Hilbert空间中一类可数维线性回归模型的参数估计

A Parameter Estimation for an Infinite Dimensional Linear Regression Model in Hilbert Space

摘要: 现有回归模型中的解释变量往往是有限多个，然而被解释变量实际上常常受到无穷多种因素的影响，这使得有限维线性回归模型很有可能遗漏某些重要的解释变量，从而导致模型模拟现实的效用大为减弱。鉴于此，本文提出了一类可数维线性回归模型的参数估计问题，并利用实Hilbert空间中的正交分解定理，以及泛函分析中的压缩映射原理和相关定理，在一定的条件下，通过Picard迭代程序，寻求可数维线性回归模型参数估计的唯一逼近解。

Abstract: The existing linear regression models usually include finite explanatory variables, but in fact, the dependent variables are affected by infinite factors so that some important explanatory variables are probably omitted, which would lead to weaker efficacy of the models. In this article, we pro-posed a parameter estimation for a countable dimensional linear regression model under certain conditions by using the orthogonal decomposition theorem in real Hilbert space. In other words, the Picard iteration program is used to find the unique approximate solution for parameter estimation for a countable dimensional linear regression model under certain conditions by using the orthogonal decomposition theorem in real Hilbert space and the compression mapping theorem in functional analysis and some related lemmas.