The extremal index θ, a number in the interval [0, 1], is known to be a measure of primal importance for analyzing the extremes of a stationary time series. New rank-based estimators for θ are proposed which rely on the construction of approximate samples from the exponential distribution with parameter θ that is then to be fitted via the method of moments. The new estimators are analyzed both theoretically as well as empirically through a large-scale simulation study. In specific scenarios, in particular for time series models with θ ≈ 1, they are found to be superior to recent competitors from the literature.

The extremal index θ, a number in the interval [0, 1], is known to be a measure of primal importance for analyzing the extremes of a stationary time series. New rankbased estimators for θ are proposed which rely on the construction of approximate samples from the exponential distribution with parameter θ that is then to be fitted via the method of moments. The new estimators are analyzed both theoretically as well as empirically through a large-scale simulation study. In specific scenarios, in particular for time series models with θ ≈ 1, they are found to be superior to recent competitors from the literature.

No abstract

Einmahl, de Haan and Zhou (2016, Journal of the Royal Statistical Society: Series B, 78(1), 31-51) recently introduced a stochastic model that allows for heteroscedasticity of extremes. The model is extended to the situation where the observations are serially dependent, which is crucial for many practical applications. We prove a local limit theorem for a kernel estimator for the scedasis function, and a functional limit theorem for an estimator for the integrated scedasis function. We further prove consistency of a bootstrap scheme that allows to test for the null hypothesis that the extremes are homoscedastic. Finally, we propose an estimator for the extremal index governing the dynamics of the extremes and prove its consistency. All results are illustrated by Monte Carlo simulations. An important intermediate result concerns the sequential tail empirical process under serial dependence.

No abstract