Empirical Copula Process Research Articles

The empirical beta copula

Given a sample from a continuous multivariate distribution F , the uniform random variates generated independently and rearranged in the order specified by the componentwise ranks of the original sample look like a sample from the copula of F . This idea can be regarded as a variant on Baker’s [J. Multivariate Anal. 99 (2008) 2312–2327] copula construction and leads to the definition of the empirical beta copula. The latter turns out to be a particular case of the empirical Bernstein copula, the degrees of all Bernstein polynomials being equal to the sample size. Necessary and sufficient conditions are given for a Bernstein polynomial to be a copula. These imply that the empirical beta copula is a genuine copula. Furthermore, the empirical process based on the empirical Bernstein copula is shown to be asymptotically the same as the ordinary empirical copula process under assumptions which are significantly weaker than those given in Janssen, Swanepoel and Veraverbeke [J. Statist. Plann. Inference 142 (2012) 1189–1197]. A Monte Carlo simulation study shows that the empirical beta copula outperforms the empirical copula and the empirical checkerboard copula in terms of both bias and variance. Compared with the empirical Bernstein copula with the smoothing rate suggested by Janssen et al., its finite-sample performance is still significantly better in several cases, especially in terms of bias.

Testing exchangeability of copulas in arbitrary dimension

ABSTRACTA test for exchangeability of copulas for arbitrary dimensions is proposed, generalising and extending a result by Genest et al. [(2012), ‘Tests of Symmetry for Bivariate Copulas’, Annals of the Institute of Statistical Mathematics, 64, 811–834]. Three test statistics together with some modifications are presented and their asymptotical behaviour is analysed. Empirical p-values are computed by using a bootstrap-procedure proposed by Rémillard and Scaillet [(2009), ‘Testing for Equality between Two Copulas’, Journal of Multivariate Analysis, 100, 377–386] and suggested by Bücher and Dette [(2010), ‘A Note on Bootstrap Approximations for the Empirical Copula Process’, Statistics & Probability Letters, 80, 1925–1932], based on a multiplier central limit theorem by van der Vaart and Wellner [(1996), Weak Convergence and Empirical Processes, Springer Series in Statistics, New York: Springer]. Finally a simulation study compares various versions of the proposed tests.

Some applications of the strong approximation of the integrated empirical copula processes

The purpose of the present paper is to provide a strong invariance principle for the integrated empirical copula process [introduced in a series of papers by Henze and Nikitin in the univariate setting] with the rate of the approximation for multivariate empirical processes. The applications discussed here are change-point detection in multivariate copula models and the integrated empirical copula process with estimated parameter. Finally, a general notion of bootstrapped integrated empirical copula process, constructed by exchangeably weighting sample, is presented.

A dependent multiplier bootstrap for the sequential empirical copula process under strong mixing

Two key ingredients to carry out inference on the copula of multivariate observations are the empirical copula process and an appropriate resampling scheme for the latter. Among the existing techniques used for i.i.d. observations, the multiplier bootstrap of R\'{e}millard and Scaillet (J. Multivariate Anal. 100 (2009) 377-386) frequently appears to lead to inference procedures with the best finite-sample properties. B\"{u}cher and Ruppert (J. Multivariate Anal. 116 (2013) 208-229) recently proposed an extension of this technique to strictly stationary strongly mixing observations by adapting the dependent multiplier bootstrap of B\"{u}hlmann (The blockwise bootstrap in time series and empirical processes (1993) ETH Z\"{u}rich, Section 3.3) to the empirical copula process. The main contribution of this work is a generalization of the multiplier resampling scheme proposed by B\"{u}cher and Ruppert along two directions. First, the resampling scheme is now genuinely sequential, thereby allowing to transpose to the strongly mixing setting many of the existing multiplier tests on the unknown copula, including nonparametric tests for change-point detection. Second, the resampling scheme is now fully automatic as a data-adaptive procedure is proposed which can be used to estimate the bandwidth parameter. A simulation study is used to investigate the finite-sample performance of the resampling scheme and provides suggestions on how to choose several additional parameters. As by-products of this work, the validity of a sequential version of the dependent multiplier bootstrap for empirical processes of B\"{u}hlmann is obtained under weaker conditions on the strong mixing coefficients and the multipliers, and the weak convergence of the sequential empirical copula process is established under many serial dependence conditions.

Asymptotic total variation tests for copulas

We propose a new platform of goodness-of-fit tests for copulas, based on empirical copula processes and nonparametric bootstrap counterparts. The standard Kolmogorov–Smirnov type test for copulas that takes the supremum of the empirical copula process indexed by orthants is extended by test statistics based on the empirical copula process indexed by families of $L_{n}$ disjoint boxes, with $L_{n}$ slowly tending to infinity. Although the underlying empirical process does not converge, the critical values of our new test statistics can be consistently estimated by nonparametric bootstrap techniques, under simple or composite null assumptions. We implemented a particular example of these tests and our simulations confirm that the power of the new procedure is oftentimes higher than the power of the standard Kolmogorov–Smirnov or the Cramér–von Mises tests for copulas.

Bernstein estimation for a copula derivative with application to conditional distribution and regression functionals

Bernstein estimators attracted considerable attention as smooth nonparametric estimators for distribution functions, densities, copulas and copula densities. The present paper adds a parallel result for the first-order derivative of a copula function. This result then leads to Bernstein estimators for a conditional distribution function and its important functionals such as the regression and quantile functions. Results of independent interest have been derived such as an almost sure oscillation behavior of the empirical copula process and a Bahadur-type almost sure asymptotic representation for the Bernstein estimator of a regression quantile function. Simulations demonstrate the good performance of the proposed estimators.

A New Measure of Vector Dependence, with an Application to Financial Contagion

We propose a new nonparametric measure of association between an arbitrary number of random vectors. The measure is based on the empirical copula process for the multivariate marginals, corresponding to the vectors, and is insensitive to the within-vector dependence. It is bounded by the [0, 1] interval, covering the entire range of dependence from vector independence to a vector version of a monotone relationship. We study the properties of the new measure under several well-known copulas and provide a nonparametric estimator of the measure, along with its asymptotic theory, under fairly general assumptions. To illustrate the applicability of the new measure, we use it to assess the degree of interdependence between equity markets in North and South America, Europe and Asia, surrounding the financial crisis of 2008. We find strong evidence of previously unknown contagion patterns, with selected regions exhibiting little dependence before and after the crisis and a lot of dependence during the crisis period.

Nonparametric tests for constant tail dependence with an application to energy and finance

New tests for detecting structural breaks in the tail dependence of multivariate time series using the concept of tail copulas are presented. To obtain asymptotic properties, we derive a new limit result for the sequential empirical tail copula process. Moreover, consistency of both the tests and a break-point estimator are proven. We analyze the finite sample behavior of the tests by Monte Carlo simulations. Finally, and crucial from a risk management perspective, we apply the new findings to datasets from energy and financial markets.

Nonparametric Tests for Constant Tail Dependence with an Application to Energy and Finance

The present paper proposes new tests for detecting structural breaks in the tail dependence of multivariate time series using the concept of tail copulas. To obtain asymptotic properties, we derive a new limit result for the sequential empirical tail copula process. Moreover, consistency of both the tests and a break-point estimator are proven. We analyze the finite sample behavior of the tests by Monte Carlo simulations. Finally, and crucial from a risk management perspective, we apply the new findings to datasets from energy and financial markets.

Non-parametric weighted tests for independence based on empirical copula process

We propose a class of flexible non-parametric tests for the presence of dependence between components of a random vector based on weighted Cramér–von Mises functionals of the empirical copula process. The weights act as a tuning parameter and are shown to significantly influence the power of the test, making it more sensitive to different types of dependence. Asymptotic properties of the test are stated in the general case, for an arbitrary bounded and integrable weighting function, and computational formulas for a number of weighted statistics are provided. Several issues relating to the choice of the weights are discussed, and a simulation study is conducted to investigate the power of the test under a variety of dependence alternatives. The greatest gain in power is found to occur when weights are set proportional to true deviations from independence copula.

Detecting changes in cross-sectional dependence in multivariate time series

Classical and more recent tests for detecting distributional changes in multivariate time series often lack power against alternatives that involve changes in the cross-sectional dependence structure. To be able to detect such changes better, a test is introduced based on a recently studied variant of the sequential empirical copula process. In contrast to earlier attempts, ranks are computed with respect to relevant subsamples, with beneficial consequences for the sensitivity of the test. For the computation of p-values we propose a multiplier resampling scheme that takes the serial dependence into account. The large-sample theory for the test statistic and the resampling scheme is developed. The finite-sample performance of the procedure is assessed by Monte Carlo simulations. Two case studies involving time series of financial returns are presented as well.

Multiplier bootstrap of tail copulas with applications

For the problem of estimating lower tail and upper tail copulas, we propose two bootstrap procedures for approximating the distribution of the corresponding empirical tail copulas. The first method uses a multiplier bootstrap of the empirical tail copula process and requires estimation of the partial derivatives of the tail copula. The second method avoids this estimation problem and uses multipliers in the two-dimensional empirical distribution function and in the estimates of the marginal distributions. For both multiplier bootstrap procedures, we prove consistency. For these investigations, we demonstrate that the common assumption of the existence of continuous partial derivatives in the the literature on tail copula estimation is so restrictive, such that the tail copula corresponding to tail independence is the only tail copula with this property. Moreover, we are able to solve this problem and prove weak convergence of the empirical tail copula process under nonrestrictive smoothness assumptions that are satisfied for many commonly used models. These results are applied in several statistical problems, including minimum distance estimation and goodness-of-fit testing.

Strong approximation of empirical copula processes by Gaussian processes

In this paper, we provide the strong approximation of normalized empirical copula processes by a Gaussian process. In addition, we establish a strong approximation of the smoothed empirical copula processes and a law of iterated logarithm.

Asymptotic behavior of weighted multivariate Cramér-von Mises-type statistics under contiguous alternatives

In this paper, we study the behavior of the weighted quadratic functionals of the multivariate empirical copula processes under sequences of contiguous alternatives. The Karhunen-Loeve expansions of the corresponding limiting Gaussian processes are derived by using the results, in a series of papers, by Deheuvels, which are used to obtain the asymptotic distribution of the weighted multivariate Cramer-von Mises-type statistics. These results are applied to compute the relative local asymptotic efficiency of the considered statistics, in the spirit of Genest et al. (2006, 2007), and discuss briefly some aspects regarding the power of these statistical tests. Finally, we give some additional results concerning the integrated copula processes.

Simulated Method of Moments Estimation for Copula-Based Multivariate Models

This article considers the estimation of the parameters of a copula via a simulated method of moments (MM) type approach. This approach is attractive when the likelihood of the copula model is not known in closed form, or when the researcher has a set of dependence measures or other functionals of the copula that are of particular interest. The proposed approach naturally also nests MM and generalized method of moments estimators. Drawing on results for simulation-based estimation and on recent work in empirical copula process theory, we show the consistency and asymptotic normality of the proposed estimator, and obtain a simple test of overidentifying restrictions as a specification test. The results apply to both iid and time series data. We analyze the finite-sample behavior of these estimators in an extensive simulation study. We apply the model to a group of seven financial stock returns and find evidence of statistically significant tail dependence, and mild evidence that the dependence between these assets is stronger in crashes than booms. Supplementary materials for this article are available online.

Nonparametric Identification of Copula Structures

We propose a unified framework for testing a variety of assumptions commonly made about the structure of copulas, including symmetry, radial symmetry, joint symmetry, associativity and Archimedeanity, and max-stability. Our test is nonparametric and based on the asymptotic distribution of the empirical copula process. We perform simulation experiments to evaluate our test and conclude that our method is reliable and powerful for assessing common assumptions on the structure of copulas, particularly when the sample size is moderately large. We illustrate our testing approach on two datasets.

Empirical and sequential empirical copula processes under serial dependence

Empirical and sequential empirical copula processes play a central role for statistical inference on copulas. However, as pointed out by Johan Segers [J. Segers, Asymptotics of empirical copula processes under non-restrictive smoothness assumptions, Bernoulli 18 (3) (2012) 764–782] the usual assumptions under which these processes have been studied so far are too restrictive. In this paper, we provide a unified approach to the analysis of empirical and sequential empirical copula processes that circumvents those restrictive assumptions in a very general setting. In particular, our methods allow for an easy analysis of copula processes and appropriate bootstrap approximations in the setting of sequentially dependent data. One particularly useful finding is that certain sequential empirical copula processes converge without any smoothness assumptions on the copula.

A fluctuation test for constant Spearman’s rho with nuisance-free limit distribution

A CUSUM type test for constant correlation that goes beyond a previously suggested correlation constancy test by considering Spearman’s rho in arbitrary dimensions is proposed. Since the new test does not require the existence of any moments, the applicability on usually heavy-tailed financial data is greatly improved. The asymptotic null distribution is calculated using an invariance principle for the sequential empirical copula process. The limit distribution is free of nuisance parameters and critical values can be obtained without bootstrap techniques. A local power result and an analysis of the behavior of the test in small samples are provided.

A Strong Invariance Theorem of the Tail Empirical Copula Processes

We study the behavior of bivariate empirical copula process 𝔾 n (·, ·) on pavements [0, k n /n]2 of [0, 1]2, where k n is a sequence of positive constants fulfilling some conditions. We provide a upper bound for the strong approximation of 𝔾 n (·, ·) by a Gaussian process when k n /n↘γ as n → ∞, where 0 ≤ γ ≤1.

An Overview of the Goodness-of-Fit Test Problem for Copulas

We review the main “omnibus procedures” for goodness-of-fit (GOF) testing for copulas: tests based on the empirical copula process, on probability integral transformations (PITs), on Kendall’s dependence function, etc., and some corresponding reductions of dimension techniques. The problems of finding asymptotic distribution-free test statistics and the calculation of reliable p-values are discussed. Some particular cases, like convenient tests for time-dependent copulas, for Archimedean or extreme-value copulas, etc., are dealt with. Finally, the practical performances of the proposed approaches are briefly summarized.