On the Exceedance Point Process for a Stationary Sequence.

T., Hsing,; Leadbetter, M. R.

doi:10.21236/ada152827

Cited by 48 publications

(86 citation statements)

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“…(See [31,Chapter 5],[28] and references therein for more information on the subject).Similarly to Theorems 1 and 2, we show that if there exists a limiting continuous time stochastic process for the HTPP, when properly normalised, then the same holds for the EPP and vice-versa. In the sequel d − → denotes convergence in distribution.Theorem 3 Let(X , B, µ, f ) be a dynamical system where µ is an acip and consider ζ ∈ X for which Lebesgue's Differentiation Theorem holds.…”

supporting

confidence: 74%

Hitting time statistics and extreme value theory

Freitas

Todd

2009

Probab. Theory Relat. Fields

174

259

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We consider discrete time dynamical systems and show the link between Hitting Time Statistics (the distribution of the first time points land in asymptotically small sets) and Extreme Value Theory (distribution properties of the partial maximum of stochastic processes). This relation allows to study Hitting Time Statistics with tools from Extreme Value Theory, and vice versa. We apply these results to non-uniformly hyperbolic systems and prove that a multimodal map with an absolutely continuous invariant measure must satisfy the classical extreme value laws (with no extra condition on the speed of mixing, for example). We also give applications of our theory to higher dimensional examples, for which we also obtain classical extreme value laws and exponential hitting time statistics (for balls). We extend these ideas to the subsequent returns to asymptotically small sets, linking the Poisson statistics of both processes.J. M.

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supporting

confidence: 74%

Hitting time statistics and extreme value theory

Freitas

Todd

2009

Probab. Theory Relat. Fields

174

259

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“…Section 1 gives a formal definition of the extremal index; an alternative characterisation, provided by Hsing et al (1988), is that θ −1 is the limiting mean cluster size in the point process of exceedance times over a high threshold. This suggests that a suitable way to estimate the extremal index can be found through methods which identify clusters of extremes, the estimate itself being found as the reciprocal of the mean cluster size.…”

Section: Cluster Size Methodsmentioning

confidence: 99%

Bayesian inference for clustered extremes

Fawcett

Walshaw

2008

Extremes

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We consider Bayesian inference for the extremes of dependent stationary series. We discuss the virtues of the Bayesian approach to inference for the extremal index, and for related characteristics of clustering behaviour. We develop an inference procedure based on an automatic declustering scheme, and using simulated data we implement and assess this procedure, making inferences for the extremal index, and for two cluster functionals. We then apply our procedure to a set of real data, specifically a time series of windspeed measurements, where the clusters correspond to storms. Here the two cluster functionals selected previously correspond to the mean storm length and the mean inter-storm interval. We also consider inference for long-period return levels, advocating the posterior predictive distribution as being most representative of the information required by engineers interested in design level specifications.

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“…Hence, if the extremal index θ is strictly positive, the maximum converges to some nondegenerate limit distribution that is of the same type as the limit distribution in the case of independence. Moreover, Hsing et al (1988) proved that under weak additional assumptions [including the slightly stronger mixing condition (u n )] the point process n t=1 ε t/n 1 (a n x+b n ,∞) (X t ) of standardized time points at which exceedances occur converges to a compound Poisson process. Then, typically, the extremal index θ equals the reciprocal value of its mean cluster size (although in general one only knows that θ is a lower bound for this value).…”

Section: Comparison Of Model-based and Direct Extreme Quantile Estimamentioning

confidence: 99%

Some aspects of extreme value statistics under serial dependence

Drees

2008

Extremes

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On the occasion of Laurens de Haan's 70th birthday, we discuss two aspects of the statistical inference on the extreme value behavior of time series with a particular emphasis on his important contributions. First, the performance of a direct marginal tail analysis is compared with that of a modelbased approach using an analysis of residuals. Second, the importance of the extremal index as a measure of the serial extremal dependence is discussed by the example of solutions of a stochastic recurrence equation.

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On the Exceedance Point Process for a Stationary Sequence.

Cited by 48 publications

References 1 publication

Hitting time statistics and extreme value theory

Hitting time statistics and extreme value theory

Bayesian inference for clustered extremes

Some aspects of extreme value statistics under serial dependence

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