Goodness-of-fit Tests for Functional Linear Models Based on Integrated Projections

Abstract

Functional linear models are one of the most fundamental tools to assess the relation between two random variables of a functional or scalar nature. This contribution proposes a goodness-of-fit test for the functional linear model with functional response that neatly adapts to func-tional/scalar responses/predictors. In particular, the new goodness-of-fit test extends a previous proposal for scalar response. The test statistic is based on a convenient regularized estimator, is easy to compute, and is calibrated through an efficient bootstrap resampling. A graphical diagnostic tool, useful to visualize the deviations from the model, is introduced and illustrated with a novel data application. The R package goffda implements the proposed methods and allows for the reproducibility of the data application. 1 Functional linear models 1.1 Formulation Given two separable Hilbert spaces H 1 and H 2 , we consider the regression setting with centered H 2-valued response Y and centered H 1-valued predictor X : Y = m(X) + E, (1) where m : X ∈ H 1 → E [Y|X = X ] ∈ H 2 is the regression operator and the H 2-valued error E is such that E [E|X ] = 0. When H 1 = L 2 ([a, b]) and H 2 = L 2 ([c, d]), the Functional Linear Model with Functional Response (FLMFR; see, e.g., Ramsay and Silverman (2005, Chapter 16)) is the most well-known parametric instance of (1). If the regression operator is assumed to be Hilbert-Schmidt, m is parametrizable as m β (X) = b a β(s, ·)X (s) ds =: β, X XX, (2) for β ∈ H 1 ⊗ H 2 = L 2 ([a, b] × [c, d]) a square-integrable kernel. The present work considers this framework and is concerned with the goodness-of-fit of the family of H 2-valued and H 1-conditioned linear models L := {{{β, ··· : β ∈ H 1 ⊗ H 2 }. (3)