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

Abstract

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.