Copula Distributions

Copula distributions: elliptical and Archimedean families.

Copula Base

The universal abstract base class shared by every copula family (Mean-Variance, Archimedean).

CopulAX shared copula base class.

Houses CopulaBase, the universal abstract base for every copula family in copulax (Archimedean, mean-variance, future extensions). It carries the Sklar / marginal-fitting / sampling machinery common to all copulas.

Mean-variance copulas (Gaussian, Student-T, GH, Skewed-T) live in _mv_copulas.py; Archimedean copulas live in _archimedean.py.

class copulax._src.copulas._distributions.CopulaBase(name)[source]

Base class for all copula distributions.

Provides Sklar’s theorem implementations for the joint distribution, common marginal-fitting logic, and sampling via inverse-transform of copula samples.

Parameters:

name (str)

property dist_type: str

Distribution family type identifier.

support(params=None)[source]

Support of the joint distribution.

Return type:

Array

Parameters:

params (dict)

get_u(x, params=None)[source]

Compute marginal CDF values u = (F_1(x_1), …, F_d(x_d)).

Parameters:
Return type:

Array

Returns:

Array of shape (n, d) with values in [0, 1].

abstractmethod copula_logpdf(u, params=None, **kwargs)[source]

Log-density of the copula (subclasses must implement).

Return type:

Array

Parameters:
copula_pdf(u, params=None, **kwargs)[source]

Density of the copula: c(u) = exp(copula_logpdf(u)).

Return type:

Array

Parameters:
abstractmethod copula_rvs(size, params, key=None)[source]

Generate random samples from the copula (subclasses must implement).

Return type:

Array

Parameters:
copula_sample(size, params=None, key=None)[source]

Alias for copula_rvs.

Return type:

Array

Parameters:
logpdf(x, params=None, **kwargs)[source]

Joint log-PDF via Sklar’s theorem.

log f(x) = log c(F_1(x_1),…,F_d(x_d)) + sum log f_i(x_i)

Parameters:
Return type:

Array

Returns:

Array of shape (n, 1).

pdf(x, params=None, **kwargs)[source]

Joint PDF.

Return type:

Array

Parameters:
rvs(size, params=None, key=None, brent=False, nodes=100)[source]

Sample from the joint distribution.

  1. Sample u from copula

  2. Transform u to x via marginal PPFs

Parameters:

  • size (Union[float, int, Array, ndarray, bool, number, bool, complex]) – Number of samples.

  • params (dict) – Distribution parameters.

  • key (Array) – JAX random key.

  • brent (bool) – Forwarded to the marginal Univariate.ppf(). False (default) uses the analytical inverse CDF when available and otherwise the Chebyshev cubic spline; True forces per-quantile Brent root-finding (slower but machine-epsilon accurate).

  • nodes (int) – Number of Chebyshev-Lobatto nodes when the cubic path is used. Ignored for analytical marginals and when brent=True.

Return type:

Array

Returns:

Array of shape (size, d).

fit_marginals(x, univariate_fitter_options=None)[source]

Fit univariate marginal distributions to each dimension.

Parameters:

  • x (Union[Array, ndarray, bool, number, bool, int, float, complex]) – Input data of shape (n, d).

  • univariate_fitter_options (tuple[dict] | dict) – Options for the univariate fitter. Dict applies same options to all dimensions; tuple of dicts applies per-dimension options.

Return type:

dict

Note

Not jitable.

Return type:

dict

Returns:

dict with key ‘marginals’ containing fitted distributions.

Parameters:
abstractmethod fit_copula(u, **kwargs)[source]

Fit copula parameters (subclasses must implement).

Return type:

dict

Parameters:

u (Array | ndarray | bool | number | bool | int | float | complex)

fit(x, univariate_fitter_options=None, name=None, **kwargs)[source]

Fit marginals and copula to the data.

Equivalent to calling fit_marginals then fit_copula.

Parameters:

  • x (Union[Array, ndarray, bool, number, bool, int, float, complex]) – Input data of shape (n, d).

  • univariate_fitter_options (tuple[dict] | dict) – Options for marginal fitting.

  • name (str) – Optional custom name for the fitted instance.

  • **kwargs – Additional arguments forwarded to fit_copula (method, lr, maxiter, tol, patience).

Return type:

dict

Note

Not jitable.

Return type:

dict

Returns:

dict with keys ‘marginals’ and ‘copula’.

Parameters:

Mean-Variance Copulas

Copulas derived from normal mixture distributions (McNeil, Frey, Embrechts 2005, §3.2). MeanVarianceCopulaBase is the umbrella; EllipticalCopula covers normal variance mixtures (Gaussian, Student-T; γ=0, strictly elliptical); MeanVarianceCopula covers normal mean-variance mixtures (GH, Skewed-T; γ≠0).

CopulAX implementation of mean-variance (normal-mixture) copulas.

Houses the umbrella MeanVarianceCopulaBase plus its two taxonomic sub-bases — EllipticalCopula (γ=0; pure normal variance mixtures: Gaussian, Student-T) and MeanVarianceCopula (γ≠0; normal mean-variance mixtures: GH, Skewed-T) — and all four concrete copula classes.

The shared CopulaBase (Sklar / marginal fitting / sampling machinery used by every copula family, including Archimedean) lives in _distributions.py and is imported here.

Reference:

McNeil, Frey, Embrechts (2005) Quantitative Risk Management, §3.2.1 (variance mixtures), §3.2.2 (mean-variance mixtures), §3.2.4 Algorithm 3.14 (EM / ECME for GH).

class copulax._src.copulas._mv_copulas.MeanVarianceCopulaBase(name, mvt, uvt, *, marginals=None, copula=None)[source]

Umbrella base class for normal-mixture copula distributions.

Holds the shared fit_copula dispatcher, _METHOD_KWARGS validation, correlation estimation, and other machinery common to both normal variance mixture copulas (true elliptical, γ=0; see EllipticalCopula) and normal mean-variance mixture copulas (with skewness γ; see MeanVarianceCopula).

Reference:

McNeil, Frey, Embrechts (2005) Quantitative Risk Management, §3.2.1 (variance mixtures) and §3.2.2 (mean-variance mixtures).

Parameters:

  • mvt (Multivariate)

  • uvt (Univariate)

example_params(dim=3, *args, **kwargs)[source]

Example parameters for the copula distribution.

Generates example marginal and copula parameters for the overall joint distribution.

Parameters:

dim (int) – int, number of dimensions of the copula distribution. Default is 3.

get_x_dash(u, params, brent=False, nodes=100)[source]

Computes x’ values, which represent the mappings of the independent marginal cdf values (U) to the domain of the joint multivariate distribution.

Routes through Univariate.ppf(), so distributions with an analytical inverse CDF (Normal, Gamma, LogNormal, IG, Uniform, Gen-Normal) use the closed-form path automatically and ignore nodes.

Note

If you intend to jit wrap this function, both brent and nodes must be static arguments.

Parameters:

  • u (Union[Array, ndarray, bool, number, bool, int, float, complex]) – The independent univariate marginal cdf values (U) for each dimension, shape (n, d).

  • params (dict) – The copula and marginal distribution parameters.

  • brent (bool) – If False (default), use the analytical inverse CDF when available and otherwise the Chebyshev-node cubic spline approximation. If True, force per-quantile Brent root-finding (machine-epsilon accurate, slower).

  • nodes (int) – Number of Chebyshev-Lobatto nodes used by the cubic spline path. Ignored for analytical marginals and when brent=True.

Return type:

Array

Returns:

x' values of shape (n, d).

copula_logpdf(u, params=None, brent=False, nodes=100)[source]

Computes the log-pdf of the copula distribution.

Note

If you intend to jit wrap this function, both brent and nodes must be static arguments.

Parameters:

  • u (Union[Array, ndarray, bool, number, bool, int, float, complex]) – The independent univariate marginal cdf values (u) for each dimension.

  • params (dict) – The copula and marginal distribution parameters.

  • brent (bool) – Forwarded to get_x_dash(). False (default) uses the analytical inverse CDF when available and otherwise the Chebyshev cubic spline; True forces per-quantile Brent root-finding.

  • nodes (int) – Number of Chebyshev-Lobatto nodes used by the cubic spline path. Ignored for analytical marginals and when brent=True.

Returns:

The log-pdf values of the copula

distribution.

Return type:

Array

copula_rvs(size, params=None, key=None)[source]

Generates random samples from the copula distribution.

Note

If you intend to jit wrap this function, ensure that ‘size’ is a static argument.

Parameters:

  • size (Union[float, int, Array, ndarray, bool, number, bool, complex]) – size (Scalar): The size / shape of the generated output array of random numbers. Must be scalar. Generates an (size, d) array of random numbers, where d is the number of dimensions inferred from the provided distribution parameters.

  • params (dict) – The copula and marginal distribution parameters.

  • key (Array) – The Key for random number generation.

Return type:

Array

fit_copula(u, corr_method='rm_pp_kendall', method='fc_mle', **kwargs)[source]

Fit copula parameters from pseudo-observations.

Two-stage estimation following McNeil, Frey & Embrechts (2005), Section 5.5:

Stage 1 — Estimate the copula correlation matrix P from the pseudo-observations u using rank correlation. The default rm_pp_kendall computes Kendall’s tau, applies the inversion \(\hat\rho_{ij} = \sin(\tfrac{\pi}{2}\,\hat\tau_{ij})\) (Proposition 5.37), and ensures positive semi-definiteness via eigenvalue clamping.

Stage 2 — Estimate remaining parameters (e.g. ν, γ) by maximising the copula log-likelihood, with P either held fixed at the Stage 1 estimate or jointly re-optimised, depending on method.

Parameters:

  • u (Union[Array, ndarray, bool, number, bool, int, float, complex]) – Pseudo-observations of shape (n, d) in [0, 1].

  • corr_method (str) – Correlation estimation method for Stage 1. Default 'rm_pp_kendall'. See copulax.multivariate.corr for all methods.

  • method (str) – Fitting algorithm for Stage 2. One of: 'fc_mle' — Fixed-Correlation MLE: shape parameters optimised via projected gradient with Σ held at the Stage 1 Kendall-τ estimate. Available for every concrete subclass. 'mle' — Full joint MLE: all parameters (shape + Σ off-diagonals) optimised together via Adam, with Σ tanh-parameterised onto the correlation manifold. Mean-variance subclasses (GH, SkewedT) only. 'ecme' — Inner EM updates (P, γ); outer gradient descent on the copula log-likelihood for the remaining shape parameters (McNeil §3.2.4 ECME variant). Mean-variance subclasses only. 'ecme_double_gamma' — Like ecme but γ is additionally re-optimised in the outer numerical M-step (so γ is updated twice per outer iteration). Mean-variance subclasses only. 'ecme_outer_gamma' — Inner EM updates Σ only (γ frozen); outer MLE on all shape parameters including γ. Mean-variance subclasses only.

  • **kwargs – Method-specific keyword arguments. Each method accepts only its own set of kwargs; Common kwargs: lr (float, all methods), maxiter (int, all), brent (bool, all except fc_mle), nodes (int, all except fc_mle).

Return type:

dict

Returns:

dict with key 'copula' containing fitted parameters.

Raises:

ValueError – If method is not accepted by this subclass, or if kwargs contains a key not accepted by the chosen method.

class copulax._src.copulas._mv_copulas.EllipticalCopula(name, mvt, uvt, *, marginals=None, copula=None)[source]

True elliptical copulas (normal variance mixtures, γ=0).

Concrete subclasses (GaussianCopula, StudentTCopula) only support the 'fc_mle' Stage 2 fitting method.

Reference:

McNeil, Frey, Embrechts (2005) Quantitative Risk Management, §3.2.1 Normal Variance Mixtures.

Parameters:

  • mvt (Multivariate)

  • uvt (Univariate)

class copulax._src.copulas._mv_copulas.MeanVarianceCopula(name, mvt, uvt, *, marginals=None, copula=None)[source]

Normal mean-variance mixture copulas with skewness γ.

Concrete subclasses (GHCopula, SkewedTCopula) additionally implement γ-aware fitting methods (mle, ecme, ecme_double_gamma, ecme_outer_gamma) on top of fc_mle.

Note

MeanVarianceCopulaBase is the broader umbrella covering this class and EllipticalCopula (the γ=0 special case). This class is the proper γ≠0 specialisation.

Reference:

McNeil, Frey, Embrechts (2005) Quantitative Risk Management, §3.2.2 Normal Mean-Variance Mixtures, §3.2.4 Algorithm 3.14.

Parameters:

  • mvt (Multivariate)

  • uvt (Univariate)

class copulax._src.copulas._mv_copulas.GaussianCopula(name, mvt, uvt, *, marginals=None, copula=None)[source]

The Gaussian Copula is a copula that uses the multivariate normal distribution to model the dependencies between random variables.

The copula is parameterised by the correlation matrix P only. Fitting estimates P from pseudo-observations via rank correlation (McNeil et al. 2005, Example 5.58).

https://en.wikipedia.org/wiki/Copula_(statistics)

Parameters:

  • mvt (Multivariate)

  • uvt (Univariate)

class copulax._src.copulas._mv_copulas.StudentTCopula(name, mvt, uvt, *, marginals=None, copula=None)[source]

The Student-T Copula is a copula that uses the multivariate Student-T distribution to model the dependencies between random variables.

The copula is parameterised by degrees of freedom ν and correlation matrix P. Fitting estimates P via Kendall’s tau inversion and ν by maximising the copula log-likelihood (McNeil et al. 2005, Examples 5.54 and 5.59).

https://en.wikipedia.org/wiki/Copula_(statistics)

Parameters:

  • mvt (Multivariate)

  • uvt (Univariate)

class copulax._src.copulas._mv_copulas.GHCopula(name, mvt, uvt, *, marginals=None, copula=None)[source]

The GH Copula is a copula that uses the multivariate generalized hyperbolic (GH) distribution to model the dependencies between random variables.

The copula is parameterised by (λ, χ, ψ, γ) and correlation matrix P. Fitting estimates P via Kendall’s tau inversion and the remaining parameters via ML or EM (McNeil et al. 2005, Section 5.5).

https://en.wikipedia.org/wiki/Copula_(statistics)

Parameters:

  • mvt (Multivariate)

  • uvt (Univariate)

class copulax._src.copulas._mv_copulas.SkewedTCopula(name, mvt, uvt, *, marginals=None, copula=None)[source]

The Skewed-T Copula is a copula that uses the multivariate skewed-T distribution to model the dependencies between random variables.

The copula is parameterised by degrees of freedom ν, skewness vector γ, and correlation matrix P. Fitting estimates P via Kendall’s tau inversion and (ν, γ) via ML or EM (McNeil et al. 2005, Section 5.5).

https://en.wikipedia.org/wiki/Copula_(statistics)

Parameters:

  • mvt (Multivariate)

  • uvt (Univariate)

Archimedean Copulas

Copulas defined by a generator function φ and its inverse ψ.

CopulAX implementation of Archimedean copula distributions.

An Archimedean copula is defined by a generator function φ: [0,1] → [0,∞) with φ(1) = 0, and its pseudo-inverse ψ = φ⁻¹:

C(u₁,…,u_d) = ψ(φ(u₁) + … + φ(u_d))

References

Nelsen, R. B. (2006). An Introduction to Copulas, 2nd ed.

Springer Series in Statistics.

McNeil, A. J., Frey, R., & Embrechts, P. (2005).

Quantitative Risk Management. Princeton University Press.

McNeil, A. J. & Nešlehová, J. (2009). Multivariate Archimedean

copulas, d-monotone functions and l₁-norm symmetric distributions. Annals of Statistics, 37(5B), 3059-3097.

Hofert, M. (2011). Efficiently Sampling Nested Archimedean

Copulas. Computational Statistics & Data Analysis, 55(1), 57-70.

class copulax._src.copulas._archimedean.ArchimedeanCopula(name, *, marginals=None, copula=None)[source]

Base class for Archimedean copula distributions.

An Archimedean copula is defined by a generator function φ: [0,1] → [0,∞) with φ(1) = 0, and its pseudo-inverse ψ = φ⁻¹:

C(u₁,…,u_d) = ψ(φ(u₁) + … + φ(u_d))

The copula density is:

\[c(u) = \bigl|\psi^{(d)}\bigl(\textstyle\sum_i \phi(u_i)\bigr)\bigr| \cdot \prod_i \bigl|\phi'(u_i)\bigr|\]

where \(\psi^{(d)}\) is the d-th derivative of the inverse generator.

References

Nelsen, R. B. (2006). An Introduction to Copulas, 2nd ed.

Springer Series in Statistics.

McNeil, A. J. & Nešlehová, J. (2009). Multivariate Archimedean

copulas, d-monotone functions and l₁-norm symmetric distributions. Annals of Statistics, 37(5B), 3059-3097.

generator(t, theta)[source]

Generator function φ(t; θ).

Must satisfy φ(1) = 0, φ is strictly decreasing and convex.

Return type:

Union[float, int, Array, ndarray, bool, number, bool, complex]

Parameters:
generator_inv(s, theta)[source]

Inverse generator ψ(s; θ) = φ⁻¹(s; θ).

Also known as the Laplace-Stieltjes transform.

Return type:

Union[float, int, Array, ndarray, bool, number, bool, complex]

Parameters:
example_params(dim=3, *args, **kwargs)[source]

Example parameters for the Archimedean copula distribution.

Parameters:

dim (int) – Number of dimensions. Default is 3.

Return type:

dict

Returns:

dict with keys ‘marginals’ and ‘copula’.

copula_cdf(u, params=None)[source]

Copula CDF: C(u₁,…,u_d) = ψ(φ(u₁) + … + φ(u_d)).

Parameters:
Return type:

Array

Returns:

Array of shape (n, 1).

copula_logpdf(u, params=None, **kwargs)[source]

Copula log-density via generator derivatives.

Uses the formula:

log c(u) = log|ψ⁽ᵈ⁾(∑ φ(uᵢ))| + ∑ log|φ’(uᵢ)|

where ψ⁽ᵈ⁾ is computed via d nested applications of jax.grad.

Parameters:
Return type:

Array

Returns:

Array of shape (n, 1).

copula_rvs(size, params=None, key=None)[source]

Sample from the copula using the Marshall-Olkin algorithm.

Algorithm (Marshall & Olkin, 1988):

  1. Sample V ~ frailty distribution F_θ

  2. Sample E₁,…,E_d iid ~ Exp(1)

  3. Uᵢ = ψ(Eᵢ / V)

Parameters:
Return type:

Array

Returns:

Array of shape (size, d) with values in (0, 1).

aic(x, params=None)[source]

Akaike Information Criterion.

Return type:

float

Parameters:
bic(x, params=None)[source]

Bayesian Information Criterion.

Return type:

float

Parameters:
fit_copula(u, method='kendall', **kwargs)[source]

Fit the copula parameter θ.

Parameters:
Returns:

fitted_theta}.

Return type:

dict

Raises:

ValueError – If method is not 'kendall', or if kwargs contains any keys (the Kendall method takes no tuning parameters).

class copulax._src.copulas._archimedean.ClaytonCopula(name, *, marginals=None, copula=None)[source]

Clayton copula with generator φ(t) = t^{-θ} - 1.

Parameters:

∈ (θ)

Properties:

  • Lower tail dependence: λ_L = 2^{-1/θ}

  • Upper tail dependence: λ_U = 0

  • Kendall’s tau: τ = θ/(θ+2)

References

Clayton, D. G. (1978). A model for association in bivariate

life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika, 65(1), 141-151.

Nelsen (2006), Example 4.2.

generator(t, theta)[source]

Generator function φ(t; θ).

Must satisfy φ(1) = 0, φ is strictly decreasing and convex.

generator_inv(s, theta)[source]

Inverse generator ψ(s; θ) = φ⁻¹(s; θ).

Also known as the Laplace-Stieltjes transform.

copula_logpdf(u, params=None, **kwargs)[source]

Closed-form Clayton copula log-density.

log c(u) = ∑_{k=1}^{d-1} log(1 + kθ)

  • (1/θ + d) · log(∑ uᵢ^{-θ} - (d-1))

  • (θ + 1) · ∑ log(uᵢ)

Derivation from ψ⁽ᵈ⁾(s) = (-1)^d · ∏_{k=0}^{d-1}(1/θ+k) · (1+s)^{-1/θ-d}.

class copulax._src.copulas._archimedean.FrankCopula(name, *, marginals=None, copula=None)[source]

Frank copula with generator φ(t) = -ln((e^{-θt}-1)/(e^{-θ}-1)).

Parameters:

{0} (θ ∈ ℝ)

Properties:

  • No tail dependence: λ_L = λ_U = 0

  • Allows negative dependence (θ < 0)

  • Kendall’s tau: τ = 1 - 4/θ · (1 - D₁(θ)) where D₁ is the first Debye function

References

Frank, M. J. (1979). On the simultaneous associativity of

F(x,y) and x + y - F(x,y). Aequationes Mathematicae, 19, 194-226.

Nelsen (2006), Example 4.5.

generator(t, theta)[source]

Generator function φ(t; θ).

Must satisfy φ(1) = 0, φ is strictly decreasing and convex.

generator_inv(s, theta)[source]

Inverse generator ψ(s; θ) = φ⁻¹(s; θ).

Also known as the Laplace-Stieltjes transform.

class copulax._src.copulas._archimedean.GumbelCopula(name, *, marginals=None, copula=None)[source]

Gumbel copula with generator φ(t) = (-ln t)^θ.

Parameters:

  • [1 (θ ∈)

  • ∞)

Properties:

  • No lower tail dependence: λ_L = 0

  • Upper tail dependence: λ_U = 2 - 2^{1/θ}

  • Kendall’s tau: τ = 1 - 1/θ

  • θ = 1 gives the independence copula

References

Gumbel, E. J. (1960). Distributions des valeurs extrêmes en

plusieurs dimensions. Publications de l’Institut de Statistique de l’Université de Paris, 9, 171-173.

Nelsen (2006), Example 4.4.

generator(t, theta)[source]

Generator function φ(t; θ).

Must satisfy φ(1) = 0, φ is strictly decreasing and convex.

generator_inv(s, theta)[source]

Inverse generator ψ(s; θ) = φ⁻¹(s; θ).

Also known as the Laplace-Stieltjes transform.

class copulax._src.copulas._archimedean.JoeCopula(name, *, marginals=None, copula=None)[source]

Joe copula with generator φ(t) = -ln(1 - (1-t)^θ).

Parameters:

  • [1 (θ ∈)

  • ∞)

Properties:

  • No lower tail dependence: λ_L = 0

  • Upper tail dependence: λ_U = 2 - 2^{1/θ}

  • Kendall’s tau: τ = 1 - 4·∑_{k=1}^∞ 1/(k(kθ+2)((k-2)θ+2))

References

Joe, H. (1993). Parametric families of multivariate

distributions with given margins. Journal of Multivariate Analysis, 46(2), 262-282.

Nelsen (2006), Example 4.10.

generator(t, theta)[source]

Generator function φ(t; θ).

Must satisfy φ(1) = 0, φ is strictly decreasing and convex.

generator_inv(s, theta)[source]

Inverse generator ψ(s; θ) = φ⁻¹(s; θ).

Also known as the Laplace-Stieltjes transform.

class copulax._src.copulas._archimedean.AMHCopula(name, *, marginals=None, copula=None)[source]

Ali-Mikhail-Haq copula with generator φ(t) = ln((1-θ(1-t))/t).

Parameters:

  • [-1 (θ ∈)

  • 1)

Properties:

  • Only valid for dimension d ≤ 2

  • Weak dependence: τ ∈ [-0.1817, 1/3)

  • No tail dependence: λ_L = λ_U = 0

Note

The AMH generator is only completely monotone (hence valid for all dimensions) when θ ∈ [0, 1). For θ ∈ [-1, 0), the copula is only valid for d = 2 (2-monotone but not completely monotone). The Marshall-Olkin sampling uses a Geometric frailty distribution which requires θ ∈ [0, 1). For θ < 0, frailty V is set to 1, providing approximate sampling.

References

Ali, M. M., Mikhail, N. N., & Haq, M. S. (1978). A class

of bivariate distributions including the bivariate logistic. Journal of Multivariate Analysis, 8(3), 405-412.

Nelsen (2006), Example 4.8.

generator(t, theta)[source]

Generator function φ(t; θ).

Must satisfy φ(1) = 0, φ is strictly decreasing and convex.

generator_inv(s, theta)[source]

Inverse generator ψ(s; θ) = φ⁻¹(s; θ).

Also known as the Laplace-Stieltjes transform.

example_params(dim=2, *args, **kwargs)[source]

Example parameters for AMH copula (d=2 only).

Parameters:

dim (int) – Must be 2 for AMH copula.

Raises:

ValueError – If dim != 2.

Return type:

dict

class copulax._src.copulas._archimedean.IndependenceCopula(name, *, marginals=None, copula=None)[source]

Independence copula (product copula) C(u₁,…,u_d) = ∏ uᵢ.

The independence copula corresponds to the Archimedean generator φ(t) = −ln(t) with inverse ψ(s) = e^{−s}. It has no free parameters and represents the case of stochastically independent margins.

Properties:

  • No tail dependence: λ_L = λ_U = 0

  • Kendall’s tau: τ = 0

  • Copula density: c(u) = 1 for all u ∈ (0,1)^d

  • No parameters to fit

  • Valid for any dimension d ≥ 2

The independence copula is useful as a null / benchmark model for comparing against parametric copula fits.

References

Nelsen, R. B. (2006). An Introduction to Copulas, 2nd ed.

Springer Series in Statistics, Section 2.5.

generator(t, theta)[source]

Generator function φ(t; θ).

Must satisfy φ(1) = 0, φ is strictly decreasing and convex.

generator_inv(s, theta)[source]

Inverse generator ψ(s; θ) = φ⁻¹(s; θ).

Also known as the Laplace-Stieltjes transform.

example_params(dim=3, *args, **kwargs)[source]

Example parameters for the independence copula.

Parameters:

dim (int) – Number of dimensions. Default is 3.

Return type:

dict

Returns:

dict with keys ‘marginals’ and ‘copula’ (empty dict).

copula_cdf(u, params=None, **kwargs)[source]

Independence copula CDF: C(u₁,…,u_d) = ∏ uᵢ.

Parameters:

  • u – Uniform marginal values of shape (n, d).

  • params – Ignored (no copula parameters).

Returns:

Array of shape (n, 1).

copula_logpdf(u, params=None, **kwargs)[source]

Independence copula log-density: log c(u) = 0.

Parameters:

  • u – Uniform marginal values of shape (n, d).

  • params – Ignored (no copula parameters).

Returns:

Array of zeros with shape (n, 1).

copula_rvs(size, params=None, key=None)[source]

Sample independent uniform margins.

Parameters:

  • size – Number of samples to generate.

  • params – Must contain ‘marginals’ (for dimension inference).

  • key – JAX random key.

Returns:

Array of shape (size, d) with iid Uniform(0,1) entries.

fit_copula(u, method='kendall', **kwargs)[source]

No parameters to fit for the independence copula.

Validates method (only 'kendall' is accepted) and rejects any kwargs for parity with the rest of the family, then returns an empty copula-params dict.

Returns:

dict with key ‘copula’ → {} (empty).

Parameters:

method (str)

aic(x, params=None)[source]

Akaike Information Criterion with k=0 free parameters.

bic(x, params=None)[source]

Bayesian Information Criterion with k=0 free parameters.