A0799
Title: Minimax estimation for FPCA on discretized data
Authors: Nassim Bourarach - Université Paris Dauphine - PSL (France) [presenting]
Vincent Rivoirard - Paris Dauphine University (France)
Angelina Roche - Universite Paris Dauphine (France)
Franck Picard - CNRS Lyon (France)
Abstract: The purpose is to consider $p$ noisy evaluations of $n$ realizations of random functions on a common design $(t_j)^p_{j=1}\in [0,1]$,$Y_i\left(t_j\right):=X_i\left(t_j\right)+\varepsilon_{i, j}$ for $(i,j)\in[\![1,n]\!]\times[\![1,p]\!],$ where $\varepsilon_{i,j}$ are i.i.d as $\mathcal{N}\left(0,\sigma^2\right)$ with $\sigma^2>0$. The $\varepsilon_{i,j}$'s are independent of the random functions $X_i$ which are also i.i.d. and defined on $[0,1]$. The interest is in the estimation of $(\psi_\ell^*,\lambda_{\ell}^{*})$, respectively the $\ell$-th eigenfunction and eigenvalue of the covariance integral operator associated to $X$. The first contributions are non-asymptotic minimax lower-bounds for the estimation of these eigenelements when the covariance kernel is $m$-Holder regular (for all $m\in\mathbb{R}_+^*$) and when the spectrum of the covariance operator obeys some constraints. The class of processes used for the minimax study allows us to analyze the impact of the spectrum of the covariance operator on the estimation rates and obtain inconsistent results if the constraints are not satisfied. Then, simple estimators of the eigenelements are presented based on a projection onto a wavelet basis. The obtained estimators are minimax optimal under additional assumptions and attain rates (in $n$ and $p$) of the form $n^{-1} + p^{-2m}$. Surprisingly enough, even if the problem is non-parametric in nature, there is actually no need for data smoothing.
