This paper compiles the research and experiments I carried out during my research project, as part of my penultimate year of engineering studies at Grenoble INP Ensimag. In this paper, we will present elements of the copula theory, including dependence coefficients in order to study copula properties on several examples. Then we will focus on risk management applications of copulas and particul...

Convergence of a sequence of bivariate Archimedean copulas to another Archimedean copula or to the comonotone copula is shown to be equivalent with convergence of the corresponding sequence of Kendall distribution functions. No extra differentiability conditions on the generators are needed. r 2007 Elsevier B.V. All rights reserved.

According to Markowitz (1952) portfolio theory assumed that the investor has a concave utility function that expresses an attitude of risk aversion and managed to put portfolio selection based on two criteria, mean and variance. Other studies have improved this approach and following Basel II recommendations by using Value-at-Risk (VaR) as a standard risk measure in finance, Alexander & Baptist...

The question as to the role that correlated activity plays in the coding of information in the brain continues to be one of the most important in neuroscience. One approach to understanding this role is to formally model the ensemble responses as multivariate probability distributions. We have previously introduced alternatives to linear assumptions of multivariate Gaussian dependence for spike...

Multivariate regular variation describes the relative decay rates of joint tail probabilities of a random vector with respect to tail probabilities of a norm (any norm) of this random vector, and it is often used in studying heavy-tail phenomena observed in data analysis in various fields, such as finance and insurance. Multivariate regular variation can be analyzed in terms of the intensity me...

The aim of this paper is the derivation of the maximum likelihood estimators of the Marshal-Olkin copula. This copula comes from the Marshall-Olkin Bivariate Exponential (MOBE) distribution, that has been proposed in reliability analysis to study complex systems in which the components are not independent and it is also used in the extreme value theory. We find the likelihood estimators conside...

We consider the problem of testing hypotheses on the copula density from n bidimensional observations. We wish to test the null hypothesis characterized by a parametric class against a composite nonparametric alternative. Each density under the alternative is separated in the L2-norm from any density lying in the null hypothesis. The copula densities under consideration are supposed to belong t...

This research estimates portfolio VaR (Value-at-Risk) on G7 exchange rates using a GJR-GARCH-EVT (extreme value theory)-Copula based approach. We first extracts the filtered residuals from each return series via an asymmetric GJR-GARCH model, then constructs the semi-parametric empirical marginal cumulative distribution function (CDF) of each asset using a Gaussian kernel estimate for the inter...

In this work, we propose a new framework for learning mixture models from continuous data. Gaussian Mixture Models (GMMs) are commonly used for this task and are popular among practitioners because of their sound statistical foundation and the availability of an efficient learning algorithm [2]. However, the underlying assumption about the normally distributed mixing components, is often too ri...

This chapter constitutes a survey on copula-based measures of multivari­ ate association i.e. association in a d-dimensional random vector X = (XI1""Xd) where d ~ 2. Some of the measures discussed are multivariate extensions of well­ known bivariate measures such as Spearman's rho, Kendall's tau, Blomqvist's beta or Gini's gamma. Others rely on information theory or are based on Lp-distances of...