For discretely observed functional data, estimating eigenfunctions with diverging index is essential in nearly all methods based on functional principal components analysis. In this paper, we propose a new approach to handle each term appeared in the perturbation series and overcome the summability issue caused by the estimation bias. We obtain the moment bounds for eigenfunctions and eigenvalues for a wide range of the sampling rate. We show that under some mild assumptions, the moment bound for the eigenfunctions with diverging indices is optimal in the minimax sense. This is the first attempt at obtaining an optimal rate for eigenfunctions with diverging index for discretely observed functional data. Our results fill the gap in theory between the ideal estimation from fully observed functional data and the reality that observations are taken at discrete time points with noise, which has its own merits in models involving inverse problem and deserves further investigation.
翻译:对于不同观察的功能数据,在基于功能主要组成部分分析的几乎所有方法中,估算使用不同指数的机能几乎都是必要的。在本文件中,我们建议采用一种新的方法来处理在扰动序列中出现的每个术语,并克服估算偏差引起的共性问题。我们获得大量抽样率的机能功能和机能值的时空界限。我们表明,根据一些轻度假设,使用不同指数的机能的机能的时空在微缩意义上是最佳的。这是第一次尝试为独立观察的功能数据获得使用不同指数的机能的最佳比率。我们的结果填补了从充分观察的功能数据中得出的理想估计值与在与噪音不同时点进行观测之间的理论差距,而噪音在涉及反向问题的模型中具有其本身的优点,值得进一步调查。