A0192
Title: Testing optimal dimension for suitable Hilbert-valued processes
Authors: Enea Bongiorno - Universita del Piemonte Orientale (Italy)
Jean-Baptiste Aubin - Insa-Lyon (France) [presenting]
Abstract: The small-ball probability (SmBP) of a Hilbert-valued process is considered. Recent works have shown that, for a fixed number $d$ and as the radius epsilon of the ball tends to zero, the SmBP is asymptotically proportional to (a) the joint density of the first $d$ principal components (PCs) evaluated at the center of the ball, (b) the volume of the $d$-dimensional ball with radius epsilon, and (c) a correction factor weighting the use of a truncated version of the process expansion. Under suitable assumptions on the decay rate of the eigenvalues of the covariance operator of the process, it has been shown that the correction factor in (c) tends to 1 as the dimension increases. The properties of the correction factor are studied and a consistent estimator is introduced. Features of such estimator allow to conservatively test whenever the correction factor equals 1. This implicitly implies that, for the class of processes whose eigenvalues of the covariance operator decay hyper-exponentially, an optimal dimension can be defined allowing to use a ``finite-dimensional'' approach in approximating the SmBP and, hence, providing a natural model advantage.
