SPECIAL ISSUE LinStat: a new proposal for robust estimation of the extremal index

Abstract

The extremal index (EI) is a parameter defined in the framework of extreme value theory, which measures the degree of dependence among exceedances above high fixed thresholds. When the EI exists and those exceedances occur in clusters, the EI is related to the dimension of the clusters and, in the limit distribution, it coincides with the reciprocal of the mean clusters dimension. There are different EI estimators according to the method used in the identification of those clusters. Some of them use the sample mean in the estimation of the cluster dimension. The present proposal is based on the ‘Runs’ estimator. It considers a negative binomial as the limit distribution of the number of exceedances that occur before a non-exceedance observation. Then its mean is estimated through the constant term of a negative binomial regression. The reciprocal of that value gives the EI estimate. The procedure makes use of robust estimators for counting processes, with known properties. Thus the negative binomial distribution is integrated in the estimation of the mean, while small deviations from the assumptions are controlled, including the occurrence of atypical cluster size values. A simulation study explores and compares the present robust proposal with other estimators.

Keywords:

• Extremal index
• robustness
• estimation
• negative binomial regression
• simulation

Acknowledgments

This work was supported by The Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT – Fundação para a Ciência e a Tecnologia), references https://doi.org/10.54499/UIDB/04106/2020 and https://doi.org/10.54499/UIDP/04106/2020.

Disclosure statement

No potential conflict of interest was reported by the author(s).