标题： 尖锐模板估计在移位曲线模型中 显示英文标题

标题： Sharp template estimation in a shifted curves model

摘要： 本文研究了随机平移曲线模型中模板的自适应估计问题。通过使用数据的Fourier变换，我们证明了该问题可以转化为一个随机线性反问题。我们的目标是在Fourier域中定义的有限线性估计器集合上，寻求在真实模板上的风险最小的估计器。基于无偏经验风险最小化原则，在已知随机平移分布律的情况下，我们推导出了一个非渐近的oracle不等式。然后可以利用这个不等式在观察到的曲线数量趋于无穷时，得到关于Sobolev空间的自适应结果。一些数值实验被给出以说明我们方法的表现。 显示英文摘要

摘要： 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.