Extreme Value family

Extreme-value copulas are max-stable dependence models used throughout multivariate extreme-value theory. In dimension $d$, an extreme-value copula $C$ satisfies

$$C(u_1^t,\dots ,u_d^t)=C(u_1,\dots ,u_d{)}^t,t>0.$$

The natural dimension-free representation is the stable tail dependence function (STDF) $\ell$:

$$C(\mathit{u})=\exp \{-\ell (-\log u_1,\dots ,-\log u_d)\}.$$

A valid STDF is convex and one-homogeneous and satisfies

$$\underset{i}{max}{x}_i\le \ell (\mathit{x})\le \sum _{i=1}^d{x}_i.$$

By homogeneity, $\ell$ can be restricted to the simplex

$${\mathrm{\Delta }}_{d-1}=\{\mathit{w}\in [0,1{]}^d:\sum _{i=1}^d{w}_i=1\}$$

to obtain the multivariate Pickands dependence function $A:{\mathrm{\Delta }}_{d-1}\to \mathbb{R}$. For $r=\sum _i{x}_i$ and $\mathit{w}=\mathit{x}/r$,

$$A(\mathit{w})=\ell (\mathit{w}),\ell (\mathit{x})=rA\left(\frac{\mathit{x}}{r}\right),$$

with $\underset{i}{max}{w}_i\le A(\mathit{w})\le 1$. Thus the Pickands representation itself is not restricted to dimension two.

What is specifically bivariate is the scalar parametrization of the simplex. For $d=2$, write $\mathit{w}=(t,1-t)$; then $A:[0,1]\to [1/2,1]$ and

$$\ell (x_1,x_2)=(x_1+x_2)A\left(\frac{x_1}{x_1+x_2}\right),$$

where

$$max\{t,1-t\}\le A(t)\le 1,A(0)=A(1)=1.$$

Accordingly, Copulas.jl uses the STDF $\ell$ as the dimension-independent EV interface and also supports the scalar Pickands representation in dimension two.

The bivariate copula therefore has the classical representation

$$C(u_1,u_2)=\exp \{\log (u_1{u}_2)A\left(\frac{\log u_1}{\log (u_1{u}_2)}\right)\}.$$

Bivariate and multivariate EV copulas

Copulas.jl uses $\ell$ as the mathematical EV interface in arbitrary dimension while retaining optimized bivariate formulas and sampling algorithms where available.

Choose the family, not the algorithm

A public constructor identifies the mathematical family. It does not ask the user to choose a bivariate or multivariate numerical algorithm.

What is new in the EV subsystem?

The current EV implementation goes beyond the historical bivariate-only design:

• the core representation is a dimension-aware STDF ℓ;

• several classical EV families now use the same public constructor in $d=2$ and $d>2$;

• a few matrix and vector parameterizations infer the dimension when a single parameter determines it immediately;

• general Hüsler-Reiss variograms and extremal-$t$ correlation matrices are supported in addition to exchangeable submodels;

• Tawn and asymmetric Galambos expose structured multivariate subset parameterizations;

• BC2 and Marshall-Olkin have multivariate spectral/shock representations;

• multivariate empirical EV estimation is available through a shape-constrained discrete spectral projection;

• multivariate densities and exact sampling are available for the documented families.

Constructors and dimensional conventions

The canonical constructor always makes the dimension part of the type:

FamilyCopula{d}(params...)

The equivalent runtime-dimension form

FamilyCopula(d, params...)

is convenient but is not type-stable with respect to d.

No implicit bivariate dimension for scalar parameters

Scalar/exchangeable EV families require an explicit dimension. For example, write GalambosCopula{2}(2.3) or GalambosCopula(2, 2.3); GalambosCopula(2.3) is intentionally not a constructor.

Scalar or exchangeable families therefore have canonical forms

LogCopula{d}(θ)
GalambosCopula{d}(θ)
MixedCopula{d}(θ)
CuadrasAugeCopula{d}(θ)
HuslerReissCopula{d}(θ)
tEVCopula{d}(ν, ρ)

with FamilyCopula(d, ...) as runtime sugar.

Structured parameterizations follow exactly the same rule:

HuslerReissCopula{d}(Γ)
tEVCopula{d}(ν, R)
TawnCopula{d}(α, weights)
AsymGalambosCopula{d}(α, weights)
BC2Copula{d}(a)
MOCopula{d}(λ)
EmpiricalEVCopula{d}(U)

Because these objects determine their own dimension, the following additional convenience forms are also available:

HuslerReissCopula(Γ)
BC2Copula(a)
MOCopula(λ)
EmpiricalEVCopula(U)

The full subset representations are likewise available as

TawnCopula{d}(dep, asy)
AsymGalambosCopula{d}(dep, asy)

together with FamilyCopula(d, dep, asy).

Family	Canonical constructor	Supported dimension	Interpretation
Logistic	LogCopula{d}(θ)	$d\ge 2$	exchangeable
Galambos	GalambosCopula{d}(θ)	$d\ge 2$	exchangeable negative logistic
Mixed	MixedCopula{d}(θ)	$d\ge 2$	scalar Copulas.jl extension of the bivariate mixed model
Cuadras-Augé	CuadrasAugeCopula{d}(θ)	$d\ge 2$	scalar
Hüsler-Reiss	HuslerReissCopula{d}(θ)	$d\ge 2$	exchangeable variogram
Hüsler-Reiss	HuslerReissCopula{d}(Γ)	$d=\mathrm{size}(\Gamma ,1)$	general variogram
extremal-$t$	tEVCopula{d}(ν, ρ)	$d\ge 2$	equicorrelation
extremal-$t$	tEVCopula{d}(ν, R)	$d=\mathrm{size}(R,1)$	general correlation matrix
Tawn	TawnCopula{d}(α, weights)	$d=\mathrm{length}(weights)$	full-set logistic component + singleton remainders
Tawn	TawnCopula{d}(dep, asy)	$d\ge 2$	full subset representation
Asymmetric Galambos	AsymGalambosCopula{2}(α, θ₁, θ₂)	2	bivariate form of the unified subset model
Asymmetric Galambos	AsymGalambosCopula{d}(α, weights)	$d=\mathrm{length}(weights)$	full-set negative-logistic component + singleton remainders
Asymmetric Galambos	AsymGalambosCopula{d}(dep, asy)	$d\ge 2$	full subset representation
BC2	BC2Copula{2}(a, b)	2	classical bivariate representation
BC2	BC2Copula{d}(a::AbstractVector)	$d=\mathrm{length}(a)$	two-atom spectral representation
Marshall-Olkin	MOCopula{2}(λ₁, λ₂, λ₁₂)	2	classical three-shock representation
Marshall-Olkin	MOCopula{d}(λ)	$\mathrm{length}(\lambda )=2^d-1$	full subset-shock representation
Empirical EV	EmpiricalEVCopula{2}(U)	2	bivariate Pickands/CFG/OLS estimator
Empirical EV	EmpiricalEVCopula{d}(U)	$d=\mathrm{size}(U,1)$	shape-constrained multivariate spectral estimator
Asymmetric logistic	AsymLogCopula{2}(...)	2	bivariate
Asymmetric mixed	AsymMixedCopula{2}(...)	2	bivariate

Exchangeable versus general Hüsler-Reiss and extremal-t

Hüsler-Reiss has two public parameterizations. In the package convention,

$$\gamma ={\left(\frac{2}{\theta }\right)}^2$$

maps the exchangeable scalar parameter to the common off-diagonal variogram entry. Thus

HuslerReissCopula{d}(θ)

is an exchangeable submodel, while

HuslerReissCopula(Γ)

accepts a general valid variogram matrix. In dimension two a matrix parameterization has the same mathematical meaning as the corresponding bivariate scalar form.

The extremal-$t$ family follows the same pattern:

tEVCopula{d}(ν, ρ) # equicorrelation
tEVCopula{d}(ν, R) # general correlation matrix

A valid $2\times 2$ correlation matrix likewise defines the bivariate model through its off-diagonal correlation.

Tawn and asymmetric Galambos subset models

The multivariate Tawn model follows the asymmetric logistic construction of Tawn [ DocumenterCitations.CitationSiteNode("tawn1990multivariate-cite-1")

]. The full representation associates components with every nonempty subset of $\{1,\dots ,d\}$. There are

$$2^d-1$$

nonempty subsets and

$$2^d-d-1$$

non-singleton subsets.

Accordingly,

TawnCopula{d}(dep, asy)

contains one dependence parameter per non-singleton subset and one asymmetry weight vector per nonempty subset. The weights involving each margin must sum to one.

TawnCopula{d}(α, weights) is a convenient lower-dimensional parameterization implemented in Copulas.jl: one logistic component acts on the full set and singleton components carry the remaining marginal mass.

Asymmetric Galambos uses the analogous negative-logistic subset construction; the multivariate min-stable framework is described by Joe [ DocumenterCitations.CitationSiteNode("Joe1990-cite-1")

]. The convenience constructor

AsymGalambosCopula{d}(α, weights)

is a Copulas.jl parameterization of that valid subset model, not a separate literature family.

Copulas.jl Mixed extension

The historical Mixed model is bivariate [ DocumenterCitations.CitationSiteNode("tawn1988bivariate-cite-1")

]. The $d$-dimensional extension used by Copulas.jl is obtained from the identity

$${\ell }_{\mathrm{Mixed},\theta }(\mathit{x})=(1-\theta )\sum _{i=1}^d{x}_i+\theta {\ell }_{\mathrm{Galambos},1}(\mathit{x}).$$

Both terms are valid STDFs, so their convex combination is a valid STDF. For $d=2$ the identity reduces exactly to

$$A(t)=1-\theta t(1-t),$$

which is the historical Mixed Pickands model.

What is literature and what is derived here?

Tawn [ DocumenterCitations.CitationSiteNode("tawn1988bivariate-cite-2")

] is the reference for the original bivariate Mixed model, and Galambos [ DocumenterCitations.CitationSiteNode("galambos1975order-cite-1")

] for the negative-logistic component. The dimension-free convex-combination identity above is the extension defined by Copulas.jl; we do not attribute that exact $d$-dimensional parameterization to either paper.

Multivariate empirical EV estimation

EmpiricalEVCopula selects the estimator from the sample dimension. In two dimensions it preserves the historical Pickands/CFG/OLS estimator of $A$.

For $d\ge 3$,

EmpiricalEVCopula(U; method=:ols)

first constructs a multivariate Pickands pilot estimator and then projects it onto the class induced by a finite discrete spectral measure. This matters because convexity and the elementary Pickands bounds are no longer sufficient to characterize validity when $d\ge 3$. The multivariate estimators follow Gudendorf and Segers [ DocumenterCitations.CitationSiteNode("gudendorf2011nonparametric-cite-1")

], and the shape-constrained discrete spectral projection follows Gudendorf and Segers [ DocumenterCitations.CitationSiteNode("gudendorf2012multivariate-cite-1")

].

The fitted STDF is valid by construction and exact sampling is available.

Advanced Concepts

Here, we present some important concepts from the theory of extreme value copulas that are useful for the development of this package.

Let $(X,Y)\sim C$ where $C$ is a bivariate extreme value copula. We have the following result from [ DocumenterCitations.CitationSiteNode("ghoudi1998proprietes-cite-1")

]:

Property: Ghoudi 1998

Let $(X,Y)\sim C$, where $C$ is an extreme value copula. The joint distribution of $X$ and $Z=\frac{\log (X)}{\log (XY)}$ is given by:

$$P(Z\le z,X\le x)=G(z,x)=(z+z(1-z)\frac{A^{\prime }(z)}{A(z)})x^{A(z)/z},0\le x,z\le 1$$

where $A^{\prime }(z)$ denotes the derivative of $A(z)$ at point $z.$

Since $A$ is a convex function defined on $[0,1]$ and satisfies $-1\le A^{\prime }(z)\le 1$, by extension, we define $A^{\prime }(1)$ as the supremum of $A^{\prime }(z)$ over $(0,1)$. By setting $x=1$ in the previous result, we obtain the marginal distribution of $Z$: $P(Z\le z)=G_Z(z)=z+z(1-z)\frac{A^{\prime }(z)}{A(z)},0\le z\le 1.$

This result was demonstrated by Deheuvels (1991) [ DocumenterCitations.CitationSiteNode("deheuvels1991limiting-cite-1")

] in the case where $A$ admits a second derivative.

Simulation of Bivariate Extreme Value Distributions

To simulate a bivariate extreme value distribution $C(x,y)$, note that if $F_1$ and $F_2$ are univariate extreme value distributions, then the pair $(F_1^{-1}(X),F_2^{-1}(Y))$ is distributed according to a bivariate extreme value distribution. The proposed algorithm in Ghoudi, 1998 [ DocumenterCitations.CitationSiteNode("ghoudi1998proprietes-cite-2")

] allows simulating such a distribution.

Assume $A$ has a second derivative, making the distribution absolutely continuous. In this case, $Z$ is also absolutely continuous and has a density $g_Z(z)$ given by:

$$g_Z(z)=1+(1-2z)\frac{A^{\prime }(z)}{A(z)}+z(1-z)[\frac{A^{″}(z)}{A(z)}-{\left(\frac{A^{\prime }(z)}{A(z)}\right)}^2].$$

The conditional distribution of $W$ given $Z$ is:

$$F(w|z)=\frac{1}{g_Z(z)}\frac{d}{dz}F(z,w),$$

which simplifies to:

$$F(w|z)=w\frac{z(1-z)A^{\prime }(z)}{A(z)g_Z(z)}+(w-w\log w)(1-\frac{z(1-z)A^{″}(z)}{A(z)g_Z(z)})$$

Given $Z$, the distribution of $W$ is uniform on $(0,1)$ with probability $p(Z)$ and equals the product of two independent uniforms on $(0,1)$ with probability $1-p(Z)$, where:

$$p(z)=\frac{z(1-z)A^{″}(z)}{A(z)g_Z(z)}.$$

Since $g_Z(z)$ is the derivative of the cumulative distribution function of $Z$, it holds that $0\le p(z)\le 1$.

For the class of Extreme Value Copulas, We follow the methodology proposed by Ghoudi,1998. page 191. [ DocumenterCitations.CitationSiteNode("ghoudi1998proprietes-cite-3")

]. Here, is a detailed algorithm for sampling from bivariate Extreme Value Copulas:

Algorithm: Bivariate Extreme Value Copulas sampling

• Simulate $U_1,U_2\sim \mathcal{U}[0,1]$

• Simulate $Z\sim G_Z(z)$

• Select $W=U_1$ with probability $p(Z)$ and $W=U_1{U}_2$ with probability $1-p(Z)$

• Return $X=W^{Z/A(Z)}$ and $Y=W^{(1-Z)/A(Z)}$

Note that all functions present in the algorithm were previously defined to ensure that the implemented methodology has a solid theoretical basis.

Multivariate sampling

The bivariate Ghoudi construction above remains an important part of the EV implementation. It is not replaced by multivariate sampling.

The public interface is

rand(C, n)

and Copulas.jl selects an appropriate exact algorithm for each documented family and dimension.

You never select the sampler yourself

rand(LogCopula(2, θ), n) and rand(LogCopula(10, θ), n) have the same public API. The same is true for Galambos, Hüsler-Reiss, Mixed, and extremal-$t$.

The numerical implementation remains free to evolve without changing that public behavior.

See the canonical Public API entries for Tail, ExtremeValueCopula.

Conditionals and distortions

For any copula $C$, the conditional copula and the univariate conditional distortions are given by partial-derivative ratios. If we condition on a set $J$ with $m=|J|$ components and write $I=\{1,\dots ,d\}\setminus J$, then

$$C_{I\mid J}({\mathit{u}}_I\mid {\mathit{u}}_J)=\frac{{\partial }^m}{\partial {\mathit{u}}_J}C({\mathit{u}}_I,{\mathit{u}}_J)/\frac{{\partial }^m}{\partial {\mathit{u}}_J}C({\mathbf{1}}_I,{\mathit{u}}_J),$$

and, for a single coordinate $i\in I$ (setting the other coordinates in $I$ to 1), the conditional distortion is

$$H_{i\mid J}(u\mid {\mathit{u}}_J)=\frac{{\partial }^m}{\partial {\mathit{u}}_J}C(\dots ,u_i=u,\dots ,{\mathit{u}}_J)/\frac{{\partial }^m}{\partial {\mathit{u}}_J}C(\mathbf{1},{\mathit{u}}_J).$$

For a bivariate extreme value copula with Pickands function $A$, the copula is

$$C(u_1,u_2)=\exp \{\log (u_1{u}_2)A\left(\frac{\log u_1}{\log (u_1{u}_2)}\right)\},$$

so the above derivatives can be written explicitly in terms of $A$ (and $A^{\prime }$ when it exists) by the chain rule. In the implementation, these derivatives are obtained directly from this representation, using analytic formulas when available and automatic differentiation otherwise.

Visual illustrations

Pickands dependence functions A(t)

using Copulas, Plots, Distributions
ts = range(0.0, 1.0; length=401)
tails = (
    Copulas.GalambosTail(0.8),    # upper tail dep.
    Copulas.HuslerReissTail(1.0), # intermediate
    Copulas.LogTail(1.6),         # logistic
)
labels = ("Galambos(0.8)", "Hüsler–Reiss(1.0)", "Log(1.6)")
plot(size=(700, 300))
for (i, tail) in enumerate(tails)
    plot!(ts, Copulas.A.(Ref(tail), ts); label=labels[i])
end
plot!(ts, max.(ts, 1 .- ts); label="bounds", ls=:dash, color=:black)
plot!(ts, ones(length(ts)); label="1", ls=:dot, color=:gray)

Sample scatter (uniform scale)

C = GalambosCopula(2, 1.0)
plot(C, title="Galambos copula sample")

Conditional distortion (EV example)

C = HuslerReissCopula(2, 1.2)
u2 = 0.4
D = condition(C, 2, u2)
ts = range(0.0, 1.0; length=401)
plot(ts, cdf.(Ref(D), ts); xlabel="u", ylabel="H_{1|2}(u|u₂=0.4)",
     title="Conditional distortion for Hüsler–Reiss")

Rosenblatt sanity check (EV)

using StatsBase
U = rand(C, 2000)
S = reduce(hcat, (rosenblatt(C, U[:, i]) for i in 1:size(U,2)))
ts = range(0.0, 1.0; length=401)
EC = [ecdf(S[k, :]) for k in 1:2]
plot(ts, ts; label="Uniform", color=:blue, alpha=0.6, size=(650,300))
plot!(ts, EC[1].(ts); seriestype=:steppost, label="s₁", color=:black)
plot!(ts, EC[2].(ts); seriestype=:steppost, label="s₂", color=:gray)

Available models

An extreme-value copula is determined by $C(\mathit{u})=\exp [-\ell (-\log \mathit{u})]$. In dimension two, $\ell (x,y)=(x+y)A(x/(x+y))$. The table therefore defines each model through its STDF $\ell$ or Pickands function $A$ and gives both the component and copula constructors.

Model	Mathematical definition and parameter domain	Public constructors
Logistic	$\ell (\mathit{x})=(\sum _i{x}_i^{\theta }{)}^{1/\theta }$, $\theta \ge 1$	LogTail(θ); LogCopula{d}(θ); LogCopula(d, θ)
Galambos	$\ell (\mathit{x})=\sum _i{x}_i-(\sum _i{x}_i^{-\theta }{)}^{-1/\theta }$, $\theta \ge 0$	GalambosTail(θ); GalambosCopula{d}(θ); GalambosCopula(d, θ)
Mixed	$A(t)=1-\theta t(1-t)$, $0\le \theta \le 1$; in higher dimensions Copulas.jl uses the convex combination with the Galambos $\theta =1$ STDF described above	MixedTail(θ); MixedCopula{d}(θ); MixedCopula(d, θ)
Cuadras–Augé	$\ell (\mathit{x})=(1-\theta )\sum _i{x}_i+\theta \underset{i}{max}{x}_i$, $0\le \theta \le 1$	CuadrasAugeTail(θ); CuadrasAugeCopula{d}(θ); CuadrasAugeCopula(d, θ)
Hüsler–Reiss	Brown–Resnick/Hüsler–Reiss STDF determined either by the exchangeable variogram ${\gamma }_{ij}=(2/\theta {)}^2$ with $\theta \ge 0$, or by a finite, symmetric, conditionally negative-definite variogram matrix $\mathrm{\Gamma }$ with zero diagonal	HuslerReissTail(θ) / HuslerReissTail(Γ); HuslerReissCopula{d}(θ) / HuslerReissCopula(Γ)
extremal-$t$	extremal-$t$ STDF with $\nu >0$ and correlation $R$; the exchangeable form uses $\rho \in (-1/(d-1),1]$	tEVTail(ν, ρ) / tEVTail(ν, R); tEVCopula{d}(ν, ρ) / tEVCopula{d}(ν, R) / runtime-dimension forms
Tawn asymmetric logistic	sum of logistic STDF components over nonempty coordinate subsets; non-singleton dependence parameters satisfy ${\alpha }_S\ge 1$ and asymmetry weights lie in $[0,1]$ and sum to one for each margin	TawnTail(α, weights) / TawnTail(d, dep, asy); corresponding TawnCopula forms
Asymmetric Galambos	sum of negative-logistic components over coordinate subsets; ${\alpha }_S\ge 0$ with normalized asymmetry weights	AsymGalambosTail(α, weights) / AsymGalambosTail(dep, asy); corresponding AsymGalambosCopula forms
Asymmetric logistic (bivariate)	$A(t)=[{\theta }_1^{\alpha }(1-t{)}^{\alpha }+{\theta }_2^{\alpha }{t}^{\alpha }{]}^{1/\alpha }+({\theta }_1-{\theta }_2)t+1-{\theta }_1$; $\alpha \ge 1$, ${\theta }_i\in [0,1]$	AsymLogTail(α, θ₁, θ₂); AsymLogCopula{2}(α, θ₁, θ₂); AsymLogCopula(2, α, θ₁, θ₂)
Asymmetric mixed (bivariate)	$A(t)={\theta }_2{t}^3+{\theta }_1{t}^2-({\theta }_1+{\theta }_2)t+1$; ${\theta }_1\ge 0$, ${\theta }_1+{\theta }_2\le 1$, ${\theta }_1+2{\theta }_2\le 1$, ${\theta }_1+3{\theta }_2\ge 0$	AsymMixedTail(θ₁, θ₂); AsymMixedCopula{2}(θ₁, θ₂); AsymMixedCopula(2, θ₁, θ₂)
BC2 spectral	two-atom spectral model; in dimension two $A(t)=max(at,b(1-t))+max((1-a)t,(1-b)(1-t))$ with $a,b\in [0,1]$	BC2Tail(a, b) / BC2Tail(a_vector); corresponding BC2Copula forms
Marshall–Olkin	common-shock model with one non-negative intensity for every nonempty subset of coordinates; the vector therefore has length $2^d-1$	MOTail(λ₁, λ₂, λ₁₂) / MOTail(λ); corresponding MOCopula forms
Empirical EV (bivariate)	shape-constrained estimate of $A$ from pseudo-observations using the Pickands, CFG, or OLS criterion	EmpiricalEVTail(U; method=:ols); EmpiricalEVCopula{2}(U; method=:ols)
Empirical EV (multivariate)	finite spectral projection defining a valid homogeneous convex STDF	EmpiricalEVMultivariateTail(U; method=:ols, degree=3); EmpiricalEVCopula{d}(U; method=:ols, degree=3)

Symmetric parametric models

Logistic. This is the extreme-value form of the Gumbel copula. The parameter increases monotonically from independence at $\theta =1$ to comonotonicity as $\theta \to \mathrm{\infty }$; in dimension two, ${\lambda }_U=2-2^{1/\theta }$ and $\tau =1-1/\theta$. It is exchangeable, so a single parameter cannot represent heterogeneous pairwise extremal dependence in high dimension.

Galambos. Galambos is the negative-logistic extreme-value family, despite describing positive upper-tail association. Zero is independence and infinity is the comonotonic limit; bivariately, ${\lambda }_U=2^{-1/\theta }$ for $\theta >0$. Its parameter is not on the same scale as Logistic's, so raw values should not be compared across the two families.

Mixed. The bivariate Pickands curve interpolates linearly between independence and the Galambos model with parameter one, giving ${\lambda }_U=\theta /2$. Thus $\theta =1$ is not comonotonic. In dimensions greater than two, the convex-combination STDF is a Copulas.jl extension of the historical bivariate model; results should not be attributed automatically to a multivariate “Mixed” family in the cited literature.

Cuadras–Augé. This family is a direct mixture, at the STDF level, of independence and comonotonicity. The endpoints $\theta =0$ and $\theta =1$ give those two limits, and bivariately ${\lambda }_U=\theta$. Every nonzero parameter introduces a singular component, so an ordinary Lebesgue density does not describe the whole law even away from the fully comonotonic endpoint.

Gaussian and Student extremal models

Hüsler–Reiss. The scalar constructor uses an exchangeable variogram $\gamma =(2/\theta {)}^2$: $\theta =0$ is independence and $\theta \to \mathrm{\infty }$ is comonotonicity. The matrix constructor instead accepts the variogram directly, for which an all-zero matrix is comonotonic; confusing the scalar $\theta$ with a variogram entry reverses the interpretation. The matrix must satisfy conditional negative definiteness, not merely symmetry and non-negative entries.

Extremal-$t$. Both $\nu$ and the correlation structure affect extremal dependence. In particular, zero correlation does not generally mean independence because the common Student scale still couples extremes, while $\rho =1$ gives comonotonicity. Equicorrelation must satisfy the dimension-dependent lower bound, and a general matrix must be strictly positive definite; estimates near either boundary can be numerically delicate.

Asymmetric subset models

Tawn. Tawn decomposes the STDF into logistic components attached to coordinate subsets. The dependence parameters control each active subset, whereas the asymmetry weights allocate its contribution among margins; the per-margin normalization is essential for uniform margins. The compact (α, weights) constructor retains only a full-set component plus singleton remainders, while (dep, asy) is the full subset model—these forms should not be assumed to have the same number of free parameters.

Asymmetric Galambos. This is the analogous subset construction with negative-logistic components. Zero dependence parameters deactivate the corresponding components; an infinite full-set parameter reaches comonotonicity only when its weights are all one and no competing subset is active. As with Tawn, the ordering of dep and asy follows the documented nonempty-subset convention, so constructing long vectors manually is prone to indexing mistakes.

Asymmetric logistic. The bivariate parameters ${\theta }_1$ and ${\theta }_2$ allocate extremal dependence asymmetrically to the two margins, while $\alpha$ controls the logistic component. $\alpha =1$ or either zero weight gives independence, making the remaining parameters unidentifiable on those boundaries. Comonotonicity is reached only jointly as $\alpha \to \mathrm{\infty }$ with both weights equal to one.

Asymmetric Mixed. Its feasible parameter set is the quadrilateral defined by the four inequalities in the table, not a rectangular box. Independent unconstrained bounds on ${\theta }_1$ and ${\theta }_2$ are therefore insufficient during optimization. The origin is independence and ${\theta }_2=0$ gives the symmetric Mixed subfamily; close to that line the asymmetry parameter can be weakly identified.

Discrete spectral and empirical models

BC2. BC2 is most naturally understood through its two spectral atoms, not as two independent “strength” parameters. The parameters must produce a valid spectral measure with the required marginal moments, and relabeling the atoms can give equivalent descriptions. Discrete spectral mass may create singular components, so users should rely on the generalized distribution interface rather than assume a globally smooth density.

Marshall–Olkin. Each parameter is the intensity of a shock hitting one specific nonempty subset of margins. Simultaneous shocks generate singular mass and are precisely the feature of the model, not a numerical artifact. Multiplying every intensity by the same positive constant leaves the copula unchanged, so only relative rates are identifiable; zero rates are allowed, but every margin still needs positive total shock intensity.

Bivariate empirical EV. Pickands, CFG, and OLS are different estimators of the same Pickands curve and need not agree in finite samples. The fitted curve is projected to satisfy the shape constraints, so it is not simply the raw pointwise estimate. Grid resolution and endpoint trimming affect numerical accuracy, and pseudo_values=false must be used when the input has not already been transformed to ranks.

Multivariate empirical EV. The higher-dimensional estimator projects onto a finite spectral basis; degree trades approximation flexibility against optimization cost and potential instability. The result supports cdf and sampling but can contain singular components, so no global Lebesgue pdf is promised. Reproducibility and fit quality should be assessed together with the agreement between the fitted tail and the empirical target on a diagnostic grid, rather than from the constructor succeeding alone.

The canonical Public API documents complete call forms, validation, limiting cases, and numerical restrictions.

References

1. J. A. Tawn. Modelling multivariate extreme value distributions. Biometrika 77, 245–253 (1990).

2. H. Joe. Families of min-stable multivariate exponential and multivariate extreme value distributions. Statistics & probability letters 9, 75–81 (1990).

3. J. A. Tawn. Bivariate extreme value theory: models and estimation. Biometrika 75, 397–415 (1988).

4. J. Galambos. Order statistics of samples from multivariate distributions. Journal of the American Statistical Association 70, 674–680 (1975).

5. G. Gudendorf and J. Segers. Nonparametric estimation of an extreme-value copula in arbitrary dimensions. Journal of multivariate analysis 102, 37–47 (2011).

6. G. Gudendorf and J. Segers. Nonparametric estimation of multivariate extreme-value copulas. Journal of Statistical Planning and Inference 142, 3073–3085 (2012).

7. K. Ghoudi, A. Khoudraji and E. L.-P. Rivest. Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles. Canadian Journal of Statistics 26, 187–197 (1998).

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