发表机构
University of Mississippi Medical Center; Millsaps College（密西西比大学医学中心; 米尔斯普斯学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一个不依赖内积结构的统一框架，用于Banach空间值预测变量的非参数回归，处理测量误差，建立大样本理论，并扩展至误差变量和部分污染模型。
AI 中文摘要
我们考虑具有Banach空间值预测变量的非参数回归问题，其中协变量取值于一般可分Banach空间。我们的主要目标是开发一个统一的框架来估计回归算子，而不依赖于内积结构，从而将经典和函数型非参数回归扩展到存在测量误差的Euclid空间和Hilbert空间之外。我们建立了所提出估计量的基本性质，并在基于Banach值预测变量的几何形状和局部集中性的条件下发展了其大样本理论。特别地，我们研究了逐点估计和推断，并讨论了逐点置信区间和一致置信带的构造。我们提出的框架进一步扩展到误差变量设置，其中预测变量观测时带有污染。我们还研究了由逆加权函数构造引起的相关反问题，并考虑了部分污染模型。这些发展为具有复杂无限维预测变量的非参数回归和推断提供了一个通用框架，并识别了当通常的Hilbert空间结构不可用时出现的若干统计和数学挑战。
英文摘要
We consider nonparametric regression with a Banach-space-valued predictor where the covariate takes values in a general separable Banach space. Our primary objective is to develop a unified framework for estimating the regression operator without relying on an inner-product structure, thereby extending classical and functional nonparametric regression beyond Euclidean and Hilbert spaces in the presence of measurement error. We establish the fundamental properties of the proposed estimator and develop its large-sample theory under conditions formulated in terms of the geometry and local concentration of the Banach-valued predictor. In particular, we study pointwise estimation and inference and discuss the construction of pointwise confidence intervals and uniform confidence bands. Our proposed framework is further extended to accommodate error-in-variables settings, where the predictor is observed with contamination. We also investigate related inverse problems arising from the construction of inverse weighting functions and consider partial contamination models. These developments provide a general framework for nonparametric regression and inference with complex infinite-dimensional predictors and identify several statistical and mathematical challenges that arise when the usual Hilbert-space structure is unavailable.
