All Block Maxima method for estimating the extreme value index

Chen Zhou; Jochem Oorschot

arxiv: 2010.15950 · v2 · submitted 2020-10-29 · 🧮 math.ST · stat.TH

All Block Maxima method for estimating the extreme value index

Jochem Oorschot , Chen Zhou This is my paper

Pith reviewed 2026-05-24 13:41 UTC · model grok-4.3

classification 🧮 math.ST stat.TH

keywords extreme value indexblock maximaall block maxima estimatorasymptotic variancegeneralized extreme value distributiontail empirical processpermutation invariance

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The pith

The All Block Maxima estimator for the extreme value index is permutation invariant and attains the lowest asymptotic variance among block maxima estimators.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the All Block Maxima (ABM) estimator, which fits the generalized extreme value distribution using maxima from every possible block in the sample. This construction makes the estimator unchanged under any reordering of the observations. Asymptotic theory establishes that the ABM estimator achieves the smallest variance among all estimators that rely on the block maxima approach. Simulation experiments match the predicted variance reduction. The proofs rest on new asymptotic expansions for the tail empirical process that incorporate weights on higher-order statistics.

Core claim

The ABM estimator is obtained by using all potential blocks from the sample to fit the GEV distribution, leading to an estimator that is invariant under permutations of the sample and has the lowest asymptotic variance among all BM-based estimators for the extreme value index.

What carries the argument

The All Block Maxima (ABM) estimator, which maximizes the likelihood using every possible block maximum of fixed length drawn from the sample.

If this is right

The ABM estimator is asymptotically normal with an explicit variance formula smaller than that of competing BM estimators.
The estimator remains consistent and asymptotically normal under standard second-order conditions on the tail.
Permutation invariance removes the need to choose a specific blocking scheme or ordering.
Simulation results confirm that the finite-sample behavior aligns with the asymptotic variance advantage.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The approach could improve efficiency when the total number of observations is modest but many overlapping blocks can still be formed.
The weighted tail-process expansions developed here may apply directly to other weighted statistics in extreme-value theory.
Implementation would require care with computational cost when the number of blocks grows quadratically with sample size.

Load-bearing premise

The sample distribution permits asymptotic expansions for the tail empirical process based on higher order statistics with weights.

What would settle it

A Monte Carlo experiment with large samples in which the empirical variance of the ABM estimator exceeds the variance of any other block-maxima estimator would falsify the minimal-variance claim.

Figures

Figures reproduced from arXiv: 2010.15950 by Chen Zhou, Jochem Oorschot.

**Figure 2.** Figure 2: Single sample performance, Student-t(ν = 2) 0 00 00 00 00 000 0 00 0 00 0 00 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Figure 3: MSE for ABM and BM estimator (a) Student-t (ν = 2) [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

**Figure 4.** Figure 4: Bias for ABM and BM estimator (a) Student-t (ν = 2) [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗

**Figure 8.** Figure 8: MSE under GARCH (a) λ1 = 0.08, λ2 = 0.91 [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗

**Figure 9.** Figure 9: MSE under Scale Heterogeneity (a) r = 2 [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗

read the original abstract

The block maxima (BM) approach in extreme value analysis fits a sample of block maxima to the Generalized Extreme Value (GEV) distribution. We consider all potential blocks from a sample, which leads to the All Block Maxima (ABM) estimator. Different from existing estimators based on the BM approach, the ABM estimator is permutation invariant. We show the asymptotic behavior of the ABM estimator, which has the lowest asymptotic variance among all estimators using the BM approach. Simulation studies justify our asymptotic theories. A key step in establishing the asymptotic theory for the ABM estimator is to obtain asymptotic expansions for the tail empirical process based on higher order statistics with weights.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

ABM is a new permutation-invariant estimator that uses every possible block maximum, claiming lowest asymptotic variance among BM methods, but the optimality rests on expansions for the weighted tail process under overlapping dependence.

read the letter

The paper introduces the All Block Maxima estimator, which computes the tail index from maxima over all possible blocks rather than a single partition into non-overlapping blocks. This makes the estimator invariant to sample order, a property the usual BM estimators lack. The authors derive its limiting distribution and state that it achieves the smallest asymptotic variance among all estimators that stay within the block-maxima framework. Simulations are included to check the asymptotics in finite samples. That is the concrete contribution: a different way to aggregate the block information that buys invariance and, they argue, efficiency. The approach is straightforward to implement once the blocks are enumerated, and the invariance property is immediately useful when the data order is arbitrary or when one wants to avoid arbitrary choices of block starts. The soft spot is the justification for the limiting variance. The abstract identifies the key technical step as asymptotic expansions of the tail empirical process based on weighted higher-order statistics. Overlapping blocks create dependence among the maxima, so the standard independent-block arguments do not apply directly. If the paper supplies a rigorous expansion that accounts for this dependence, the variance comparison stands; if the dependence is handled only heuristically or left implicit, the claim that ABM is optimal among BM estimators does not go through. The simulations can only confirm behavior under the model, not the validity of the expansion itself. This is a paper for researchers already working inside extreme-value theory who care about block-maxima estimators and their efficiency. It is not aimed at a broader audience. The work is coherent on its own terms and engages the literature directly, so it deserves a serious referee who can check the expansion in detail rather than a desk rejection.

Referee Report

2 major / 2 minor

Summary. The paper introduces the All Block Maxima (ABM) estimator for the extreme value index, constructed by using all possible blocks from a sample rather than disjoint blocks. It claims the estimator is permutation invariant, derives its asymptotic behavior via expansions of the weighted tail empirical process on higher-order statistics, and asserts that this yields the lowest asymptotic variance among all block-maxima-based estimators. Simulation studies are cited to support the asymptotic results.

Significance. If the asymptotic expansions and resulting variance comparison hold under the dependence induced by overlapping blocks, the ABM estimator would represent a theoretically optimal choice within the block maxima framework, offering improved efficiency for extreme value index estimation while maintaining permutation invariance.

major comments (2)

[Abstract and asymptotic theory] Abstract (key step paragraph) and the asymptotic theory section: the claim of lowest asymptotic variance among BM estimators rests entirely on the limiting distribution obtained from the asymptotic expansions of the tail empirical process based on higher-order statistics with weights. No derivation, error bounds, or explicit handling of the dependence structure from overlapping blocks is provided in the available text, which is load-bearing for both the variance formula and the optimality comparison.
[Asymptotic theory] The manuscript states that standard independent-block expansions do not apply directly due to overlapping blocks, yet the justification for the weighted higher-order expansion under this dependence is not detailed; this gap prevents verification of the central minimal-variance result.

minor comments (2)

[Abstract] The abstract mentions simulation studies justifying the theories, but no details on sample sizes, block lengths, or comparison estimators are visible in the provided text.
[Introduction] Notation for the weights in the tail empirical process and the precise definition of the ABM estimator should be introduced earlier for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below.

read point-by-point responses

Referee: [Abstract and asymptotic theory] Abstract (key step paragraph) and the asymptotic theory section: the claim of lowest asymptotic variance among BM estimators rests entirely on the limiting distribution obtained from the asymptotic expansions of the tail empirical process based on higher-order statistics with weights. No derivation, error bounds, or explicit handling of the dependence structure from overlapping blocks is provided in the available text, which is load-bearing for both the variance formula and the optimality comparison.

Authors: We agree that the current manuscript presents the asymptotic expansions and resulting variance comparison in a condensed form without a full step-by-step derivation or explicit error bounds for the overlapping-blocks dependence. The expansions are obtained by adapting the theory of weighted tail empirical processes for higher-order statistics, with the dependence structure incorporated via the limiting covariance of the process induced by the overlapping maxima. In the revision we will add a dedicated appendix containing the detailed derivation, including the covariance calculations and error bounds, to support verification of the minimal-variance claim. revision: yes
Referee: [Asymptotic theory] The manuscript states that standard independent-block expansions do not apply directly due to overlapping blocks, yet the justification for the weighted higher-order expansion under this dependence is not detailed; this gap prevents verification of the central minimal-variance result.

Authors: The justification proceeds by replacing the independent-block assumption with the joint distribution of the overlapping block maxima and deriving the corresponding weighted empirical process limit; however, we acknowledge that this step is not expanded sufficiently in the text. We will revise the asymptotic theory section (and add supporting lemmas in an appendix) to provide the explicit justification under dependence, thereby allowing direct verification of the central result. revision: yes

Circularity Check

0 steps flagged

No circularity: central claims rest on newly derived asymptotic expansions for weighted tail empirical processes, not on self-definition, fitted inputs renamed as predictions, or load-bearing self-citations.

full rationale

The paper constructs the ABM estimator from all possible blocks and derives its limiting distribution and minimal-variance property directly from fresh asymptotic expansions of the tail empirical process on higher-order statistics with weights. These expansions are presented as the key technical contribution required to handle overlapping-block dependence; they are not obtained by fitting parameters to the target quantity, by renaming known results, or by invoking prior self-citations whose content reduces to the present claim. The abstract and reader's summary explicitly flag the expansions as an independent step rather than a tautology or imported uniqueness theorem. Consequently the optimality comparison among BM estimators follows from the new limiting variance formula rather than from any definitional reduction. This places the work in the default non-circular category.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the work rests on standard extreme value theory assumptions not detailed here.

pith-pipeline@v0.9.0 · 5630 in / 941 out tokens · 19198 ms · 2026-05-24T13:41:57.187059+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Extreme Value Analysis based on Blockwise Top-Two Order Statistics
math.ST 2025-02 unverdicted novelty 6.0

A consistent bias-corrected estimator based on blockwise top-two order statistics is developed for extreme value analysis after showing the naive independence-likelihood approach is inconsistent.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · cited by 1 Pith paper

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B \"u cher, A. and J. Segers (2018a). Inference for heavy tailed stationary time series based on sliding blocks. Electron. J. Statist.\/ 12\/ (1), 1098--1125

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B \"u cher, A. and J. Segers (2018b). Maximum likelihood estimation for the F r \'e chet distribution based on block maxima extracted from a time series. Bernoulli\/ 24\/ (2), 1427--1462

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Chernick, M. R., T. Hsing, and W. P. McCormick (1991). Calculating the extremal index for a class of stationary sequences. Advances in Applied Probability\/ 23\/ (4), 835--850

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de Haan, L. and A. Ferreira (2006). Extreme value theory: an introduction . Springer

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de Haan, L. and S. Resnick (1998). On asymptotic normality of the H ill estimator. Stochastic Models\/ 14\/ (4), 849--866

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Dombry, C. and A. Ferreira (2019). Maximum likelihood estimators based on the block maxima method. Bernoulli\/ 25\/ (3), 1690--1723

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de Haan, and D

Drees, H., L. de Haan, and D. Li (2003). On large deviation for extremes. Statistics & probability letters\/ 64\/ (1), 51--62

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Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Mathematica\/ 131 , 207--248

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Biard, C

Mefleh, A., R. Biard, C. Dombry, and Z. Khraibani (2019). Permutation bootstrap and the block maxima method. Communications in Statistics-Simulation and Computation\/ , forthcoming

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Segers, J. (2001). Extreme of a random sample: limit theorems and statistical applications . Ph.\ D. thesis, Katholieke Universiteit Leuven

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Sun, P. and C. Zhou (2014). Diagnosing the distribution of GARCH innovations. Journal of Empirical Finance\/ 29 , 287--303

work page 2014
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Wager, S. (2014). Subsampling extremes: From block maxima to smooth tail estimation. Journal of Multivariate Analysis\/ 130 , 335--353

work page 2014

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B \"u cher, A. and J. Segers (2017). On the maximum likelihood estimator for the generalized extreme-value distribution. Extremes\/ 20\/ (4), 839--872

work page 2017

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B \"u cher, A. and J. Segers (2018a). Inference for heavy tailed stationary time series based on sliding blocks. Electron. J. Statist.\/ 12\/ (1), 1098--1125

work page

[3] [3]

B \"u cher, A. and J. Segers (2018b). Maximum likelihood estimation for the F r \'e chet distribution based on block maxima extracted from a time series. Bernoulli\/ 24\/ (2), 1427--1462

work page

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Chernick, M. R., T. Hsing, and W. P. McCormick (1991). Calculating the extremal index for a class of stationary sequences. Advances in Applied Probability\/ 23\/ (4), 835--850

work page 1991

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Cs \"o rg \"o , M. and L. Horv \'a th (1993). Weighted approximations in probability and statistics . J. Wiley & Sons

work page 1993

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Cs \"o rgo, M. and P. R \'e v \'e sz (2014). Strong approximations in probability and statistics . Academic Press

work page 2014

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de Haan, L. and A. Ferreira (2006). Extreme value theory: an introduction . Springer

work page 2006

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de Haan, L. and S. Resnick (1998). On asymptotic normality of the H ill estimator. Stochastic Models\/ 14\/ (4), 849--866

work page 1998

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Dombry, C. and A. Ferreira (2019). Maximum likelihood estimators based on the block maxima method. Bernoulli\/ 25\/ (3), 1690--1723

work page 2019

[10] [10]

de Haan, and D

Drees, H., L. de Haan, and D. Li (2003). On large deviation for extremes. Statistics & probability letters\/ 64\/ (1), 51--62

work page 2003

[11] [11]

Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Mathematica\/ 131 , 207--248

work page 1973

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Biard, C

Mefleh, A., R. Biard, C. Dombry, and Z. Khraibani (2019). Permutation bootstrap and the block maxima method. Communications in Statistics-Simulation and Computation\/ , forthcoming

work page 2019

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Segers, J. (2001). Extreme of a random sample: limit theorems and statistical applications . Ph.\ D. thesis, Katholieke Universiteit Leuven

work page 2001

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Sun, P. and C. Zhou (2014). Diagnosing the distribution of GARCH innovations. Journal of Empirical Finance\/ 29 , 287--303

work page 2014

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Wager, S. (2014). Subsampling extremes: From block maxima to smooth tail estimation. Journal of Multivariate Analysis\/ 130 , 335--353

work page 2014