The integrated periodogram of a dependent extremal event sequence

Thomas Mikosch and Yuwei Zhao

Stochastic Processes and their Applications, 2015, vol. 125, issue 8, 3126-3169

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 covariance structure is in general unfamiliar, but in the i.i.d. 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.

Keywords: Extreme value theory; Functional central limit theorem; Stationary bootstrap; Goodness-of-fit test; Spectral analysis (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (4)

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DOI: 10.1016/j.spa.2015.02.017

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