# Estimating the conditional distribution in functional regression problems

• Thomas Kuenzer (Speaker)

Activity: Talk or presentationTalk at conference or symposiumScience to science

## Description

We consider the problem of consistently estimating the conditional distribution $P(Y \in A |X)$ of a functional data object $Y=(Y(t): t\in[0,1])$ given covariates $X$ in a general space, assuming that $Y$ and $X$ are related by a functional linear regression model. Two natural estimation methods for this problem are proposed, based on either bootstrapping the estimated model residuals, or fitting functional parametric models to the model residuals and estimating $P(Y \in A |X)$ via simulation. We show that under general consistency conditions on the regression operator estimator, which hold for certain functional principal component based estimators, consistent estimation of the conditional distribution can be achieved, both when $Y$ is an element of a separable Hilbert space, and when $Y$ is an element of the Banach space of continuous functions on the unit interval. The latter results imply that sets $A$ that specify path properties of $Y$ that are of interest in applications can be considered, such as the maximum of the curve.
Our methods have numerous applications in the context of constructing prediction sets, quantile regression and VaR estimation. Compared to direct modelling these curve properties using scalar-on-function regression, modelling the whole response distribution and extracting the curve properties in a second step allows us to harness the full information contained in the functional data to fit the regression model and achieve better results. We study the proposed methods in several simulation experiments and real data analysis of electricity price curves and show that they outperform both the non-parametric kernel estimator and functional binary regression.
Period 28 Sep 2021 15th German Probability and Statistics Days: GPSD 2021 Conference Virtuell, Germany