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, while BivariatePickandsTail names the additional scalar Pickands capability used by specialized $d=2$ algorithms.

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 now uses $\ell$ as the mathematical EV interface in arbitrary dimension while preserving the mature bivariate Pickands machinery. Bivariate formulas based on A, dA, d²A, Ghoudi sampling, conditional distortions, and family-specific kernels remain available whenever the tail provides them.

Think family first, backend second

A public constructor identifies the mathematical family. It does not ask the user to choose a bivariate or multivariate algorithm. Copulas.jl selects the appropriate density and sampling representation internally.

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;

• BivariatePickandsTail identifies a tail with a native scalar bivariate Pickands kernel and is a computational capability, not necessarily a mathematical restriction to dimension two;

• 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 can be built from mixed partial derivatives of $\ell$;

• rand(C, n) uses internal backend routing, so optimized bivariate and multivariate samplers coexist behind one public API.

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	scalar Pickands fast path 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 keeps its matrix representation, while dispatch selects the specialized bivariate kernel after recovering the scalar parameter.

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 keeps its matrix representation, while dispatch selects the specialized bivariate kernel from its off-diagonal correlation.

One public family, parameter-driven dispatch

Scalar and matrix constructors of each family share one tail type. The stored parameter type selects the exchangeable or general representation, and dimension-specific methods retain the fast bivariate kernels.

Tawn and asymmetric Galambos subset models

The multivariate Tawn model follows the asymmetric logistic construction of Tawn [45]. 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 [46]. The convenience constructor

AsymGalambosCopula{d}(α, weights)

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

Implementation-derived Mixed extension

The historical Mixed model is bivariate [47]. 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 [47] is the reference for the original bivariate Mixed model, and Galambos [48] for the negative-logistic component. The dimension-free convex-combination identity above is the extension used and derived in the Copulas.jl implementation; 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 [49], and the shape-constrained discrete spectral projection follows Gudendorf and Segers [50].

The resulting object stores a DiscreteSpectralTail; consequently the fitted STDF is valid by construction and exact spectral sampling is available.

One public constructor, two internal representations

EmpiricalEVCopula uses the lightweight historical implementation in two dimensions and the shape-constrained spectral representation in higher dimensions.

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 [51]:

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) [52] 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 [51] 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. [51]. 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 and backend routing

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

The public interface is always

rand(C, n)

and Copulas.jl chooses the backend internally. A tail with a native bivariate Pickands kernel can use the Ghoudi route in $d=2$; a family with a faster or more natural exact multivariate representation can transparently use that representation even in dimension two.

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-$t$. Backend routing is an implementation detail.

This separation is useful because the best algorithm is family-specific: specialized bivariate Pickands sampling is excellent for some tails, whereas spectral or max-stable constructions can be dramatically faster for others.

Copulas.Tail Type

Tail

Abstract type. Implements the API for stable tail dependence functions (STDFs) of extreme-value copulas in dimension d.

A STDF is a function $\ell :{\mathbb{R}}_{+}^d\to [0,\mathrm{\infty })$ that is 1-homogeneous ($\ell (t·x)=t·\ell (x)$ for all $t\ge 0$), convex, and satisfies the bounds $max(x_1,\dots ,x_d)\le \ell (x)\le x_1+\cdots +x_d$ (in particular $\ell (e_i)=1$).

Pickands representation. By homogeneity, for $x\ne 0$ let ${\Vert x\Vert }_1=x_1+\cdots +x_d$ and $\omega =x/{\Vert x\Vert }_1\in {\mathrm{\Delta }}_{d-1}$. There exists a Pickands dependence function $A:{\mathrm{\Delta }}_{d-1}\to [0,1]$ (convex, $max({\omega }_i)\le A(\omega )\le 1$) such that $\ell (x)={\Vert x\Vert }_1·A(\omega )$. For $d=2$, $A$ reduces to a convex function on $[0,1]$ with $max(t,1-t)\le A(t)\le 1$ and $A(0)=A(1)=1$.

Interface. A concrete tail must implement either A or ℓ; each representation is derived from the other by homogeneity.

• A(tail::Tail, ω::NTuple{d,Real}) — Pickands function on the simplex \Delta_{d-1}. (For d=2, a convenience A(tail::Tail, t::Real) may be provided.)

• ℓ(tail::Tail, x::NTuple{d,Real}) — STDF. By default the package defines ℓ(tail, x) = ‖x‖₁ * A(tail, x/‖x‖₁) when A is available.

We do not algorithmically verify convexity/bounds; implementers are responsible for validity.

Additional helpers (with defaults).

• For d=2: dA, d²A via AD; stable logpdf/rand (Ghoudi sampler).

• In any d: cdf(u) = exp(-ℓ(-log.(u))).

References:

• Pickands (1981); Gudendorf & Segers (2010); Ghoudi, Khoudraji & Rivest (1998); de Haan & Ferreira (2006).

• Rasell

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Copulas.ExtremeValueCopula Type

ExtremeValueCopula{d, TT}

Constructor

ExtremeValueCopula(d, tail::Tail)
ExtremeValueCopula{d}(tail::Tail)

Extreme-value copulas model tail dependence via a stable tail dependence function (STDF) $\ell$ or, equivalently, via a Pickands dependence function $A$. In any dimension $d$, the copula cdf is

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

For $d=2$, write $x=-\log u$, $y=-\log v$, $s=x+y$, and $t=x/s$. The relation between $\ell$ and $A$ is

$$\ell (x,y)=sA(t),A:[0,1]\to [1/2,1],A(0)=A(1)=1,A\text{ convex}.$$

Usage

• Provide any valid tail tail::Tail (which implements A and/or ℓ) to construct the copula.

• Sampling, cdf, and logpdf follow the standard Distributions.jl API.

Example

C = ExtremeValueCopula(2, GalambosTail(θ))
U = rand(C, 1000)
logpdf.(Ref(C), eachcol(U))

References:

• [53] G., & Segers, J. (2010). Extreme-value copulas. In Copula Theory and Its Applications (pp. 127-145). Springer.

• [4] Joe, H. (2014). Dependence Modeling with Copulas. CRC Press.

• [54] Mai, J. F., & Scherer, M. (2014). Financial engineering with copulas explained (p. 168). London: Palgrave Macmillan.

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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)
Cs = (
    GalambosCopula(2, 0.8),    # upper tail dep.
    HuslerReissCopula(2, 1.0), # intermediate
    LogCopula(2, 1.6),         # asymmetric
)
labels = ("Galambos(0.8)", "Hüsler–Reiss(1.0)", "Log(1.6)")
plot(size=(700, 300))
for (i, C) in enumerate(Cs)
    plot!(ts, Copulas.A.(C.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

MTail

Copulas.MTail Type

MTail

Corresponds to the MCopula viewed as an etreme value copula.

source

NoTail

Copulas.NoTail Type

NoTail

Corresponds to the case where the pickads function is identically One, which means no particular tail behavior.

source

TawnTail

Copulas.TawnTail Type

TawnTail(d, dep, asy)
TawnTail(α, weights)

Multivariate asymmetric-logistic stable tail dependence function. The full subset representation follows Tawn's multivariate extreme-value construction [45]:

$$\ell (x)=\sum _{\varnothing \ne C\subseteq \{1,\dots ,d\}}{[\sum _{i\in C}({\beta }_{i,C}{x}_i{)}^{{\alpha }_C}]}^{1/{\alpha }_C},$$

with α_C ≥ 1, β_{i,C} ≥ 0, β_{i,C}=0 for i ∉ C, and

$$\sum _{C\ni i}{\beta }_{i,C}=1$$

for every margin.

TawnTail(d, dep, asy) exposes the full subset model. TawnTail(α, weights) is a Copulas.jl convenience parameterization with one full-set logistic component plus singleton remainders; it is a structured submodel of the same valid Tawn representation, not a separate literature family.

References:

• [47] for the bivariate precursor.

• [45] for the multivariate model.

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AsymGalambosTail

Copulas.AsymGalambosTail Type

AsymGalambosTail{T}, AsymGalambosCopula{d,T}

AsymGalambosCopula{2}(α, θ₁, θ₂)
AsymGalambosCopula(2, α, θ₁, θ₂)
AsymGalambosCopula{d}(α, weights)
AsymGalambosCopula(d, α, weights)
AsymGalambosCopula{d}(dep, asy)
AsymGalambosCopula(d, dep, asy)

Asymmetric Galambos (negative-logistic) extreme-value family.

The family uses the subset-based negative-logistic/min-stable construction of Joe [46]. For nonempty subsets C,

$$\ell (x)=\sum _C{\ell }_{\mathrm{Galambos},{\alpha }_C}(({\beta }_{i,C}{x}_i{)}_{i\in C}),$$

with nonnegative asymmetry weights satisfying the marginal normalization constraints.

AsymGalambosCopula(d, dep, asy) exposes the full subset representation. AsymGalambosCopula{d}(α, weights) is a convenience parameterization with one full-set negative-logistic component and singleton remainders. In d=2, it is equivalent to the historical (α, θ₁, θ₂) parameterization and retains the specialized scalar Pickands formulas.

Literature model versus package parameterization

[46] supports the multivariate min-stable/negative-logistic construction. The one-full-set-plus-singletons weights constructor is a convenience parameterization introduced at the implementation level in Copulas.jl.

References:

• [48] Order statistics of samples from multivariate distributions. JASA, 1975.

• [46] Families of min-stable multivariate exponential and multivariate extreme value distributions. Statist. Probab, 1990.

source

AsymLogTail

Copulas.AsymLogTail Type

AsymLogTail{T}, AsymLogCopula{d,T}

Fields:

• α::Real — dependence parameter (α ≥ 1)

• θ₁::Real — asymmetry weight in [0,1]

• θ₂::Real — asymmetry weight in [0,1]

Constructor

AsymLogCopula{2}(α, θ₁, θ₂)
AsymLogCopula(2, α, θ₁, θ₂)
ExtremeValueCopula(2, AsymLogTail(α, θ₁, θ₂))

The (bivariate) asymmetric logistic extreme–value copula is parameterized by α ∈ [1, ∞) and θ₁, θ₂ ∈ [0,1]. Its Pickands dependence function is

$$A(t)=({\theta }_1^{\alpha }(1-t{)}^{\alpha }+{\theta }_2^{\alpha }{t}^{\alpha }{)}^{1/\alpha }+({\theta }_1-{\theta }_2)t+1-{\theta }_1,t\in [0,1].$$

Special cases:

• θ₁ = θ₂ = 1 ⇒ symmetric Logistic (Gumbel) copula

• α = 1 or either asymmetry weight is zero ⇒ independence (A(t) ≡ 1)

References:

• [47] : Tawn, Jonathan A. "Bivariate extreme value theory: models and estimation." Biometrika 75.3 (1988): 397-415.

source

AsymMixedTail

Copulas.AsymMixedTail Type

AsymMixedTail{T}, AsymMixedCopula{d,T}

Fields:

• θ₁::Real — parameter

• θ₂::Real — parameter

Constructor

AsymMixedCopula{2}(θ₁, θ₂) AsymMixedCopula(2, θ₁, θ₂) ExtremeValueCopula(2, AsymMixedTail(θ₁, θ₂))

The (bivariate) asymmetric Mixed extreme-value copula is parameterized by two parameters ${\theta }_1$, ${\theta }_2$ subject to the following constraints:

• θ₁ ≥ 0

• θ₁ + θ₂ ≤ 1

• θ₁ + 2θ₂ ≤ 1

• θ₁ + 3θ₂ ≥ 0

Its Pickands dependence function is

$$A(t)={\theta }_2{t}^3+{\theta }_1{t}^2-({\theta }_1+{\theta }_2)t+1,t\in [0,1].$$

Special cases:

• θ₁ = θ₂ = 0 ⇒ IndependentCopula

• θ₂ = 0 ⇒ symmetric Mixed copula

References:

• [47] : Tawn, Jonathan A. "Bivariate extreme value theory: models and estimation." Biometrika 75.3 (1988): 397-415.

source

BC2Tail

Copulas.BC2Tail Type

BC2Tail{T}, BC2Copula{d,T}

BC2Copula{2}(a, b)
BC2Copula(2, a, b)
BC2Copula{d}(a::AbstractVector)
BC2Copula(d, a::AbstractVector)
BC2Copula(a::AbstractVector)

BC2 extreme-value family with a finite two-atom spectral representation.

For the classical bivariate model [55],

$$A(t)=max\{at,b(1-t)\}+max\{(1-a)t,(1-b)(1-t)\}.$$

Copulas.jl also accepts a vector a=(a₁,…,a_d) and uses the direct d-dimensional two-atom spectral extension

$$\ell (x)=\underset{i}{max}(a_i{x}_i)+\underset{i}{max}((1-a_i)x_i).$$

The vector length determines d; in dimension two the same representation activates the specialized Pickands methods.

Copulas.jl multivariate parameterization

The bivariate BC2 model is documented in [55]. The vector constructor is the direct higher-dimensional two-atom spectral construction used by Copulas.jl; general finite spectral constructions are discussed in [56].

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CuadrasAugeTail

Copulas.CuadrasAugeTail Type

CuadrasAugeTail{T}, CuadrasAugeCopula{d,T}

CuadrasAugeCopula{d}(θ)
CuadrasAugeCopula(d, θ)

Cuadras-Augé extreme-value copula in dimension d ≥ 2, with θ ∈ [0,1]. Copulas.jl uses the stable tail dependence function

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

For d = 2 this yields the usual Pickands dependence function

$$A(t)=max\{t,1-t\}+(1-\theta )min\{t,1-t\}.$$

The model has a finite discrete-spectral representation.

Special cases:

• θ = 0 represents IndependentCopula(d).

• θ = 1 represents MCopula(d).

References:

• [56] Mai, J. F., & Scherer, M. (2012). Simulating copulas: stochastic models, sampling algorithms, and applications. World Scientific.

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GalambosTail

Copulas.GalambosTail Type

GalambosTail{T}, GalambosCopula{d,T}

GalambosCopula{d}(θ)
GalambosCopula(d, θ)

Galambos (negative-logistic) extreme-value copula in dimension d ≥ 2, with θ ∈ [0, ∞]. Its stable tail dependence function is

$$\ell (x)=\sum _{\varnothing \ne I\subseteq \{1,\dots ,d\}}(-1{)}^{|I|+1}{\left(\sum _{i\in I}{x}_i^{-\theta }\right)}^{-1/\theta }.$$

For d = 2, the equivalent Pickands dependence function is

$$A(t)=1-{(t^{-\theta }+(1-t{)}^{-\theta })}^{-1/\theta },$$

and the implementation uses the native bivariate derivatives when beneficial.

Special cases:

• θ = 0 represents IndependentCopula(d).

• θ = ∞ represents MCopula(d).

References:

• [48] Galambos, J. (1975). Order statistics of samples from multivariate distributions. Journal of the American Statistical Association, 70(351a), 674-680.

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HuslerReissTail

Copulas.HuslerReissTail Type

HuslerReissTail{T}, HuslerReissCopula{d,T}

HuslerReissCopula{d}(θ)
HuslerReissCopula(d, θ)
HuslerReissCopula{d}(Γ)
HuslerReissCopula(d, Γ)
HuslerReissCopula(Γ)

Hüsler-Reiss extreme-value copula.

HuslerReissCopula(d, θ) is the exchangeable representation with θ ∈ [0,∞]. For d > 2, it corresponds to a variogram with constant off-diagonal entry

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

HuslerReissCopula(Γ) is the general variogram representation. The square matrix Γ determines the dimension. It must be finite and symmetric, with zero diagonal, and must satisfy the Hüsler-Reiss variogram validity conditions. For a non-degenerate d ≥ 3 representation, off-diagonal entries are strictly positive and the variogram is strictly conditionally negative definite.

Scalar and matrix parameters are stored by the same HuslerReissTail type; its parameter type selects the exchangeable or general-variogram algorithms. A 2×2 variogram uses the specialized bivariate kernel after recovering $\theta =2/\sqrt{{\mathrm{\Gamma }}_{12}}$. In dimension two,

$$A(t)=t\mathrm{\Phi }({\theta }^{-1}+\frac{\theta }{2}\log \frac{t}{1-t})+(1-t)\mathrm{\Phi }({\theta }^{-1}+\frac{\theta }{2}\log \frac{1-t}{t}).$$

Special cases:

• θ = 0 represents IndependentCopula(d).

• θ = ∞, or an all-zero variogram, represents MCopula(d).

References:

• [57] Hüsler, J., & Reiss, R. D. (1989). Maxima of normal random vectors: between independence and complete dependence. Statistics & Probability Letters, 7(4), 283-286.

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LogTail

Copulas.LogTail Type

LogTail{T}, LogCopula{d,T}

LogCopula{d}(θ)
LogCopula(d, θ)

Logistic (Gumbel-Hougaard) extreme-value copula in dimension d ≥ 2, with θ ∈ [1, ∞]. Its stable tail dependence function is

$$\ell (x_1,\dots ,x_d)={\left(\sum _{i=1}^d{x}_i^{\theta }\right)}^{1/\theta }.$$

For d = 2 this is the usual logistic extreme-value model and is equivalent to GumbelCopula(2, θ). The same mathematical tail is used in every supported dimension, while dimension two retains specialized analytic kernels.

Special cases:

• θ = 1 represents IndependentCopula(d).

• θ = ∞ represents MCopula(d).

References:

• [47] Tawn, J. A. (1988). Bivariate extreme value theory: models and estimation. Biometrika, 75(3), 397-415.

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MixedTail

Copulas.MixedTail Type

MixedTail{T}, MixedCopula{d,T}

MixedCopula{d}(θ)
MixedCopula(d, θ)

Mixed extreme-value model with θ ∈ [0,1].

In dimension two its Pickands dependence function is

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

The original bivariate model is described by Tawn [47].

For d ≥ 2, Copulas.jl uses the dimension-free extension

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

This is a convex combination of the independence STDF and the multivariate Galambos STDF with parameter one, hence it is a valid STDF in every supported dimension. In d=2 it reduces exactly to the historical Mixed Pickands model.

Copulas.jl implementation derivation

The cited Tawn paper supports the original bivariate Mixed family and [48] supports the negative-logistic component. The dimension-free convex-combination identity above is the extension derived and used in Copulas.jl; it is not attributed here as a formula from either source.

Special case:

• θ = 0 represents IndependentCopula(d).

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MOTail

Copulas.MOTail Type

MOTail{T}, MOCopula{d,T}

MOCopula{2}(λ₁, λ₂, λ₁₂)
MOCopula(2, λ₁, λ₂, λ₁₂)
MOCopula{d}(λ::AbstractVector)
MOCopula(d, λ::AbstractVector)
MOCopula(λ::AbstractVector)

Marshall-Olkin extreme-value family.

The bivariate parameterization uses private-shock intensities λ₁, λ₂ ≥ 0 and common-shock intensity λ₁₂ ≥ 0.

The multivariate representation assigns one nonnegative shock intensity λ_S to every nonempty subset S ⊆ {1,…,d}. Therefore λ has length 2^d-1, ordered by subset cardinality and then lexicographically. If only λ is supplied, the dimension is inferred from its length.

With

$$r_i=\sum _{S\ni i}{\lambda }_S,$$

the stable tail dependence function is

$$\ell (x)=\sum _{\varnothing \ne S}\underset{i\in S}{max}\left(\frac{{\lambda }_S}{r_i}{x}_i\right).$$

Every margin must have positive total shock rate. Multiplying all shock intensities by the same positive constant leaves the copula unchanged.

References:

• [56] Mai, J. F., & Scherer, M. (2012). Simulating copulas: stochastic models, sampling algorithms, and applications. World Scientific.

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tEVTail

Copulas.tEVTail Type

tEVTail{T,P}, tEVCopula{d,T,P}

tEVCopula{d}(ν, ρ)
tEVCopula(d, ν, ρ)
tEVCopula{d}(ν, R)
tEVCopula(d, ν, R)

Extremal-t extreme-value copula with degrees of freedom ν > 0.

tEVCopula(d, ν, ρ) uses an exchangeable correlation matrix with common off-diagonal correlation ρ. For a non-degenerate d-dimensional model,

$$-\frac{1}{d-1}<\rho <1.$$

tEVCopula{d}(ν, R) uses a general correlation matrix R. R must be d×d, finite, symmetric, have unit diagonal, and be strictly positive definite in the non-degenerate general representation. Scalar and matrix parameters are stored by the same tEVTail type. A valid 2×2 matrix retains its matrix representation while the specialized bivariate analytic kernel recovers ρ from its off-diagonal entry.

For d = 2, the Pickands dependence function is

$$A(x)=x{t}_{\nu +1}(Z_x)+(1-x)t_{\nu +1}(Z_{1-x}),$$

where

$$Z_x=\sqrt{\frac{1+\nu }{1-{\rho }^2}}[{\left(\frac{x}{1-x}\right)}^{1/\nu }-\rho ].$$

Special case:

• ρ = 1 represents MCopula(d).

References:

• [58] Nikoloulopoulos, A. K., Joe, H., & Li, H. (2009). Extreme value properties of multivariate t copulas. Extremes, 12, 129-148.

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EmpiricalEVTail

Copulas.EmpiricalEVTail Type

EmpiricalEVTail

Fields:

• tgrid::Vector{Float64} — evaluation grid in (0,1)

• Ahat::Vector{Float64} — estimated Pickands function values on tgrid

• slope::Vector{Float64} — per-segment slopes for linear interpolation

Constructor

EmpiricalEVTail(u; method=:ols, grid=401, eps=1e-3, pseudo_values=true) ExtremeValueCopula(2, EmpiricalEVTail(u; ...))

The empirical extreme-value (EV) copula (bivariate) is defined from pseudo-observations u = (U₁, U₂) and a nonparametric estimator of the Pickands dependence function. Supported estimators are:

• :pickands — classical Pickands estimator

• :cfg — Capéraà–Fougères–Genest (CFG) estimator

• :ols — OLS-intercept estimator

For stability, the estimated function is always projected onto the class of valid Pickands functions (convex, bounded between max(t,1-t) and 1, with endpoints fixed at 1).

Its Pickands function is

Â(t),  t ∈ (0,1),

evaluated via piecewise linear interpolation on the grid tgrid.

References

• [caperaa1997nonparametric] Capéraà, Fougères, Genest (1997) Biometrika

• [gudendorf2011nonparametric] Gudendorf, Segers (2011) Journal of Multivariate Analysis

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EmpiricalEVMultivariateTail

Copulas.EmpiricalEVMultivariateTail Type

EmpiricalEVMultivariateTail(u; method=:ols, degree=nothing,
                            pseudo_values=true)

Shape-constrained nonparametric extreme-value tail in arbitrary dimension.

The pilot Pickands estimator is one of :ols, :cfg, or :pickands. The multivariate CFG/OLS construction follows Gudendorf and Segers [49].

For d ≥ 3, ordinary convexification is not sufficient to characterize a valid Pickands dependence function. The pilot is therefore projected by least squares onto a class generated by a finite spectral measure supported on a simplex grid, following the shape-constrained projection methodology of Gudendorf and Segers [50].

If the spectral grid has atoms v₁,…,vₘ and masses h₁,…,hₘ, the projected Pickands function is

$$A(w)=\sum _{k=1}^m{h}_k\underset{i}{max}(w_i{v}_{ik}),$$

subject to nonnegative masses and the spectral moment constraints. The resulting tail is stored as a DiscreteSpectralTail, so STDF validity and exact spectral simulation follow by construction.

degree controls the simplex-grid resolution. The automatic dimension-adaptive degree and the numerical projection details are Copulas.jl implementation choices; the statistical estimator and shape-constrained spectral projection are literature-based.

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References

2. H. Joe. Dependence Modeling with Copulas (CRC press, 2014).

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

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

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

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

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

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

9. 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).

10. P. Deheuvels. On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions. Statistics & Probability Letters 12, 429–439 (1991).

11. G. Gudendorf and J. Segers. Extreme-value copulas. In: Copula Theory and Its Applications: Proceedings of the Workshop Held in Warsaw, 25-26 September 2009 (Springer, 2010); pp. 127–145.

12. J.-F. Mai and M. Scherer. Financial engineering with copulas explained (Springer, 2014).

13. J.-F. Mai and M. Scherer. Bivariate extreme-value copulas with discrete Pickands dependence measure. Extremes 14, 311–324 (2011).

14. J.-F. Mai and M. Scherer. Simulating copulas: stochastic models, sampling algorithms, and applications. Vol. 4 (World Scientific, 2012).

15. J. Hüsler and R.-D. Reiss. Maxima of normal random vectors: between independence and complete dependence. Statistics & Probability Letters 7, 283–286 (1989).

16. A. K. Nikoloulopoulos, H. Joe and H. Li. Extreme value properties of multivariate t copulas. Extremes 12, 129–148 (2009).