线性回归模型参数有偏估计的研究

发布时间：2018-01-10 05:22

本文关键词：线性回归模型参数有偏估计的研究　出处：《东北农业大学》2014年硕士论文　论文类型：学位论文

【摘要】：线性模型作为统计模型的一类,由于其应用广泛、形式简单和易于处理,得到了广泛的关注与研究。在线性模型中,由于参数估计的基础性地位,作为线性模型首先要面临的问题,决定了估计回归参数具有重要意义。参数估计最基本、最常见的方法是最小二乘估计。近些年来当数据存在复共线性时最小二乘估计均方误差异常大导致估计值变坏,提出了有偏估计的概念。有偏估计在一定条件下减小了均方误差,改进了最小二乘估计的不足。参数有偏估计对于线性模型理论的发展和完善具有非常重要的意义。本文结合国内外参数有偏估计的相关研究理论、存在的问题,主要在相对效率意义下推导优于最小二乘估计效率上界,针对已有的有偏估计只从局部改进最小二乘估计,提出了带有约束的有偏估计类,主要工作如下：首先,用有偏估计代替最小二乘估计后虽然减小了估计的均方误差,但是估计的精度受到一些损失,相对效率能更好的度量有偏估计代替最小二乘估计损失的大小。因此本文重点在相对效率评价准则意义下分别推导出广义岭估计和Liu估计优于最小二乘估计的效率上界。为推导有偏估计优于最小二乘估计的条件提供了新的思路。其次,已有的有偏估计虽然改进了最小二乘估计,但得出的均方误差是未知参数的函数,有偏估计得到的结果是探索性的,而不是确证性的。当变量存在约束情况下,已有的有偏估计并不适用,得出的估计值很不理想。针对上述这些问题本文从函数差异性角度出发,采用新的度量函数,在变量有约束时最终提出I-divergence估计。并在Kuhn-Tucker理论基础上,设计迭代算法,得出迭代解,并证明了迭代过程的收敛性。如果所涉及的数据均为非负实值约束,并且数据呈正相关性,则I-divergence准则是仅有的一致性选择。进一步用仿真数据验证新估计的优劣,计算岭估计、Liu估计的均方误差和新估计比较充分证明了在变量非负约束情况下新估计比已有的有偏估计更好的减小了均方误差。I-divergence估计理论的提出又进一步丰富和发展了参数估计理论。最后,结合股票定价模型实例给出了I-divergence估计在股票定价中的应用,分析结果说明了该估计的可行性和优越性。该估计能够帮助投资人有效描述和跟踪市场变化,将逐渐为证券界认可和接受,其意义重大,必将为如何更有效地为企业价值评估提供一些有益的思考。在金融领域初次应用该估计有力说明了统计理论知识的实用性和创新性。带约束参数有偏估计对于线性模型理论的发展和完善具有非常重要的意义。做为线性模型理论有益补充,将广泛应用于农业、管理,经济、军事,工程技术等领域,进一步丰富了统计理论。对社会作出巨大的贡献。
[Abstract]:As a kind of statistical model, linear model has been widely studied because of its wide application, simple form and easy to deal with. In the linear model, because of the basic position of parameter estimation. As the first problem of linear model, it is important to estimate regression parameters, which is the most basic. The most common method is the least square estimation. In recent years, when the data is complex collinearity, the mean square error of the least squares estimation is very large, which leads to the deterioration of the estimation value. The concept of biased estimation is proposed, which reduces the mean square error under certain conditions. The parameter biased estimation is very important for the development and improvement of the linear model theory. In this paper, combined with the domestic and foreign research theory of biased estimation of parameters, the existing problems, mainly in the sense of relative efficiency than the least square estimation of the efficiency of the upper bound. In this paper, a class of constrained biased estimators is proposed, which only improves the least square estimators from the local level. The main work is as follows: First, though the mean square error of estimation is reduced by using biased estimation instead of least square estimation, the accuracy of estimation is reduced. The relative efficiency can better measure the loss of biased estimation instead of least square estimation. Therefore, the generalized ridge estimate and Liu estimate are derived respectively in the sense of relative efficiency evaluation criterion, which is superior to the least square estimate. A new idea is provided for the derivation of the condition that the biased estimation is superior to the least square estimation. Secondly, although the existing biased estimation improves the least square estimation, the mean square error obtained is a function of unknown parameters, and the results obtained by the biased estimation are exploratory. When the variables are constrained, the existing biased estimation is not applicable, and the estimated value is not ideal. In view of these problems, this paper starts from the point of view of function difference. Using a new metric function, the I-divergence estimation is proposed when the variables are constrained. Based on the Kuhn-Tucker theory, an iterative algorithm is designed and the iterative solution is obtained. The convergence of the iterative process is proved if the data involved are non-negative real value constraints and the data are positively correlated. The I-divergence criterion is the only consistent choice. Further, the new estimation is verified by simulation data, and the ridge estimation is calculated. The mean square error of Liu estimator and the new estimator fully prove that the new estimator can reduce the mean square error better than the existing biased estimator in the case of variable nonnegative constraint. The theory of parameter estimation is further enriched and developed. Finally, the application of I-divergence estimation in stock pricing is given with an example of stock pricing model. The analysis results show the feasibility and superiority of this estimate, which can help investors to effectively describe and track market changes, and will gradually be recognized and accepted by the securities industry, which is of great significance. It will provide some useful thoughts on how to evaluate the enterprise value more effectively. The first application of this estimate in the field of finance can explain the practicability and innovation of statistical theory knowledge. The biased estimation of constrained parameters is of great significance to the development and improvement of linear model theory. As a useful supplement of linear model theory, it will be widely used in agriculture, management, economy and military. Engineering technology and other fields have further enriched statistical theory and made great contributions to society.
【学位授予单位】：东北农业大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：O212.1;F830.91

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