【摘要】：半?yún)?shù)回歸模型在統(tǒng)計學理論中扮演著極其重要的角色,該模型為了充分利用數(shù)據(jù)中存在的相關(guān)信息,不僅引入?yún)?shù)分量還引入了非參數(shù)函數(shù),極大的優(yōu)化了模型的擬合效果.作為半?yún)?shù)模型的一種,部分線性模型結(jié)合了線性模型和非參數(shù)模型的特點,既具有線性模型容易解釋的特點,又具有非參數(shù)模型的穩(wěn)健性和靈活性,更重要的是它可以避免“維數(shù)災(zāi)禍”.這些特點使得該模型更加廣泛的應(yīng)用于實際問題.但是在解決實際問題時,我們通常不能直接獲得變量的真實值,得到的數(shù)據(jù)是變量的觀測值,這些觀測值中往往存在各種誤差,例如因為抽樣引起的誤差、因為實驗過程中使用的工具或者是操作不規(guī)范引起的誤差等等.我們稱觀測到的數(shù)據(jù)為真實協(xié)變量的代理變量,稱協(xié)變量含有測量誤差的問題為“測量誤差問題”,稱用于擬合帶有測量誤差數(shù)據(jù)的模型為“測量誤差模型”在本文中,我們主要研究含有可加和非可加測量誤差的半?yún)?shù)部分線性模型,其中參數(shù)部分協(xié)變量的觀測值含有非可加測量誤差,其真值需要借助輔助變量估計得到,非參數(shù)部分協(xié)變量的觀測值含有可加測量誤差.我們首先校正參數(shù)部分協(xié)變量,然后利用半?yún)?shù)最小二乘方法估計參數(shù)分量和非參數(shù)函數(shù).在適當?shù)募僭O(shè)條件下,證明了參數(shù)分量和非參數(shù)函數(shù)的漸近正態(tài)性、誤差方差的漸近正態(tài)性,并進行了參數(shù)分量的廣義似然比檢驗.最后,通過數(shù)值模擬與實際例子驗證了文中所提出的估計方法以及檢驗方法.
[Abstract]:The semi-parametric regression model plays an important role in the statistical theory. In order to make full use of the relevant information in the data, the semi-parametric regression model not only introduces the parameter component but also the non-parametric function, which greatly optimizes the fitting effect of the model. As a semi-parametric model, the partial linear model combines the characteristics of the linear model and the non-parametric model, which has the characteristics of easy interpretation of the linear model, robustness and flexibility of the non-parametric model. More importantly, it can avoid the "dimensionality disaster". These characteristics make the model more widely used in practical problems. But when we solve practical problems, we usually can't get the real value of variables directly. The data we get are the observations of variables. There are often various errors in these observations, for example, because of the errors caused by sampling. Because of the tools used in the experiment or the error caused by the nonstandard operation, etc. We call the observed data the proxy variable of the real covariable, the problem that the covariable contains the measurement error as the "measurement error problem", and the model used to fit the data with the measurement error is called the "measurement error model" in this paper. We mainly study the semi-parametric partial linear model with additive and non-additive measurement errors, in which the observed values of some parameter covariables contain non-additive measurement errors, and the true values need to be obtained by the estimation of auxiliary variables. The observed values of nonparametric partial covariables contain additive measurement errors. We first calibrate some parameter covariables and then estimate parametric components and nonparametric functions by semi-parametric least square method. Under appropriate assumptions, the asymptotic normality of parametric components and nonparametric functions and the asymptotic normality of error variance are proved, and the generalized likelihood ratio test of parametric components is given. Finally, the proposed estimation method and the test method are verified by numerical simulation and practical examples.
【學位授予單位】：南京理工大學
【學位級別】：碩士
【學位授予年份】：2016
【分類號】：O212.1

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