Statistics Seminar – Dr. Tobias Kley, Ruhr-Universitaet Bochum
“Quantile-Based Spectral Analysis of Time Series”

Date            Friday, October 03, 2014
Time            3:30 pm – 5:00 pm
Location        7 David Kinley Hall

Abstract:
In this talk an alternative approach to the spectral analysis of time series is presented. First, a “new” spectrum is defined for stationary processes {Y_t}_{t in Z}. We call the Fourier transform of the differences between copulas of the pairs (Y_t,Y_t−k) and the independence copula the copula spectral density kernel. This object allows to separate marginal and serial aspects of a time series. The cop- ula intrinsically provides more information about the conditional distribution than the covariance. Thus, the copula spectral density kernel is more informative than the spectral density obtained from the autocovariances. In particular, a complete description of the distributions of all pairs (Y_t,Y_{t−k}) is possible. Two rank-based estimators of the copula spectral density kernel are provided. We establish their asymptotic properties. The results are applicable without restrictive distributional assumptions such as the existence of finite moments and only weak mixing assumptions are necessary. To allow non-stationarity in the analysis, localized versions of the copula spectrum and of the estimators are introduced. We define the concept of local strict stationarity and show consis- tency of a localized estimator under this condition. The R package quantspec will be introduced. A demonstration on how it can be used in the analysis of an empirical example will be given. This is joint work with H. Dette, M. Hallin, S. Skowronek, and S. Volgushev. Dette, H., Hallin, M., Kley, T. and Volgushev, S. (2014+). Of copulas, quantiles, ranks and spectra: An L1-approach to spectral analysis. To appear in Bernoulli. Kley, T., Volgushev, S., Dette, H. and Hallin, M. (2014). Quantile spectral pro- cesses: Asymptotic analysis and inference (arxiv.org/abs/1401.8104). Skowronek, S, Volgushev, S., Kley, T., Dette, H. and Hallin, M. (2014). Quantile spectral analysis for locally stationary time series (arxiv.org/abs/1404.4605). Kley,T. (2014). Quantile-based spectral analysis in an object-oriented frame- work and a reference implementation in R: The quantspec Package (arxiv.org/abs/1408.6755).