Issue |
ESAIM: PS
Volume 18, 2014
|
|
---|---|---|
Page(s) | 881 - 899 | |
DOI | https://doi.org/10.1051/ps/2014009 | |
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
Grenoble,
France
karim.benhenni@upmf-grenoble.fr
2 Department of Mathematical Sciences,
DePaul University, Chicago
60614, Illinois, USA
ddegrasv@depaul.edu
Received:
20
December
2012
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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