線性模型的估計(jì)比較和預(yù)測(cè)理論研究
[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.
【學(xué)位授予單位】:湖南大學(xué)
【學(xué)位級(jí)別】:博士
【學(xué)位授予年份】:2015
【分類號(hào)】:O212
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