本文選題：半?yún)?shù)回歸模型 + 非線性函數(shù)��； 參考：《西北工業(yè)大學(xué)》2015年博士論文

【摘要】：含非線性函數(shù)的半?yún)?shù)回歸模型具有參數(shù)模型的可解釋性和非參數(shù)模型的靈活性,并且克服了線性函數(shù)在表述客觀模型方面的局限性.在應(yīng)用中經(jīng)常遇到缺失數(shù)據(jù)和測量誤差數(shù)據(jù)等復(fù)雜數(shù)據(jù).經(jīng)驗(yàn)似然方法作為一種重要的非參數(shù)統(tǒng)計(jì)方法,廣泛應(yīng)用于興趣參數(shù)的置信域構(gòu)造.因此,在復(fù)雜數(shù)據(jù)下研究含非線性函數(shù)的半?yún)?shù)回歸模型的經(jīng)驗(yàn)似然推斷具有一定的理論意義和實(shí)用價(jià)值.本論文研究了部分非線性模型和變系數(shù)部分非線性模型的經(jīng)驗(yàn)似然推斷,還研究了時(shí)空變系數(shù)模型的非平穩(wěn)性檢驗(yàn)問題.本文的主要研究成果如下:(1)首次給出了部分非線性模型的經(jīng)驗(yàn)似然推斷.構(gòu)造了模型中未知參數(shù)和未知函數(shù)的對數(shù)經(jīng)驗(yàn)似然比函數(shù),證明了其漸近χ2分布,給出了非線性函數(shù)中未知參數(shù)的置信域和未知系數(shù)函數(shù)的置信帶.同時(shí),得到了未知參數(shù)和未知函數(shù)的極大經(jīng)驗(yàn)似然估計(jì),證明了估計(jì)量的漸近正態(tài)性.模擬結(jié)果和實(shí)例分析表明在構(gòu)造未知參數(shù)的置信區(qū)間和未知函數(shù)的置信帶方面,經(jīng)驗(yàn)似然方法優(yōu)于近似正態(tài)方法.(2)探索了復(fù)雜數(shù)據(jù)下部分非線性模型中未知參數(shù)的經(jīng)驗(yàn)似然置信域估計(jì)問題.對于響應(yīng)變量隨機(jī)缺失的部分非線性模型,為了避免已有文獻(xiàn)中權(quán)重因子和調(diào)整因子的估計(jì),提出了一種局部糾偏的線性插補(bǔ)技術(shù),提高了估計(jì)的精度.對于變量帶測量誤差的部分非線性模型,借助替代數(shù)據(jù)和核實(shí)樣本,給出了未知參數(shù)的兩種估計(jì).構(gòu)造了未知參數(shù)的對數(shù)似然比函數(shù),證明了其漸近于加權(quán)的χ2分布之和.模擬結(jié)果表明在構(gòu)造未知參數(shù)的置信區(qū)間方面,經(jīng)驗(yàn)似然方法優(yōu)于近似正態(tài)方法.(3)首次給出了變系數(shù)部分非線性模型的經(jīng)驗(yàn)似然推斷.構(gòu)造了未知參數(shù)的對數(shù)經(jīng)驗(yàn)似然比函數(shù),證明了其漸近χ2分布,得到了未知參數(shù)的極大經(jīng)驗(yàn)似然估計(jì),并證明了其漸近正態(tài)性.對于非參數(shù)部分帶測量誤差的變系數(shù)部分非線性模型,提出了未知參數(shù)的局部糾偏的剖面最小二乘估計(jì),證明了其漸近正態(tài)性,構(gòu)造了未知參數(shù)的對數(shù)經(jīng)驗(yàn)似然比函數(shù),并證明了其漸近χ2分布.模擬結(jié)果驗(yàn)證了方法的有效性.(4)檢驗(yàn)了時(shí)空變系數(shù)回歸模型的非平穩(wěn)性.構(gòu)造了廣義似然比統(tǒng)計(jì)量,對整個(gè)回歸關(guān)系進(jìn)行關(guān)于時(shí)間和空間的非平穩(wěn)性檢驗(yàn),利用Bootstrap方法計(jì)算檢驗(yàn)的p值;構(gòu)造了合適的統(tǒng)計(jì)量對各個(gè)回歸系數(shù)進(jìn)行關(guān)于時(shí)間和空間的非平穩(wěn)性檢驗(yàn),利用三階矩χ2逼近方法計(jì)算檢驗(yàn)的p值.模擬算例和實(shí)際例子表明檢驗(yàn)方法的有效性.
[Abstract]:The semi-parametric regression model with nonlinear function has the interpretability of parametric model and the flexibility of non-parametric model, and overcomes the limitation of linear function in describing objective model.Complex data such as missing data and measuring error data are often encountered in applications.As an important nonparametric statistical method, empirical likelihood method is widely used in constructing confidence regions of parameters of interest.Therefore, it is of theoretical significance and practical value to study the empirical likelihood inference of semi-parametric regression models with nonlinear functions under complex data.In this paper, empirical likelihood inference for partial nonlinear models and partial nonlinear models with variable coefficients is studied, and the nonstationarity test of spatio-temporal models with variable coefficients is also studied.The main results of this paper are as follows: 1) for the first time, the empirical likelihood inference of partial nonlinear model is given.The logarithmic empirical likelihood ratio function of unknown parameter and unknown function in the model is constructed, and its asymptotic 蠂 2 distribution is proved. The confidence region of unknown parameter and the confidence band of unknown coefficient function in nonlinear function are given.At the same time, the maximum empirical likelihood estimators of unknown parameters and unknown functions are obtained, and the asymptotic normality of the estimators is proved.The results of simulation and the analysis of examples show that in the construction of confidence intervals of unknown parameters and confidence bands of unknown functions,The empirical likelihood method is superior to the approximate normal method. 2) the empirical likelihood confidence region estimation problem of unknown parameters in partial nonlinear models with complex data is explored.In order to avoid the estimation of weight factor and adjustment factor in some nonlinear models with random missing response variables, a local correction linear interpolation technique is proposed to improve the accuracy of the estimation.For the partial nonlinear model with variable measurement error, two kinds of estimations of unknown parameters are given by means of substitute data and verified samples.The logarithmic likelihood ratio function of unknown parameters is constructed, and it is proved that it is asymptotically equal to the sum of weighted 蠂 ~ 2 distributions.The simulation results show that the empirical likelihood method is superior to the approximate normal method in constructing confidence intervals of unknown parameters.The logarithmic empirical likelihood ratio function of unknown parameters is constructed, its asymptotic 蠂 ~ 2 distribution is proved, the maximum empirical likelihood estimation of unknown parameters is obtained, and its asymptotic normality is proved.For the nonparametric partial nonlinear model with variable coefficients with measurement errors, the least square estimation of local correction of unknown parameters is presented, its asymptotic normality is proved, and the logarithmic empirical likelihood ratio function of unknown parameters is constructed.The asymptotic 蠂 2 distribution is proved.The simulation results verify the validity of the method and test the nonstationarity of the spatiotemporal variable coefficient regression model.The generalized likelihood ratio statistic is constructed and the non-stationary test of time and space is carried out for the whole regression relationship. The p value of the test is calculated by using Bootstrap method.The nonstationary test of time and space for each regression coefficient is constructed, and the p value of the test is calculated by using the third-order moment 蠂 ~ 2 approximation method.Simulation examples and practical examples show the effectiveness of the test method.
【學(xué)位授予單位】：西北工業(yè)大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2015
【分類號】：O212.1

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本文編號：1762757