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arxiv: 2502.15036 · v3 · submitted 2025-02-20 · 🧮 math.ST · stat.TH

Extreme Value Analysis based on Blockwise Top-Two Order Statistics

Axel B\"ucher , Erik Haufs This is my paper

Pith reviewed 2026-05-23 02:25 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords extreme value theoryblock maximaorder statisticstime seriesbias correctionreturn levelsconsistency
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The pith

A bias-corrected estimator from blockwise top-two order statistics consistently estimates extreme-value parameters in time series and improves efficiency over standard block maxima methods.

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

The paper establishes that the parameters of the limiting extreme-value distribution appear not only in the block maxima but also in the joint limits of the top two order statistics within each block. A direct likelihood approach treating these as independent fails to be consistent under time series dependence, so the authors introduce an explicit bias correction that restores consistency. Theoretical analysis and simulations then show the corrected estimator outperforms the classical block maxima approach when the target is a large return level or return period. This matters because environmental and financial applications often rely on accurate tail estimates from dependent data where collecting more blocks is costly. The approach keeps the block structure intact while extracting extra information from the second-largest observation in each block.

Core claim

The target parameters of the extreme-value distribution also appear in the limiting joint distribution of the top-two order statistics per block. Maximizing an independence log-likelihood based on these statistics produces an inconsistent estimator under typical time-series dependence, but an explicit bias correction yields a consistent estimator that is asymptotically more efficient than the classical block-maxima estimator for functionals such as high return levels.

What carries the argument

The bias-corrected maximum-likelihood estimator constructed from the joint limiting distribution of the blockwise top-two order statistics.

If this is right

  • The estimator remains consistent for the extreme-value parameters under standard mixing conditions on the time series.
  • It achieves lower asymptotic variance than the block-maxima estimator when estimating high quantiles or return periods.
  • Finite-sample performance improves for moderate block sizes in both independent and dependent data.
  • The method extends the classical block-maxima framework without requiring a change in block length or data collection.

Where Pith is reading between the lines

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

  • The same correction idea might apply to the top three or top k order statistics per block for further variance reduction.
  • Practitioners estimating return levels from annual maxima could reprocess existing data to extract the second-largest value and obtain tighter intervals.
  • The approach could be combined with existing declustering techniques when dependence within blocks is strong.

Load-bearing premise

The joint limiting distribution of the top two order statistics within blocks must match the form implied by the extreme-value parameters under the dependence present in the time series.

What would settle it

A simulation study in which the proposed estimator fails to converge in probability to the true extreme-value parameters as the number of blocks grows, while the classical block-maxima estimator does converge, would falsify the consistency claim.

Figures

Figures reproduced from arXiv: 2502.15036 by Axel B\"ucher, Erik Haufs.

Figure 1
Figure 1. Figure 1: Different ρ functions. The examples ‘linear’, ‘power’ and ‘ARMAX’ correspond to Example 2.3 [a] (c = 0.6), [b] (c = 0.4) and [c] (c = 0.6), respectively. to the case of Fr´echet marginals, we are interested in fitting the standard Fr´echet-Welsch distribution SW(α, σ) to z. In view of its absolute continuity, we may rely on standard maximum likelihood estimation, with the respective independence log-likeli… view at source ↗
Figure 2
Figure 2. Figure 2: Left: graph of ρ0 7→ ϖρ0 . Right: graph of its derivative. with α1 = α0 if and only if ρ ∈ {ρ⊥⊥, ρpd}. It will turn out that the ML estimator for α0 converges to α1 in probability; it is hence inconsistent unless ρ ∈ {ρ⊥⊥, ρpd}. We now make the required convergence of empirical moments more precise. For 0 < α− < α+ < ∞, consider the class of functions from (0, ∞) 2 into R defined as F1(α−, α+) := {(x, y) 7… view at source ↗
Figure 3
Figure 3. Figure 3: Standardized asymptotic variance of shape (left) and scale (right) estimators, that is, the diagonal entries of the asymptotic covariance matrices Σ(mb) TopTwo(1, ρ) and Σ (mb) max (1) from (4.14) and (5.10), respectively, at α0 = 1 and as a function of ρ0. The examples “linear”, “power” and “kink” correspond to Example 2.3 [a], [b] and [c], respectively. For the disjoint blocks version, the respective cur… view at source ↗
Figure 4
Figure 4. Figure 4: Standardized asymptotic bias of shape (left) and scale (right) estimators as a func￾tion of ρ0, for α0 = 1 and λ1 = 1. More precisely, the depicted values correspond to the mean of the asymptotic distributions of √ kn(αen −α0) and √ kn(σen/σn −1), respectively, under the assumption that √ kn/rn = λ1 + o(1) for n → ∞. 101 102 r 10−2 AMSE AMSE(α˜n) vs r, ρ0 = 0.2 max db max sb tt db tt sb 101 102 r AMSE(α˜n)… view at source ↗
Figure 5
Figure 5. Figure 5: Asymptotic MSE of αe (mb) n as a function of the block size r, for fixed α0 = 1, n = 1000 and three choices of ρ(η) = c · (1 − η), c ∈ {0.2, 0.5, 0.9}. extreme-value distribution based on block maxima has also been considered in [Dom15; FH15; OZ20]. Because of the serial dependence, the conditions from the previous section can be simplified considerably. For instance, weak convergence of the two largest or… view at source ↗
Figure 6
Figure 6. Figure 6: Asymptotic expansions in the IID case. Top row: AMSE(ˆα (mb) n ) as a function of the block size r, for fixed α0 = 5, c = −1, ¯τ = 1, and three sample sizes. Middle row: minr AMSE(ˆα (mb) n ) for fixed n = 1000 and as a function of ¯τ (left), c (middle) and α0 (right), keeping the other parameters fixed at α0 = 5, c = −1, τ¯ = 1 where applicable. Bottom row: AMSE(ˆα (mb) n ) as a function of the number of … view at source ↗
Figure 7
Figure 7. Figure 7: Estimation of the shape parameter α0 for fixed block size r = 100. Top row: mean squared error. Bottom row: relative mean squared error with respect to the disjoint block maxima estimator, MSE(·)/MSE(ˆα (db) max). The results are summarized in [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Estimation of RL(100, 100), that is, the 100-block return level with fixed block size r = 100. Top row: mean squared error. Bottom row: relative mean squared error with respect to the disjoint block maxima estimator, MSE(·)/MSE(RLc (db) max). The all block maxima estimator is not displayed on the right-hand side. as it is outside the range approach to assess the bootstrap error distribution, i.e., the dist… view at source ↗
Figure 9
Figure 9. Figure 9: Estimation of α0 for fixed n = 10 000. −0.4 −0.2 0.0 0.2 0.4 Error 0.0 0.5 1.0 1.5 2.0 Density Circular Bootstrap, Shape True error distribution Bootstrap error distribution True error KDE Bootstrap error KDE −10 −5 0 5 10 Error Circular Bootstrap, RL [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Histograms of estimation error (blue) and (circular block) bootstrap estimation errors (green) together with associated kernel density estimates. Left: shape estimation. Right: RL(100,100)-estimation. return level RL(365, 100) of about 112mm. Respective results for the max-only and the top-two estimators can be found in [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Left: annual top-two sample of daily precipitation amounts at K¨oln-Stammheim. Right: The estimated mapping T 7→ RLc botw(365, T) together with its bootstrap confidence region. Furthermore, it is demonstrated that taking into account overlapping ‘sliding’ blocks leads to even more efficient estimators. As the estimator’s asymptotic variance is unknown in practice, the adaptation of a circular bootstrap ap… view at source ↗
Figure 12
Figure 12. Figure 12: Revisiting the left-hand side of [PITH_FULL_IMAGE:figures/full_fig_p081_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Revisiting the left-hand side of [PITH_FULL_IMAGE:figures/full_fig_p081_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Asymptotic MSE for different choices of ρ0, as a function of the block size r for a fixed total sample size n = 1000. Top: AMSE( ˜αn), bottom: AMSE(˜σn). 101 102 103 k = n/r 10−3 10−2 AMSE AMSE(α˜n), r = 30 max db max sb tt db tt sb 101 102 103 k = n/r AMSE(α˜n), r = 90 101 10 2 10 3 k = n/r AMSE(α˜n), r = 365 101 102 103 k = n/r 10−3 10−2 10−1 AMSE AMSE(σ˜n), r = 30 max db max sb tt db tt sb 101 102 103 … view at source ↗
Figure 15
Figure 15. Figure 15: Asymptotic MSE for different choices of (fixed) block size r = 30, 90, 365, as a function of the effective sample size k, fixed ρ0 = 0.5. Top: AMSE(˜αn), bottom: AMSE(˜σn). 82 [PITH_FULL_IMAGE:figures/full_fig_p082_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Simulated (1,000 repetitions) vs. theoretical rescaled variance of the disjoint blocks top-two and max-only estimators for the shape and scale parameter. The effective sample size is k = 200 and the block size is r = 100. 0.00 0.25 0.50 0.75 1.00 ρ0 0.000 0.005 0.010 0.015 Bias Bias of α˜n TT Max Max (theoretical) TT (theoretical) 0.00 0.25 0.50 0.75 1.00 ρ0 0.000 0.005 0.010 0.015 0.020 0.025 Bias Bias o… view at source ↗
Figure 17
Figure 17. Figure 17: Simulated (1,000 repetitions) vs. theoretical rescaled bias of the disjoint blocks top-two and max-only estimators for the shape and scale parameter. The effective sample size is k = 200 and the block size is r = 100. G.2. Bias correction In this section we study the effect of the additional estimation step needed for the bias￾correction. We only consider the iid model and the ARMAX model, for which we kn… view at source ↗
Figure 18
Figure 18. Figure 18: Shape estimation based on estimated bias correction (black) and oracle bias cor￾rection (red). Top row: r = 50 and r ′ = 25. Bottom row: r = 100 and r ′ = 50. r ′ = 25 and r ′ = 50, respectively; see Section 4.3 for the definition of r ′ . It can be seen that the oracle and the estimator perform quite similar, with small advantages for the estimated bias correction in some of the models. G.3. Further resu… view at source ↗
Figure 19
Figure 19. Figure 19: Scale estimation for fixed block size r = 100. The estimation variance is shown here. mostly consistent with those presented in Section 6.1: unless the serial dependence is very strong, the top-two sliding estimator is best for shape estimation and the botw-estimator is best for return level estimation. For very strong serial dependence, the sliding max-only estimator wins. This can be explained by the fa… view at source ↗
Figure 20
Figure 20. Figure 20: Shape estimation for the iid model with fixed block size. Top row: mean squared error. Bottom row: relative mean squared error with respect to the disjoint block maxima estimator, MSE(·)/MSE(ˆα (db) max). 10−3 10−2 MSE r = 50 tt, db tt, sb max, db max, sb max, ab r = 100 r = 200 200 400 Number of blocks 0.3 0.4 0.6 1 2 Relative MSE 200 400 Number of blocks 200 400 Number of blocks [PITH_FULL_IMAGE:figure… view at source ↗
Figure 21
Figure 21. Figure 21: Shape estimation for the AR(0.5)-model with fixed block size. Top row: mean squared error. Bottom row: relative mean squared error with respect to the disjoint block maxima estimator, MSE(·)/MSE(ˆα (db) max). 86 [PITH_FULL_IMAGE:figures/full_fig_p086_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Shape estimation for fixed block size r = 100. The curves represent the rela￾tive mean squared error with respect to the disjoint block maxima estimator, MSE(·)/MSE(ˆα (db) max). Top row: AR-models. Bottom row: ARMAX-models. The ABM estimator is only depicted on the left, as it is otherwise outside the plotting range. 0.6 0.7 0.8 0.9 1 1.2 Relative MSE AR(0.2) tt, db tt, sb max, db max, sb BOTWE AR(0.5) A… view at source ↗
Figure 23
Figure 23. Figure 23: Return level estimation for fixed block size r = 100 and for T = 100. The curves represent the relative mean squared error with respect to the disjoint block maxima estimator, MSE(·)/MSE(RLc (db) max). Top row: AR-models. Bottom row: ARMAX-models. 87 [PITH_FULL_IMAGE:figures/full_fig_p087_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Return level estimation for the AR(0.5)-model with fixed block size r = 100 and T ∈ {50, 100, 200}. 25 50 100 150 200 250 300 350 400 450 500 number of blocks 500 450 400 350 300 250 200 150 100 50 25 block size tt, db max, sb max, db max, ab 2550 100 150 200 250 300 350 400 450 500 number of blocks 500 450 400 350 300 250 200 150 100 50 25 block size 25 50 100 150 200 250 300 350 400 450 500 number of bl… view at source ↗
Figure 25
Figure 25. Figure 25: Shape estimation in the iid model for various combinations of the block size and the number of blocks ranging from 25 to 500. Depicted is the relative MSE, i.e., the MSE of the estimator indicated on right divided by the MSE of the estimator indicated at the top. Red color means that the top estimator performs better. 88 [PITH_FULL_IMAGE:figures/full_fig_p088_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Shape estimation in the AR(0.5)-model for various combinations of the block size and the number of blocks ranging from 25 to 500. Depicted is the relative MSE, i.e., the MSE of the estimator indicated on right divided by the MSE of the estimator indicated at the top. Red color means that the top estimator performs better. 89 [PITH_FULL_IMAGE:figures/full_fig_p089_26.png] view at source ↗
read the original abstract

Extreme value analysis for time series is often based on the block maxima method, in particular for environmental applications. In the classical univariate case, the latter is based on fitting an extreme-value distribution to the sample of (annual) block maxima. Mathematically, the target parameters of the extreme-value distribution also show up in limit results for other high order statistics, which suggests estimation based on blockwise large order statistics. It is shown that a naive approach based on maximizing an independence log-likelihood yields an estimator that is inconsistent in general. A consistent, bias-corrected estimator is proposed, and is analyzed theoretically and in finite-sample simulation studies. The new estimator is shown to be more efficient than traditional counterparts, for instance for estimating large return levels or return periods.

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.

Referee Report

2 major / 2 minor

Summary. The paper proposes using blockwise top-two order statistics for extreme value analysis of time series, rather than only block maxima. It shows that a naive estimator maximizing an independence log-likelihood is inconsistent in general, derives a bias correction from limit results for order statistics to obtain a consistent estimator, analyzes its theoretical properties, and demonstrates via finite-sample simulations that it is more efficient than standard block-maxima approaches for estimating large return levels and return periods.

Significance. If the joint limiting distribution assumption holds under the paper's dependence conditions, the bias-corrected estimator could improve efficiency by extracting additional information from the second-largest value per block without introducing extra parameters, offering a practical advance for environmental time-series applications where block maxima alone are inefficient.

major comments (2)
  1. [derivation of bias correction (following abstract claim on limit results for other high order statistics)] The consistency proof for the bias-corrected estimator requires that the joint asymptotic distribution of the top-two order statistics per block depends on the extreme-value parameters and known constants only, without residual unknown dependence features from the time series mixing or clustering structure. This is invoked via general limit results for high-order statistics but needs explicit verification that the correction formula restores consistency under the serial dependence assumed in the model; without this, the central claim that the estimator is consistent does not follow.
  2. [simulation studies] Simulation designs must be checked to ensure they do not inadvertently satisfy the joint tail dependence assumption used for the correction; if the data-generating processes used in §5 are such that the top-two joint limit matches the independence case or a specific form, the reported efficiency gains may not generalize to the broader class of dependent series where the naive estimator is inconsistent.
minor comments (2)
  1. [introduction and notation] Clarify notation for the blockwise top-two statistics (e.g., denote the second order statistic explicitly) to avoid ambiguity when referring to the joint distribution.
  2. [theoretical analysis] Add a reference or brief derivation sketch for the specific joint limit law of the top two under the mixing conditions used, even if citing a standard result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

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

read point-by-point responses

  1. Referee: The consistency proof for the bias-corrected estimator requires that the joint asymptotic distribution of the top-two order statistics per block depends on the extreme-value parameters and known constants only, without residual unknown dependence features from the time series mixing or clustering structure. This is invoked via general limit results for high-order statistics but needs explicit verification that the correction formula restores consistency under the serial dependence assumed in the model; without this, the central claim that the estimator is consistent does not follow.

    Authors: We appreciate this observation. Under the mixing and clustering conditions of the manuscript, the referenced general limit theorems for high-order statistics ensure the joint limiting distribution of the top-two per block is determined solely by the extreme-value parameters and the extremal index (a known model constant). The bias correction is derived directly from this distribution. To address the request for explicit verification, the revised manuscript will include an added remark in Section 3 (or a short appendix) confirming that the correction eliminates inconsistency without residual unknown dependence terms. revision: partial

  2. Referee: Simulation designs must be checked to ensure they do not inadvertently satisfy the joint tail dependence assumption used for the correction; if the data-generating processes used in §5 are such that the top-two joint limit matches the independence case or a specific form, the reported efficiency gains may not generalize to the broader class of dependent series where the naive estimator is inconsistent.

    Authors: We agree this verification is important. The DGPs in Section 5 are stationary dependent processes (with clustering) chosen precisely so that the naive independence-likelihood estimator is inconsistent, consistent with the theoretical results. The joint limits follow the general dependent form rather than independence. In the revision we will add a short confirmation (e.g., a table or paragraph) showing that the chosen DGPs produce inconsistency of the naive estimator, supporting generalizability of the reported efficiency gains. revision: yes

Circularity Check

0 steps flagged

No circularity: estimator derived from asymptotic limit theorems for order statistics

full rationale

The paper starts from known limit results in which EV parameters appear for high-order statistics, shows via direct argument that the independence-likelihood estimator is inconsistent under serial dependence, then constructs a bias correction whose form follows from those same limits. No step reduces the final estimator to a re-expression of fitted values, no load-bearing self-citation is invoked, and the consistency claim rests on external asymptotic theory rather than on the estimator itself. The derivation is therefore self-contained against the stated limit theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on standard extreme-value limit theorems for order statistics within blocks; no free parameters, invented entities, or ad-hoc axioms are identifiable from the provided text.

axioms (1)
  • domain assumption Limit theorems for high-order statistics in extreme value theory hold for the underlying time series and allow the extreme-value parameters to appear in the joint limiting distribution of the top two order statistics per block.
    Invoked in the abstract to justify that the top-two statistics carry information about the same limiting distribution as the block maxima.

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Reference graph

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