Simultaneous confidence bands for nonparametric regression with functional data

We consider nonparametric regression in the context of functional data, that is, when a random sample of functions is observed on a fine grid. We obtain a functional asymptotic normality result allowing to build simultaneous confidence bands (SCB) for various estimation and inference tasks. Two applications to a SCB procedure for the regression function and to a goodness-of-fit test for curvilinear regression models are proposed. The first one has improved accuracy upon the other available methods while the second can detect local departures from a parametric shape, as opposed to the usual goodness-of-fit tests which only track global departures. A numerical study of the SCB procedures and an illustration with a speech data set are provided.

💡 Research Summary

The paper addresses the problem of constructing simultaneous confidence bands (SCBs) for non‑parametric regression when the data consist of a random sample of functions observed on a dense grid—a setting commonly referred to as functional data. Traditional non‑parametric regression theory assumes independent scalar observations; however, in functional data each experimental unit is a curve, and observations across the same curve are strongly dependent. Ignoring this dependence leads to overly conservative or inaccurate inference.

To overcome this, the authors model each observed function (X_i(t)) as a random element in the Hilbert space (L^2