Extreme value estimation for discretely sampled continuous processes

Abstract

In environmental applications of extreme value statistics, the underlying stochastic process is often modeled either as a max-stable process in continuous time/space or as a process in the domain of attraction of such a max-stable process. In practice, however, the processes are typically only observed at discrete points and one has to resort to interpolation to fill in the gaps. We discuss the influence of such an interpolation on estimators of marginal parameters as well as estimators of the exponent measure. In particular, natural conditions on the fineness of the observational scheme are developed which ensure that asymptotically the interpolated estimators behave in the same way as the estimators which use fully observed continuous processes.

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Acknowledgements

L. de Haan and F. Turkman have been partly funded by FCT - Fundação para a Ciência e a Tecnologia, Portugal, through the project UID/MAT/00006/2013. H. Drees has been partly supported by DFG project DR 271/6-2 within the research unit FOR 1735. We thank two anonymous referees whose constructive remarks led to an improvement of the presentation.

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Correspondence to Holger Drees.

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Drees, H., de Haan, L. & Turkman, F. Extreme value estimation for discretely sampled continuous processes. Extremes 21, 533–550 (2018). https://doi.org/10.1007/s10687-018-0313-0

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Keywords

  • Discrete and continuous sampling
  • Interpolation
  • Max-stable process

AMS 2000 Subject Classifications

  • Primary—62G32
  • Secondary—62G05, 62M30