Testing Copula Hypothesis with Copula Entropy

Testing Copula Hypothesis with Copula Entropy Jian MA∗ Hitachi China Research Laboratory November 6, 2025 Abstract Testing copula hypothesis is of fundamental importance in the applications of copula theory. In this paper we proposed a copula hypothesis testing with copula entropy. Since copula entropy is a unified theory in probability and therefore testing copula hypothesis based on it can be applied to any types of copula function. The test statistic is defined as the difference of copula entropy of copula hypothesis and true copula entropy. We propose the estimation method of the proposed statistic and two special cases for Gaussian copula hypothesis and Archimedean copula hypothesis. We test the effectiveness of the proposed method with simulation experiments. Keywords: Copula Entropy; Copula; Hypothesis Test; Gaussian Copula; Archimedean Copula 1 Introduction Evaluating the fitness of models to data is a common practice in scientific activities. Testing hypothesis is one of the fundamental problems in statistics and and has wide applications in every branch of sciences. Copula theory is about representing multivariate dependence with copula functions [1, 2]. As the core result of copula theory, Sklar’s theorem [3] states that multivariate density function can be represented as a copula function with marginal functions as its inputs. There are many copula function families available for real applications, such as Gaussian copula, t Copula [4], Archimedean copula, Archimax copula [5], Sibuya Copula [6], among others. Modeling with copula function is a widely used methods in many scientific fields [7, 8, 9, 10, 11, 12] and hence testing copula hypothesis is important in those practices [13]. Many research have been contributed to copula hypothesis testing, such as testing Gaussian copula hypothesis [14, 15], testing Archimedeanity [16, 17], testing symmetry of copula [18]. These work on testing copula hypothesis are mainly focusing on special types of copula function and a general method for any types of copula is needed. Kole, et al [19] suggest using Kolmogorov-Smirnov test and Anderson-Darling test to ∗Email: ma[email protected] 1 select copulas. Grønneberg and Hjort [20] proposed an AIC-like criteria for copula model selection, named Copula Information Criteria (CIC). Genest and R´emillard [21] proposed to use Cram´er-von Mises test and Kolmogorov-Smirmov test for Goodness-of-fit testing of copula models. Copula Entropy (CE) is a recently proposed theory in probability. It defined the concept of CE as a special kind of Shannon entropy with copula function [22]. Copula function represents the dependence relationship between random variables while CE measures such relationship in a unified way. Contrast to other dependence measures based on copula, such as Spearman’s ρand Kendall’s τ, CE has many good properties, including non-negative, invariance to monotonic transformation, and equivalent to correlation matrix under Gaussianity. CE has been applied to hypothesis testing recently, including multivariate normality test [23], two-sample test [24], change point detection [25], and symmetry test [26]. In this paper, we proposed a copula hypothesis testing with copula entropy. It can be used for any types of copula hypothesis testing. The test statistic is defined as the difference of CE of copula hypothesis and true CE. We give the estimation method of the proposed statistic and two cases for Gaussian copula hypothesis and Archimdean copula hypotheses. We test the effectiveness of the proposed method with simulation experiments. This paper is organized as follows: Section 2 introduces the basic theory of CE, Section 3 presents the proposed testing method, Section 4 gives the estimation method of the proposed statistic, simulation experiments will be presented in Section 5, Section 6 concludes the paper. 2 Copula Entropy With copula theory, Ma and Sun [22] defined the concept of Copula Entropy as follows: Definition 1 (Copula Entropy).Let Xbe random variables with marginals u and copula density function c. The CE of Xis defined as Hc(x) = −Zu c(u) log c(u)du.(1) They also proposed a non-parametric estimator of CE [22] comprising of two simple steps: 1. estimating empirical copula density function with rank statistic; 2. estimating the entropy of the estimated empirical copula density with the kNN entropy estimator[27]. If the copula density function cis given, the CE can also be estimated with the following way: Hc(x) = −E(log c(u)).(2) 3 Testing Copula Hypothesis Given random variables X∈Rnand its samples XTassociated with copula density function cx(u). Our goal is to test whether cbelong to a hypothesis 2 c(u), the null hypothesis of the problem is H0:cx(u) = c(u); (3) alternative hypothesis is H0:cx(u)=c(u).(4) We propose to test copula hypothesis with CE. The principle of testing is to compare the CE of the copula hypothesis with the true CE: Tc(XT|c) = Hc(XT|c)−Hc(XT|cx),(5) The first term is the CE under the hypothesis of copula cand the second term is true CE. If H0is true, then Tcshould be 0; otherwise, Tcshould be large value. Since CE is a unified theory for copula function, testing copula hypothesis based on CE can be used for any types of copula function. The only work needed is to select the copula family c. 4 Estimation The statistic in (5) can be estimated as two part. The second term is true CE and therefore can be estimated directly from data with the nonparametric estimator of CE. The first term is the CE of copula hypothesis which can be estimated in the following 3 steps: 1. estimate empirical copula density ˆu from XT; 2. estimate the parameters αof copula cwith ˆu; 3. calculate the CE of the copula hypothesis with the following equation: Hc(XT|c) = −E(log c(ˆu;α)).(6) In the first step, empirical copula density can be estimated with rank statistic and in the second step, the parameters αof copula density can be estimated with the likelihood method [2]. Here we give two special cases of estimating CE of copula hypothesis: Gaussian Copula Gaussian copula density function can be written as the following [28, 29]: cn(u) = |Σρ|− 1 2exp −1 2Φ(u)(Σ−1 ρ−I)ΦT(u),(7) where Σρis correlation matrix, Φ is quantile normal function, and Iis identity matrix. So estimating CE of Gaussian copula hypothesis can be done as follows: 1. estimate Σρfrom XT; 2. calculate the value of Gaussian copula with 7; 3. calculate CE of copula hypothesis with 6. 3 Archimedean Copula Gumbel copula, Frank copula, and Clayton copula are three main members of Archimedean copula family, and bivariate Gumbel copula density function is cg(u) = exp    −"2 X i=1 (−ln ui)α# 1 α    "2 X i=1 (−ln ui)α# 1 α −1  2 X i=1 (−ln ui)α ui! , (8) bivariate Frank copula density function is cf(u) = α(1 −e−α)e−α(u1+u2) {1−e−α−(1 −e−αu1)(1 −e−αu2)}2,(9) and bivariate Clayton copula density function is cc(u) = (α+ 1)(u1u2)−α−1(u−α 1+u−α 2+ 1)−2,(10) where αis the parameter of Gumbel copula. So estimating CE of Archimedean copula hypothesis can be done as follows: 1. estimate αwith the likelihood method [2]; 2. calculate the value of Archimedean copula with (8), (9) or (10); 3. calculate CE of Gumbel copula with (6). 5 Simulations We test the proposed method with two simulation experiments 1. The first experiment simulate bivariate Gaussian copula with correlation coefficient ρ range from 0.1 to 0.9 by the step 0.1. The second experiment simulate bivariate Gumbel, Frank, and Clayton copula with the parameter αchanging from 2 to 10 under the margins being standard normal distribution and exponential distribution. All the sample size of simulations is 300. We applied the above testing method of Gaussian copula hypothesis and Archimedean copula hypotheses to simulated data and derived four estimated statistics from each sample set. The experimental results is shown in Figure 1, Figure 2, Figure 3, and Figure 4. It can be learned that in the first simulation experiment, the estimated statistics of Gaussian copula hypothesis is smaller than those of Archimedean copula hypotheses and that in the second simulation experiment, the estimated statistics of Gumbel, Frank, or Clayton copula hypothesis is smaller than those of Gaussian copula hypothesis and the other Archimedean copulas. These mean that Gaussian copula hypothesis is true and Gumbel, Frank, or Clayton copula hypothesis is true in two experiments respectively. The Rpackage copula [30] was used for Archimedean copula. The Rpackage mvtnorm [31] waw used for simulating Gaussian distribution. The Rpackage copent [32] was used as the implementation of CE estimator. 1The code is available at https://github.com/majianthu/tch 4 0.2 0.4 0.6 0.8 −0.2 −0.1 0.0 0.1 0.2 0.3 Gaussian ρ statistic Gaussian Gumbel Frank Clayton Figure 1: Results of Gaussian copula hypothesis simulation experiments. 2 4 6 8 10 0123456 Gumbel α statistic Gaussian Gumbel Frank Clayton Figure 2: Results of Gumbel copula hypothesis simulation experiments. 5 2 4 6 8 10 0.0 0.5 1.0 1.5 Frank α statistic Gaussian Gumbel Frank Clayton Figure 3: Results of Frank copula hypothesis simulation experiments. 2 4 6 8 10 012345 Clayton α statistic Gaussian Gumbel Frank Clayton Figure 4: Results of Clayton copula hypothesis simulation experiments. 6 6 Conclusions In this paper, we proposed a copula hypothesis testing with copula entropy. The test statistic is defined. 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