Title: Recovering covariance from functional fragments
Authors: Marie-Helene Descary - University of Quebec in Montreal (Canada) [presenting]
Victor Panaretos - EPFL (Switzerland)
Abstract: The problem of nonparametric estimation of a covariance function on the unit square is considered given a sample of discretely observed fragments of functional data. When each sample path is only observed on a subinterval of length $\delta<1$, one has no statistical information on the unknown covariance outside a $\delta$-band around the diagonal. A priori, the problem seems unidentifiable without parametric assumptions, but we nevertheless show that nonparametric estimation is feasible under suitable smoothness and rank conditions on the unknown covariance. This remains true even when observation is discrete, and we give precise deterministic conditions on how fine the observation grid needs to be relative to the rank and fragment length for identifiability to hold. We show that our conditions translate the estimation problem to a low-rank matrix completion problem, and construct a nonparametric estimator in this vein. Our estimator is seen to be consistent in a fully functional sense, and indeed we obtain convergence rates demonstrating that even a parametric rate is attainable provided the grid is sufficiently dense. We illustrate the performance of our method in a real and simulated examples.