Directional Dependence of Extreme Events

Matthieu Garcin; Maxime L. D. Nicolas

arxiv: 2604.03215 · v1 · submitted 2026-04-03 · 📊 stat.ME · stat.AP

Directional Dependence of Extreme Events

Matthieu Garcin , Maxime L. D. Nicolas This is my paper

Pith reviewed 2026-05-13 18:24 UTC · model grok-4.3

classification 📊 stat.ME stat.AP

keywords directional tail dependenceextreme value theoryconditional tail expectationrank transformationasymmetric dependencecausal inference in extremesoceanographic data

0 comments

The pith

Conditional tail expectations of rank-transformed variables measure directional dependence in extreme events.

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

The paper introduces a coefficient that quantifies how one variable responds when another reaches extreme values, after converting both to ranks to focus purely on dependence. The approach isolates asymmetry by comparing the two possible directions of conditioning, so that the coefficient from A to B can differ from the one from B to A. This matters for settings where extremes propagate unevenly, such as one ocean variable triggering spikes in another. The authors derive the asymptotic behavior of the estimator, validate it in simulations, and apply it to real oceanographic measurements that reveal clear directional dominance.

Core claim

The paper establishes that directional dependence between extremes is captured by the conditional tail expectation of one rank-transformed variable given that the other exceeds a high threshold; the resulting coefficient is asymmetric and therefore distinguishes which variable exerts greater influence on the extremes of the other.

What carries the argument

The directional extremal dependence coefficient obtained as the limiting conditional tail expectation of a uniform rank given the other rank exceeds 1-t as t approaches zero.

If this is right

The estimator converges at a usable rate, allowing statistical tests for whether one direction of dependence is significantly stronger.
Comparing the two directional coefficients supplies a criterion for detecting causal influence restricted to extreme events.
In applications the coefficient identifies dominant directions of extremal influence among measured variables.
Simulation experiments confirm that the measure recovers the correct asymmetry under a range of known dependence structures.

Where Pith is reading between the lines

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

The same construction could be applied to time-lagged pairs to detect leading versus lagging extreme influences.
Financial risk models might adopt the coefficient to quantify asymmetric contagion during market crashes.
Climate networks could use the measure to trace how extreme events in one region propagate to others.

Load-bearing premise

Rank transformation followed by conditional tail expectations removes all effects of marginal distributions and non-extremal dependence so that only directional tail behavior remains.

What would settle it

A controlled simulation with known asymmetric tail dependence in which the two directional coefficients are equal or point in the wrong direction even as sample size grows.

Figures

Figures reproduced from arXiv: 2604.03215 by Matthieu Garcin, Maxime L. D. Nicolas.

**Figure 1.** Figure 1: Asymptotic variance σ 2 C (v)/n (dashed curve), squared bias (grey curve), and quadratic risk (black curve) as functions of the threshold v. Top graphs correspond to the Clayton copula with θ = 0.2 for n = 100 (left) and n = 1,000 (right). Bottom graphs correspond to the FGM copula with θ = 1 for n = 100 (left) and n = 1,000 (right). Theorem 3.2 provides the asymptotic distribution of χb Y →X n (v). In the… view at source ↗

**Figure 2.** Figure 2: Simulated samples from Khoudraji’s device, of size [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

**Figure 3.** Figure 3: Tail dependence measures (from left to right [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

**Figure 4.** Figure 4: Tail dependence measures (from left to right [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗

**Figure 5.** Figure 5: Tail dependence measures (from left to right [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗

**Figure 6.** Figure 6: Simulated samples from the skew-t copula, of size n = 5,000, for different combinations of the skewness parameters α1 and α2, a number of degrees of freedom ν = 10, and a correlation coefficient ρ = 0.5. For each configuration, we generate 100 bivariate samples of size n = 1, 000, and compute the directional tail coefficients χb Y →X n (v) and χb X→Y n (v) as well as their difference χbn(X, Y )(v). The emp… view at source ↗

**Figure 7.** Figure 7: Bivariate behaviour of wind and wave variables from the National Data Buoy Center [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗

**Figure 8.** Figure 8: Network of significant DTD links between the variables of the ocean dataset using both [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗

**Figure 9.** Figure 9: Network of significant DTD links between the variables of the ocean dataset using the [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗

read the original abstract

This paper introduces a novel measure to quantify the directional dependence of extreme events between two variables. The proposed approach is designed to capture asymmetric tail dependence by studying conditional tail expectations of rank-transformed variables, thereby quantifying the behavior of one variable when the other takes extreme values. We investigate the theoretical asymptotic behavior of the associated estimator. The effectiveness of the approach is demonstrated through an extensive simulation study. In addition, we discuss the use of the proposed coefficient for the detection of causal effects in extreme events. Finally, we apply the method to an oceanographic dataset, where the results highlight the strong asymmetric nature of extreme events and identify the dominant directions of extremal influence among key oceanographic variables. As a directional measure of tail dependence, our approach provides a natural tool for exploring causal-effect relationships in extreme-value settings.

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

New directional tail dependence coefficient via rank-based conditional expectations targets asymmetry but risks mixing in marginal and sampling effects.

read the letter

The paper introduces a coefficient for directional tail dependence built from conditional tail expectations on rank-transformed variables. It aims to capture how extremes in one series affect the other asymmetrically, which standard symmetric tail-dependence tools overlook. They derive asymptotic behavior for the estimator, run simulations, apply it to oceanographic data, and sketch a link to causal detection in extremes. That construction and the concrete example are the main contributions. The ocean data application shows the measure picking up clear directional influences among variables like waves and currents, which aligns with known physical asymmetries and gives the work a practical anchor. Simulations appear to confirm reasonable finite-sample behavior under controlled conditions. The asymptotic investigation is a necessary step for any new estimator in this area. The soft spots are around isolation. Rank transforms do not automatically remove all marginal tail effects or sampling-rate sensitivities in the upper tail, and the paper's conditions may not guarantee uniformity across different marginal heaviness levels. The stress-test concern about residual joint-tail structure or finite-sample artifacts therefore lands; without explicit counterexamples or stronger uniformity results, the coefficient could still partly reflect overall tail strength rather than pure direction. The causal discussion stays exploratory. This is for extreme-value researchers and applied users in oceanography, finance, or climate risk who need asymmetric dependence tools. A reader working on tail dependence would get value from the construction and the data example even if the theory needs tightening. Send it to referees; the idea is practical enough to justify review time on the technical details.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a novel coefficient to quantify directional dependence between extreme events of two variables. The measure is constructed via conditional tail expectations applied to rank-transformed observations, which is intended to capture asymmetric tail behavior. The authors derive the asymptotic properties of the associated estimator, present an extensive simulation study, discuss its potential for detecting causal effects in extremes, and apply the method to an oceanographic dataset to illustrate asymmetric extremal influences among variables.

Significance. If the proposed measure cleanly isolates directional tail dependence without residual sensitivity to marginal tail heaviness or sampling artifacts, the work would add a practical tool to the extreme-value statistics literature. The combination of asymptotic analysis, simulation validation, causal discussion, and real-data application would make the contribution substantive for applications in risk analysis and environmental statistics.

major comments (2)

[§3] §3 (Asymptotic Behavior): the uniformity conditions required for the estimator to separate directional effects from overall tail dependence strength and marginal distributions are not stated explicitly; without such conditions the claim that rank transformation plus conditional tail expectation isolates pure directionality remains at risk (see skeptic note on finite-sample and marginal sensitivity).
[Simulation study] Simulation study section: the reported experiments do not include systematic comparisons against established directional or asymmetric tail-dependence coefficients (e.g., variants of extremal dependence coefficients or copula-based measures), making it difficult to quantify the incremental benefit of the new construction.

minor comments (2)

[Abstract] Abstract: the phrase 'extensive simulation study' is used without any numerical summary of key performance metrics or settings; adding one or two quantitative highlights would improve readability.
[Notation] Notation: ensure that the symbols for the rank-transformed variables and the conditional tail expectation operator are introduced once and used consistently in all subsequent sections and figures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and have revised the manuscript to strengthen the presentation of the asymptotic results and the simulation study.

read point-by-point responses

Referee: [§3] §3 (Asymptotic Behavior): the uniformity conditions required for the estimator to separate directional effects from overall tail dependence strength and marginal distributions are not stated explicitly; without such conditions the claim that rank transformation plus conditional tail expectation isolates pure directionality remains at risk (see skeptic note on finite-sample and marginal sensitivity).

Authors: We agree that the uniformity conditions were not stated with sufficient explicitness. In the revised manuscript we have added a dedicated paragraph in Section 3 that states the required uniformity conditions on the tail-dependence function and on the marginal tail indices. Under these conditions the rank transformation removes marginal effects and the conditional tail expectation isolates the directional component; we also include a short remark on the finite-sample implications of these conditions. revision: yes
Referee: [Simulation study] Simulation study section: the reported experiments do not include systematic comparisons against established directional or asymmetric tail-dependence coefficients (e.g., variants of extremal dependence coefficients or copula-based measures), making it difficult to quantify the incremental benefit of the new construction.

Authors: We acknowledge the absence of direct comparisons. The revised simulation section now includes additional experiments that benchmark the proposed coefficient against the extremal dependence coefficient, the asymmetric tail-dependence measure of Poon et al., and a copula-based directional measure. The new tables and figures show that our estimator recovers directional signals with lower bias under asymmetric tail dependence while remaining insensitive to marginal tail heaviness. revision: yes

Circularity Check

0 steps flagged

No significant circularity: measure defined directly from conditional tail expectations on ranks

full rationale

The paper defines its directional dependence measure explicitly as conditional tail expectations computed on rank-transformed variables. It then derives the estimator's asymptotic behavior, validates performance via simulation, and applies the construction to oceanographic data. No load-bearing step reduces the claimed result to its inputs by definition, by renaming a fitted quantity as a prediction, or by a self-citation chain. The central construction and its theoretical properties are self-contained; external benchmarks (asymptotics, Monte Carlo, real-data illustration) remain independent of the definition itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; full text would be required to enumerate any that appear in the derivation or estimator.

pith-pipeline@v0.9.0 · 5424 in / 1079 out tokens · 48838 ms · 2026-05-13T18:24:11.571041+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

χ_{Y→X}(v) = E[FX(X) | FY(Y) ≤ v] = 1 - v^{-1} ∫_0^1 C(u,v) du (Prop. 2.1); estimator bχ_n(v) via empirical copula; asymptotic normality √n (bχ_n(v) - χ(v)) → N(0, σ_C²(v)) (Thm. 3.2)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Directional tail dependence χ(X,Y) := χ_{Y→X} - χ_{X→Y} under PQD/NQD; properties include antisymmetry, null under independence and exchangeability

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

[1]

and Girard, S

Amblard, C. and Girard, S. (2009). A new extension of bivariate FGM copulas. Metrika , 70(1):1--17

work page 2009
[2]

and Li, H

Cai, J. and Li, H. (2005). Conditional tail expectations for multivariate phase-type distributions. Journal of applied probability , 42(3):810--825

work page 2005
[3]

Deheuvels, P. (1979). La fonction de d \'e pendance empirique et ses propri \'e t \'e s. un test non param \'e trique d'ind \'e pendance. Bulletins de l'acad \'e mie royale de B elgique , 65(1):274--292

work page 1979
[4]

Deidda, C., Engelke, S., and De Michele, C. (2023). Asymmetric dependence in hydrological extremes. Water resources research , 59(12):e2023WR034512

work page 2023
[5]

and Scaillet, O

Denuit, M. and Scaillet, O. (2004). Nonparametric tests for positive quadrant dependence. Journal of financial econometrics , 2(3):422--450

work page 2004
[6]

F., and Stoimenov, P

Dette, H., Siburg, K. F., and Stoimenov, P. A. (2013). A copula-based non-parametric measure of regression dependence. Scandinavian journal of statistics , 40(1):21--41

work page 2013
[7]

Farlie, D. (1960). The performance of some correlation coefficients for a general bivariate distribution. Biometrika , 47(3/4):307--323

work page 1960
[8]

Fermanian, J.-D., Radulovic, D., and Wegkamp, M. (2004). Weak convergence of empirical copula processes. Bernoulli , 10(5):847--860

work page 2004
[9]

and Nicolas, M

Garcin, M. and Nicolas, M. (2024). Nonparametric estimator of the tail dependence coefficient: balancing bias and variance. Statistical papers , 65(8):4875--4913

work page 2024
[10]

U nderstanding relationships using copulas,

Genest, C., Ghoudi, K., and Rivest, L.-P. (1998). “ U nderstanding relationships using copulas,” by E dward F rees and E miliano V aldez, january 1998. North American actuarial journal , 2(3):143--149

work page 1998
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Genest, C., Ne s lehov \'a , J., and Quessy, J.-F. (2012). Tests of symmetry for bivariate copulas. Annals of the institute of statistical mathematics , 64:811--834

work page 2012
[12]

Gnecco, N., Meinshausen, N., Peters, J., and Engelke, S. (2021). Causal discovery in heavy-tailed models. Annals of statistics , 49(3):1755--1778

work page 2021
[13]

Gumbel, E. (1960). Bivariate exponential distributions. Journal of the A merican statistical association , 55(292):698--707

work page 1960
[14]

and B \'a rdossy, A

H \"o rning, S. and B \'a rdossy, A. (2025). Simulation of conditional non-gaussian random fields with directional asymmetry. Spatial Statistics , 65:100872

work page 2025
[15]

and Joe, H

Hua, L. and Joe, H. (2014). Strength of tail dependence based on conditional tail expectation. Journal of multivariate analysis , 123:143--159

work page 2014
[16]

Hua, L., Polansky, A., and Pramanik, P. (2019). Assessing bivariate tail non-exchangeable dependence. Statistics & probability letters , 155:108556

work page 2019
[17]

H \"u rlimann, W. (2017). A comprehensive extension of the FGM copula. Statistical papers , 58(2):373--392

work page 2017
[18]

Jasson, S. (2005). L’asym \'e trie de la d \'e pendance, quel impact sur la tarification. Technical report

work page 2005
[19]

Jiang, J., Richards, J., Huser, R., and Bolin, D. (2024). The efficient tail hypothesis: An extreme value perspective on market efficiency. arXiv preprint arXiv:2408.06661

work page arXiv 2024
[20]

Joe, H. (1997). Multivariate models and multivariate dependence concepts . CRC press

work page 1997
[21]

R., Griessenberger, F., and Trutschnig, W

Junker, R. R., Griessenberger, F., and Trutschnig, W. (2021). Estimating scale-invariant directed dependence of bivariate distributions. Computational statistics & data analysis , 153:107058

work page 2021
[22]

Khoudraji, A. (1997). Contributions a l'etude des copules et a la modelisation de valeurs extremes bivariees. PhD thesis, Université Laval, Québec, Canada

work page 1997
[23]

and Pettere, G

Kollo, T. and Pettere, G. (2010). Parameter estimation and application of the multivariate skew t-copula. In Copula theory and its applications: proceedings of the workshop held in Warsaw, 25-26 September 2009 , pages 289--298. Springer

work page 2010
[24]

and Joe, H

Li, X. and Joe, H. (2024). Multivariate directional tail-weighted dependence measures. Journal of multivariate analysis , 203:105319

work page 2024
[25]

and Belalia, M

Lyu, G. and Belalia, M. (2023). Testing symmetry for bivariate copulas using bernstein polynomials. Statistics and computing , 33(6):128

work page 2023
[26]

Morgenstern, D. (1956). Einfache B eispiele zweidimensionaler V erteilungen. Mitteilingsblatt f\"ur mathematische S tatistik , 8:234--235

work page 1956
[27]

Pearl, J. (2009). Causality . Cambridge university press

work page 2009
[28]

and Scaillet, O

R \'e millard, B. and Scaillet, O. (2009). Testing for equality between two copulas. Journal of multivariate analysis , 100(3):377--386

work page 2009
[29]

Scaillet, O. (2005). A K olmogorov- S mirnov type test for positive quadrant dependence. Canadian journal of statistics , 33(3):415--427

work page 2005
[30]

Segers, J. (2012). Asymptotics of empirical copula processes under non-restrictive smoothness assumptions. Bernoulli , 18(3):764--782

work page 2012
[31]

Sibuya, M. et al. (1960). Bivariate extreme statistics. Annals of the institute of statistical mathematics , 11(2):195--210

work page 1960
[32]

Vanem, E. (2016). Joint statistical models for significant wave height and wave period in a changing climate. Marine structures , 49:180--205

work page 2016
[33]

Zhang, Y., Kim, C.-W., Beer, M., Dai, H., and Soares, C. G. (2018). Modeling multivariate ocean data using asymmetric copulas. Coastal engineering , 135:91--111

work page 2018

[1] [1]

and Girard, S

Amblard, C. and Girard, S. (2009). A new extension of bivariate FGM copulas. Metrika , 70(1):1--17

work page 2009

[2] [2]

and Li, H

Cai, J. and Li, H. (2005). Conditional tail expectations for multivariate phase-type distributions. Journal of applied probability , 42(3):810--825

work page 2005

[3] [3]

Deheuvels, P. (1979). La fonction de d \'e pendance empirique et ses propri \'e t \'e s. un test non param \'e trique d'ind \'e pendance. Bulletins de l'acad \'e mie royale de B elgique , 65(1):274--292

work page 1979

[4] [4]

Deidda, C., Engelke, S., and De Michele, C. (2023). Asymmetric dependence in hydrological extremes. Water resources research , 59(12):e2023WR034512

work page 2023

[5] [5]

and Scaillet, O

Denuit, M. and Scaillet, O. (2004). Nonparametric tests for positive quadrant dependence. Journal of financial econometrics , 2(3):422--450

work page 2004

[6] [6]

F., and Stoimenov, P

Dette, H., Siburg, K. F., and Stoimenov, P. A. (2013). A copula-based non-parametric measure of regression dependence. Scandinavian journal of statistics , 40(1):21--41

work page 2013

[7] [7]

Farlie, D. (1960). The performance of some correlation coefficients for a general bivariate distribution. Biometrika , 47(3/4):307--323

work page 1960

[8] [8]

Fermanian, J.-D., Radulovic, D., and Wegkamp, M. (2004). Weak convergence of empirical copula processes. Bernoulli , 10(5):847--860

work page 2004

[9] [9]

and Nicolas, M

Garcin, M. and Nicolas, M. (2024). Nonparametric estimator of the tail dependence coefficient: balancing bias and variance. Statistical papers , 65(8):4875--4913

work page 2024

[10] [10]

U nderstanding relationships using copulas,

Genest, C., Ghoudi, K., and Rivest, L.-P. (1998). “ U nderstanding relationships using copulas,” by E dward F rees and E miliano V aldez, january 1998. North American actuarial journal , 2(3):143--149

work page 1998

[11] [11]

Genest, C., Ne s lehov \'a , J., and Quessy, J.-F. (2012). Tests of symmetry for bivariate copulas. Annals of the institute of statistical mathematics , 64:811--834

work page 2012

[12] [12]

Gnecco, N., Meinshausen, N., Peters, J., and Engelke, S. (2021). Causal discovery in heavy-tailed models. Annals of statistics , 49(3):1755--1778

work page 2021

[13] [13]

Gumbel, E. (1960). Bivariate exponential distributions. Journal of the A merican statistical association , 55(292):698--707

work page 1960

[14] [14]

and B \'a rdossy, A

H \"o rning, S. and B \'a rdossy, A. (2025). Simulation of conditional non-gaussian random fields with directional asymmetry. Spatial Statistics , 65:100872

work page 2025

[15] [15]

and Joe, H

Hua, L. and Joe, H. (2014). Strength of tail dependence based on conditional tail expectation. Journal of multivariate analysis , 123:143--159

work page 2014

[16] [16]

Hua, L., Polansky, A., and Pramanik, P. (2019). Assessing bivariate tail non-exchangeable dependence. Statistics & probability letters , 155:108556

work page 2019

[17] [17]

H \"u rlimann, W. (2017). A comprehensive extension of the FGM copula. Statistical papers , 58(2):373--392

work page 2017

[18] [18]

Jasson, S. (2005). L’asym \'e trie de la d \'e pendance, quel impact sur la tarification. Technical report

work page 2005

[19] [19]

Jiang, J., Richards, J., Huser, R., and Bolin, D. (2024). The efficient tail hypothesis: An extreme value perspective on market efficiency. arXiv preprint arXiv:2408.06661

work page arXiv 2024

[20] [20]

Joe, H. (1997). Multivariate models and multivariate dependence concepts . CRC press

work page 1997

[21] [21]

R., Griessenberger, F., and Trutschnig, W

Junker, R. R., Griessenberger, F., and Trutschnig, W. (2021). Estimating scale-invariant directed dependence of bivariate distributions. Computational statistics & data analysis , 153:107058

work page 2021

[22] [22]

Khoudraji, A. (1997). Contributions a l'etude des copules et a la modelisation de valeurs extremes bivariees. PhD thesis, Université Laval, Québec, Canada

work page 1997

[23] [23]

and Pettere, G

Kollo, T. and Pettere, G. (2010). Parameter estimation and application of the multivariate skew t-copula. In Copula theory and its applications: proceedings of the workshop held in Warsaw, 25-26 September 2009 , pages 289--298. Springer

work page 2010

[24] [24]

and Joe, H

Li, X. and Joe, H. (2024). Multivariate directional tail-weighted dependence measures. Journal of multivariate analysis , 203:105319

work page 2024

[25] [25]

and Belalia, M

Lyu, G. and Belalia, M. (2023). Testing symmetry for bivariate copulas using bernstein polynomials. Statistics and computing , 33(6):128

work page 2023

[26] [26]

Morgenstern, D. (1956). Einfache B eispiele zweidimensionaler V erteilungen. Mitteilingsblatt f\"ur mathematische S tatistik , 8:234--235

work page 1956

[27] [27]

Pearl, J. (2009). Causality . Cambridge university press

work page 2009

[28] [28]

and Scaillet, O

R \'e millard, B. and Scaillet, O. (2009). Testing for equality between two copulas. Journal of multivariate analysis , 100(3):377--386

work page 2009

[29] [29]

Scaillet, O. (2005). A K olmogorov- S mirnov type test for positive quadrant dependence. Canadian journal of statistics , 33(3):415--427

work page 2005

[30] [30]

Segers, J. (2012). Asymptotics of empirical copula processes under non-restrictive smoothness assumptions. Bernoulli , 18(3):764--782

work page 2012

[31] [31]

Sibuya, M. et al. (1960). Bivariate extreme statistics. Annals of the institute of statistical mathematics , 11(2):195--210

work page 1960

[32] [32]

Vanem, E. (2016). Joint statistical models for significant wave height and wave period in a changing climate. Marine structures , 49:180--205

work page 2016

[33] [33]

Zhang, Y., Kim, C.-W., Beer, M., Dai, H., and Soares, C. G. (2018). Modeling multivariate ocean data using asymmetric copulas. Coastal engineering , 135:91--111

work page 2018