线性模型的估计比较和预测理论研究

发布时间：2018-10-23 11:20

【摘要】：线性模型是现代统计学中一类重要的模型,在经济、金融等领域有着广泛的应用背景.在其建模分析过程,模型的参数估计理论相当重要,得到统计学家的高度重视.一方面,统计学家研究模型参数估计理论和方法,并对各种估计进行比较；另一方面,他们利用参数估计结果研究未来观察值的预测.在此背景下,本文主要基于统计决策理论对线性模型中参数估计的比较和有限总体回归系数的预测方法进行研究.对于误差服从多元t分布的线性模型,我们在平衡损失下对回归系数的Stein-rule (SR)估计,Positive-part Stein-rule (PSR)估计,可行最小均方误差估计和改进可行最小均方误差估计的优良性进行研究.首先基于预检验估计思想给出了这四个估计的统一表达式,并在此基础上得到了各个估计的显式风险.其次,基于风险显式表达式理论上对PSR估计和SR估计的优良性进行分析.最后考虑到风险显式表达式的复杂性,我们采用数值分析的方法进一步对估计的优良性进行研究.对于具有椭球等高分布误差的误定线性模型,我们在误差平方损失下对误差方差的最小二乘估计,约束最小二乘估计,预检验估计和Stein型估计进行比较.首先,基于椭球等高分布的性质得到了各个估计风险的显式表达式.其次,基于估计风险的显式表达式在理论上分析了影响预检验估计风险大小的因素,并进一步考察了预检验估计风险与最小二乘估计风险、约束最小二乘估计风险的关系,同时研究了预检验估计的最优临界值.最后,考虑到估计风险依赖于未知参数,且在结构上非常复杂,为此在多元t分布特例下,我们采用数值分析和自助法分别对这四个估计进一步进行比较.对于有限总体回归系数的预测问题,在超总体观点下,我们在平衡损失下分别对误差不具有正态假定和具有正态假定总体中有限总体回归系数的可容许预测进行研究.首先,对于不具有正态假定的总体,我们得到了齐次线性预测在齐次线性预测类中可容许的充分必要条件,并给出了有限总体回归系数的最佳线性无偏预测,同时分析了最佳线性无偏预测在齐次线性预测类中的可容许性.其次,我们在正态总体下讨论了齐次线性可容许预测在一切预测类中是否可容许的问题,得到了齐次线性预测在一切预测类中可容许性的充分条件,并证明了在适当的条件下,该充分条件也是齐次线性预测在一切预测类中可容许的必要条件.最后,针对具有正态假定的总体,我们给出了有限总体回归系数的最佳无偏预测,并分析了它在一切预测类中的可容许性.最后,我们在改进平衡损失函数的基础上进一步对误差不具有正态假定和具有正态假定总体中有限总体回归系数的Minimax预测进行研究.一方面,我们在非正态总体下得到了齐次线性预测类中有限总体回归系数的线性Minimax预测,并对该预测在齐次线性预测类中的可容许性进行分析,同时将其和Bolfarine提出的最佳线性无偏预测进行比较.另一方面,我们在正态总体下探讨了有限总体回归系数在一切预测类中的线性Minimax预测,并对其在一切预测类中的可容许性进行了分析,同时将其和简单投影预测进行比较.
[Abstract]:The linear model is a kind of important model in modern statistics, and has a wide application background in the fields of economy, finance and so on. In the process of modeling and analyzing, the parameter estimation theory of the model is very important, and it is highly attached importance to the theory of parameter estimation. On the one hand, the parameter estimation theory and method of model parameter estimation are studied, and various estimates are compared; on the other hand, they use parameter estimation results to study the prediction of future observation values. In this context, the paper mainly studies the comparison of parameter estimation and the prediction method of the finite global regression coefficient based on the statistical decision-making theory. In this paper, we study Stein-rule (SR), Positive-Rule Stein-rule (Bernstein), feasible and minimum mean square error estimates of regression coefficients under the equilibrium loss and improve the good benignity of feasible and least square error estimation. First, the unified expressions of these four estimates are given based on the pre-inspection estimation idea, and the explicit risk of each estimate is obtained. Secondly, on the basis of the theory of risk explicit expression, we analyze the merit of estimation and SR estimation. Finally, considering the complexity of the risk explicit expression, we use the method of numerical analysis to further study the estimation superiority. In this paper, the least squares estimate of error variance, the constrained least squares estimate, the pre-test estimate and the Stein type estimate are compared with the error square loss for the misset linear model with a high distribution error such as an ellipsoid. First, the explicit expression of each estimation risk is obtained based on the properties of the high distribution of the ellipsoid and the like. Secondly, based on the explicit expression of the estimated risk, the factors which influence the pre-inspection estimated risk size are analyzed, and the relationship between the pre-inspection estimated risk and the least two-multiplied estimation risk, the constrained least two-multiplied estimation risk is further investigated. At the same time, the optimal critical value of pre-test estimation is studied. Finally, taking into account that the estimated risk is dependent on unknown parameters and is very complicated in structure, in the case of multivariate t-distribution, we use numerical analysis and self-help method to compare these four estimates. For the prediction of the limited overall regression coefficient, under the super-general view, we study the allowable prediction of the error not having the positive state assumption and the finite overall regression coefficient with the positive state hypothesis under the equilibrium loss, respectively. First of all, we get the sufficient and necessary conditions which can be tolerated in homogeneous linear prediction class for the whole population without the assumption of positive state, and give the best linear unbiased prediction of the finite overall regression coefficient. At the same time, the admissible property of the best linear unbiased prediction in homogeneous linear prediction class is analyzed. Secondly, we discuss whether the homogeneous linear admissible prediction can be tolerated in all kinds of prediction classes in the normal state, and obtain sufficient conditions for the admissible property of homogeneous linear prediction in all prediction classes, and prove that under proper conditions, This sufficient condition is also a necessary condition for homogeneous linear prediction in all prediction classes. Finally, we give the best unbiased prediction of the finite global regression coefficient, and analyze its admissibility in all prediction classes. Finally, on the basis of improving the balance loss function, we further study the Minimax prediction with the assumption that the error does not have the positive state assumption and the finite overall regression coefficient with positive state assumption. On the one hand, we get the linear Minimax prediction of the finite general regression coefficient in homogeneous linear prediction class in the non-positive state, and analyze the admissible property of the prediction in homogeneous linear prediction class, and compare it with the best linear unbiased prediction proposed by Bolfarine. On the other hand, we discuss the linear Minimax prediction of the finite overall regression coefficient in all prediction classes, and compare it with simple projection prediction.
【学位授予单位】：湖南大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O212

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