Bootstrap-based Inference for Bivariate Heteroscedastic Extremes with a Changing Tail Copula

Yanxi Hou; Yifan Hu

REVIEW 2 major objections 5 minor 13 references

Bootstrap-based Inference for Bivariate Heteroscedastic Extremes with a Changing Tail Copula

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A single empirical process makes bootstrap inference valid for bivariate extremes whose marginal tails and tail dependence both change.

desk verdict A solid, genuinely new bivariate tail-process framework; the third null hypothesis is mis-stated and must be corrected, and the proofs are in the supplement. read the letter →

arxiv 2411.15819 v2 pith:4GKPEQ7Y submitted 2024-11-24 stat.ME

Yifan Hu , Yanxi Hou This is my paper · ORCID

classification stat.ME MSC 62G3262G0962G20

keywords extremevaluetheoryheteroscedasticextremeschangingtailcopulabivariatesequentialempiricalprocessbootstrapscedasisfunctionHillestimatorfunctionaldeltamethod

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that one process, the bivariate sequential tail empirical process (B-STEP), carries valid bootstrap inference for data whose marginal tail heaviness and dependence between extremes both change across observations. The authors prove that the B-STEP and its multiplier-bootstrap version converge weakly to the same Gaussian process, then use the functional delta method to derive asymptotic normality for the quasi-tail copula, integrated scedasis functions, and Hill estimators. From these results, they build bootstrap tests for equal extreme value indices, equal scedasis functions, and an unchanging tail copula, and prove that the rejection probabilities converge to the nominal significance level. The payoff is simultaneous inference on marginal tail risk and changing dependence without needing to know the complicated asymptotic variances, which matters for climate and financial data where both features drift together.

What carries the argument

The B-STEP is the weighted empirical process $F_n(x,y,z) = (\tilde R'(x,y,z)-R'(x,y,z))/q(x,y)$, where $\tilde R'$ counts exceedances of high marginal thresholds up to time $\lfloor nz\rfloor$, the quasi-tail copula $R'(x,y,z) = \int_0^z R(c_1(t)x,c_2(t)y,t)\,dt$ is its expectation limit, and the weight $q(x,y)=(x\vee y)^\eta$ with $0\le\eta<1/2$ tames the tails. The bootstrap B-STEP replaces the counting indicators with $\xi_{bi}$-weighted indicators, where the $\xi_{bi}$ are iid positive multipliers with mean and variance one, so that margins and copula are resampled jointly. The proof routes through the functional delta method, using the Hadamard-differentiable maps $\Phi$ for the quasi-tail copula and $\Psi$ for the Hill estimator, making the process convergence transfer to all three estimators and their bootstrap counterparts.

What would settle it

Simulate a triangular array whose copulas approach the tail limit only logarithmically, for example $t C_{n,i}(x/t,y/t)=R(x,y,i/n)+(\log t)^{-1}$, and check whether the bootstrap tests still hold their nominal level as the sample size grows; this violates Assumption 2's uniform $t^{-\alpha}$ rate, so the central convergence should break.

Watch

Extended reading notes

Core claim

The central discovery is a functional limit theorem for non-identically distributed bivariate extremes: under Assumptions 1-5, the normalized B-STEP, $\sqrt{k}F_n$, converges weakly to $W/q$, and its bootstrap version $\sqrt{k}F_n^b$ converges conditionally given the data to the same $W/q$ in $\ell^\infty(D_T)$, where $W$ is a Gaussian process with covariance $R'(x_1\wedge x_2, y_1\wedge y_2, z_1\wedge z_2)$. This single convergence result is the engine of the paper. From it, the quasi-tail copula estimator, the integrated scedasis estimators, and the Hill estimators are shown to be asymptotically normal with explicit Gaussian limits, and the bootstrap versions of these estimators are asymptotically valid. Consequently, bootstrap-based Kolmogorov-Smirnov and Cramér-von Mises tests for equal extreme value indices, equal scedasis functions, and constant tail dependence have rejection probabilities converging to the nominal level as the sample size and number of bootstrap replications diverge.

Load-bearing premise

The true dependence between extreme events must converge to its limiting tail copula at a uniform polynomial rate across all observations and sample sizes, a condition that cannot be checked from the data and whose failure would break the central convergence theorem.

Editorial extensions

If this is right

Theorem 1 implies that every Hadamard-differentiable functional of the joint tail, not just the three estimators written out, can be bootstrapped under this model of changing marginal and dependence heterogeneity.
The three bootstrap tests are asymptotically of the nominal level; the simulations show size control at $k=200$ and rejection frequency rising in $k$ under alternatives.
For the equal-scedasis test, the bootstrap is necessary because $\hat C_1-\hat C_2$ does not converge to a Brownian bridge when the copula changes across samples; the bootstrap handles the unknown covariance.
For the non-changing-tail-copula test with identical scedasis functions, the proposed statistic converges to a Brownian bridge limit at root-$k$ rate, faster than integrated angular measures, so less data is needed than in the earlier test it generalizes.
When the two margins are asymptotically tail independent, the two Hill estimators become asymptotically independent, so the test for equal extreme value indices remains valid in that boundary case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same multiplier-bootstrap B-STEP could produce simultaneous confidence bands for the quasi-tail copula and for differences of scedasis functions, since the limiting Gaussian covariance is explicit.
A natural next step, which the authors leave open, is extending the process convergence to weakly dependent triangular arrays; the process-centric proof structure suggests block-multiplier versions would be the route.
Because the key convergence-rate condition on the true copula sequence is uncheckable, a practical diagnostic comparing bootstrap-based quantiles across several intermediate orders $k$ could reveal whether that rate assumption is credible for a given dataset.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know upfront. The paper does something real: it builds a bivariate sequential tail empirical process for independent, non-identically distributed extremes, lets the tail copula change across the sample, and proves weak convergence for the process and its bootstrap. That unifies marginal heterogeneity (scedasis) with dependence heterogeneity, which Einmahl-Zhou and Drees treat separately. The bootstrap tests for equal EVI, equal scedasis, and constant tail copula follow as functionals, and the simulations look reasonable.

The main soft spot is the formal statement of the third null. In (5.3), H30 is written as R'(x,y,z) = R(x,y,1) for all (x,y,z). That cannot be the intended null. Under a tail copula that does not depend on z, R'(x,y,z) = R(x,y,1) C(z) when the two scedasis functions are equal. So the stated H30 forces C(z) = z, i.e., c ≡ 1, and even then it only holds at z = 1. The test statistic itself is centered for the correct null — the ratio R'(k1/k, k2/k, z)/R'(k1/k, k2/k, 1) cancels the C(z) — so the test likely works as intended. But the hypothesis has to be restated as R(x,y,z) = R(x,y,1) for all z. As written, Proposition 3(c) is false because the condition is overstrong; a literal reader would get the wrong limit. That is a fixable but real error.

The other wart is Table 1: the text says rows 1–4 are equivalent EVI, but row 4 has lambda1 = 2.5 and lambda2 = 2. The same row is later treated as an alternative. That needs cleaning up.

I could not verify the core theorems: the proofs are in the supplementary material, not posted. So my confidence rests on the standard nature of the assumptions and the examples. Assumption 2's uniform polynomial rate is untestable but typical for this area.

The paper deserves a serious referee. The framework is a genuine step forward for heteroscedastic extremes, and the bootstrap results are useful. A referee should insist on a corrected H30, a fixed Table 1, and ideally a supplement that can be checked. I would send it to review, not desk-reject.

Referee Report

2 major / 5 minor

Summary. This paper develops a bivariate extension of heteroscedastic extremes for independent but non-identically distributed data, allowing both the marginal tails (via scedasis functions) and the copula (via a changing tail copula R(x,y,z)) to vary across the sample. The main theoretical object is a bivariate sequential tail empirical process (B-STEP) and its weighted bootstrap counterpart. The paper states weak convergence of both processes to the same Gaussian limit (Theorem 1), derives asymptotic distributions for the quasi-tail copula, integrated scedasis functions, and Hill estimators as functionals of the process (Theorems 2 and 3), and constructs bootstrap tests for equal extreme value indices, equal scedasis functions, and a constant tail copula under identical scedasis functions, together with a simulation study. The central proofs are deferred to the supplementary material, and the formal statement of the third null hypothesis in (5.3) is inconsistent with the intended null, which affects Proposition 3(c) as written.

Significance. The proposed framework is a natural and potentially useful unification of two strands of recent EVT literature (heteroscedastic margins and changing dependence), and the process-centric bootstrap approach is attractive because it avoids explicit estimation of complicated covariance structures. The paper also provides readily implementable tests and supplies simulation evidence. Its main strengths are the unifying B-STEP convergence result, the bootstrap construction that jointly resamples margins and copulas, and the explicit treatment of three testing problems. The main limitations are that the proofs of Theorems 1-3 are not in the posted text, so the central derivations could not be independently verified, and that the displayed null hypothesis in (5.3) does not correspond to the null actually used in the test statistic and simulations. These issues are correctable, but they need to be fixed before the paper can be accepted.

major comments (2)

[§5.3, Eq. (5.3) and Proposition 3(c)] The formal null hypothesis in (5.3) is R'(x,y,z) = R(x,y,1) for all (x,y,z) in D_T. Under the intended null (no change in the tail copula) and with c1=c2, the homogeneity of R gives R'(x,y,z) = \int_0^z R(c(t)x,c(t)y,t) dt = R(x,y,1) C(z), where C(z) is the common integrated scedasis function. Since C(z)<1 for z<1 when c is positive and C(1)=1, the displayed equality R'(x,y,z)=R(x,y,1) cannot hold for all z unless c is identically 1, which is far stronger than a constant tail copula. Under the literal H30, T30(z) converges in probability to 2C(z)-2, so sqrt(k) T30(z) diverges for z<1 and the rejection probability tends to 1, contradicting Proposition 3(c). The simulation evidence in Table 3 is consistent with the intended null (constant R, arbitrary c1=c2) rather than the displayed H30, since the rows with a1,a1,C5 and C6 have non-constant scedasis. The intended null should be stated as R(x,y,z)=R(x,y,1) for all z (equivalently, R'(x,y,z)=R(x,y,1)C(z) when C1=C2). This is not a purely typographical issue: the centering of T30 is derived from the intended null, and a reader implementing the displayed H30 will obtain invalid inference.
[§4, Theorems 1-3 and Remark 3] The main convergence results are stated without proof: Theorem 1 (weak convergence of the B-STEP and its bootstrap version), Theorem 2, and Theorem 3 are all deferred to the supplementary material. Since Proposition 3 and the simulation interpretations rest on these theorems, the posted text is not self-contained. In particular, the bias bound described in Remark 3 depends on Assumption 3 in a way that is not demonstrated in the main text, and the conditional weak convergence in (4.1) is only defined, not established. I could not verify from the posted material that Assumptions 1-5 indeed imply (4.3). Please ensure the supplementary proofs are included in the review package and are complete.

minor comments (5)

[§5.4, Table 1] The text states that rows 1-4 of Table 1 have equivalent EVIs, but row 4 has lambda1=2.5 and lambda2=2, which differ; the text later correctly refers to row 4 as an alternative case. The null rows are rows 1-3, and this inconsistency should be corrected.
[Assumption 5, Eq. (3.1)] The weight function q in (3.1) allows eta=0, but Assumption 5 requires E|1-xi|^{1/eta}<infty, which is undefined for eta=0. Either restrict eta>0 in (3.1) or state that the moment condition is vacuous when eta=0.
[§4, before Eq. (4.2)] The text says 'Weiner process'; this should be 'Wiener process'. Moreover, the process W defined by the covariance in (4.2) is not a classical Wiener process on the product space but a centered Gaussian process with that covariance, so the terminology should be adjusted.
[§5.1] The paragraph discussing R(1,1,z)=0 is outside the scope of Assumption 2, which requires R(1,1,z)>0. If the authors intend to cover asymptotic tail independence, this needs to be stated as a separate assumption or clearly marked as a heuristic discussion.
[§5.3] The limiting statement 'sup_{z in [0,1]} sqrt(k)|T30(z)| -> sup_{z in [0,1]} |{...} B(z)|' is asserted without derivation. Given the covariance structure in (4.2) and the process limits in Theorem 2, this is not immediate and should be proved or explicitly referenced in the supplementary material.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is a CLT + delta-method + bootstrap-consistency chain under explicit assumptions; the H30 null is mis-stated but that is a correctness issue, not circularity.

full rationale

Under Assumptions 1–5 the paper proves a functional CLT for the centered bivariate sequential tail empirical process, sqrt(k) F_n => W/q with cov W = R' (Theorem 1, Eq. 4.3), and then uses the delta method to transfer this to functionals (quasi-tail copula, integrated scedasis, Hill) and to a bootstrap process whose conditional limit is the same W/q. This is standard empirical-process theory: the limiting covariance is the probability limit of the empirical covariance, and the bootstrap is shown consistent by comparing it with the original process, not by fitting any parameter to the data. The test statistics in Section 5 are centered at their null targets and bootstrapped by recentering (T_b - T); asymptotic level alpha follows from the same CLT, not from calibration. I found no load-bearing self-citation: the cited external results (Einmahl–Zhou, Drees, Bücher–Dette, Einmahl et al.) are used as benchmarks or building blocks, and the paper's own theorems are proved in the supplement rather than imported from the authors' prior work. One non-circular correctness concern: the stated null in (5.3), H30: R'(x,y,z)=R(x,y,1), is stronger than 'the tail copula R does not depend on z' whenever the common scedasis c is not identically 1, since R' = integral_0^z R(c(t)x,c(t)y,t)dt then equals R(·,·,1)C(z) under the intended null. This makes the literal wording of H30 and hence Proposition 3(c) problematic if read formally, but it is a hypothesis-specification error, not a circular reduction, so it does not increase the circularity score.

Assumptions & free parameters 2 free parameters · 8 assumptions · 0 invented entities

The central claim rests on the assumed model for independent non-identically distributed data (marginal scedasis functions and a changing tail copula), on uniform rate and smoothness conditions for the copula convergence, on intermediate-order rate conditions, and on standard empirical process and bootstrap theory. No new physical entities are postulated; the quasi-tail copula is a derived functional, not an added entity.

free parameters (2)

intermediate order sequences k, k1, k2 = k=50, 100, 200 in simulations; in theory any sequence satisfying Assumption 4
Chosen by the analyst, not estimated. The asymptotic approximations and test power depend critically on k; small k leads to under-rejection in the reported simulations.
weight exponent eta in q(x,y) = not specified in simulations; constrained 0 <= eta < 1/2
Hand-chosen parameter controlling the weight function in the process domain and the moment condition on bootstrap weights in Assumption 5.

assumptions (8)

domain assumption Sklar's representation and the triangular-array model (1.1)-(1.3): each observation has survival copula C_{n,i} converging to R(x,y,i/n).
This is the data-generating model; the paper does not test it, and all theorems are conditional on it.
domain assumption Assumption 1: marginal scedasis functions c_j exist, are positive and continuous, with second-order von Mises condition on G_j and rates A_j and B_j.
Needed for the univariate STEP and Hill estimator asymptotics; inherited from Einmahl et al. (2014).
domain assumption Assumption 2: uniform O(t^{-alpha}) convergence of t C_{n,i}(x/t, y/t) to R(x,y,i/n), with R(1,1,z) > 0 and continuity of partial derivatives.
Controls the bias of B-STEP; this is the weakest and most fragile assumption.
domain assumption Assumption 3: smoothing conditions linking the mesh 1/n to sqrt(k) for c_j and R.
Technical condition ensuring the discretization bias vanishes; per Remark 3 it is not needed for the bootstrap limit.
domain assumption Assumption 4: k, k1, k2 are intermediate orders with k/n -> 0, k/k_j -> s_j, and rates sqrt(k) A_j(n/k) -> 0, sqrt(k) B_j(n/k) -> 0, sqrt(k)(n/k)^{-alpha} -> 0.
Defines the admissible threshold sequences; the choice of k matters for practical performance.
domain assumption Assumption 5: bootstrap weights satisfy E(ξ)=1, E(ξ-1)^2=1, and E|1-ξ|^{1/eta} < infinity.
Standard multiplier bootstrap condition ensuring the bootstrap variance matches the process variance.
standard math Functional delta method and weak convergence in ell-infinity(D_T), including conditional bootstrap weak convergence theory from van der Vaart and Wellner (1996) and Kosorok (2003).
Used to turn process convergence into estimator asymptotics.
standard math Hadamard differentiability of the functionals Φ and Ψ (Propositions 1 and 2).
Proofs are deferred to the supplementary material; if these differentiability results fail, the delta-method results for the estimators would not follow.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Bootstrap-based Inference for Bivariate Heteroscedastic Extremes with a Changing Tail Copula." pith.science (2026). https://pith.science/paper/4GKPEQ7Y

@misc{pith2026241115819,
  author       = {Pith},
  title        = {Pith review of: Bootstrap-based Inference for Bivariate Heteroscedastic Extremes with a Changing Tail Copula},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4GKPEQ7Y}},
  note         = {Machine review of arXiv:2411.15819}
}

read the original abstract

This paper introduces a copula-based model for independent but non-identically distributed data with heteroscedastic extremes marginal and changing tail dependence structures. We establish a unified framework for inference by proving the weak convergence of the bivariate sequential tail empirical process and its empirical bootstrap counterpart. We derive the asymptotic properties of several estimators on the tail, including the quasi-tail copula, integrated scedasis function, and Hill estimator, treating them as functionals of the bivariate sequential tail empirical process. This process-centric approach enables the development of bootstrap-based methods and ensures the theoretical validity of the derived statistics. As an application of our inference method, we propose bootstrap-based tests for the equivalence of extreme value indices, the equivalence of scedasis functions, and non-changing tail dependence when marginal scedasis functions are identical. Our simulations validate the robustness and efficiency of the bootstrap-based tests.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 9 canonical work pages

[5]

URL https://doi.org/10.1080/01621459.2022.2120400

doi: 10.1080/01621459.2022.2120400 . URL https://doi.org/10.1080/01621459.2022.2120400. H. Drees. Statistical inference on a changing extreme value depen dence structure. The Annals of Statistics , 51(4):1824–1849,

arXiv 2022
[6]

URL https://doi.org/10.1007/978-1-4757-2545-2_6 . N. Zou, S. Volgushev, and A. B¨ ucher. Multiple block sizes and overla pping blocks for multi- variate time series extremes. Annals of Statistics , 49(1):295–320, 2021

work page doi:10.1007/978-1-4757-2545-2_6 2021
[7]

doi: https://doi.org/10.1016/j.ecosta.2 024.09.004

ISSN 2452-3062. doi: https://doi.org/10.1016/j.ecosta.2 024.09.004. URL https://www.sciencedirect.com/science/article/pii/S2452306224000698. J. H. J. Einmahl, L. Haan, and C. Zhou. Statistics of Heteroscedas tic Extremes. Journal of the Royal Statistical Society Series B: Statistical Meth odology, 78(1):31–51,

work page doi:10.1016/j.ecosta.2
[1996]

doi: 10.1007/978-1-4757-2545-2

ISBN 978-1-4757-2545-2. doi: 10.1007/978-1-4757-2545-2

doi:10.1007/978-1-4757-2545-2
[2003]

doi: https://doi.org/10.1016/S0047-259 X(02)00040-4

ISSN 0047-259X. doi: https://doi.org/10.1016/S0047-259 X(02)00040-4. URL https://www.sciencedirect.com/science/article/pii/S0047259X02000404. A. Meﬂeh, R. Biard, C. Dombry, and Z. Khraibani. Trend detection f or heteroscedastic ex- tremes. Extremes, 23(1):85–115,

doi:10.1016/s0047-259
[2013]

URL https://doi.org/10.3150/12-BEJ425

doi: 10.3150/12-BEJ425. URL https://doi.org/10.3150/12-BEJ425. A. B¨ ucher and T. Jennessen. Statistics for heteroscedastic tim e series extremes. Bernoulli, 30 (1):46–71,

work page doi:10.3150/12-bej425
[2014]

doi: 10.1111/rssb.12099

ISSN 1369-7412. doi: 10.1111/rssb.12099. URL https://doi.org/10.1111/rssb.12099. Y. He and J. H. J. Einmahl. Extreme value inference for general het erogeneous data. SSRN Electronic Journal,

work page doi:10.1111/rssb.12099
[2018]

URL https://doi.org/10.1214/18-EJS1415

doi: 10.1214/18-EJS1415. URL https://doi.org/10.1214/18-EJS1415. A. B¨ ucher and I. Kojadinovic. A Note on Conditional Versus Joint U ncondi- tional Weak Convergence in Bootstrap Consistency Results. Journal of Theo- retical Probability , 32(3):1145–1165,

work page doi:10.1214/18-ejs1415

Show all 13 references

[2019]

UR L https://ideas.repec.org/a/spr/jotpro/v32y2019i3d10.1007_s10959-018-0823-3.html

doi: 10.1007/s10959-018-0823-3. UR L https://ideas.repec.org/a/spr/jotpro/v32y2019i3d10.1007_s10959-018-0823-3.html. L. de Haan and C. Zhou. Bootstrapping extreme value estimators. Journal of the American Statistical Association, 119(545):382–393,

doi
[2020]

doi: 10.1007/s10687-01 9-00363-1

ISSN 1572-915X. doi: 10.1007/s10687-01 9-00363-1. URL https://doi.org/10.1007/s10687-019-00363-1 . A. W. van der Vaart and J. A. Wellner. Weak Convergence and Empirical Processes: With Applications to Statistics . Springer, New York, NY,

doi
[2021]

URL https://doi.org/10.3150/20-BEJ1279

doi: 10.3150/20-BEJ1279. URL https://doi.org/10.3150/20-BEJ1279. M. R. Kosorok. Bootstraps of sums of independent but not identic ally dis- tributed stochastic processes. Journal of Multivariate Analysis , 84(2):299–318,

doi
[2023]

URL https://doi.org/10.1214/23-AOS2294

doi: 10.1214/23-AOS2294. URL https://doi.org/10.1214/23-AOS2294. J. H. Einmahl and C. Zhou. Tail copula estimation for heteroscedas tic extremes. Econometrics and Statistics ,

doi
[2024]

URL https://doi.org/10.3150/22-BEJ1560

doi: 10.3150/22-BEJ1560. URL https://doi.org/10.3150/22-BEJ1560. A. B¨ ucher and J. Segers. Inference for heavy tailed stationary time series based on sliding blocks. Electronic Journal of Statistics , 12(1):1098–1125,

doi

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [5]

URL https://doi.org/10.1080/01621459.2022.2120400

doi: 10.1080/01621459.2022.2120400 . URL https://doi.org/10.1080/01621459.2022.2120400. H. Drees. Statistical inference on a changing extreme value depen dence structure. The Annals of Statistics , 51(4):1824–1849,

arXiv 2022

[2] [6]

URL https://doi.org/10.1007/978-1-4757-2545-2_6 . N. Zou, S. Volgushev, and A. B¨ ucher. Multiple block sizes and overla pping blocks for multi- variate time series extremes. Annals of Statistics , 49(1):295–320, 2021

work page doi:10.1007/978-1-4757-2545-2_6 2021

[3] [7]

doi: https://doi.org/10.1016/j.ecosta.2 024.09.004

ISSN 2452-3062. doi: https://doi.org/10.1016/j.ecosta.2 024.09.004. URL https://www.sciencedirect.com/science/article/pii/S2452306224000698. J. H. J. Einmahl, L. Haan, and C. Zhou. Statistics of Heteroscedas tic Extremes. Journal of the Royal Statistical Society Series B: Statistical Meth odology, 78(1):31–51,

work page doi:10.1016/j.ecosta.2

[4] [1996]

doi: 10.1007/978-1-4757-2545-2

ISBN 978-1-4757-2545-2. doi: 10.1007/978-1-4757-2545-2

doi:10.1007/978-1-4757-2545-2

[5] [2003]

doi: https://doi.org/10.1016/S0047-259 X(02)00040-4

ISSN 0047-259X. doi: https://doi.org/10.1016/S0047-259 X(02)00040-4. URL https://www.sciencedirect.com/science/article/pii/S0047259X02000404. A. Meﬂeh, R. Biard, C. Dombry, and Z. Khraibani. Trend detection f or heteroscedastic ex- tremes. Extremes, 23(1):85–115,

doi:10.1016/s0047-259

[6] [2013]

URL https://doi.org/10.3150/12-BEJ425

doi: 10.3150/12-BEJ425. URL https://doi.org/10.3150/12-BEJ425. A. B¨ ucher and T. Jennessen. Statistics for heteroscedastic tim e series extremes. Bernoulli, 30 (1):46–71,

work page doi:10.3150/12-bej425

[7] [2014]

doi: 10.1111/rssb.12099

ISSN 1369-7412. doi: 10.1111/rssb.12099. URL https://doi.org/10.1111/rssb.12099. Y. He and J. H. J. Einmahl. Extreme value inference for general het erogeneous data. SSRN Electronic Journal,

work page doi:10.1111/rssb.12099

[8] [2018]

URL https://doi.org/10.1214/18-EJS1415

doi: 10.1214/18-EJS1415. URL https://doi.org/10.1214/18-EJS1415. A. B¨ ucher and I. Kojadinovic. A Note on Conditional Versus Joint U ncondi- tional Weak Convergence in Bootstrap Consistency Results. Journal of Theo- retical Probability , 32(3):1145–1165,

work page doi:10.1214/18-ejs1415