Nonparametric Regression with Measurement Error in Banach Spaces

RESUMO

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.