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# Theory in functional principal components of discretely observed data

Sep 2022

Convergence of eigenfunctions with diverging index is essential in nearly allmethods based on functional principal components analysis. The main goal ofthis work is to establish the unified theory for such eigencomponents indifferent types of convergence based on discretely observed functional data. Weobtain the moment bounds for eigenfunctions and eigenvalues for a wide range ofthe sampling rate and show that under some mild assumptions, the$\mathcal{L}^{2}$ bound of eigenfunctions estimator with diverging indices isoptimal in the minimax sense as if the curves are fully observed. This is thefirst attempt at obtaining an optimal rate for eigenfunctions with divergingindex for discretely observed functional data. We propose a double truncationtechnique in handling the uniform convergence of function data and establishthe uniform convergence of covariance function as well as the eigenfunctionsfor all sampling scheme under mild assumptions. The technique route proposed inthis work provides a new tool in handling the perturbation series withdiscretely observed functional data and can be applied in most problems basedon functional principal components analysis and models involving inverse issue.

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