線性模型的估計(jì)比較和預(yù)測(cè)理論研究
發(fā)布時(shí)間：2018-10-23 11:20
【摘要】：線性模型是現(xiàn)代統(tǒng)計(jì)學(xué)中一類重要的模型,在經(jīng)濟(jì)、金融等領(lǐng)域有著廣泛的應(yīng)用背景.在其建模分析過(guò)程,模型的參數(shù)估計(jì)理論相當(dāng)重要,得到統(tǒng)計(jì)學(xué)家的高度重視.一方面,統(tǒng)計(jì)學(xué)家研究模型參數(shù)估計(jì)理論和方法,并對(duì)各種估計(jì)進(jìn)行比較；另一方面,他們利用參數(shù)估計(jì)結(jié)果研究未來(lái)觀察值的預(yù)測(cè).在此背景下,本文主要基于統(tǒng)計(jì)決策理論對(duì)線性模型中參數(shù)估計(jì)的比較和有限總體回歸系數(shù)的預(yù)測(cè)方法進(jìn)行研究.對(duì)于誤差服從多元t分布的線性模型,我們?cè)谄胶鈸p失下對(duì)回歸系數(shù)的Stein-rule (SR)估計(jì),Positive-part Stein-rule (PSR)估計(jì),可行最小均方誤差估計(jì)和改進(jìn)可行最小均方誤差估計(jì)的優(yōu)良性進(jìn)行研究.首先基于預(yù)檢驗(yàn)估計(jì)思想給出了這四個(gè)估計(jì)的統(tǒng)一表達(dá)式,并在此基礎(chǔ)上得到了各個(gè)估計(jì)的顯式風(fēng)險(xiǎn).其次,基于風(fēng)險(xiǎn)顯式表達(dá)式理論上對(duì)PSR估計(jì)和SR估計(jì)的優(yōu)良性進(jìn)行分析.最后考慮到風(fēng)險(xiǎn)顯式表達(dá)式的復(fù)雜性,我們采用數(shù)值分析的方法進(jìn)一步對(duì)估計(jì)的優(yōu)良性進(jìn)行研究.對(duì)于具有橢球等高分布誤差的誤定線性模型,我們?cè)谡`差平方損失下對(duì)誤差方差的最小二乘估計(jì),約束最小二乘估計(jì),預(yù)檢驗(yàn)估計(jì)和Stein型估計(jì)進(jìn)行比較.首先,基于橢球等高分布的性質(zhì)得到了各個(gè)估計(jì)風(fēng)險(xiǎn)的顯式表達(dá)式.其次,基于估計(jì)風(fēng)險(xiǎn)的顯式表達(dá)式在理論上分析了影響預(yù)檢驗(yàn)估計(jì)風(fēng)險(xiǎn)大小的因素,并進(jìn)一步考察了預(yù)檢驗(yàn)估計(jì)風(fēng)險(xiǎn)與最小二乘估計(jì)風(fēng)險(xiǎn)、約束最小二乘估計(jì)風(fēng)險(xiǎn)的關(guān)系,同時(shí)研究了預(yù)檢驗(yàn)估計(jì)的最優(yōu)臨界值.最后,考慮到估計(jì)風(fēng)險(xiǎn)依賴于未知參數(shù),且在結(jié)構(gòu)上非常復(fù)雜,為此在多元t分布特例下,我們采用數(shù)值分析和自助法分別對(duì)這四個(gè)估計(jì)進(jìn)一步進(jìn)行比較.對(duì)于有限總體回歸系數(shù)的預(yù)測(cè)問(wèn)題,在超總體觀點(diǎn)下,我們?cè)谄胶鈸p失下分別對(duì)誤差不具有正態(tài)假定和具有正態(tài)假定總體中有限總體回歸系數(shù)的可容許預(yù)測(cè)進(jìn)行研究.首先,對(duì)于不具有正態(tài)假定的總體,我們得到了齊次線性預(yù)測(cè)在齊次線性預(yù)測(cè)類中可容許的充分必要條件,并給出了有限總體回歸系數(shù)的最佳線性無(wú)偏預(yù)測(cè),同時(shí)分析了最佳線性無(wú)偏預(yù)測(cè)在齊次線性預(yù)測(cè)類中的可容許性.其次,我們?cè)谡龖B(tài)總體下討論了齊次線性可容許預(yù)測(cè)在一切預(yù)測(cè)類中是否可容許的問(wèn)題,得到了齊次線性預(yù)測(cè)在一切預(yù)測(cè)類中可容許性的充分條件,并證明了在適當(dāng)?shù)臈l件下,該充分條件也是齊次線性預(yù)測(cè)在一切預(yù)測(cè)類中可容許的必要條件.最后,針對(duì)具有正態(tài)假定的總體,我們給出了有限總體回歸系數(shù)的最佳無(wú)偏預(yù)測(cè),并分析了它在一切預(yù)測(cè)類中的可容許性.最后,我們?cè)诟倪M(jìn)平衡損失函數(shù)的基礎(chǔ)上進(jìn)一步對(duì)誤差不具有正態(tài)假定和具有正態(tài)假定總體中有限總體回歸系數(shù)的Minimax預(yù)測(cè)進(jìn)行研究.一方面,我們?cè)诜钦龖B(tài)總體下得到了齊次線性預(yù)測(cè)類中有限總體回歸系數(shù)的線性Minimax預(yù)測(cè),并對(duì)該預(yù)測(cè)在齊次線性預(yù)測(cè)類中的可容許性進(jìn)行分析,同時(shí)將其和Bolfarine提出的最佳線性無(wú)偏預(yù)測(cè)進(jìn)行比較.另一方面,我們?cè)谡龖B(tài)總體下探討了有限總體回歸系數(shù)在一切預(yù)測(cè)類中的線性Minimax預(yù)測(cè),并對(duì)其在一切預(yù)測(cè)類中的可容許性進(jìn)行了分析,同時(shí)將其和簡(jiǎn)單投影預(yù)測(cè)進(jìn)行比較.
[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
本文編號(hào)：2289071
[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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