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Local polynomial estimation of the mean function and its derivatives based on functional data and regular designs. (English) Zbl 1305.62168

Summary: We study the estimation of the mean function of a continuous-time stochastic process and its derivatives. The covariance function of the process is assumed to be nonparametric and to satisfy mild smoothness conditions. Assuming that \(n\) independent realizations of the process are observed at a sampling design of size \(N\) generated by a positive density, we derive the asymptotic bias and variance of the local polynomial estimator as \(n,N\) increase to infinity. We deduce optimal sampling densities, optimal bandwidths, and propose a new plug-in bandwidth selection method. We establish the asymptotic performance of the plug-in bandwidth estimator and we compare, in a simulation study, its performance for finite sizes \(n,N\) to the cross-validation and the optimal bandwidths. A software implementation of the plug-in method is available in the R environment.

MSC:

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference

Software:

R
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