Title: Estimating the conditional distribution in functional regression problems
Authors: Siegfried Hoermann - Graz University of Technology (Austria) [presenting]
Gregory Rice - University of Waterloo (Canada)
Thomas Kuenzer - Graz University of Technology (Austria)
Abstract: The problem of consistently estimating the conditional distribution of a functional data object given covariates in a general space is considered. Thereby we assume that the response and the covariate are related by a linear regression model. Two natural estimation methods are proposed, based on either bootstrapping the estimated model residuals, or fitting functional parametric models to the model residuals and estimating the conditional distribution via simulation. Whether either of these methods lead to consistent estimators depends on the consistency properties of the regression operator estimator, and the space within which the response is viewed. We show that under general consistency conditions on the regression operator estimator, consistent estimation of the conditional distribution can be achieved, both when the response is an element of a separable Hilbert space, and when it is an element of the Banach space of continuous functions. The latter result implies that we can estimate the conditional probability of certain path properties, which are of interest in applications. The proposed methods are studied in several simulation experiments, and data analyses of electricity price and pollution curves. We also demonstrate how our method can be used for constructing confidence regions and in the context of functional quantile regression.