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Estimation of a functional single index model

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Single index models have been mostly studied as an alternative dimension reduction technique for nonparametric regression with multivariate covariates. The index parameter appearing in the model summarizes the effect of the covariates in a finite dimensional vector. We consider an extension to a functional single index parameter which is of infinite dimensional, as a summary of the effect of a functional explanatory variable on a scalar response variable and propose a new estimator based on the idea of functional derivative estimation.