Empirical Copula: Definition & Applications

Updated 22 November 2025

Empirical Copula is a nonparametric, rank-based estimator that maps the dependence structure of multivariate distributions using normalized ranks.
Smoothed variants—such as the empirical beta and Bernstein copulas—reduce bias and variance, ensuring genuine copula properties for improved inference.
Extensions to high-dimensional and discrete data, including weighted processes, enhance its use in robust statistical testing, resampling, and generative modeling.

An empirical copula is a nonparametric, fully rank-based estimator of the copula of a multivariate distribution, central to dependence modeling, nonparametric inference, and numerous statistical procedures in both theoretical and applied domains. The estimator is defined for both continuous and discrete (count or finite-support) data, with formal convergence theory under weak assumptions, and multiple smoothed variants for improved finite-sample properties. Recent developments extend empirical copula techniques to high dimensions, weighted process topologies, serial dependence, and discrete contingency tables, establishing their role as a backbone of modern nonparametric dependence analysis.

1. Formal Definition and Variants

The classical empirical copula is constructed from an $\mathbf{X}_1, \dots, \mathbf{X}_n$ sample from a $d$ -variate distribution with continuous margins $F_1, \dots, F_d$ . For each margin, compute the normalized ranks $R_{ij}$ : $R_{ij} = \sum_{k=1}^n \mathbf{1}\{X_{kj} \le X_{ij}\}.$ The pseudo-observations are $U_{ij} = R_{ij}/n$ , and the empirical copula $C_n:[0,1]^d \to [0,1]$ is

$C_n(u_1, \dots, u_d) = \frac{1}{n} \sum_{i=1}^n \prod_{j=1}^d \mathbf{1}\{U_{ij} \le u_j\}.$

This is a (piecewise constant) step-function estimator of the copula $C$ (Coblenz et al., 2023).

For discrete or finite-support data, the naive rank-based empirical copula fails to be a genuine copula (i.e., fails to have exactly uniform margins and is not always $d$ -increasing). In this setting, the multilinear or checkerboard extension is used: starting from the empirical distribution, multilinear interpolation yields a continuous $C_n^*$ such that for any margin, $u \mapsto C_n^*(\dots, u, \dots)$ is linear between adjacent observed values, and the whole function is a proper copula (Genest et al., 2014). For count data, the checkerboard/multilinear empirical copula has well-defined limiting theory only on open subsets of $[0,1]^d$ away from the discretization grid.

Smoothing the empirical copula yields further estimators---the empirical beta copula, empirical Bernstein copula, and their data-adaptive generalizations. The empirical beta copula, for instance, replaces the indicator $\mathbf{1}\{U_{ij} \le u_j\}$ by the Beta CDF $F_{n,R_{ij}}(u_j) = \mathrm{P}(\text{Bin}(n, u_j) \ge R_{ij})$ , so

$C_n^\beta(u_1, \dots, u_d) = \frac{1}{n} \sum_{i=1}^n \prod_{j=1}^d F_{n,R_{ij}}(u_j).$

Both $C_n$ and $C_n^\beta$ have exactly uniform margins for all $n$ , and $C_n^\beta$ is a genuine copula (Segers et al., 2016, Coblenz et al., 2023, Berghaus et al., 2017).

2. Asymptotic Theory and Weak Convergence

Under mild assumptions---existence and continuity of first partial derivatives of the true copula $C$ on interior subsets of $[0,1]^d$ ---the empirical copula process

$\mathbb{C}_n(u) = \sqrt{n}\, \bigl\{ C_n(u) - C(u) \bigr\}$

satisfies

$\mathbb{C}_n \rightsquigarrow \mathbb{C} \quad \text{in } \ell^\infty([0,1]^d)$

where $\mathbb{C}$ is a centered Gaussian process: $\mathbb{C}(u) = \alpha(u) - \sum_{j=1}^d \dot{C}_j(u)\, \alpha_j(u_j),$ with $\alpha$ the $C$ -Brownian bridge (covariance $C(u \wedge v) - C(u)C(v)$ ), and $\alpha_j(u_j) = \alpha(1,\dots,1,u_j,1,\dots,1)$ (Segers, 2010, Bücher et al., 2011, Bücher et al., 2014).

Smoothed variants such as the empirical beta copula process share the same limiting law: $\sqrt{n}\bigl(C_n^\beta(u) - C(u)\bigr) = \sqrt{n}\bigl(C_n(u) - C(u)\bigr) + o_p(1),$ with weak convergence in $\ell^\infty([0,1]^d)$ to the same Gaussian process (Segers et al., 2016, Berghaus et al., 2017, Kojadinovic et al., 2018).

When $d$ is allowed to increase with $n$ exponentially, Stute's representation shows that all fixed $k$ -margins of $\mathbb{C}_n$ linearize simultaneously, with rates $O(n^{-1/4} (\log n)^{3/4})$ , provided that $\log d = o(n^{1/3})$ (Bücher et al., 9 May 2024).

For discrete/finite-support data, the multilinear empirical copula process converges only in $\mathcal{C}(K)$ for compacts $K$ within the smooth subset $\mathcal{O}$ of $[0,1]^d$ , reflecting discontinuities inherited from the atom structure of the marginals (Genest et al., 2014).

3. Smoothing, Adaptivity, and Weighted Processes

To address the discontinuity and step-function bias of $C_n$ , multiple smoothing approaches have been developed:

Empirical beta copula: Automatic, parameter-free smoothing with optimal $O(n^{-1/2})$ bandwidth, always a genuine copula, often lower bias and variance than checkerboard or Bernstein copulas (Segers et al., 2016).
Empirical Bernstein copula: Uses Bernstein polynomial basis, but produces a proper copula only if the polynomial degree divides $n$ ; otherwise may require data-adaptive degree selection via empirical Bayes or plug-in criteria (Lu et al., 2021, Kojadinovic et al., 2021).
Weighted empirical processes: In applications focusing on tail dependence or boundary-sensitive functionals, the empirical copula process is normalized by a weight function $g(u)$ vanishing at the boundary, e.g., $g(u) = \min_j \{u_j \wedge \max_{k \neq j}(1-u_k)\}$ , and weak convergence is established for $\mathbb{C}_n(u)/g(u)^\omega$ for $\omega \in [0,1/2)$ (Bücher et al., 2014, Berghaus et al., 2017, Kojadinovic et al., 2018).

Weighted/smoothed empirical copulas permit robust implementation of Anderson–Darling type functionals and Pickands estimator for extreme-value copulas, even when $C$ or its score functions are singular or explode near the edges of $[0,1]^d$ .

4. Empirical Copula for Discrete Data

For joint distributions with finite supports, the continuous-rank-based definitions break down. The empirical discrete copula, as formalized by the minimum-Kullback–Leibler (Csiszár's I-projection) of the empirical joint count array onto uniform-margins polytopes, yields a canonical estimator: $\hat{\gamma}_n = \operatorname*{argmin}_{\gamma \in \Gamma} I(\gamma \| \hat{p}_n),$ where $\hat{p}_n$ is the Laplace-smoothed empirical frequency array, $\Gamma$ is the polytope of arrays with all margins uniform, and $I(\cdot \|\cdot)$ is the Kullback–Leibler divergence. The solution is computed with the Sinkhorn (IPF) algorithm (Geenens et al., 14 Jun 2025).

Main properties:

Strong $L^1$ consistency and root- $n$ asymptotic Gaussianity with explicit sandwich covariance.
Margin-free inference for discrete analogues of rank correlation (e.g., Yule's $\rho$ ).
Chi-square testing for (quasi-)independence using the linear parameterization of copula arrays.

The construction is directly analogous to entropic regularized optimal transport between the empirical joint and the set of margins-uniform arrays, uniting copula inference with the OT and minimum-divergence modeling frameworks (log-linear/exponential families) (Geenens et al., 14 Jun 2025).

For count data, the multilinear (checkerboard) extension (Genest et al., 2014) and the empirical discrete copula (Geenens et al., 14 Jun 2025) provide foundations for robust dependence testing in contingency tables, including scenarios where the sample size itself varies with the dimension.

5. Practical Applications and Computational Aspects

Empirical copula techniques underpin a vast array of procedures:

Goodness-of-fit and independence testing: Weighted Cramér–von Mises statistics with region-specific weights enhance sensitivity to tail or median deviations in copula structure. Empirical total-variation tests based on the supremum over growing families of boxes offer power against local departures, with nonparametric bootstrap providing consistent critical values (Medovikov, 2013, Fermanian et al., 2012).
Resampling and Inference: The empirical beta copula simplifies bootstrapping, as it is always a genuine copula and easy to sample from via mixture of beta laws conditioned on ranks. Confidence intervals, power, and coverage properties for rank-based functionals (Kendall's tau, Spearman's rho) uniformly benefit from the beta smoothing (Kiriliouk et al., 2019).
Streaming Algorithms: Memory-efficient, streaming empirical copula summaries (e.g., copula-quantile summaries for bivariate data) permit online computation and real-time dependence analytics, with provable error bounds, extensible to higher dimensions via vine constructions (Gregory, 2018).
High-dimensional Inference: Type-I error control and familywise error rate (FWER) for mass independence tests are ensured in high dimensions by multiplier bootstrapping and simultaneous linearization of all fixed $k$ -dimensional margins, with Gumbel limit theory for max-type statistics (Bücher et al., 9 May 2024).
Generative Modeling: In machine learning, empirical beta copula models have been deployed for nonparametric generative modeling of latent spaces in autoencoders, providing sample-efficient, easily conditioned generation within the convex hull of observed data (Coblenz et al., 2023).

Table: Common Empirical Copula Estimators and Their Key Features

Estimator	Copula for all $n$ ?	Smoothing Parameter	Asymptotic Law
Empirical copula $C_n$	Yes (if no ties)	None	Gaussian
Beta copula $C_n^\beta$	Yes	None	Gaussian, same as $C_n$
Checkerboard $C_n^\sharp$	Yes	None	Gaussian
Bernstein $B_m(C_n)$	Only if $m\|n$	$m$ (must tune)	If $m \gg \sqrt{n}$ : Gaussian
Discrete copula (Sinkhorn)	Yes (finite supports)	None (via IPF)	Sandwich-form Gaussian

6. Advanced Topics and Extensions

Serial and Long-Range Dependence: Extensions to stationary time series under $\alpha$ -mixing, strong mixing or even long-range dependence are available with corresponding functional CLTs, where the weight function and process topology must be carefully chosen (Bücher et al., 2011, Simayi, 2018).
Indexing by Functions: Weak convergence of the empirical copula process indexed by smooth or bounded-variation function classes (e.g., for generalized rank statistics) leverages novel multivariate integration-by-parts techniques, permitting inference beyond local box or step-function functionals (Radulovic et al., 2014).
Subsampling: Subsampling (with finite-population corrections) is asymptotically valid for empirical copula processes and their smooth/weighted variants, providing simple, tie-free inference in both i.i.d. and dependent (e.g., AR(1)) scenarios (Kojadinovic et al., 2018).
Adaptive and Mixture Smoothing: New classes of smooth, data-adaptive copula estimators combine conditional beta kernels with pilot copulas or shape control parameters, systematically reducing integrated mean-squared error relative to beta copulas in finite samples (Kojadinovic et al., 2021).

7. Connections to Optimal Transport and Log-linear Models

Fundamentally, the empirical discrete copula for finite-support data is the minimum-divergence (I-projection) plan between the empirical joint and uniform margins, coinciding exactly with the entropic regularized optimal transport solution when the cost $C=0$ (Geenens et al., 14 Jun 2025). Both Sinkhorn and iterative proportional fitting (IPF) algorithms solve the same convex optimization, uniting minimum-divergence models across copulas, log-linear contingency table models, and regularized OT. This formalism connects empirical copulas to exponential families and broader statistical inference frameworks, with empirical copula arrays corresponding to projected log-linear models with uniform-margins constraints.

References:

(Geenens et al., 14 Jun 2025) Geenens, Kojadinović, and Martini, "The empirical discrete copula process" (2025)
(Segers et al., 2016) Segers, Sibuya & Tsukahara, "The Empirical Beta Copula" (2016)
(Berghaus et al., 2017) Berghaus & Segers, "Weak convergence of the weighted empirical beta copula process" (2017)
(Coblenz et al., 2023) Empirical-Beta-Copula Autoencoder (2023)
(Bücher et al., 9 May 2024) "The empirical copula process in high dimensions: Stute's representation and applications" (2024)
(Segers, 2010) Segers, "Asymptotics of empirical copula processes under non-restrictive smoothness assumptions" (2010)
(Genest et al., 2014) Genest, Nešlehová & Rémillard, "On the empirical multilinear copula process for count data" (2014)
(Kojadinovic et al., 2018) Kojadinović & Stemikovskaya, "Subsampling (weighted smooth) empirical copula processes" (2018)