A Note on the Adaptive Choice of the Optimal Threshold in Extreme Value Analysis
M. Ivette Gomes1, Dinis Pestana1
1Department of Statistics and Operations Research (DEIO), Faculty of Science, University of Lisbon, Lisbon, Portugal; 2Department of Mathematics, Faculty of Science and Technology, New University of Lisbon, Costa de Caparica, Portugal

The main objective of Statistics of Extremes is the prediction of rare events, and thus the need for an adequate estimation of parameters related to natural disasters. The primary parameter is the extreme value index (EVI). One of the most recent and general approaches is the semi-parametric one, where it is merely assumed that the model underlying the available random sample is in the domain of attraction of a (unified) extreme value (EV) distribution. The EVI estimation is then based on the largest k order statistics in the sample or on the excesses over a high level u. The question that has been often addressed in the practical applications of Extreme Value Theory is the choice of either k or u. A great variety of semi-parametric EVI-estimators have the same type of problems, consistency for intermediate ranks, high variance for small values of k, and high asymptotic bias for large values of k. In this paper we shall concentrate on heavy tails, i.e., a positive EVI. The most common methods of adaptive choice of the threshold k, among which we mention the bootstrap based methods, are based on the minimization of some kind of mean squared error estimator. Here we advance with a methodology based on bias properties for the selection of the optimal sample fraction. The methodology depends on a tuning parameter, and we provide a choice for such a tuning parameter. In order to achieve our objectives we derive the asymptotic behavior of an adequate linear combination of the scaled log-spacings, suggest the use of two alternative auxiliary statistics, and draw some overall comments.

Keywords: Statistics of extremes; Semi-parametric inference; Threshold selection; Bias properties and resampling-based methodologies

Biography: M. Ivette Gomes is a Full Professor at the Department of Statistics and Operations Research, Faculty of Science, University of Lisbon.

She is also a full member of the Research Centre on Statistics and Applications, University of Lisbon.

She got her Ph.D. in Statistics from the University of Sheffield, United Kingdom, and her Habilitation Degree in Applied Mathematics from the University of Lisbon, Portugal.

Her main research interests lie in the areas of Order Statistics and Extreme Value Theory, Computational Statistics, Monte-Carlo Simulation and Statistical Quality Control.