半参数回归分析的随机函数法及其高维情形

A random-function approach based on Gaussian stochastic process is proposed for inference in a semiparametric regression model in which the nuisance parameter is an unknown fixed function. By this approach, the unknown fixed function is modelled as a Gaussian process on its domain, that is, a random function, so that the semiparametric regression problem becomes a parametric regression problem whose estimation method is also given.The consistency and asymptotic normality of the estimator of the regression parameter are built under mild conditions;some practical data analysis and simulations are done to confirm this random-function approach's advantages and applicability. This project then applies the random-function approach to the high dimensional cases of the simeparametric regression problem, finds the corresponding estimation method, builds its corresponding theory and does some corresponding practical data analysis and simulations. This project also explores the efficiency problems of the estimators, given by the random-function approach, of regression parameters in semiparametric regression model and its high dimensional cases. Finally this project applies the random-function approach to the other commonly encountered semiparametric problems and achieves the corresponding results.

基于高斯随机过程的随机函数法被提出用来对半参数回归（Semiparametric regression）模型作统计推断，此模型中的多余参数(Nuisance parameter)为未知非随机函数。按此方法，该未知非随机函数被模拟为其定义域上的高斯过程，也就是一随机函数，从而该半参数回归问题变成参数回归问题，其估计方法也被给出。回归参数估计量的相容性（Consistency）和渐近正态性（Asymptotic normality）在温和条件下被建立；一些实际数据分析和模拟被做来确证此方法的优势和实用性。本项目然后应用随机函数法于半参数回归问题的高维情形，找到相应的估计方法，建立其相应的理论，并做一些实际数据分析和模拟。本项目也将探索随机函数法应用于半参数回归问题及其高维情形所得到的回归参数估计量的有效形（Efficiency）问题；最后本项目应用随机函数法于其它常见半参数问题并得到相应结果。

基于高斯随机过程的随机函数法被提出用来对半参数回归（Semiparametric regression）模型作统计推断，此模型中的多余参数(Nuisance parameter)为未知非随机函数。按此方法，该未知非随机函数被模拟为其定义域上的高斯过程，也就是一随机函数，从而该半参数回归问题变成参数回归问题，其估计方法也被给出。回归参数估计量的相容性（Cons istency）和渐近正态性（Asymptotic normality）在温和条件下被建立；一些实际数据分析和模拟被做来确证此方法的优势和实用性。本项目也可应用随机函数法于半参数回归问题的高维情形，找到相应的估计方法，建立其相应的理论，并做一些实际数据分析和模拟。本项目也可偿试探索随机函数法应用于半参数回归问题及其高维情形所得到的回归参数估计量的有效形（Efficiency）问题。此项目也可探索半参数随机函数法在大数据分布式计算环境下的理论优势与计算便利。