Empirical Copula Process Research Articles

A class of smooth, possibly data-adaptive nonparametric copula estimators containing the empirical beta copula

A broad class of smooth, possibly data-adaptive nonparametric copula estimators that contains empirical Bernstein copulas introduced by Sancetta and Satchell (and thus the empirical beta copula proposed by Segers, Sibuya and Tsukahara) is studied. Within this class, a subclass of estimators that depend on a scalar parameter determining the amount of marginal smoothing and a functional parameter controlling the shape of the smoothing region is specifically considered. Empirical investigations of the influence of these parameters suggest to focus on two particular data-adaptive smooth copula estimators that were found to be uniformly better than the empirical beta copula in all of the considered Monte Carlo experiments. Finally, with future applications to change-point detection in mind, conditions under which related sequential empirical copula processes converge weakly are provided.

Tie-Break Bootstrap for Nonparametric Rank Statistics

In this article, we propose a new bootstrap procedure for the empirical copula process. The procedure involves taking pseudo samples of normalized ranks in the same fashion as the classical bootstrap and applying small perturbations to break ties in the normalized ranks. Our procedure is a simple modification of the usual bootstrap based on sampling with replacement, yet it provides noticeable improvement in the finite sample performance. We also discuss how to incorporate our procedure into the time series framework. Since nonparametric rank statistics can be treated as functionals of the empirical copula, our proposal is useful in approximating the distribution of rank statistics in general. As an empirical illustration, we apply our bootstrap procedure to test the null hypotheses of positive quadrant dependence, tail monotonicity, and stochastic monotonicity, using U.S. Census data on spousal incomes in the past 15 years.

Conditional empirical copula processes and generalized measures of association

We study the weak convergence of conditional empirical copula processes indexed by general families of conditioning events that have non zero probabilities. Moreover, we also study the case where the conditioning events are chosen in a data-driven way. The validity of several bootstrap schemes is stated, including the exchangeable bootstrap. We define general multivariate measures of association, possibly given some fixed or random conditioning events. By applying our theoretical results, we prove the asymptotic normality of the estimators of such measures. We illustrate our results with financial data.

On the weighted tests of independence based on Bernstein empirical copula

In this article, we propose a flexible weighted test of independence based on Cramér-von Mises statistic of the Bernstein empirical copula process. We investigate the asymptotic properties of the new weighted test. A Monte Carlo study is conducted to measure the power performance of the new test under various dependence structures and also to compare the performance with some other alternative tests. The effects of the weights and the selection of optimal Bernstein polynomial degree are significantly influencing the power of the test according to the particular type of dependence. Finally, the procedure is also applied to a dataset on diabetes for illustrative purpose.

Empirical tail copulas for functional data

For multivariate distributions in the domain of attraction of a max-stable distribution, the tail copula and the stable tail dependence function are equivalent ways to capture the dependence in the upper tail. The empirical versions of these functions are rank-based estimators whose inflated estimation errors are known to converge weakly to a Gaussian process that is similar in structure to the weak limit of the empirical copula process. We extend this multivariate result to continuous functional data by establishing the asymptotic normality of the estimators of the tail copula, uniformly over all finite subsets of at most D points (D fixed). An application for testing tail copula stationarity is presented. The main tool for deriving the result is the uniform asymptotic normality of all the D-variate tail empirical processes. The proof of the main result is nonstandard.

Empirical Tail Copulas for Functional Data

For multivariate distributions in the domain of attraction of a max-stable distribution, the tail copula and the stable tail dependence function are equivalent ways to capture the dependence in the upper tail. The empirical versions of these functions are rank-based estimators whose inflated estimation errors are known to converge weakly to a Gaussian process that is similar in structure to the weak limit of the empirical copula process. We extend this multivariate result to continuous functional data by establishing the asymptotic normality of the estimators of the tail copula,uniformly over all finite subsets of at most D points (D fixed). As a special case we obtain the uniform asymptotic normality of all estimated uppertail dependence coeffcients. The main tool for deriving the result is the uniform asymptotic normality of all the D-variate tail empirical processes. The proof of the main result is non-standard.

A goodness-of-fit test for copulas based on martingale transformation

This paper proposes an asymptotically distribution-free test for copulas with dynamic marginal distributions, such as GARCH and ARMA processes. The test is based on the empirical copula process with parametrically estimated marginal distributions. By applying the Khmaladze (1982, 1988, 1993) martingale transformation method, the transformed empirical process converges to a standard Gaussian process, so the resulting test statistics are asymptotically distribution-free. Monte Carlo simulations show that the test performs well in finite samples. An empirical application to test copulas between EUR/USD and GBP/USD exchange rates is provided.

Rank-based inference tools for copula regression, with property and casualty insurance applications

Rank-based procedures are commonly used for inference in copula models for continuous responses whose behavior does not depend on covariates. This paper describes how these procedures can be adapted to the broader framework in which (possibly non-linear) regression models for the marginal responses are linked by a copula that does not depend on covariates. The validity of many of these techniques can be derived from the asymptotic equivalence between the classical empirical copula process and its analog based on suitable residuals from the marginal models. Moment-based parameter estimation and copula goodness-of-fit tests are shown to remain valid under weak conditions on the marginal error term distributions, even when the residual-based empirical copula process fails to converge weakly. The performance of these procedures is evaluated through simulation in the context of two general insurance applications: micro-level multivariate insurance claims, and dependent loss triangles.

Subsampling (weighted smooth) empirical copula processes

A key tool to carry out inference on the unknown copula when modeling a continuous multivariate distribution is a nonparametric estimator known as the empirical copula. One popular way of approximating its sampling distribution consists of using the multiplier bootstrap. The latter is however characterized by a high implementation cost. Given the rank-based nature of the empirical copula, the classical empirical bootstrap of Efron does not appear to be a natural alternative, as it relies on resamples which contain ties. The aim of this work is to investigate the use of subsampling in the aforementioned framework. The latter consists of basing the inference on statistic values computed from subsamples of the initial data. One of its advantages in the rank-based context under consideration is that the formed subsamples do not contain ties. Another advantage is its asymptotic validity under minimalistic conditions. In this work, we show the asymptotic validity of subsampling for several (weighted, smooth) empirical copula processes both in the case of serially independent observations and time series. In the former case, subsampling is observed to be substantially better than the empirical bootstrap and equivalent, overall, to the multiplier bootstrap in terms of finite-sample performance.

Testing for independence in arbitrary distributions

Statistics are proposed for testing the hypothesis that arbitrary random variables are mutually independent. The tests are consistent and well behaved for any marginal distributions; they can be used, for example, for contingency tables which are sparse or whose dimension depends on the sample size, as well as for mixed data. No regularity conditions, data jittering, or binning mechanisms are required. The statistics are rank-based functionals of Cramér–von Mises type whose asymptotic behaviour derives from the empirical multilinear copula process. Approximate |$p$|-values are computed using a wild bootstrap. The procedures are simple to implement and computationally efficient, and maintain their level well in moderate to large samples. Simulations suggest that the tests are robust with respect to the number of ties in the data, can easily detect a broad range of alternatives, and outperform existing procedures in many settings. Additional insight into their performance is provided through asymptotic local power calculations under contiguous alternatives. The procedures are illustrated on traumatic brain injury data.

On the asymptotic covariance of the multivariate empirical copula process

Abstract Genest and Segers (2010) gave conditions under which the empirical copula process associated with a random sample from a bivariate continuous distribution has a smaller asymptotic covariance than the standard empirical process based on a random sample from the underlying copula. An extension of this result to the multivariate case is provided.

Weak convergence of the weighted empirical beta copula process

The empirical copula has proved to be useful in the construction and understanding of many statistical procedures related to dependence within random vectors. The empirical beta copula is a smoothed version of the empirical copula that enjoys better finite-sample properties. At the core lie fundamental results on the weak convergence of the empirical copula and empirical beta copula processes. Their scope of application can be increased by considering weighted versions of these processes. In this paper we show weak convergence for the weighted empirical beta copula process. The weak convergence result for the weighted empirical beta copula process is stronger than the one for the empirical copula and its use is more straightforward. The simplicity of its application is illustrated for weighted Cramér–von Mises tests for independence and for the estimation of the Pickands dependence function of an extreme-value copula.

Weak convergence of empirical copula processes indexed by functions

Weak convergence of the empirical copula process indexed by a class of functions is established. Two scenarios are considered in which either some smoothness of these functions or smoothness of the underlying copula function is required. A novel integration by parts formula for multivariate, right-continuous functions of bounded variation, which is perhaps of independent interest, is proved. It is a key ingredient in proving weak convergence of a general empirical process indexed by functions of bounded variation.

Recent Developments in Copula Models

Copula models have become very popular and well studied among the scientific community.[...]

Asymptotic behavior of the empirical multilinear copula process under broad conditions

The empirical checkerboard copula is a multilinear extension of the empirical copula, which plays a key role for inference in copula models. Weak convergence of the corresponding empirical process based on a random sample from the underlying multivariate distribution is established here under broad conditions which allow for arbitrary univariate margins. It is only required that the underlying checkerboard copula has continuous first-order partial derivatives on an open subset of the unit hypercube. This assumption is very weak and always satisfied when the margins are discrete. When the margins are continuous, one recovers the limit of the classical empirical copula process under conditions which are comparable to the weakest ones currently available in the literature. A multiplier bootstrap method is also proposed to replicate the limiting process and its validity is established. The empirical checkerboard copula is further shown to be a more precise estimator of the checkerboard copula than the empirical copula based on jittered data. Finally, the weak convergence of the empirical checkerboard copula process is shown to be sufficiently strong to derive the asymptotic behavior of a broad class of functionals that are directly relevant for the development of rigorous statistical methodology for copula models with arbitrary margins.

Testing independence based on Bernstein empirical copula and copula density

ABSTRACTIn this paper we provide three nonparametric tests of independence between continuous random variables based on the Bernstein copula distribution function and the Bernstein copula density function. The first test is constructed based on a Cramér-von Mises divergence-type functional based on the empirical Bernstein copula process. The two other tests are based on the Bernstein copula density and use Cramér-von Mises and Kullback–Leibler divergence-type functionals, respectively. Furthermore, we study the asymptotic null distribution of each of these test statistics. Finally, we consider a Monte Carlo experiment to investigate the performance of our tests. In particular we examine their size and power which we compare with those of the classical nonparametric tests that are based on the empirical distribution function.

Goodness-of-Fit Tests for Copulas of Multivariate Time Series

In this paper, we study the asymptotic behavior of the sequential empirical process and the sequential empirical copula process, both constructed from residuals of multivariate stochastic volatility models. Applications for the detection of structural changes and specification tests of the distribution of innovations are discussed. It is also shown that if the stochastic volatility matrices are diagonal, which is the case if the univariate time series are estimated separately instead of being jointly estimated, then the empirical copula process behaves as if the innovations were observed; a remarkable property. As a by-product, one also obtains the asymptotic behavior of rank-based measures of dependence applied to residuals of these time series models.

Weak convergence of the empirical copula process with respect to weighted metrics

The empirical copula process plays a central role in the asymptotic analysis of many statistical procedures which are based on copulas or ranks. Among other applications, results regarding its weak convergence can be used to develop asymptotic theory for estimators of dependence measures or copula densities, they allow to derive tests for stochastic independence or specific copula structures, or they may serve as a fundamental tool for the analysis of multivariate rank statistics. In the present paper, we establish weak convergence of the empirical copula process (for observations that are allowed to be serially dependent) with respect to weighted supremum distances. The usefulness of our results is illustrated by applications to general bivariate rank statistics and to estimation procedures for the Pickands dependence function arising in multivariate extreme-value theory.

A New Measure of Vector Dependence, with Applications to Financial Risk and 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 robust to the within-vector dependence. It is confined to the [0,1] interval and covers the entire range of dependence from vector independence to a monotone relationship element-wise. 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 in applications to financial contagion, systemic risk, and portfolio choice. Specifically, we test for contagion effects between equity markets in North and South America, Europe and Asia, surrounding the financial crisis of 2008 and find strong evidence of previously unknown contagion patterns. In the context of sovereign bonds and credit default swaps, we study the evolution of systemic risk in European financial markets and uncover large differences from previous estimates. Finally, based on a real-time portfolio utilizing the new systemic risk estimates, we illustrate the implications of the new measure for portfolio choice and risk management.

Kac’s representation for empirical copula process from an asymptotic viewpoint

The purpose of the present paper is to obtain some approximations of the multivariate empirical copula process with Poisson bridges. The construction of the Poisson bridges is based on Kac’s representation of empirical processes of Horváth (1995) combined with Bahadur–Kiefer representation of the empirical copula process.