Copula Entropy Two-Sample Tests

• Copula Entropy-Based Two-Sample Tests are nonparametric procedures that recast the equality-of-distributions problem as a dependence challenge between pooled data and sample labels.
• They use a rank-transformed empirical copula density and the KSG entropy estimator to quantify the contrast between null and actual labelings.
• The method is multivariate and applicable to change-point detection and symmetry testing, though it features limited inferential calibration.

Copula entropy-based two-sample tests are nonparametric procedures that use copula entropy (CE) to compare two samples by recasting the equality-of-distributions problem as a dependence problem between pooled observations and a sample-membership label. In the formulation introduced in "Two-Sample Test with Copula Entropy" (Ma, 2023) and reviewed in "Copula Entropy: Theory and Applications" (Ma, 20 Dec 2025), the key object is not a direct distance between two empirical copulas, but a contrast between CE under a null labeling and CE under the actual two-sample labeling. This places CE-based two-sample tests at the intersection of copula theory, information-theoretic dependence measurement, and multivariate nonparametric testing.

1. Conceptual setting and scope

The standard setting considers two samples

$$\mathbf{X}_1=\{X_{11},\cdots,X_{1m}\}\sim P_1,\qquad \mathbf{X}_2=\{X_{21},\cdots,X_{2n}\}\sim P_2,$$

with $\mathbf{X}_1,\mathbf{X}_2\in \mathbb{R}^d$, and the hypotheses

$H_0: P_1 = P_2,\qquad H_1: P_1 \neq P_2.$

In the original CE two-sample paper, the problem is explicitly framed as equality of the full multivariate distributions $P_1$ and $P_2$, not merely equality of copulas or of a scalar dependence coefficient (Ma, 2023).

This formulation differs from two neighboring literatures. First, it differs from one-sample copula goodness-of-fit testing, where one tests whether a single sample follows a specified copula $c(\mathbf{u})$; "Testing Copula Hypothesis with Copula Entropy" develops precisely that one-sample problem, not a two-sample problem (Ma, 26 Oct 2025). Second, it differs from full copula-equality tests such as empirical-copula randomization tests, Bernstein-copula tests, or pattern-frequency tests, all of which target

$H_0:C_1=C_2$

for the entire dependence structure (Seo, 2018). CE-based two-sample tests instead compare distributions through a CE contrast built from pooled data and labels.

The broader CE framework gives the conceptual motivation. CE is introduced as a dependence quantity defined on the copula representation of a random vector, and the monograph treatment emphasizes that CE-based methodology is multivariate, symmetric, continuous, margin-free, invariant to monotonic transformations, and designed to measure all-order dependence (Ma, 20 Dec 2025). In the two-sample setting, these properties are exploited by interpreting sample labels as variables whose statistical dependence with the pooled observations should vanish under $H_0$ and appear under $H_1$.

2. Copula entropy and the label-based test statistic

The formal CE definition used throughout this literature is

$$H_c(\mathbf{x})=-\int_{\mathbf{u}} c(\mathbf{u})\log c(\mathbf{u})\,d\mathbf{u},$$

where $\mathbf{X}_1,\mathbf{X}_2\in \mathbb{R}^d$0 denotes the marginally uniform variables associated with $\mathbf{X}_1,\mathbf{X}_2\in \mathbb{R}^d$1, and $\mathbf{X}_1,\mathbf{X}_2\in \mathbb{R}^d$2 is the copula density (Ma, 2023). The same literature states that CE is symmetric, non-positive, equals $\mathbf{X}_1,\mathbf{X}_2\in \mathbb{R}^d$3 iff independent, is invariant to monotonic transformation, and is equivalent to correlation coefficient in Gaussian cases; it also states that mutual information is equivalent to negative CE (Ma, 2023).

The two-sample construction begins with the pooled sample

$\mathbf{X}_1,\mathbf{X}_2\in \mathbb{R}^d$4

together with two label vectors. The actual label vector is

$\mathbf{X}_1,\mathbf{X}_2\in \mathbb{R}^d$5

while the null label vector is

$\mathbf{X}_1,\mathbf{X}_2\in \mathbb{R}^d$6

The CE-based dependence quantity between pooled data and a label variable is written as

$\mathbf{X}_1,\mathbf{X}_2\in \mathbb{R}^d$7

Since $\mathbf{X}_1,\mathbf{X}_2\in \mathbb{R}^d$8 is common to both hypotheses, the proposed statistic is

$\mathbf{X}_1,\mathbf{X}_2\in \mathbb{R}^d$9

The intended interpretation is direct: $H_0: P_1 = P_2,\qquad H_1: P_1 \neq P_2.$0 should be a small value if $H_0: P_1 = P_2,\qquad H_1: P_1 \neq P_2.$1 is true and a large value if $H_0: P_1 = P_2,\qquad H_1: P_1 \neq P_2.$2 is true (Ma, 2023).

The monograph presents the same idea with slightly different notation. There, the labels are written as a null label $H_0: P_1 = P_2,\qquad H_1: P_1 \neq P_2.$3 and an alternative label $H_0: P_1 = P_2,\qquad H_1: P_1 \neq P_2.$4, and the statistic is

$H_0: P_1 = P_2,\qquad H_1: P_1 \neq P_2.$5

The interpretation remains unchanged: true two-group labels should create stronger dependence with the pooled observations than the null all-one labeling when the samples come from different distributions (Ma, 20 Dec 2025).

3. Estimation methodology and algorithmic realization

The original estimator is built from the previously proposed nonparametric CE estimator of Ma and Sun. The procedure is described as two steps: first, estimating empirical copula density function; second, estimating the entropy of the estimated empirical copula density. The empirical copula density in the first step is derived with rank statistic, and the second step is tackled with the KSG estimation method (Ma, 2023).

Operationally, the two-sample procedure is as follows. One pools the two samples into $H_0: P_1 = P_2,\qquad H_1: P_1 \neq P_2.$6, constructs the two augmented datasets

$H_0: P_1 = P_2,\qquad H_1: P_1 \neq P_2.$7

rank-transforms each coordinate to empirical-uniform pseudo-observations, estimates the CE of each augmented dataset, and returns

$H_0: P_1 = P_2,\qquad H_1: P_1 \neq P_2.$8

The paper presents this estimator as nonparametric and hyperparameter-free, contrasting it with kernel methods that require a bandwidth or kernel-scale choice (Ma, 2023).

The method is multivariate by construction, since the pooled observations may lie in $H_0: P_1 = P_2,\qquad H_1: P_1 \neq P_2.$9 and the augmented sample simply adds one label coordinate. At the same time, the implementation description is intentionally high-level. The paper does not provide pseudocode, a formal asymptotic analysis of the estimator, or a detailed treatment of the binary or constant label coordinate. It explicitly notes no special tie-handling procedure, even though the label coordinate necessarily contains ties in both the actual-label and null-label constructions (Ma, 2023).

The same CE estimator underlies downstream procedures. In change-point detection, the two-sample statistic is repeatedly recomputed on left/right temporal splits, again using empirical copula construction from ranks and entropy estimation via KSG (Ma, 2024). The monograph likewise treats the nonparametric CE estimator as the practical engine for two-sample testing, change-point detection, and symmetry testing (Ma, 20 Dec 2025).

4. Inferential status, calibration, and identification issues

A central feature of the CE two-sample literature is the asymmetry between a clearly defined statistic and a less fully developed inferential calibration. The 2023 two-sample paper does not provide a formal asymptotic null distribution, permutation calibration, or explicit p-value construction. Its empirical study compares the magnitude of estimated statistics across simulated scenarios rather than reporting rejection rates at a fixed significance level (Ma, 2023). The monograph review retains the same large-value rejection logic but, in the extracted treatment, likewise does not supply an explicit asymptotic null law for the CE two-sample statistic (Ma, 20 Dec 2025).

This limited calibration is not unique to the two-sample setting. The one-sample copula specification paper based on CE also defines a discrepancy statistic,

$P_1$0

but does not provide a limit distribution, consistency theorem, critical values, p-value formulas, bootstrap calibration, or permutation calibration (Ma, 26 Oct 2025). In both cases, CE is used primarily as a discrepancy score whose inferential threshold is left largely implicit.

A second issue concerns identification. Entropy-only comparisons are weaker than full copula comparisons. The cumulative-copula paper emphasizes that equality of a scalar entropy functional does not imply equality of copulas; distinct copulas can share the same cumulative copula entropy, so an entropy-only two-sample test can miss alternatives that alter copula shape while preserving $P_1$1 (Arshad et al., 2024). The one-sample CE copula-hypothesis paper makes the same point in a different form: its statistic is a difference of two entropies, but the paper does not prove nonnegativity or identify the discrepancy with a known divergence such as KL divergence (Ma, 26 Oct 2025).

For this reason, the strongest copula-level identification result in the broader CE literature currently appears not in the original CE two-sample paper but in the cumulative-copula framework. That paper defines the cumulative copula Kullback–Leibler divergence

$P_1$2

and proves

$P_1$3

That result is given for cumulative copula functionals and a one-sample goodness-of-fit problem, but it provides a sharper copula-comparison target than entropy difference alone (Arshad et al., 2024).

5. Empirical behavior and downstream applications

The original empirical study for the CE two-sample statistic uses three simulation settings with sample size $P_1$4 in all simulations and compares the CE statistic against an MI-based test, a kernel-based test, and an energy statistics-based test (Ma, 2023). The first simulation uses bi-variate normal data with mean shifts, the second uses bi-variate normal data with changing covariance/correlation parameter, and the third uses bi-variate Gaussian copula data with one standard normal marginal and one exponential marginal. Across these scenarios, the paper reports that the CE statistic is close to $P_1$5 under $P_1$6 in the first two simulations, increases with distributional difference, is more stable than the MI-based test, is similar to the kernel-based test in Gaussian settings, and in the non-Gaussian Gaussian-copula setting shows increasing statistics while the MI-based and energy-statistics-based tests fail to present a reasonable result (Ma, 2023).

A major downstream application is offline change-point detection. "Change Point Detection with Copula Entropy based Two-Sample Test" imports the CE two-sample statistic and uses it as a scan statistic over candidate split points in a time series (Ma, 2024). For a segment $P_1$7, the single change point is taken as the split maximizing

$P_1$8

and multiple change points are obtained by combining this scan with binary segmentation. The paper verifies the method on simulated univariate and multivariate data and on the Nile data, and in its multivariate copula experiment—built from Frank copula and Gaussian copula segments with normal and exponential marginals—the CE-based method detects a genuine change point at $P_1$9 whereas the kernel baseline detects no meaningful one (Ma, 2024).

The monograph situates this use within a larger CE testing family. It reviews two-sample testing, change-point detection, and symmetry testing as closely related constructions driven by the same CE machinery (Ma, 20 Dec 2025). In the symmetry formulation, the centered sample $P_2$0 is compared with its reflection $P_2$1 through

$P_2$2

showing that the two-sample CE statistic functions not only as a direct homogeneity measure but also as a reusable building block for broader nonparametric inference (Ma, 20 Dec 2025).

6. Relation to full copula two-sample tests and current directions

CE-based two-sample tests must be distinguished from procedures that test equality of entire copulas. The randomization framework of Seo tests

$P_2$3

through an $P_2$4 distance between empirical copulas,

$P_2$5

and develops a modified randomization procedure that is asymptotically exact under the null and consistent under the alternative (Seo, 2018). Pattern-based tests for bivariate copulas compare the full vector of permutation-pattern frequencies and prove a functional CLT together with a bootstrap-calibrated two-sample Cramér–von Mises statistic

$P_2$6

again with asymptotic level and consistency (Baringhaus et al., 13 May 2026). Bernstein-polynomial tests replace empirical copulas by empirical Bernstein copulas and study $P_2$7-type and sup-norm discrepancies together with multiplier bootstrap and subsampling calibration (Lyu et al., 2023).

These procedures target the full dependence structure. CE-based two-sample tests, by contrast, compare pooled data and labels through a scalar CE contrast. That scalarization has clear advantages—nonparametric multivariate applicability, margin invariance in the copula construction, and direct information-theoretic interpretation—but it is intrinsically weaker than equality of full copulas. The literature explicitly warns that equal entropy does not imply equal copula (Arshad et al., 2024). This point also underlies the assessment of the one-sample CE copula-hypothesis paper as only a conceptual precursor to a genuine CE-based two-sample copula test (Ma, 26 Oct 2025).

Two extension paths are already visible in the literature. One is entropy-centered: compare copula entropies or cumulative copula entropies estimated separately from two samples. The cumulative-copula paper supports this direction with an empirical beta copula estimator and almost sure consistency for $P_2$8, but it also makes clear that entropy equality remains only equality of a one-dimensional functional (Arshad et al., 2024). The second path is divergence-centered: replace entropy difference by a copula divergence such as $P_2$9, which is nonnegative and vanishes iff the two copulas are equal (Arshad et al., 2024). The one-sample CE copula-hypothesis paper suggests the need for precisely such a step when moving from a discrepancy score to a fully identifying two-sample copula test (Ma, 26 Oct 2025).

Within current arXiv literature, copula entropy-based two-sample testing is therefore best understood as a specific nonparametric, label-based methodology for testing equality of distributions, together with a set of conceptual extensions toward copula-level comparison that remain less fully developed than the corresponding randomization, pattern-based, or Bernstein-copula alternatives.