Asymptotics of empirical copula processes under non-restrictive smoothness assumptions

By Johan Segers. 2010

254 citations recorded by the source corpus.

Weak convergence of the empirical copula process is shown to hold under the assumption that the first-order partial derivatives of the copula exist and are continuous on certain subsets of the unit hypercube. The assumption is non-restrictive in the sense that it is needed anyway to ensure that the candidate limiting process exists and has continuous trajectories. In addition, resampling methods based on the multiplier central limit theorem, which require consistent estimation of the first-order derivatives, continue to be valid. Under certain growth conditions on the second-order partial derivatives that allow for explosive behavior near the boundaries, the almost sure rate in Stute’s representation of the empirical copula process can be recovered. The conditions are verified, for instance, in the case of the Gaussian copula with full-rank correlation matrix, many Archimedean copulas, and many extreme-value copulas.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c0e1bde965e5c36615f0ddd07c7d695c66e434b7

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Citations in the corpus

On the empirical multilinear copula process for count data (2014, result): “ms of a C-Brownian bridge BC, for all u 1,...,ud ∈[0,1], by Cb(u 1,...,ud)=BC(u 1,...,ud)− Xd j=1 C˙ j(u 1,...,ud)BC(1,...,1,uj,1,...,1). This result has since been reﬁned in various ways; see Segers [18] and references therein. 4. Proof of Theorem 3.1 The proof of the main result is quite involved.”
Multivariate nonparametric estimation of the Pickands dependence function using Bernstein polynomials (2014, methodology): “st, are obtained with: independence ( 0= 1), strong dependence ( 0= 0:3), weak dependence ( = 0:9) and medium dependence (0= 0:5;0:7). These results are compared with those provided in Gudendorf and Segers (2012) that are obtained with the discrete spectral measure projection method proposed by the ”
An Overview of the Goodness-of-Fit Test Problem for Copulas (2012, background): “Recently, [87] has shown that such a result is true when, for every j = 1, .”
Goodness-of-Fit Tests for Copulas of Multivariate Time Series (2017, unknown): “In order to work on the space of continuous functions on [0, 1]d, one assumes from now on the following additional technical assumption [21] (Condition 2.”
Probit Transformation for Nonparametric Kernel Estimation of the Copula Density (2014, methodology): “Second, when deriving the asymptotic properties of the estimators, the effect of resorting to pseudo-observations is carefully analyzed through Segers’s [47] recent developments on the empirical copula process (of course, the effect of pseudo-observations was not relevant to Geenens’s [22] estimator”
A Note on Conditional Versus Joint Unconditional Weak Convergence in Bootstrap Consistency Results (2017, background): “The latter seems all the more true for instance for empirical processes based on estimated or serially dependent observations (see, e.g., Rémillard and Scaillet, 2009; Segers, 2012; Bücher and Kojadinovic, 2016a).”
A Robust-Equitable Measure for Feature Ranking and Selection (2017, background): “This Hölder condition, while being a standard condition for density estimations, does not hold for any commonly used copula (Omelka et al., 2009; Segers, 2012).”
Estimation of a Copula when a Covariate Affects only Marginal Distributions (2015, background): “: Note that assumption (Cp) is the weakest assumption assumed so far to establish the weak convergence of the empirical copula process (Segers, 2012).”
Large‐sample tests of extreme‐value dependence for multivariate copulas (2011, unknown): “The second section of the paper is devoted to recent results on the weak convergence of the empirical copula process obtained in Segers (2011).”
The Copula Information Criteria (2014, methodology): “Proposition 3.1 of Segers (2012) implies the desired result using the general change of variables formula provided in Appendix A.3 of the Supporting information.”
The Cross-Quantilogram: Measuring Quantile Dependence and Testing Directional Predictability between Time Series (2014, background): “c measures of dependence, such as Spearman’s rho and Kendall’s tau. In this special case, the asymptotic results for the empirical copula, which are found in Stute (1984), Fermanian et al. (2004) and Segers (2012) among others, can apply for the cross-quantilogram. Generally, however, the cross-quan”
Nonparametric estimation of multivariate extreme-value copulas (2011, background): “Following Zhanget al. (2008) and Gudendorf and Segers (2011), random samples were generated from a trivariate extreme-value distribution with asymmetric logistic dependence functionA (Tawn, 1990): A(w) = (θ1/αw1/α1 + φ 1/αw1/α2 ) α + (θ1/αw1/α2 + φ 1/αw1/α3 ) α + (θ1/αw1/α3 + φ 1/αw1/α1 ) α +…”