Methods for estimating the upcrossings index: improvements and comparison

A. P. Martins () and J. R. Sebastião ()
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A. P. Martins: Universidade da Beira Interior
J. R. Sebastião: Escola Superior de Gestão

Statistical Papers, 2019, vol. 60, issue 4, No 14, 1317-1347

Abstract: Abstract The upcrossings index $$0\le \eta \le 1,$$ 0 ≤ η ≤ 1 , as a measure of the degree of local dependence in the upcrossings of a high level by a stationary process, plays, together with the extremal index $$\theta ,$$ θ , an important role in extreme events modelling. For stationary processes, verifying a long range dependence condition, upcrossings of high thresholds in different blocks can be assumed asymptotically independent and therefore blocks estimators for the upcrossings index can be easily constructed using disjoint blocks. In this paper we focus on the estimation of the upcrossings index via the blocks method and properties such as consistency and asymptotic normality are studied. Besides this new estimation approach for this parameter, we also enlarge its family of runs estimators and improve estimation within this class by providing an empirical way of checking local dependence conditions that control the clustering of upcrossings. We compare the performance of a range of different estimators for $$\eta $$ η and illustrate the methods using simulated data and financial data.

Keywords: Upcrossings index; Blocks estimators; Runs estimators; Dependence conditions; Consistency and asymptotic normality; 60G70 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s00362-017-0876-x

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