【作者】 刘福香；

【导师】 徐文科；

【作者基本信息】 东北林业大学 ， 应用数学， 2010， 硕士

【摘要】 本文在前人研究误差服从一阶自回归的线性模型的基础上,研究误差服从二阶自回归的线性模型的统计推断问题。误差服从自回归的线性模型,在许多领域,特别是在经济、管理、工程技术、林业和生态模型等领域具有广泛的应用。对于这种模型,如果忽略了相关性的存在,按照误差服从Gauss-Markov假设的情形,用标准的最小二乘法去处理,在许多情况下将会导致参数估计精度的下降和假设检验犯错误概率的升高,及其它一系列问题。因此,长期以来这种模型的研究受到统计学家的关注。本文在考虑矩估计缺陷(只考虑相邻两个量的相关)的基础上,提出了新的方差参数的估计方法：距离偏相关系数迭代(Distance Partial correlation coefficient iteration)估计方法(DPC),对于平面数据不涉及到距离,所以简化为PC估计。以均方误差(mean square error (MSE))作为度量估计优劣的标准,通过模拟比较了矩估计(Moment estimation (MS))、Cochrane-Orcutt (CO)迭代估计与新估计的优劣。模拟显示当误差程度较高和误差具有更高阶相关性时,偏相关系数迭代估计优于其它两个估计。关于误差服从二阶自回归的线性模型的参数估计问题,在误差服从一阶自回归的线性模型的参数估计的基础上,求出了误差的协方差阵及其逆矩阵,并进行了三角分解求出了差分加权变换矩阵。对变换后的模型采用了广义循环最小二乘估计(the cycle generalized least squares (CGLS))。通过模拟,以均方误差作为标准,比较误差服从二阶自回归的线性模型的广义循环最小二乘估计与两步广义最小二乘估计(two steps generalized least squares (TGLS))。模拟结果表明,广义循环最小二乘估计优于两步广义最小二乘估计。并研究当误差系数具有特殊关系时,用模拟比较了考虑特殊关系与不考虑特殊关系时的估计精度。表明考虑特殊关系优于不考虑特殊关系时的估计精度。本文还对误差的相关性检验进行了研究,提出了误差高阶相关时的检验方法。以实际例子,阐述了误差存在高阶相关性时的处理方法。更多还原

【Abstract】 Based on studying some statistical inference of linear models with AR(1) errors, the purpose of this statistical inference of linear models with AR(2) errors. This kind of models has extensive applications in many fields, in particular, such as economics, management, engineering, forestry and ecology. For these models, an approach for statistical inference is to ignore the existence of error correlation and then to apply standard least squares method under Gauss-Markov assumption test, this will cause some series problems in parameter estimation and hypothesis test in many cases. Thus study of these models has obtained a great attention of statisticians for a long period.For parameter estimation of linear models with AR(2) errors, the paper obtains the covariance matrix of the error vector and its inverse matrix and obtains transformation matrix of weighted 2-order differences, based on studying linear models with AR(1) errors. Then estimates parameters of linear models with AR(2) errors with the cycle generalized least squares (CGLS). Under mean square error criterion, Simulation results show that efficiency of CGLS method is superior over the method of two steps generalized least squares (GLS). Furthermore the paper studies errors parameters with some special relationship in linear regression model with AR(2) errors. The result shows considering the special relationship is superior over notIn this paper we propose a new estimate, Distance Partial correlation coefficient iteration estimation of variance parameter, which takes defect of the moment estimation (only with adjacent variable information). Simulation results show that for high errors correlation case and high order errors correlation case, new estimate is superior over the moment estimate and the CO iteration estimate appeared in literature under mean square error criterion.The paper also discusses autocorrelation test of the linear regression model and studies some methods of autocorrelation test of the linear regression model with high order autocorrelation errors, and explains treatment process of the linear regression model with high order autocorrelation errors with examples.更多还原