【作者】 胡玉琴；

【导师】 薛留根；

【作者基本信息】 北京工业大学 ， 概率论与数理统计， 2003， 硕士

【摘要】 半参数回归模型是二十世纪八十年代发展起来的一种重要的统计模型。这种模型既有参数分量，又含有非参数分量，兼顾了参数回归模型和非参数回归模型的优点，较单纯的参数回归模型或非参数回归模型有更大的适应性，并具有更强的解释能力。 本论文主要研究一类重要的半参数回归模型 y_i=x′_iβ+g(x_i)+e_i，i=1，2，…，n，其中x_i=(x_i1，x_i2，…，x_id)＇，(d≥1)，g(·)是未知函数，β是未知待估参数，e_i是随机误差，且Ee_i=0，Ee_i~2=σ~2＜∞。 在许多实际问题中，我们遇到的多为x_i是非随机设计点列，即固定设计点列的情况。因而本论文主要是研究在x_i是固定设计情况下，此类半参数回归模型的大样本性质相合性。 与通常采用的两阶段估计方法即非参数权函数法结合最小二乘法不同，考虑到此模型本身的特性，本文采用最小二乘法结合一般非参数权函数估计方法，定义了未知待估参数β和未知函数g(·)及误差方差σ~2的估计量(?)_n，(?)_n(·)和(?)_n~2。其估计方法的基本思路是先基于线性模型y_i=x′_iβ+e_i，定义未知待估参数β的估计即此线性模型的最小二乘估计(?)_n；然后将所得估计(?)_n代入原半参数回归模型中，用一般的非参数权函数方法定义未知函数g(·)的估计(?)_n(·)；最后定义误差方差σ~2的估计(?)_n~2。 为讨论简单计，本文主要讨论模型在d=1时即一维的情况下的相合性。 北京工业大学理学硕士学位论文同时指出多维情况可由一维情况平行的得到推广，从而得到该类模型在固定设计下的相合性这一重要的大样本性质． 本论文在X；是固定设计的情况下，假定未知函数9（·）连续，对非参数权函数的条件更为一般和基本，并对随机误差e；的矩的要求有限，讨论并证明了在这些条件下，P；g（·）的估计量札lin（·）及误差方差a’的估计量枯相合性和叭三2）阶平均相合性．更多还原

【Abstract】 A semiparametric regression model has been an important statistics model since 1980s. This kind of model includes not only a parametric component but also a nonparametric component. So it has the advantages of the parametric regression model and the nonparametric regression model. It has the more implements and stronger explanations than the pure parametric or nonparametric regression model .This paper considers an important semiparametric regression modelwhere is an unknown function, is an unkown parameter to be estimated. ei are iid. random errors with Bei = 0 and Eei2 = 2 < .In many practical problems, Xi usually are some nonrandom design points, that is , fixed design points. So, the purpose of this paper concentrates on the semiparametric regression model’s large sample property-consitency when xi are the fixed design points.Unlike the normal two stages estimate method (the usual nonparametric weighted method combined with the least square estimate) , considering the characteristics of this model, this paper uses the least square estimate combining with the usual nonparametric weighted method and defines the estimators and n2 for the unknown parameter ,the unkown fuction g( ) and the unknown variance of errors 2 .The basic idea of the estimate method is ,firstly ,based on the linear model yi = x’i + ei,defining the least square estimator n of the linear model for the unkown parametric ;secondly, using the estimator n we ’ve got to substitute for in the original semiparametric regression model yi = x’i + g(xi) + ei and using the usual nonparametric weighted function method to define the estimator gn(-) for the unknown function g( ); finally ,defining the estimator 2 for the unknown variance of errors 2.For simplicity ,this paper focuses on the consistency of the model in one dimension case,that is d = 1. This paper also points out the consistency that can be generalized more than one dimension. So ,we achieve the large sample property -consistency of this class of model on the fixed design.In this paper,for fixed design points xi; under the assumption that the unknown function g is continuous function and the moment of random error exists and is finity,we discuss and show that the estimators n,gn and n2 for ,g and 2 have strong consistency, p th-mean consistency for more general nonparametric weighted fuction.更多还原