Open Access
June 2020 On the optimal reconstruction of partially observed functional data
Alois Kneip, Dominik Liebl
Ann. Statist. 48(3): 1692-1717 (June 2020). DOI: 10.1214/19-AOS1864


We propose a new reconstruction operator that aims to recover the missing parts of a function given the observed parts. This new operator belongs to a new, very large class of functional operators which includes the classical regression operators as a special case. We show the optimality of our reconstruction operator and demonstrate that the usually considered regression operators generally cannot be optimal reconstruction operators. Our estimation theory allows for autocorrelated functional data and considers the practically relevant situation in which each of the $n$ functions is observed at $m_{i}$, $i=1,\dots ,n$, discretization points. We derive rates of consistency for our nonparametric estimation procedures using a double asymptotic. For data situations, as in our real data application where $m_{i}$ is considerably smaller than $n$, we show that our functional principal components based estimator can provide better rates of convergence than conventional nonparametric smoothing methods.


Download Citation

Alois Kneip. Dominik Liebl. "On the optimal reconstruction of partially observed functional data." Ann. Statist. 48 (3) 1692 - 1717, June 2020.


Received: 1 September 2018; Revised: 1 May 2019; Published: June 2020
First available in Project Euclid: 17 July 2020

zbMATH: 07241608
MathSciNet: MR4124340
Digital Object Identifier: 10.1214/19-AOS1864

Primary: 62G05 , 62G08 , 62H25 , 62M20

Keywords: Functional data analysis , functional principal components , incomplete functions

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 3 • June 2020
Back to Top