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Fully nonparametric methods for regression from functional data have pooraccuracy from a statistical viewpoint, reflecting the fact that theirconvergence rates are slower than nonparametric rates for the estimation ofhigh-dimensional functions. This difficulty has led to an emphasis on theso-called functional linear model, which is much more flexible than commonlinear models in finite dimension, but nevertheless imposes structuralconstraints on the relationship between predictors and responses. Recentadvances have extended the linear approach by using it in conjunction with linkfunctions, and by considering multiple indices, but the flexibility of thistechnique is still limited. For example, the link may be modeled parametricallyor on a grid only, or may be constrained by an assumption such as monotonicity;multiple indices have been modeled by making finite-dimensional assumptions. Inthis paper we introduce a new technique for estimating the link functionnonparametrically, and we suggest an approach to multi-index modeling usingadaptively defined linear projections of functional data. We show that ourmethods enable prediction with polynomial convergence rates. The finite sampleperformance of our methods is studied in simulations, and is illustrated by anapplication to a functional regression problem.

Single and multiple index functional regression models with nonparametric link