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Quantile spectral processes: Asymptotic analysis and inference
Quantile- and copula-related spectral concepts recently have been considered
by various authors. Those spectra, in their most general form, provide a full
characterization of the copulas associated with the pairs in a
process , and account for important dynamic features,
such as changes in the conditional shape (skewness, kurtosis),
time-irreversibility, or dependence in the extremes that their traditional
counterparts cannot capture. Despite various proposals for estimation
strategies, only quite incomplete asymptotic distributional results are
available so far for the proposed estimators, which constitutes an important
obstacle for their practical application. In this paper, we provide a detailed
asymptotic analysis of a class of smoothed rank-based cross-periodograms
associated with the copula spectral density kernels introduced in Dette et al.
[Bernoulli 21 (2015) 781-831]. We show that, for a very general class of
(possibly nonlinear) processes, properly scaled and centered smoothed versions
of those cross-periodograms, indexed by couples of quantile levels, converge
weakly, as stochastic processes, to Gaussian processes. A first application of
those results is the construction of asymptotic confidence intervals for copula
spectral density kernels. The same convergence results also provide asymptotic
distributions (under serially dependent observations) for a new class of
rank-based spectral methods involving the Fourier transforms of rank-based
serial statistics such as the Spearman, Blomqvist or Gini autocovariance
coefficients.Comment: Published at http://dx.doi.org/10.3150/15-BEJ711 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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