有关线性模型参数有偏估计的研究
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摘要
众所周知,线性回归模型在现代统计学中是最重要的模型之一,它的应用遍及工业、农业、经济、保险、生物、医学、工程技术和社会科学等领域。
线性回归模型中最基本的问题是回归参数的估计。回归参数的估计方法很多,最基本的方法是最小二乘法。但是,随着容许性理论的发展和对含有较多自变量的大型回归问题的研究,人们发现在有些情况下最小二乘估计的性能可能变的很坏。针对这种情况众多的统计学家提出了各种各样的关于回归参数的有偏估计以改进最小二乘法估计,并研究了它们的一些性质。本文就对文献[4]中提出的有偏估计的性质进行了研究,得到了相应的结果。
全文分三部分:
第一部分 介绍了线性回归模型参数的有偏估计的发展历史和现状,以及本文涉及到的预备知识和符号记法。
第二部分 在[4]中,王松桂得到了在均方误差标准下双k类估计优于最小二乘估计的一个充分条件。在本文中,我们通过一个比较简洁的方法得到了在广义均方误差标准下一个相应的充分条件,从而从另一个方面证明了最小二乘估计的不可容许性。
第三部分 在这一章里我们展望了在有偏估计方面还可以进一步深入的研究的工作,即在广义均方误差准则下讨论各种有偏估计的优良性。
As we all know, linear regression model is one of the most important models in the modern statistics, which is used extensively in industry, agriculture, economy, insurance, biology, medicine, engineering technology, social science and other fields.Estimating regression parameters is a basic problem in linear regression model. These are many methods to Estimate regression parameters, in which methods the least squares estimator(LSE) is a basic methods. But with the development of the admissibility theory and the study on large-scale regression problem, we find the property of the least squares estimator (LSE)may be very bad. Because of these reason, many researchers proposed all kind of the biased estimators in order to improve the least squares estimator and studied their properties. In this paper, we studied the properties of the double k-class estimators ,which are proposed in [4].This paper consists of three follow parts:Part 1: we introduced the history and recent advances on the biased estimators of regression parameters. Besides these, we list the notations and prepared knowledge that we will use in this paper.Part 2: In [4], Song-Gui Wang gave a sufficient condition that the double k-class estimators are better than least square estimator (LSE) based on the mean square error (MSE) criterion. In this paper, we obtain a corresponding sufficient condition based on the generalized mean square error (GMSE) criterion by using a directed method and show that the LSE for the multicovariate normal mean is inadmissible in another respect.Part 3: In this part, we prospected the study that can be done deeply on the biased estimators, which are the discussion on the good properties of some biased estimators based on the generalized mean square error (GMSE) criterion.
线性回归模型中最基本的问题是回归参数的估计。回归参数的估计方法很多,最基本的方法是最小二乘法。但是,随着容许性理论的发展和对含有较多自变量的大型回归问题的研究,人们发现在有些情况下最小二乘估计的性能可能变的很坏。针对这种情况众多的统计学家提出了各种各样的关于回归参数的有偏估计以改进最小二乘法估计,并研究了它们的一些性质。本文就对文献[4]中提出的有偏估计的性质进行了研究,得到了相应的结果。
全文分三部分:
第一部分 介绍了线性回归模型参数的有偏估计的发展历史和现状,以及本文涉及到的预备知识和符号记法。
第二部分 在[4]中,王松桂得到了在均方误差标准下双k类估计优于最小二乘估计的一个充分条件。在本文中,我们通过一个比较简洁的方法得到了在广义均方误差标准下一个相应的充分条件,从而从另一个方面证明了最小二乘估计的不可容许性。
第三部分 在这一章里我们展望了在有偏估计方面还可以进一步深入的研究的工作,即在广义均方误差准则下讨论各种有偏估计的优良性。
As we all know, linear regression model is one of the most important models in the modern statistics, which is used extensively in industry, agriculture, economy, insurance, biology, medicine, engineering technology, social science and other fields.Estimating regression parameters is a basic problem in linear regression model. These are many methods to Estimate regression parameters, in which methods the least squares estimator(LSE) is a basic methods. But with the development of the admissibility theory and the study on large-scale regression problem, we find the property of the least squares estimator (LSE)may be very bad. Because of these reason, many researchers proposed all kind of the biased estimators in order to improve the least squares estimator and studied their properties. In this paper, we studied the properties of the double k-class estimators ,which are proposed in [4].This paper consists of three follow parts:Part 1: we introduced the history and recent advances on the biased estimators of regression parameters. Besides these, we list the notations and prepared knowledge that we will use in this paper.Part 2: In [4], Song-Gui Wang gave a sufficient condition that the double k-class estimators are better than least square estimator (LSE) based on the mean square error (MSE) criterion. In this paper, we obtain a corresponding sufficient condition based on the generalized mean square error (GMSE) criterion by using a directed method and show that the LSE for the multicovariate normal mean is inadmissible in another respect.Part 3: In this part, we prospected the study that can be done deeply on the biased estimators, which are the discussion on the good properties of some biased estimators based on the generalized mean square error (GMSE) criterion.
引文
[1] Stigler, S. M. Gauss and the invention of least squares, Annals of Statistics, 9(1981)465-474
[2] Stein C. Inadmissibility of the usual estimator for the mean of a multivariate normal distribution, Prob. third Berkeley Sypm. Math. Statist. Prob. Ⅰ, 1956,197-206
[3] Stein C. Inadmissibility of the usual estimator for the variance of a normal distribution with unknown mean, Annals. Inst. Statist. Math. 1964,16:155-160
[4] 王松桂,《线性模型的理论及其应用》,安徽教育出版社,1987年8月
[5] 张金槐,《线性模型参数估计及其改进》,国防科技大学出版社,1999年3月
[6] 陈希孺,王松桂,《近代回归分析——原理方法及应用》,安徽教育出版社,1987年9月
[7] Hoerl A.E. and Kennard R.D. Ridge regression, biased estimation for orthogonal problems. Technonetrios. 1970.12:55-67
[8] 汪明瑾,王静龙。岭回归中确定K值的一种方法。应用概率统计。2001.2(1)7-13
[9] 王正明。关于线性回归模型的有偏估计。系统科学与数学。1995.10.319-328
[10] Wang Songgui. On biased linear estimators in a linear model with arbitrary rank. Commu. Statist. 1982. All 1571-1581
[11] Hawkins D.W. Relations between ridge regression and eigenanaly sis of the argmented correlation matrix. technometrios 1970.12:591-612
[12] Farebother R. Further result on the mean square error of ridge regression. J. Roy. Statist. Soc. 1976B38 248-250
[13] 戴俭华等。岭估计优于最小二乘估计的条件。数理统计与应用概率。1994,2:53-58
[14] 林路。回归系数的综合岭估计。数理统计与应用概率。1996.6
[15] Sreawderman W. E. Minimac adaptive generalized ridge regression estimators J. R. Stat. Soc. B, 1978,73:623-627
[16] Judge C. G. and Bock K. E. The statistical implication of pretest and Stein rule estimators, In Econometrios. North Hollard, New York. 1978
[17] 张金槐。线性模型参数估计及其改进(第二版)。国防科技大学出版社 1999.3
[18] W. F. Massy. Principal components regression in exploratory statistical research, J. R. stat. 1965, 60:234-266
[19] 田保光。岭型主成分估计的最小方差和。纯粹数学与应用数学。1998年.4(1)93-95
[20] Tian Bao-guang. Several optimatitles of combining ridge and principal components estimator. Chinese quarterly Journal of math. 2000.3. Voll5. No1
[21] 王松桂等。主成分估计的最优性质。科学通报。1984.8
[22] 王松桂。主成分估计的最优性与广义主成分估计类。应用概率统计,1985,1:23-30
[23] Webster, J. T. et al, Latent root regression analysis, Technometrics, 16 (1974) 513-522
[24] 王正明。改进的特征根估计。应用数学。1990年第2期
[25] Nagar. A. L. Double k-class estimators of parameters in simultaneous equations and their small sample properties. International Econmic Review 1962.3:168-188
[26] Ullah. A. and Ullah. S. Double k-class estimators of coefficients inlinear regression. Econometrica. 1978.46
[27] Dwivedi, T. D. and V. K Srivastava.Exact finite sample properties of double k-class estimators in simultaneous equations. Journal of Econometrics. 1984.25:263-286
[2] Stein C. Inadmissibility of the usual estimator for the mean of a multivariate normal distribution, Prob. third Berkeley Sypm. Math. Statist. Prob. Ⅰ, 1956,197-206
[3] Stein C. Inadmissibility of the usual estimator for the variance of a normal distribution with unknown mean, Annals. Inst. Statist. Math. 1964,16:155-160
[4] 王松桂,《线性模型的理论及其应用》,安徽教育出版社,1987年8月
[5] 张金槐,《线性模型参数估计及其改进》,国防科技大学出版社,1999年3月
[6] 陈希孺,王松桂,《近代回归分析——原理方法及应用》,安徽教育出版社,1987年9月
[7] Hoerl A.E. and Kennard R.D. Ridge regression, biased estimation for orthogonal problems. Technonetrios. 1970.12:55-67
[8] 汪明瑾,王静龙。岭回归中确定K值的一种方法。应用概率统计。2001.2(1)7-13
[9] 王正明。关于线性回归模型的有偏估计。系统科学与数学。1995.10.319-328
[10] Wang Songgui. On biased linear estimators in a linear model with arbitrary rank. Commu. Statist. 1982. All 1571-1581
[11] Hawkins D.W. Relations between ridge regression and eigenanaly sis of the argmented correlation matrix. technometrios 1970.12:591-612
[12] Farebother R. Further result on the mean square error of ridge regression. J. Roy. Statist. Soc. 1976B38 248-250
[13] 戴俭华等。岭估计优于最小二乘估计的条件。数理统计与应用概率。1994,2:53-58
[14] 林路。回归系数的综合岭估计。数理统计与应用概率。1996.6
[15] Sreawderman W. E. Minimac adaptive generalized ridge regression estimators J. R. Stat. Soc. B, 1978,73:623-627
[16] Judge C. G. and Bock K. E. The statistical implication of pretest and Stein rule estimators, In Econometrios. North Hollard, New York. 1978
[17] 张金槐。线性模型参数估计及其改进(第二版)。国防科技大学出版社 1999.3
[18] W. F. Massy. Principal components regression in exploratory statistical research, J. R. stat. 1965, 60:234-266
[19] 田保光。岭型主成分估计的最小方差和。纯粹数学与应用数学。1998年.4(1)93-95
[20] Tian Bao-guang. Several optimatitles of combining ridge and principal components estimator. Chinese quarterly Journal of math. 2000.3. Voll5. No1
[21] 王松桂等。主成分估计的最优性质。科学通报。1984.8
[22] 王松桂。主成分估计的最优性与广义主成分估计类。应用概率统计,1985,1:23-30
[23] Webster, J. T. et al, Latent root regression analysis, Technometrics, 16 (1974) 513-522
[24] 王正明。改进的特征根估计。应用数学。1990年第2期
[25] Nagar. A. L. Double k-class estimators of parameters in simultaneous equations and their small sample properties. International Econmic Review 1962.3:168-188
[26] Ullah. A. and Ullah. S. Double k-class estimators of coefficients inlinear regression. Econometrica. 1978.46
[27] Dwivedi, T. D. and V. K Srivastava.Exact finite sample properties of double k-class estimators in simultaneous equations. Journal of Econometrics. 1984.25:263-286
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