Tail Spectral Density Estimation and Its Uncertainty Quantification: Another Look at Tail Dependent Time Series Analysis

Abstract

We consider the estimation and uncertainty quantification of the tail spectral density, which provide a foundation for tail spectral analysis of tail dependent time series. The tail spectral density has a particular focus on serial dependence in the tail, and can reveal dependence information that is otherwise not discoverable by the traditional spectral analysis. Understanding the convergence rate of tail spectral density estimators and finding rigorous ways to quantify their statistical uncertainty, however, still stand as a somewhat open problem. The current article aims to fill this gap by providing a novel asymptotic theory on quadratic forms of tail statistics in the double asymptotic setting, based on which we develop the consistency and the long desired central limit theorem for tail spectral density estimators. The results are then used to devise a clean and effective method for constructing confidence intervals to gauge the statistical uncertainty of tail spectral density estimators, and it can be turned into a visualization tool to aid practitioners in examining the tail dependence for their data of interest. Numerical examples including data applications are presented to illustrate the developed results.