeed.
In last chaper,there are summary and outlook.
第一章简要介绍了选题的背景,包括“回归”的特点、发展,污染数据的产生背景以及国内外研究状况.
第二章介绍了截断情形下带污染数据的模型参数估计,包括截断模型下的污染系数的估计,有两种方法:一种是基于截断数据的均值估计的想法,得到了污染系数的估计,且估计具有渐近正态性,另一种是利用研究截断数据回归分析的想法来做,同样得到了具有良好性质的估计;接着使用矩估计法得到了截断的污染数据的参数估计;最后利用最小二乘法及矩估计方法得到了截断情形下污染数据半参数回归模型的参数估计.
第三章介绍了污染数据线性回归模型的估计问题.首先介绍了一些基础知识,然后重点介绍了我所做的工作:讨论简单回归模型中响应变量受到另一随机变量序列污染时,模型参数和污染系数的估计方法.利用贝叶斯统计原理,给出了污染系数的后验置信区间及模型参数估计.由于引入先验信息,增加了被估参数的信息,它对于提高估计的性能是有益的.在实际中,有广泛的应用价值.
第四章介绍了污染数据半参数回归模型的估计方法.首先利用矩方法给出了两种污染方式下污染参数及污染系数的估计.然后介绍了我所做的另一工作:利用(线性)小波估计方法,给出了未知待估参数β,未知函数g (? )以及污染参数υ的估计,并证明了它们的弱相合性.小波方法用于回归模型,在对待估函数要求较低的情况下,得到了比较优良的性质.
第五章是总结和展望.
Contamination data is a common statistical data in Biological Statistics and Financial Statistics, it is also a incomplete data .There are much factors such as experimental design ,equipment error, restrictions and observers subjective factors,so we are given incomplete data. However, the use of incomplete data is not entirely.Although sometimes we can do a statistical data, but most of the time it is cannot be duplicated, time-consuming too long or the price too is high.Moreover, in the situation which the stationary source has not verified or eliminated,we only obtain the contaminated data.Therefore, the statistical analysis of contamination data have become an important topic. This paper importantly studies the estimation of parameter in regression model which contains contamination data.
In chapter one,the problem’s background is introduced,including the characteristics and development of regression,the background of contamination data and research at home and abroad.
In chapter two,it introduces the estimation of parameter in regression model which contains contamination data under the situation of censord data,including the estimation of contamination coefficients of censord data:one is based on the mean estimate of censord data ,the other is to use the idea to do research on the analysis of regression with censord data;the use of moment estimation method in parametric estimation of broken contamination data;estimation method of semiparametric regression model with contaminated and censord data,and this paper presents the estimations of model parametric and contamination parametric respectively by using least-squares method and moment estimation method.
In chaper three,it introduces the estimation method of linear regression model with contaminated data.First,much basic knowledge is introduced.Then,it importantly introduces my perspective : in this paper the estimations of the parameters and coefficient of contamination for the simple regression model and studied when its response variables are contaminated by another random variable sequence it discusses that the interval estimation of posterior confidence probability of parameters on coefficient of contamination by using the method of Bayes inference. And it gives the point estimation of the parameters in the model.
As a result of the introduction of priori,it increased the information of estimated parameter, which is useful for improving the performance of the estimates. Application of this method have been a lot of attention to other area.
In chaper four, it introduces the estimation method of semiparametric regression model with contaminated data.First , it presents the estimations of model parametric and contamination parametric respectively by using moment estimation method. Then, it importantly introduces my another perspective:we apply (linear) wavelet estimation method to model and give the definition of the unknown parameters to be estimatedβ, unknown function g (? ) and contaminated parameterυ, then prove their weak consistency.When wavelet method is used in regression model, the treatment of conditions that require lower , good nature will be succeed.
In last chaper,there are summary and outlook.
引文
[1]Rice,J.Covergence rates for partial linear models.Statist.and Probab.letters,1986,4:203-208.
[2]Engel R,Granger C,Rice J,etal.Nonparametric Estimation of the Ratetion Between Weather and Electricity Sales.J.amer.Statist.Assoc,1988,81:310-320.
[3]Davis,D.J..An analyasis of some failure data.JASA ,1952,47:113-150.
[4]Huber, Peter J.Robust estimation of a location parameter.Annals of Mathematical Statistics,1964,35:73-101.
[5]From,S.G..Optimal spacing of quantiles for the estimation of the parameters in mixture of two exponential distributions.Commun Statist(Theory Method), 1989,18:2001-2223.
[6]郑祖康,丁邦俊,杨瑛等.关于两类数据回归分析的参数估计.高校应用数学学报(A辑),1996,11:31-40.
[7]YU K F.A note on the estimation of the mixing parameter in mixture of two distribution.Carolina:University of South Carolina,1990.
[8]湛敏.截断数据情况下的混合分布的混合参数估计.复旦大学硕士学位论文,1990.
[9]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,1998,14:73-78.
[10]任哲,陈明华.污染数据回归分析中参数的最小一乘估计.应用概率统计,2000,16:262-268.
[11]郑祖康,郎春梅,张宏鹏.定数截断寿命试验中污染数据的统计分析.复旦学报(自然科学版),2001,40:125-133.
[12]胡玉萍,王霞,李学相.污染数据回归分析参数的区间估计.郑州大学学报(工学版),2003,24:99-101.
[13]钱伟民等.污染数据线性回归模型的参数估计.同济大学学报,2003,31:246-249.
[14]潘建敏.污染数据半参数回归模型的估计方法.工程数学学报,1997,14:81-85.
[15]刘丽萍.污染数据半参数回归模型中的强相合性.同济大学学报(自然科学版),2004,32:832-835.
[16]Gill,R..Large sample behavior of the product-limit estimator on the whole line.Ann.Statist,1983,11:49-58.
[17]Kaplan,E.L.,Meier.P..Nonparametric estimation from incomplete observations.JASA,1958,53:457-481.
[18]Koul,H.,Susarla.V.,Van Ryzin,J..Regression analysis with randomly right censored data,Ann.Statist,1981,9:1276-1288.
[19]Zheng Zukang.A class of estimators for the parameters in linear regression with censored data.Acta Mathematical Aplicatae Sinica,1987,3:231-241.
[20]王巍,郑祖康.截断的污染数据的参数估计,1997,复旦大学技术报告.
[21]Alshuler,B..Theory for the measurement of competing risk in animal experiments.Math.Biosic,1970,6:1-11.
[22]Zheng Zukang.Two methods of estimating the mean survival time from censored samples,Sankhy κ ,Series A,57(1995),126-136.
[23]王启华.随机截断下半参数回归模型中的相合估计.中国科学,1995,25:819-832.
[24]胡玉萍,陆宜清.截断情形下污染数据半参数回归模型估计方法.郑州大学学报(工学版),2004,25:91-94.
[25]张尧庭,陈汉峰编著.贝叶斯统计推断.北京:科学出版社,1991.
[26]James O. Berger. 统计决策论及贝叶斯分析. 第二版. 贾乃光,吴喜之. 北京:中国统计出版社,1998年. 104-113.
[27]陈希孺等著.线性模型参数的估计理论. 北京:科学出版社,1985.
[28]张金槐编著.线性模型参数估计及其改进.第二版.湖南.长沙:国防科技大学出版社,1999.56-64.
[29]Anoop Chaturvedi. Robust Bayes analysis of the linear regression model .Journal of Statistical Planning and Inference,1996,50:175-186.
[30]Berger,J.and Berliner.Robust Bayes and empirical Bayes analysis with ε ? contaminated priors.Ann Statist,1986,14:461-486.
[31]洪圣岩.一类半参数回归模型的估计理论.中国科学(A 辑),1991,12:1258-1272.
[32]高集体等.部分线性模型中估计的渐进正态性.数学学报,1994,37:256-268.
[33]柴根象,孙平,蒋泽云.半参数回归模型的二阶段估计.应用数学学报,1995,18:353-363.
[34]钱伟民,柴根象.半参数回归模型的估计的渐近性质.高校应用数学学报(A 辑),1999,14:161-168.
[35]柴根象,徐克军.半参数回归模型的线性小波光滑.应用概率统计,1999,15:97-105.
[36]钱伟民,柴根象,蒋凤瑛.半参数回归模型的误差方差的小波估计.数学年刊,2000,21A:341-350.
[37]陈明华.污染数据半参数回归模型估计的渐近正态性.工科数学,1999,15:28-32.
[38]Speckman,P.Fernel Smoothing in Partial Linear Models.J.R.Statist.Soc.B,1988,50:413-436.
[39]Eubank,R.L.,Hart,J.D.and Speckman.P.Trigowmetric Seriess Regression Estimatiors with an Application to Partially Linear Model.J.Multivariate Anal.,1990,32:70-83.
[40]Gilbert G.Walter.Wavelets and Other Orthogonal Systems With Applications,CRC Press,Inc.,1994.
[41]Antoniads , A. , Grogoire , G. , Mckeague , I.W..Wavelet methods for curve estimation.JASA,1994,89:1340-1353.
[42]Hongchang Hu.Ridge estimation of a semeparametric regression model.Journal of computational and applied mathematics,2005,176:215-222.
[43]胡宏昌.半参数回归模型的泛补偿最小二乘估计.工程数学学报,2005,23:487-492.
[44]Richard H.Glendinning.Selecting sub-set autoregressions from outlier contaminated data.Computational Statitics&Data Analysis,2001,36:179-207.
[45]钱伟民,李玉梅.纵向污染数据回归模型中污染源密度的估计[J].同济大学学报,2004,32:539-542.
[46]赵林城.线性模型的误差方差的序贯估计及其渐近性质.数学学报,1983,26:15-28.
[47]陈希孺,王松桂.近代回归分析——原理方法及其应用.合肥:安徽教育出版社,1987.
[48]陈希孺,赵林城.线性模型中的 M 方法.高效应用数学学报,1996,11:31-39.
[49]茆诗松,王静龙,濮晓龙.高等数理统计.北京:高等教育出版社,1998.
[50]郑祖康,吴雪明,饶刚.污染数据处理.应用概率统计,1998,14:307-31.
[2]Engel R,Granger C,Rice J,etal.Nonparametric Estimation of the Ratetion Between Weather and Electricity Sales.J.amer.Statist.Assoc,1988,81:310-320.
[3]Davis,D.J..An analyasis of some failure data.JASA ,1952,47:113-150.
[4]Huber, Peter J.Robust estimation of a location parameter.Annals of Mathematical Statistics,1964,35:73-101.
[5]From,S.G..Optimal spacing of quantiles for the estimation of the parameters in mixture of two exponential distributions.Commun Statist(Theory Method), 1989,18:2001-2223.
[6]郑祖康,丁邦俊,杨瑛等.关于两类数据回归分析的参数估计.高校应用数学学报(A辑),1996,11:31-40.
[7]YU K F.A note on the estimation of the mixing parameter in mixture of two distribution.Carolina:University of South Carolina,1990.
[8]湛敏.截断数据情况下的混合分布的混合参数估计.复旦大学硕士学位论文,1990.
[9]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,1998,14:73-78.
[10]任哲,陈明华.污染数据回归分析中参数的最小一乘估计.应用概率统计,2000,16:262-268.
[11]郑祖康,郎春梅,张宏鹏.定数截断寿命试验中污染数据的统计分析.复旦学报(自然科学版),2001,40:125-133.
[12]胡玉萍,王霞,李学相.污染数据回归分析参数的区间估计.郑州大学学报(工学版),2003,24:99-101.
[13]钱伟民等.污染数据线性回归模型的参数估计.同济大学学报,2003,31:246-249.
[14]潘建敏.污染数据半参数回归模型的估计方法.工程数学学报,1997,14:81-85.
[15]刘丽萍.污染数据半参数回归模型中的强相合性.同济大学学报(自然科学版),2004,32:832-835.
[16]Gill,R..Large sample behavior of the product-limit estimator on the whole line.Ann.Statist,1983,11:49-58.
[17]Kaplan,E.L.,Meier.P..Nonparametric estimation from incomplete observations.JASA,1958,53:457-481.
[18]Koul,H.,Susarla.V.,Van Ryzin,J..Regression analysis with randomly right censored data,Ann.Statist,1981,9:1276-1288.
[19]Zheng Zukang.A class of estimators for the parameters in linear regression with censored data.Acta Mathematical Aplicatae Sinica,1987,3:231-241.
[20]王巍,郑祖康.截断的污染数据的参数估计,1997,复旦大学技术报告.
[21]Alshuler,B..Theory for the measurement of competing risk in animal experiments.Math.Biosic,1970,6:1-11.
[22]Zheng Zukang.Two methods of estimating the mean survival time from censored samples,Sankhy κ ,Series A,57(1995),126-136.
[23]王启华.随机截断下半参数回归模型中的相合估计.中国科学,1995,25:819-832.
[24]胡玉萍,陆宜清.截断情形下污染数据半参数回归模型估计方法.郑州大学学报(工学版),2004,25:91-94.
[25]张尧庭,陈汉峰编著.贝叶斯统计推断.北京:科学出版社,1991.
[26]James O. Berger. 统计决策论及贝叶斯分析. 第二版. 贾乃光,吴喜之. 北京:中国统计出版社,1998年. 104-113.
[27]陈希孺等著.线性模型参数的估计理论. 北京:科学出版社,1985.
[28]张金槐编著.线性模型参数估计及其改进.第二版.湖南.长沙:国防科技大学出版社,1999.56-64.
[29]Anoop Chaturvedi. Robust Bayes analysis of the linear regression model .Journal of Statistical Planning and Inference,1996,50:175-186.
[30]Berger,J.and Berliner.Robust Bayes and empirical Bayes analysis with ε ? contaminated priors.Ann Statist,1986,14:461-486.
[31]洪圣岩.一类半参数回归模型的估计理论.中国科学(A 辑),1991,12:1258-1272.
[32]高集体等.部分线性模型中估计的渐进正态性.数学学报,1994,37:256-268.
[33]柴根象,孙平,蒋泽云.半参数回归模型的二阶段估计.应用数学学报,1995,18:353-363.
[34]钱伟民,柴根象.半参数回归模型的估计的渐近性质.高校应用数学学报(A 辑),1999,14:161-168.
[35]柴根象,徐克军.半参数回归模型的线性小波光滑.应用概率统计,1999,15:97-105.
[36]钱伟民,柴根象,蒋凤瑛.半参数回归模型的误差方差的小波估计.数学年刊,2000,21A:341-350.
[37]陈明华.污染数据半参数回归模型估计的渐近正态性.工科数学,1999,15:28-32.
[38]Speckman,P.Fernel Smoothing in Partial Linear Models.J.R.Statist.Soc.B,1988,50:413-436.
[39]Eubank,R.L.,Hart,J.D.and Speckman.P.Trigowmetric Seriess Regression Estimatiors with an Application to Partially Linear Model.J.Multivariate Anal.,1990,32:70-83.
[40]Gilbert G.Walter.Wavelets and Other Orthogonal Systems With Applications,CRC Press,Inc.,1994.
[41]Antoniads , A. , Grogoire , G. , Mckeague , I.W..Wavelet methods for curve estimation.JASA,1994,89:1340-1353.
[42]Hongchang Hu.Ridge estimation of a semeparametric regression model.Journal of computational and applied mathematics,2005,176:215-222.
[43]胡宏昌.半参数回归模型的泛补偿最小二乘估计.工程数学学报,2005,23:487-492.
[44]Richard H.Glendinning.Selecting sub-set autoregressions from outlier contaminated data.Computational Statitics&Data Analysis,2001,36:179-207.
[45]钱伟民,李玉梅.纵向污染数据回归模型中污染源密度的估计[J].同济大学学报,2004,32:539-542.
[46]赵林城.线性模型的误差方差的序贯估计及其渐近性质.数学学报,1983,26:15-28.
[47]陈希孺,王松桂.近代回归分析——原理方法及其应用.合肥:安徽教育出版社,1987.
[48]陈希孺,赵林城.线性模型中的 M 方法.高效应用数学学报,1996,11:31-39.
[49]茆诗松,王静龙,濮晓龙.高等数理统计.北京:高等教育出版社,1998.
[50]郑祖康,吴雪明,饶刚.污染数据处理.应用概率统计,1998,14:307-31.