2009/10/16 by Christophe Crambes, Crambes, Christophe, Mas, André +1 · 2 citations
Computer Science · Decision Sciences · Mathematics · #FOS: Mathematics #Mathematical Approximation and Integration #Probability and Risk Models #Statistical Methods and Inference #Statistics Theory (math.ST) #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.0910.3070
openalex publication_date 2009/10/16 · openalex created_date 2022/11/04 · openalex updated_date 2026/07/28
We study prediction in the functional linear model with functional outputs :\nY=SX+\ε where the covariates X and Y belong to some functional\nspace and S is a linear operator. We provide the asymptotic mean square\nprediction error with exact constants for our estimator which is based on\nfunctional PCA of the input and has a classical form. As a consequence we\nderive the optimal choice of the dimension kn of the projection space. The\nrates we obtain are optimal in minimax sense and generalize those found when\nthe output is real. Our main results hold with no prior assumptions on the rate\nof decay of the eigenvalues of the input. This allows to consider a wide class\nof parameters and inputs X(\⋅) that may be either very irregular or very\nsmooth. We also prove a central limit theorem for the predictor which improves\nresults by Cardot, Mas and Sarda (2007) in the simpler model with scalar\noutputs. We show that, due to the underlying inverse problem, the bare estimate\ncannot converge in distribution for the norm of the function space\n