带随机约束线性模型中参数的有偏估计
发布时间：2018-08-06 12:13
【摘要】：线性模型是重要的统计模型,广泛应用于医学、工业、经济、管理、生物等众多领域.本文主要研究了带随机约束条件的线性模型的参数估计问题.由于约束最小二乘估计在设计阵存在复共线性时不再是“好的”估计,因此学者们提出了许多有偏估计来代替约束最小二乘估计.本文提出了三种新的有偏估计,并讨论了它们的相关性质.首先,在两参数估计的基础上,结合加权混合估计,提出了一种新的加权混合两参数估计.在均方误差矩阵准则下,分别与加权混合两参数估计、加权混合估计、加权混合岭估计和两参数估计进行比较,得到了该估计优于这些估计的充要条件,并通过数值模拟验证了相关的理论结果.其次,利用了几乎无偏的思想,在加权混合两参数估计的基础上,将几乎无偏两参数估计与加权混合估计相结合,提出了一种新的加权混合几乎无偏两参数估计.在二次偏差的准则下,比较了加权混合两参数估计和加权混合几乎无偏两参数估计的偏差,得到了加权混合几乎无偏两参数估计是对加权混合两参数估计的偏差进行矫正的估计.然后在均方误差矩阵准则下,与加权混合估计、加权混合几乎无偏岭估计、几乎无偏两参数估计相比较,得到了加权混合几乎无偏两参数估计优于这些估计的充要条件,并通过数值模拟验证了相关的理论结果.最后,在广义随机约束估计的基础上,通过寻找最优算子,使得其均方误差达到最小,提出了一种两步估计——广义最优随机约束估计,并通过蒙特卡罗模拟验证了新估计的优良性.
[Abstract]:Linear model is an important statistical model, which is widely used in many fields such as medicine, industry, economy, management, biology and so on. In this paper, the problem of parameter estimation for linear models with stochastic constraints is studied. Because constrained least squares estimators are no longer "good" estimators when complex collinearity exists in design matrices, many biased estimators have been proposed to replace constrained least squares estimators. In this paper, three new biased estimators are proposed and their related properties are discussed. Firstly, based on the two-parameter estimation, a new weighted mixed two-parameter estimation is proposed. Under the criterion of mean square error matrix, compared with weighted mixed two-parameter estimation, weighted mixed ridge estimate and two-parameter estimation, the sufficient and necessary conditions for the estimator to be superior to these estimates are obtained. The relevant theoretical results are verified by numerical simulation. Secondly, using the idea of almost unbiased, a new weighted mixed almost unbiased two-parameter estimation is proposed by combining the almost unbiased two-parameter estimation with the weighted mixed two-parameter estimation. Under the criterion of quadratic deviation, the difference between weighted mixed two-parameter estimation and weighted mixed almost unbiased two-parameter estimation is compared. It is obtained that the weighted mixed almost unbiased two-parameter estimation is an estimate that corrects the deviation of the weighted mixed two-parameter estimation. Then under the mean square error matrix criterion, compared with the weighted mixed estimators, the weighted mixed almost unbiased estimators and almost unbiased two-parameter estimators, the sufficient and necessary conditions for the weighted mixed almost unbiased two-parameter estimators to be superior to these estimates are obtained. The relevant theoretical results are verified by numerical simulation. Finally, on the basis of generalized stochastic constraint estimation, the mean square error is minimized by finding the optimal operator, and a two-step estimator-generalized optimal stochastic constraint estimation is proposed. The superiority of the new estimator is verified by Monte Carlo simulation.
【学位授予单位】：华北水利水电大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212.1
本文编号：2167694
[Abstract]:Linear model is an important statistical model, which is widely used in many fields such as medicine, industry, economy, management, biology and so on. In this paper, the problem of parameter estimation for linear models with stochastic constraints is studied. Because constrained least squares estimators are no longer "good" estimators when complex collinearity exists in design matrices, many biased estimators have been proposed to replace constrained least squares estimators. In this paper, three new biased estimators are proposed and their related properties are discussed. Firstly, based on the two-parameter estimation, a new weighted mixed two-parameter estimation is proposed. Under the criterion of mean square error matrix, compared with weighted mixed two-parameter estimation, weighted mixed ridge estimate and two-parameter estimation, the sufficient and necessary conditions for the estimator to be superior to these estimates are obtained. The relevant theoretical results are verified by numerical simulation. Secondly, using the idea of almost unbiased, a new weighted mixed almost unbiased two-parameter estimation is proposed by combining the almost unbiased two-parameter estimation with the weighted mixed two-parameter estimation. Under the criterion of quadratic deviation, the difference between weighted mixed two-parameter estimation and weighted mixed almost unbiased two-parameter estimation is compared. It is obtained that the weighted mixed almost unbiased two-parameter estimation is an estimate that corrects the deviation of the weighted mixed two-parameter estimation. Then under the mean square error matrix criterion, compared with the weighted mixed estimators, the weighted mixed almost unbiased estimators and almost unbiased two-parameter estimators, the sufficient and necessary conditions for the weighted mixed almost unbiased two-parameter estimators to be superior to these estimates are obtained. The relevant theoretical results are verified by numerical simulation. Finally, on the basis of generalized stochastic constraint estimation, the mean square error is minimized by finding the optimal operator, and a two-step estimator-generalized optimal stochastic constraint estimation is proposed. The superiority of the new estimator is verified by Monte Carlo simulation.
【学位授予单位】：华北水利水电大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212.1
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