Estimation of intensity–duration–frequency curves using max-stable processes

  • Hristos Tyralis
  • Andreas Langousis
Original Paper


We present an approach to estimate intensity–duration–frequency (IDF) curves based on max-stable processes. The proposed method has been inspired by the seminal study of Nadarajah et al. (J R Stat Soc B 60(2):473–496, 1998), who used a multivariate extreme value distribution (MEVD) to estimate (IDF) curves from rainfall records. Max-stable processes are continuous extensions of MEVD (i.e. the marginal distributions of rainfall maxima at different durations are generalized extreme valued), which are more flexible, allow for extreme rainfall estimation at any arbitrary duration d (i.e. not just a discrete set, as is the case of MEVD), while preserving asymptotic dependencies. The latter characteristic of IDF estimates results from the combined effect of the statistical structure of rainfall (i.e. temporal dependencies), as well as the IDF construction process, which involves averaging of the original series to obtain rainfall maxima at different temporal resolutions. We apply the method to hourly precipitation data, and compare it to empirical estimates and the results produced by a semiparametric approach. Our findings indicate that max-stable processes fit well the statistical structure and inter-dependencies of annual rainfall maxima at different durations, produce slightly more conservative estimates relative to semiparametric methods, while allowing for extrapolations to durations and return periods beyond the range of the available data. The proposed statistical model is fully parametric and likelihood based, providing a theoretically consistent basis in solving the problem at hand.


Brown–Resnick process Generalized extreme value distribution Max-stable processes Multivariate extreme value distribution Pairwise likelihood Rainfall maxima 



We thank two anonymous reviewers for their useful suggestions.

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.

Supplementary material (345 kb)
Supplementary material 1(ZIP 346KB)


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Hellenic Air ForceEleusisGreece
  2. 2.Department of Civil Engineering, School of EngineeringUniversity of PatrasPatrasGreece

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