Volume 18, 2014
|Page(s)||881 - 899|
|Published online||29 October 2014|
Local polynomial estimation of the mean function and its derivatives based on functional data and regular designs
1 Laboratoire LJK UMR CNRS 5224,
Université de Grenoble, 38040
2 Department of Mathematical Sciences, DePaul University, Chicago 60614, Illinois, USA
Revised: 31 March 2014
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.
Mathematics Subject Classification: 62G08 / 62G20
Key words: Local polynomial smoothing / derivative estimation / functional data / sampling density / plug-in bandwidth
© EDP Sciences, SMAI 2014
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