Strong optimality of kernel functional regression in Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} norms with partial response variables and applications

This paper proposes kernel-type estimators of a regression function, with possibly unobservable response variables in a functional covariate setting, along with their rates of convergence in general Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} norms. Here, the mechanism that causes the absence of information (in the sense of having unobservable responses) is allowed to depend on both predictors and the response variables; this makes the problem particularly more challenging in those cases where model identifiability is an issue. As an immediate byproduct of these results, we propose asymptotically optimal classification rules for the challenging problem of semi-supervised learning based on the proposed estimators. Our proposed approach involves two steps: in the first step, we construct a family of models (possibly infinite dimensional) indexed by the unknown parameter of the missing probability mechanism. In the second step, a search is carried out to find the empirically optimal member of an appropriate cover (or subclass) of the underlying family in the sense of minimizing a weighted mean squared prediction error. The main focus of the paper is to look into the rates of almost complete convergence of the Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} norms of these estimators. The issue of identifiability is also addressed. As an application of our findings, we consider the classical problem of statistical classification based on the proposed regression estimators when there are a large number of missing labels in the data.