Generalized linear model with functional predictors and their derivatives

The conditional expectation E(Y|X)E(Y|X) of a generalized functional linear model with scalar response YY is given by g{〈X,ϕ〉L2}g{〈X,ϕ〉L2} where XX and ϕϕ are functions defined in L2:=L2[0,1]L2:=L2[0,1]. Let us consider that XX belongs to the Sobolev space W:=W2,1[0,1]W:=W2,1[0,1] and denote X′X′ its derivative. In this paper we focus on an extension of the previous model where E(Y|X)E(Y|X) is given by g{〈X,β〉W+〈X′,γ〉L2}g{〈X,β〉W+〈X′,γ〉L2}. With a similar approach to Cardot and Sarda (2005) or Stone (1986) for generalized additive models, we propose estimators for the unknown parameters ββ, γγ and obtain their rate of convergence. We compare numerically the prediction performance of this new model with alternative models proposed in the literature.