On the Estimation of the Extremal Index Based on Scaling and Resampling

Abstract

The extremal index parameter θ characterizes the degree of local dependence in the extremes of a stationary time series and has important applications in a number of areas, such as hydrology, telecommunications, finance, and environmental studies. In this study, a novel estimator for θ based on the asymptotic scaling of block-maxima and resampling is introduced. It is shown to be consistent and asymptotically normal for a large class of m-dependent time series. Further, a procedure for the automatic selection of its tuning parameter is developed and different types of confidence intervals that prove useful in practice proposed. The performance of the estimator is examined through simulations, which show its highly competitive behavior. Finally, the estimator is applied to three real datasets of daily crude oil prices, daily returns of the S&P 500 stock index, and high-frequency, intraday traded volumes of a stock. These applications demonstrate additional diagnostic features of statistical plots based on the new estimator. Supplementary materials including an appendix with complete proofs, the crude oil dataset, and an R-script implementing the estimators are available online.

Multivariate dynamic intensity peaks‐over‐threshold models

“…The extremogram of losses and the cross‐extremograms for losses conditional on gains reveal the most significant dependence on higher lags. Hence losses are more strongly clustered than gains, confirming findings (e.g., by Hamidieh, Stoev, & Michailidis, ; Jondeau and Rockinger, ; Olmo, ).…”

Significant reduction of cold temperature extremes at Faraday/Vernadsky station in the Antarctic Peninsula

“…the time length over which one expects extreme values to bunch together. We use the method put forward by Hamidieh et al (2010) to estimate the extremal index. This approach uses the property that the maxima of blocks of size m are proportional to the extremal index θ( j ) for dyadic block sizes m = 2 j .…”

Distribution and clustering of fast coronal mass ejections

“…[16] Equations (1) and (2) suggest a method of estimating both a and [Stoev et al, 2006;Hamidieh et al, 2009]. The inverse exponent 1/a is obtained as a slope of the line fitted to the Max-Spectrum of the data.…”

“…[10] We use the Max-Spectrum method based on investigating of averages of data maxima taken in all time intervals of fixed sizes when the size is progressively increased [Stoev et al, 2006;Hamidieh et al, 2009], see below for more detailed description. The method does not involve a fit to an empirically determined distribution function.…”