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.
Funding Statement
This work has been supported by the Collaborative Research Center “Statistical modeling of nonlinear dynamic processes” (SFB 823) of the German Research Foundation, which is gratefully acknowledged.
Acknowledgments
The authors are grateful to two unknown referees
Computational infrastructure and support were provided by the Centre for Information and Media Technology at Heinrich Heine University Düsseldorf, which is gratefully acknowledged. The authors are grateful to Chen Zhou for helpful discussions and suggestions.
Citation
Axel Bücher. Tobias Jennessen. "Statistics for heteroscedastic time series extremes." Bernoulli 30 (1) 46 - 71, February 2024. https://doi.org/10.3150/22-BEJ1560
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