  Frequency domain methods form a ubiquitous part of the statistical toolbox
for time series analysis. In recent years, considerable interest has been given
to the development of new spectral methodology and tools capturing dynamics in
the entire joint distributions and thus avoiding the limitations of classical,
$L^2$-based spectral methods. Most of the spectral concepts proposed in that
literature suffer from one major drawback, though: their estimation requires
the choice of a smoothing parameter, which has a considerable impact on
estimation quality and poses challenges for statistical inference. In this
paper, associated with the concept of copula-based spectrum, we introduce the
notion of copula spectral distribution function or integrated copula spectrum.
This integrated copula spectrum retains the advantages of copula-based spectra
but can be estimated without the need for smoothing parameters. We provide such
estimators, along with a thorough theoretical analysis, based on a functional
central limit theorem, of their asymptotic properties. We leverage these
results to test various hypotheses that cannot be addressed by classical
spectral methods, such as the lack of time-reversibility or asymmetry in tail
dynamics.
