A Comparison of Methods for Estimating the Extremal Index

The extremal index, θ (0≤θ≤1), is the key parameter when extending discussions of the limiting behavior of the extreme values from independent and identically distributed sequences to stationary sequences. As θ measures the limiting dependence of exceedances over a threshold u, as u tends to the upper endpoint of the distribution, it may not always be informative about the extremal dependence at levels of practical interest. Therefore we also consider a threshold-based extremal index, θ (u). We compare the performance of a range of different estimators for θ and θ (u) covering processes with θ < 1 and θ = 1. We find that the established methods for estimating θ actually estimate θ (u), so perform well only when θ (u)≈ θ. For Markov processes, we introduce an estimator which is as good as the established methods when θ (u)≈ θ but provides an improvement when θ (u) < θ = 1. We illustrate our methods using simulated data and daily rainfall measurements.