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.
| Original language | English |
|---|---|
| Journal | Stochastic Processes and Their Applications |
| Volume | 125 |
| Issue number | 8 |
| Pages (from-to) | 3126-3169 |
| ISSN | 0304-4149 |
| DOIs | |
| Publication status | Published - 1 Aug 2015 |