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arXiv · 2610.04677

Estimation and inference in penalized function-on-function regression with dependent residual curves

Fabian Scheipl
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Abstract

Penalized function-on-function regression, as implemented in the R package refund, treats all grid values of all curves as independent. Residual curves are smooth, so values within a curve are dependent. We study what this does to point estimates and pointwise confidence intervals, in a simulation with Gaussian, count and binary responses and in a plasmode study built on nine real datasets. Under within-curve dependence, model-based 95% intervals cover 0.63--0.82. A curve-clustered sandwich covariance with a leverage adjustment for the penalized hat matrix (CL2), computed after the usual REML fit, brings coverage close to nominal in the simulation; in the plasmode study it is a few points short on eight datasets and 0.84--0.91 for the coefficient surface on the ninth. Satterthwaite degrees of freedom recover most of the remaining shortfall except on that dataset, also with only 20 curves, at the cost of wider intervals. REML, however, undersmooths the coefficient surface: its error is two to four times that of a fit whose smoothing parameters are chosen by curve-blocked neighbourhood cross-validation (NCV), and its CL2 intervals for the surface are wide. Intervals centred at the smoother NCV estimate are narrow but undercover, the familiar tension between estimation and pointwise inference in nonparametric regression. We recommend the NCV estimate to describe the shape of the surface and REML + CL2 with Satterthwaite degrees of freedom for inference. For a surface that is zero everywhere, under dependent errors, these intervals exclude zero at about 5% of the grid points, and the NCV estimate is five to seven times closer to zero than REML's. For fitted means, intercepts and scalar effects, REML + CL2 intervals are close to nominal, except for binary intercepts and scalar effects and for misregistered counts. We state which defaults of refund the evidence supports.

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Fabian Scheipl. 2026-10-03. Estimation and inference in penalized function-on-function regression with dependent residual curves. https://arxiv.org/abs/2610.04677

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