Title: Testing for the rank of a covariance operator
Authors: Anirvan Chakraborty - IISER Kolkata (India) [presenting]
Victor Panaretos - EPFL (Switzerland)
Abstract: How we can discern whether the covariance operator of a stochastic process is of reduced rank. If so, what is its rank, and can we say so at a given confidence level? This question is central to several problems in functional data, which require low-dimensional representations. The difficulty is that the determination has to be made based on discrete observations with measurement errors. This adds a ridge to the empirical covariance, obfuscating the underlying dimension. We discuss a matrix-completion inspired test that circumvents this issue by measuring the optimum least-square fit of the empirical covariance's off-diagonal elements, over finite rank covariances. For a sufficiently large but fixed grid, we discuss the asymptotic null distribution as the sample size grows. We use it to construct a bootstrap implementation of a stepwise testing procedure controlling the FWER corresponding to the collection of hypotheses formalising the problem. Under certain regularity assumptions, we show that the procedure is consistent and its bootstrap implementation is valid. The procedure circumvents smoothing, is indifferent to hetreroskedastic errors, and does not assume a low-noise regime. We show the excellent practical performance on simulations and real data, and also demonstrate the stability across a wide range of settings.