Confidence surfaces for the mean of locally stationary functional time series
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Date
2021
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Abstract
The problem of constructing a simultaneous confidence band for the mean function of
a locally stationary functional time series {Xi,n(t)}i=1,...n is challenging as these bands can
not be built on classical limit theory. On the one hand, for a fixed argument t of the functions
Xi,n, the maximum absolute deviation between an estimate and the time dependent
regression function exhibits (after appropriate standardization) an extreme value behaviour
with a Gumbel distribution in the limit. On the other hand, for stationary functional data,
simultaneous confidence bands can be built on classical central theorems for Banach space
valued random variables and the limit distribution of the maximum absolute deviation is
given by the sup-norm of a Gaussian process. As both limit theorems have different rates of
convergence, they are not compatible, and a weak convergence result, which could be used
for the construction of a confidence surface in the locally stationary case, does not exist.
In this paper we propose new bootstrap methodology to construct a simultaneous confidence
band for the mean function of a locally stationary functional time series, which is
motivated by a Gaussian approximation for the maximum absolute deviation. We prove the
validity of our approach by asymptotic theory, demonstrate good finite sample properties by
means of a simulation study and illustrate its applicability analyzing a data example.
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Keywords
locally stationary time series, Gaussian approximation, confidence bands, functional data