Integrated Copula Spectral Densities and their Applications
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Description
Copula spectral densities are defined in terms of the copulas associated with the pairs $(X_{t+k}, X_t)$ of a process $(X_t)_{t \in \mathbb{Z}}$. Thereby they can capture a wide range of dynamic features, such as changes in the conditional skewness or dependence of extremes, that traditional spectra cannot account for. A consistent estimator for copula spectra was suggested by Kley et al. [Bernoulli 22 (2016) 1707–831] who prove a functional central limit theorem (fCLT) according to which the estimator, considered as a stochastic process indexed in the quantile levels, converges weakly to a Gaussian limit. Similar to the traditional case, no fCLT exists for this estimator when it is considered as a stochastic process indexed in the frequencies. In this talk, we consider estimation of integrated copula spectra and show that our estimator converges weakly as a stochastic process indexed in the quantile levels and frequencies. Interestingly and in contrast to the estimator considered by Kley et al., estimation of the unknown marginal distribution has an effect on the asymptotic covariance. We apply subsampling to obtain confidence intervals for the integrated copula spectra. Further, our results allow to use copula spectra to test a wide range of hypotheses. As an example, we suggest a test for the hypothesis that the underlying process is pairwise time-reversible.
(This is joint work with H. Dette, Y. Goto, M. Hallin, R. Van Hecke and S. Volgushev.)
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About Tobias Kley
Tobias Kley is interested in statistical methods for the analysis of data that defies traditional assumptions (such as Gaussianity or stationarity). He obtained degrees in Mathematics and Information Systems from the University of Münster and, in 2014, was awarded a PhD from Ruhr University Bochum. He held postdoctoral roles at the London School of Economics and at Humboldt University Berlin before he was appointed Lecturer in Statistical Science at the University of Bristol in 2018.