How can we discern whether the covariance of a stochastic process is of reduced rank, and if so,
what its precise rank is? And how can we do so at a given level of confidence? This question is
central to a great deal of methods for functional data, which require low-dimensional
representations whether by functional PCA or other methods. The difficulty is that the
determination is to be made on the basis of i.i.d. replications of the process observed discretely
and with measurement error contamination. This adds a ridge to the empirical covariance,
obfuscating the underlying dimension. We describe a matrix-completion inspired test statistic
that circumvents this issue by measuring the best possible least square fit of the empirical
covariance's off-diagonal elements, optimised over covariances of given finite rank. For a fixed
grid of sufficiently large size, we determine the statistic's asymptotic null distribution as the
number of replications grows. We then use it to construct a bootstrap implementation of a
stepwise testing procedure controlling the family-wise error rate corresponding to the collection
of hypotheses formalising the question at hand. Under minimal regularity assumptions we prove
that the procedure is consistent and that its bootstrap implementation is valid. The procedure
circumvents smoothing and associated smoothing parameters, is indifferent to measurement
error heteroskedasticity, and does not assume a low-noise regime. Based on joint work with
Anirvan Chakraborty.
10 Dicembre 2021
Victor M. Panaretos
École Polytechnique Fédérale de Lausanne