An overview of the goodness-of-fit test problem for copulas

We review the main "omnibus procedures" for goodness-of-fit testing for copulas: tests based on the empirical copula process, on probability integral transformations, 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

Single-index copulae

We introduce so-called "single-index copulae". They are semi-parametric conditional copulae whose parameter is an unknown "link" function of a univariate index only. We provide estimates of this link function and of the finite dimensional unknown parameter. The asymptotic properties of the latter estimates are stated. Thanks to some properties of conditional Kendall's tau, we illustrate our technical conditions with several usual copula families.Comment: Revised version: correction of Assumption 3 and some minor induced modification

On kernel-based estimation of conditional Kendall's tau: finite-distance bounds and asymptotic behavior

We study nonparametric estimators of conditional Kendall's tau, a measure of concordance between two random variables given some covariates. We prove non-asymptotic bounds with explicit constants, that hold with high probabilities. We provide "direct proofs" of the consistency and the asymptotic law of conditional Kendall's tau. A simulation study evaluates the numerical performance of such nonparametric estimators.Comment: 29 pages, 4 figures. arXiv admin note: text overlap with arXiv:1802.0761

Sensitivity Analysis of VaR Expected Shortfall for Portfolios Under Netting Agreements

In this paper, we characterize explicitly the first derivative of the Value at Risk and the Expected Shortfall with respect to portfolio allocation when netting between positions exists. As a particular case, we examine a simple Gaussian example in order to illustrate the impact of netting agreements in credit risk management. We further provide nonpara-metric estimators for sensitivities and derive their asymptotic distributions. An empirical application on a typical banking portfolio is finally provided.Value at Risk, Expected Shortfall, Sensitivity, Risk Management, Credit Risk, Netting.

SOME STATISTICAL PITFALLS IN COPULA MODELING FOR FINANCIAL APPLICATIONS

In this paper we discuss some statistical pitfalls that may occur in modeling cross-dependences with copulas in financial applications. In particular we focus on issues arising in the estimation and the empirical choice of copulas as well as in the design of time-dependent copulas.Copulas; Dependence Measures; Risk Management

Nonparametric Estimation of Copulas for Time Series

We consider a nonparametric method to estimate copulas, i.e. functions linking joint distributions to their univariate margins. We derive the asymptotic properties of kernel estimators of copulas and their derivatives in the context of a multivariate stationary process satisfactory strong mixing conditions. Monte Carlo results are reported for a stationary vector autoregressive process of order one with Gaussian innovations. An empirical illustration containing a comparison with the independent, comotonic and Gaussian copulas is given for European and US stock index returns.Nonparametric, Kernel; Time Series; Copulas; Dependence Measures; Risk Management; Loss Severity Distribution

Kernel estimation of Greek weights by parameter randomization

A Greek weight associated to a parameterized random variable $Z(\lambda)$ is a random variable $\pi$ such that $\nabla_{\lambda}E[\phi(Z(\lambda))]=E[\phi(Z(\lambda))\pi]$ for any function $\phi$. The importance of the set of Greek weights for the purpose of Monte Carlo simulations has been highlighted in the recent literature. Our main concern in this paper is to devise methods which produce the optimal weight, which is well known to be given by the score, in a general context where the density of $Z(\lambda)$ is not explicitly known. To do this, we randomize the parameter $\lambda$ by introducing an a priori distribution, and we use classical kernel estimation techniques in order to estimate the score function. By an integration by parts argument on the limit of this first kernel estimator, we define an alternative simpler kernel-based estimator which turns out to be closely related to the partial gradient of the kernel-based estimator of $\mathbb{E}[\phi(Z(\lambda))]$. Similarly to the finite differences technique, and unlike the so-called Malliavin method, our estimators are biased, but their implementation does not require any advanced mathematical calculation. We provide an asymptotic analysis of the mean squared error of these estimators, as well as their asymptotic distributions. For a discontinuous payoff function, the kernel estimator outperforms the classical finite differences one in terms of the asymptotic rate of convergence. This result is confirmed by our numerical experiments.Comment: Published in at http://dx.doi.org/10.1214/105051607000000186 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org