Testing Copula Hypothesis with Copula Entropy

Testing Copula Hypothesis with Copula Entropy Jian MA∗ Hitachi China Research Laboratory October 25, 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 testing copula hypothesis based on it can be applied to any types of copula function. 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 special cases for Gaussian copula hypothesis and Gumbel 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] and hence testing copula hypothesis is important in those practices. Many research have been contributed to copula hypothesis testing, such as testing Gaussian copula hypothesis [12, 13], testing Archimedeanity [14, 15], testing symmetry of copula [16]. Current 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. 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 ∗Email: ma[email protected] 1 [17]. 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 [18], two-sample test [19], change point detection [20], and symmetry test [21]. 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 Gumbel copula hypothesis. 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 [17] 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) = −∫u c(u) log c(u)du.(1) They also proposed a non-parametric estimator of CE [17] 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[22]. 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 c(u), the null hypothesis of the problem is H0:cx(u) = c(u); (3) alternative hypothesis is H0:cx(u)=c(u).(4) 2 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|cx)−Hc(XT|c),(5) The first term is true CE and the second term is the CE under the hypothesis of copula c. 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 first term is CE and therefore can be estimated directly with the nonparametric estimator of CE. The second term is the CE of copula hypothesis which can be estimated in the following 3 steps: 1. estimate 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) Here we give two cases of estimating CE of copula hypothesis: Gaussian Copula Gaussian copula density function can be written as the following [23]: cn(u) = √|Σρ| − 1 2Φ(u)(Σ−1 ρ−I)ΦT(u),(7) where Σρis correlation matrix, Φis quantile normal function, and Iis identity matrix. So estimating CE of Gaussian copula hypothesis can be done as: 1. estimate Σρfrom XT; 2. calculate the value of Gaussian copula with 7; 3. calculate CE of copula hypothesis with 6. Gumbel Copula Gumbel Copula is one member of Archimedean copula family, and bivariate Gumbel copula density is cg(u) = exp    −[2 ∑ i=1 (−ln ui)α] 1 α    [2 ∑ i=1 (−ln ui)α] 1 α −1 (2 ∑ i=1 (−ln ui)α ui) , (8) where αis the parameter of Gumbel copula. So estimating CE of Gumbel copula hypothesis can be done as follows: 3 1. estimate αwith the likelihood method; 2. calculate the value of Gumbel copula with (8); 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 copula with the parameter alpha changing 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 Gumbel copula hypothesis to simulated data and derived two estimated statistics from each sample set. The experimental results is shown in Figure 1 and Figure 2. It can be learned that in the first simulation experiment, the estimated statistics of Gaussian copula hypothesis is smaller than those of Gumbel copula hypothesis and that in the second simulation experiment, the estimated statistics of Gumbel copula hypothesis is smaller than those of Gaussian copula hypothesis. These mean that Gaussian copula hypothesis is true and Gumbel copula hypothesis is true in two experiments respectively. The Rpackage copula [24] and gumbel [25] were used for Gumbel copula. The Rpackage mvtnorm [26] were used for simulating Gaussian distribution. The Rpackage copent [27] were used as the implementation of CE estimator. 6 Conclusions In this paper, we proposed a copula hypothesis testing with copula entropy. The test statistic is defined. The estimation method of the proposed statistic is proposed and two special cases of tests for Gaussian copula hypothesis and Gumbel copula hypothesis is given. The effectiveness of the proposed method is verified with simulation experiments. References [1] Roger B Nelsen. An introduction to copulas. Springer Science & Business Media, 2007. [2] Harry Joe. Dependence modeling with copulas. CRC press, 2014. [3] Abe Sklar. Fonctions de repartition an dimensions et leurs marges. Publications de l’Institut de statistique de l’Université de Paris, 8:229–231, 1959. [4] Stefano Demarta and Alexander J. Mcneil. The tCopula and Related Copulas. 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