Sharp template estimation in a shifted curves model

TL;DR

This paper introduces an adaptive method for estimating a template in a shifted curves model by transforming the problem into a stochastic linear inverse problem using Fourier analysis, and derives risk bounds with practical numerical validation.

Contribution

It proposes a novel adaptive estimation technique for shifted curves using Fourier domain analysis and unbiased empirical risk minimization, with theoretical risk bounds and numerical experiments.

Findings

Derived a nonasymptotic oracle inequality for the estimator.

Achieved adaptive estimation results on Sobolev spaces.

Validated the approach through numerical experiments.

Abstract

This paper considers the problem of adaptive estimation of a template in a randomly shifted curve model. Using the Fourier transform of the data, we show that this problem can be transformed into a stochastic linear inverse problem. Our aim is to approach the estimator that has the smallest risk on the true template over a finite set of linear estimators defined in the Fourier domain. Based on the principle of unbiased empirical risk minimization, we derive a nonasymptotic oracle inequality in the case where the law of the random shifts is known. This inequality can then be used to obtain adaptive results on Sobolev spaces as the number of observed curves tend to infinity. Some numerical experiments are given to illustrate the performances of our approach.