This paper presents an introduction to inference for copula models, based on rank methods. By working out in detail a small, fictitious numerical example, the writers exhibit the various steps involved in investigating the dependence between two random variables and in modeling it using copulas. Simple graphical tools and numerical techniques are presented for selecting an appropriate model, estimating its parameters, and checking its goodness-of-fit. A larger, realistic application of the methodology to hydrological data is then presented.

http://quantlabs.net/academy/download/free_quant_instituitional_books_/%5BJournal%20of%20Hydrologic%20Engineering,%20Genest%5D%20Everything%20You%20Always%20Wanted%20to%20Know%20about%20Copula%20Modeling%20but%20Were%20Afraid%20to%20Ask.pdf

Everything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask

Copulas characterize the dependence among components of random vectors. Unlike marginal and joint distributions, which are directly observable, the copula of a random vector is a hidden dependence structure that links the joint distribution with its margins. Choosing a parametric copula model is thus a nontrivial task but it can be facilitated by relying on a nonparametric estimator. Here the authors propose a kernel estimator of the copula that is mean square consistent everywhere on the support. They determine the bias and variance of this estimator. They also study the effects of kernel smoothing on copula esti mation. They then propose a smoothing bandwidth selection rule based on the derived bias and variance. After confirming their theoretical findings through simulations, they use their kernel estimator to formulate a goodness-of-fit test for parametric copula models.

https://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=1042&context=stat_las_preprints

Nonparametric estimation of copula functions for dependence modelling

INTRODUCTION Copulas are a way of formalising dependence structures of random vectors. Although they have been known about for a long time (Sklar (1959)), they have been rediscovered relatively recently in applied sciences (biostatistics, reliability, biology, etc). In finance, they have become a standard tool with broad applications: multiasset pricing (especially complex credit derivatives), credit portfolio modelling, risk management, etc. For example, see Li (1999), Patton (2001) and Longin and Solnik (1995). Although the concept of copulas is well understood, it is now recognised that their empirical estimation is a harder and trickier task. Many traps and technical difficulties are present, and these are, most of the time, ignored or underestimated by practitioners. The problem is that the estimation of copulas implies usually that every marginal distribution of the underlying random vectors must be evaluated and plugged into an estimated multivariate distribution. Such a procedure produces unexpected and unusual effects with respect to the usual statistical procedures: non-standard limiting behaviours, noisy estimations, etc (eg, see the discussion in Fermanian and Scaillet, 2005). In this chapter, we focus on the practical issues practitioners are faced with, in particular concerning estimation and visualisation. 1

/pdf/the-estimation-of-copulas-theory-and-practice-3grghskibi.pdf

The estimation of copulas : theory and practice

Evading the curse of dimensionality in nonparametric density estimation with simplified vine copulas

We reconsider the existing kernel estimators for a copula function, as proposed in Gijbels and Mielniczuk [Comm. Statist. Theory Methods 19 (1990) 445--464], Fermanian, Radulovi\v{c} and Wegkamp [Bernoulli 10 (2004) 847--860] and Chen and Huang [Canad. J. Statist. 35 (2007) 265--282]. All of these estimators have as a drawback that they can suffer from a corner bias problem. A way to deal with this is to impose rather stringent conditions on the copula, outruling as such many classical families of copulas. In this paper, we propose improved estimators that take care of the typical corner bias problem. For Gijbels and Mielniczuk [Comm. Statist. Theory Methods 19 (1990) 445--464] and Chen and Huang [Canad. J. Statist. 35 (2007) 265--282], the improvement involves shrinking the bandwidth with an appropriate functional factor; for Fermanian, Radulovi\v{c} and Wegkamp [Bernoulli 10 (2004) 847--860], this is done by using a transformation. The theoretical contribution of the paper is a weak convergence result for the three improved estimators under conditions that are met for most copula families. We also discuss the choice of bandwidth parameters, theoretically and practically, and illustrate the finite-sample behaviour of the estimators in a simulation study. The improved estimators are applied to goodness-of-fit testing for copulas.

Improved kernel estimation of copulas: Weak convergence and goodness-of-fit testing

This paper deals with estimation of the density of a copula function as well as with that of the Radon-Nikodym derivative of a bivariate distribution function with respect to the product of its marginal distribution functions. Strong uniform consistency and asymptotic normality of kernel-type estimators are proved under various conditions on the bandwidth and on the smoothness of the kernel. As an application, the estimation of Neyman-Pearson curves in the testing of independence problem is discussed.

Iène Gijbels

Papers

Estimating the density of a copula function