Axel Bücher

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.

New estimators of the Pickands dependence function and a test for extreme-value dependence

We propose a new class of estimators for Pickands dependence function which is based on the best L 2 -approximation of the logarithm of the copula by logarithms of extremevalue copulas. An explicit integral representation of the best approximation is derived and it is shown that this approximation satises the boundary conditions of a Pickands dependence function. The estimators ^ A(t) are obtained by replacing the unknown copula by its empirical counterpart and weak convergence of the process p nf ^ A(t) A(t)g t2[0;1] is shown. A comparison with the commonly used estimators is performed from a theoretical point of view and by means of a simulation study. Our asymptotic and numerical results indicate that some of the new estimators outperform the rank-based versions of Pickands estimator and an estimator which was recently proposed by Genest and Segers (2009). As a by-product of our results we obtain a simple test for the hypothesis of an extreme-value copula, which is consistent against all alternatives with continuous partial derivatives of rst order satisfying C(u;v) uv.

Consistent testing for a constant copula under strong mixing based on the tapered block multiplier technique

Considering multivariate strongly mixing time series, nonparametric tests for a constant copula with specified or unspecified change point (candidate) are derived; the tests are consistent against general alternatives. A tapered block multiplier technique based on serially dependent multiplier random variables is provided to estimate p-values of the test statistics. Size and power of the tests in finite samples are evaluated with Monte Carlo simulations. The block multiplier technique might have several other applications for statistical inference on copulas of serially dependent data.

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 Remillard and Scaillet (2009) frequently appears to lead to inference procedures with the best finite-sample properties. Bucher and Ruppert (2013) recently proposed an extension of this technique to strictly stationary strongly mixing observations by adapting the dependent multiplier bootstrap of Buhlmann (1993, Section 3.3) to the empirical copula process. The main contribution of this work is a generalization of the multiplier resampling scheme proposed by Bucher and Ruppert (2013) along two directions. First, the resampling scheme is now genuinely sequential, thereby allowing to transpose to the strongly mixing setting all 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 (block length) parameter. A simulation study is used to investigate the nitesample performance of the resampling scheme and provides suggestions on how to choose several additional parameters. As by-products of this work, the weak convergence of the sequential empirical copula process is obtained under many serial dependence conditions, and the validity of a sequential version of the dependent multiplier bootstrap for empirical processes of Buhlmann is obtained under weaker conditions on the strong mixing coecients and the multipliers.

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.

A test for Archimedeanity in bivariate copula models

We propose a new test for the hypothesis that a bivariate copula is an Archimedean copula which can be used as a preliminary step before further dependence modeling. The corresponding test statistic is based on a combination of two measures resulting from the characterization of Archimedean copulas by the property of associativity and by a strict upper bound on the diagonal by the Frechet-Hoeffding upper bound. We prove weak convergence of this statistic and show that the critical values of the corresponding test can be determined by the multiplier bootstrap method. The test is shown to be consistent against all departures from Archimedeanity. A simulation study is presented which illustrates the finite-sample properties of the new test.

Nonparametric inference on Lévy measures and copulas

In this paper nonparametric methods to assess the multivariate L evy measure are introduced. Starting from high-frequency observations of a L evy process X, we construct estimators for its tail integrals and the Pareto L evy copula and prove weak convergence of these estimators in certain function spaces. Given n observations of increments over intervals of length n, the rate of convergence is k 1=2 n for kn = n n which is natural concerning inference on the L

When uniform weak convergence fails: empirical processes for dependence functions via epi- and hypographs

For copulas whose partial derivatives are not continuous everywhere on the interior of the unit cube, the empirical copula process does not converge weakly with respect to the supremum distance. This makes it hard to verify asymptotic properties of inference procedures for such copulas. To resolve the issue, a new metric for locally bounded functions is introduced and the corresponding weak convergence theory is developed. Convergence with respect to the new metric is related to epi- and hypoconvergence and is weaker than uniform convergence. Still, for continuous limits, it is equivalent to locally uniform convergence, whereas under mild side conditions, it implies Lp convergence. Even in cases where uniform convergence fails, weak convergence with respect to the new metric is established for empirical copula and tail dependence processes. No additional assumptions are needed for tail dependence functions, and for copulas, the assumptions reduce to existence and continuity of the partial derivatives almost everywhere on the unit cube. The results are applied to obtain asymptotic properties of minimum distance estimators, goodness-of-fit tests and resampling procedures.

Some comments on goodness-of-fit tests for the parametric form of the copula based on L2-distances

In a recent paper Fermanian (2005) [9] studied a goodness-of-fit test for the parametric form of a copula, which is based on an L^2-distance between a parametric and a nonparametric estimate of the copula density. In the present paper we investigate the asymptotic properties of the proposed test statistic under fixed alternatives. We also study the impact of different estimates for the parameters of the finite-dimensional family of copulas specified by the null hypothesis and illustrate the performance of a parametric bootstrap procedure for the approximation of the critical values.

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.