The classical multivariate extreme-value theory concerns the modeling of extremes in a multivariate random sample, suggesting the use of max-stable distributions. In this work, the classical theory is extended to the case where aggregated data, such as maxima of a random number of observations, are considered. We derive a limit theorem concerning the attractors for the distributions of the aggregated data, which boil down to a new family of max-stable distributions. We also connect the extremal dependence structure of classical max-stable distributions and that of our new family of max-stable distributions. Using an inversion method, we derive a semiparametric composite-estimator for the extremal dependence of the unobservable data, starting from a preliminary estimator of the extremal dependence of the aggregated data. Furthermore, we develop the large-sample theory of the composite-estimator and illustrate its finite-sample performance via a simulation study.

Hashorva, E., Padoan, S. A., Rizzelli, S., Multivariate extremes over a random number of observations, <<SCANDINAVIAN JOURNAL OF STATISTICS>>, 2021; 48 (3): 845-880. [doi:10.1111/sjos.12463] [http://hdl.handle.net/10807/193631]

### Multivariate extremes over a random number of observations

#### Abstract

The classical multivariate extreme-value theory concerns the modeling of extremes in a multivariate random sample, suggesting the use of max-stable distributions. In this work, the classical theory is extended to the case where aggregated data, such as maxima of a random number of observations, are considered. We derive a limit theorem concerning the attractors for the distributions of the aggregated data, which boil down to a new family of max-stable distributions. We also connect the extremal dependence structure of classical max-stable distributions and that of our new family of max-stable distributions. Using an inversion method, we derive a semiparametric composite-estimator for the extremal dependence of the unobservable data, starting from a preliminary estimator of the extremal dependence of the aggregated data. Furthermore, we develop the large-sample theory of the composite-estimator and illustrate its finite-sample performance via a simulation study.
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2021
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Hashorva, E., Padoan, S. A., Rizzelli, S., Multivariate extremes over a random number of observations, <<SCANDINAVIAN JOURNAL OF STATISTICS>>, 2021; 48 (3): 845-880. [doi:10.1111/sjos.12463] [http://hdl.handle.net/10807/193631]
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/10807/193631`
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