Weak convergence of a pseudo maximum likelihood estimator for the extremal index

Berghaus, Betina; Bücher, Axel

doi:10.48550/arxiv.1608.01903

Cited by 2 publications

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“…Moreover, the sample of sliding block maxima carries more information than the sample of disjoint block maxima, which suggests the possibility of more accurate inference. Robert et al (2009), Northrop (2015) and Berghaus and Bücher (2016) applied this idea to the estimation of the extremal index, a summary measure for the strength of serial dependence between extremes. They found that estimators based on sliding blocks were indeed more efficient than their counterparts based on disjoint blocks.…”

Section: Introductionmentioning

confidence: 99%

Inference for heavy tailed stationary time series based on sliding blocks

Bücher¹,

Segers²

2018

Electron. J. Statist.

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The block maxima method in extreme value theory consists of fitting an extreme value distribution to a sample of block maxima extracted from a time series. Traditionally, the maxima are taken over disjoint blocks of observations. Alternatively, the blocks can be chosen to slide through the observation period, yielding a larger number of overlapping blocks. Inference based on sliding blocks is found to be more efficient than inference based on disjoint blocks. The asymptotic variance of the maximum likelihood estimator of the Fréchet shape parameter is reduced by more than 18%. Interestingly, the amount of the efficiency gain is the same whatever the serial dependence of the underlying time series: as for disjoint blocks, the asymptotic distribution depends on the serial dependence only through the sequence of scaling constants. The findings are illustrated by simulation experiments and are applied to the estimation of high return levels of the daily log-returns of the Standard & Poor's 500 stock market index.

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Section: Introductionmentioning

confidence: 99%

Inference for heavy tailed stationary time series based on sliding blocks

Bücher¹,

Segers²

2018

Electron. J. Statist.

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“…Via (3.1), it is possible to transform between (a r , b r ) and (ã r , br ) if the extremal index θ is known or estimated. Regarding the estimation of the extremal index, a large variety of estimators has been proposed, which may itself be grouped into four categories: 1) BM-like estimators based on "blocking" techniques (Northrop, 2015;Berghaus and Bücher, 2017), 2) POT-like estimators that rely on threshold exceedances (Ferro and Segers, 2003;Süveges, 2007), 3) estimators that use both principles simultaneously (Hsing, 1993;Robert, 2009;Robert, Segers and Ferro, 2009) and 4) estimators which, next to choosing a threshold sequence, require the choice of a run-length parameter (Smith and Weissman, 1994;Weissman and Novak, 1998). Since the distance between the time points at which the maxima within two successive blocks are attained is likely to be quite large, the sample X BM can be regarded as approximately independent.…”

Section: The Bm Approach For Time Seriesmentioning

confidence: 99%

A horse racing between the block maxima method and the peak-over-threshold approach

Bücher¹,

Zhou²

2018

Preprint

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Classical extreme value statistics consists of two fundamental approaches: the block maxima (BM) method and the peak-over-threshold (POT) approach. It seems to be general consensus among researchers in the field that the POT method makes use of extreme observations more efficiently than the BM method. We shed light on this discussion from three different perspectives. First, based on recent theoretical results for the BM approach, we provide a theoretical comparison in i.i.d. scenarios. We argue that the data generating process may favour either one or the other approach. Second, if the underlying data possesses serial dependence, we argue that the choice of a method should be primarily guided by the ultimate statistical interest: for instance, POT is preferable for quantile estimation, while BM is preferable for return level estimation. Finally, we discuss the two approaches for multivariate observations and identify various open ends for future research.

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Weak convergence of a pseudo maximum likelihood estimator for the extremal index

Cited by 2 publications

References 0 publications

Inference for heavy tailed stationary time series based on sliding blocks

Inference for heavy tailed stationary time series based on sliding blocks

A horse racing between the block maxima method and the peak-over-threshold approach

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