The bootstrap in kernel regression for stationary ergodic data when both response and predictor are functions
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We consider the double functional nonparametric regression model Y=r(X)+ϵ, where the response variable Y is Hilbert space-valued and the covariate X takes values in a pseudometric space. The data satisfy an ergodicity criterion which dates back to Laib and Louani (2010) and are arranged in a triangular array. So our model also applies to samples obtained from spatial processes, e.g., stationary random fields indexed by the regular lattice Z^N for some N∈N_+. We consider a kernel estimator of the Nadaraya--Watson type for the regression operator r and study its limiting law which is a Gaussian operator on the Hilbert space. Moreover, we investigate both a naive and a wild bootstrap procedure in the double functional setting and demonstrate their asymptotic validity. This is quite useful as building confidence sets based on an asymptotic Gaussian distribution is often difficult.
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