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arXiv · 1503.04022

The integrated periodogram of a dependent extremal event sequence

Thomas Mikosch·Yuwei Zhao
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Abstract

We investigate the asymptotic properties of the integrated periodogram calculated from a sequence of indicator functions of dependent extremal events. An event in Euclidean space is extreme if it occurs far away from the origin. We use a regular variation condition on the underlying stationary sequence to make these notions precise. Our main result is a functional central limit theorem for the integrated periodogram of the indicator functions of dependent extremal events. The limiting process is a continuous Gaussian process whose covari- ance structure is in general unfamiliar, but in the iid case a Brownian bridge appears. In the general case, we propose a stationary bootstrap procedure for approximating the distribution of the limiting process. The developed theory can be used to construct classical goodness-of-fit tests such as the Grenander- Rosenblatt and Cramér-von Mises tests which are based only on the extremes in the sample. We apply the test statistics to simulated and real-life data.

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Thomas Mikosch, Yuwei Zhao. 2015-03-13. The integrated periodogram of a dependent extremal event sequence. https://doi.org/10.1016/j.spa.2015.02.017

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