Weak Convergence and Empirical Processes - A. W. van der Vaart; J. A. Wellner.

We consider a consistent test, that is similar to a Kolmogorov-Smirnov test, of the complete set of restrictions that relate to the copula representation of positive quadrant dependence. For such a test we propose and justify inference relying on a simulation based multiplier method and a bootstrap method. We also explore the finite sample behaviour of both methods with Monte Carlo experiments. A first empirical illustration is given for US insurance claim data. A second one examines the presence of positive quadrant dependence in life expectancies at birth of males and females among countries.

A Kolmogorov-Smirnov Type Test for Positive Quadrant Dependence

Starting from the characterization of extreme-value copulas based on max-stability, large-sample tests of extreme-value dependence for multivariate copulas are studied. The two key ingredients of the proposed tests are the empirical copula of the data and a multiplier technique for obtaining approximate p-values for the derived statistics. The asymptotic validity of the multiplier approach is established, and the finite-sample performance of a large number of candidate test statistics is studied through extensive Monte Carlo experiments for data sets of dimension two to five. In the bivariate case, the rejection rates of the best versions of the tests are compared with those of the test of Ghoudi et al. (1998) recently revisited by Ben Ghorbal et al. (2009). The proposed procedures are illustrated on bivariate financial data and trivariate geological data.

/pdf/large-sample-tests-of-extreme-value-dependence-for-2otf2po3bb.pdf

Large-sample tests of extreme-value dependence for multivariate copulas

This paper is concerned with studying the dependence structure between two random variables Y1 and Y2 in the presence of a covariate X, which affects both marginal distributions but not the dependence structure. This is reflected in the property that the conditional copula of Y1 and Y2 given X, does not depend on the value of X. This latter independence often appears as a simplifying assumption in pair-copula constructions. We introduce a general estimator for the copula in this specific setting and establish its consistency. Moreover, we consider some special cases, such as parametric or nonparametric location-scale models for the effect of the covariate X on the marginals of Y1 and Y2 and show that in these cases, weak convergence of the estimator, at -rate, holds. The theoretical results are illustrated by simulations and a real data example.

/pdf/estimation-of-a-copula-when-a-covariate-affects-only-32vbwjxze2.pdf

Estimation of a Copula when a Covariate Affects only Marginal Distributions

In this paper the interest is to estimate the dependence between two variables conditionally upon a covariate, through copula modelling. In recent literature nonparametric estimators for conditional copula functions in case of a univariate covariate have been proposed. The aim of this paper is to nonparametrically estimate a conditional copula when the covariate takes on values in more complex spaces. We consider multivariate covariates and functional covariates. We establish weak convergence, and bias and variance properties of the proposed nonparametric estimators. We also briefly discuss nonparametric estimation of conditional association measures such as a conditional Kendall’s tau. The case of functional covariates is of particular interest and challenge, both from theoretical as well as practical point of view. For this setting we provide an illustration with a real data example in which the covariates are spectral curves. A simulation study investigating the finite-sample performances of the discussed estimators is provided.

/pdf/multivariate-and-functional-covariates-and-conditional-18l38xqpxv.pdf

Multivariate and functional covariates and conditional copulas

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 nonrestrictive in the sense that it is needed anyway to ensure the candidate limiting process to exist and have 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. The price to pay for the weaker assumption is the loss of an explicit rate for the remainder term. Under certain growth conditions on the second-order partial derivatives, an almost sure rate can still be established. The conditions are verified for instance in the case of the Gaussian copula with full-rank correlation matrix, many Archimedean copulas, and

/pdf/weak-convergence-of-empirical-copula-processes-under-12j3nv2wf4.pdf

Weak convergence of empirical copula processes under nonrestrictive smoothness assumptions

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Weak convergence of empirical copula processes under nonrestrictive smoothness assumptions