A Mixture of Clayton, Gumbel, and Frank Copulas: A Complete Dependence Model (original) (raw)
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Copula Approach for Modelling Dependence Structure
Nwsa Physical Sciences, 2012
This paper introduces the concept of copula as a tool to describe relationships among multivariate random variables. In this study, under the assumption of data can be modelled by one of Archimedean copulas (Gumbel, Frank, Clayton), the dependence structure between two dependent random variables with Weibull marginals is modelled by using copula.
Modeling Bivariate Dependency in Insurance Data via Copula: A Brief Study
Journal of Risk and Financial Management
Copulas are a quite flexible and useful tool for modeling the dependence structure between two or more variables or components of bivariate and multivariate vectors, in particular, to predict losses in insurance and finance. In this article, we use the VineCopula package in R to study the dependence structure of some well-known real-life insurance data and identify the best bivariate copula in each case. Associated structural properties of these bivariate copulas are also discussed with a major focus on their tail dependence structure. This study shows that certain types of Archimedean copula with the heavy tail dependence property are a reasonable framework to start in terms modeling insurance claim data both in the bivariate as well as in the case of multivariate domains as appropriate.
Bivariate dependence measures and bivariate competing risks models under the generalized FGM copula
Statistical Papers, 2016
The first part of this paper reviews the properties of bivariate dependence measures (Spearman's rho, Kendall's tau, Kochar and Gupta's dependence measure, and Blest's coefficient) under the generalized Farlie-Gumbel-Morgenstern (FGM) copula. We give a few remarks on the relationship among the bivariate dependence measures, derive Blest's coefficient, and suggest simplifying the previously obtained expression of Kochar and Gupta's dependence measure. The second part of this paper derives some useful measures for analyzing bivariate competing risks models under the generalized FGM copula. We obtain the expression of sub-distribution functions under the generalized FGM copula, which has not been discussed in the literature. With the Burr III margins, we show that our expression has a closed form and generalizes the reliability measure previously obtained by Domma and Giordano (Stat Pap 54(3):807-826, 2013).
General Multivariate Dependence Using Associated Copulas
SSRN Electronic Journal, 2011
This paper studies the general multivariate dependence of a random vector using associated copulas. We extend definitions and results of positive dependence to the general dependence case. This includes associated tail dependence functions and associated tail dependence coefficients. We derive the relationships among associated copulas and study the associated copulas of the perfect dependence cases and elliptically contoured distributions. We present the expression for the associated tail dependence function of the multivariate Student-t copula, which accounts for all types of tail dependence. Date: April 20th, 2012. This paper is based on results from the first chapters of my doctorate thesis supervised by Dr. Wing Lon Ng. I would like to thank Mexico's CONACYT, for the funding during my studies. I would also like to thank my examiners Dr. Aristidis Nikoloulopoulos and Dr. Nick Constantinou for the corrections, suggestions and comments that have made this work possible.
Quantitative Finance, 2010
The t copula is often used in risk management as it allows for modelling tail dependence between risks and it is simple to simulate and calibrate. However, the use of a standard t copula is often criticized due to its restriction of having a single parameter for the degrees of freedom (dof) that may limit its capability to model the tail dependence structure in a multivariate case. To overcome this problem, grouped t copula was proposed recently, where risks are grouped a priori in such a way that each group has a standard t copula with its specific dof parameter. In this paper we propose the use of a grouped t copula, where each group consists of one risk factor only, so that a priori grouping is not required. The copula characteristics in the bivariate case are studied. We explain simulation and calibration procedures, including a simulation study on finite sample properties of the maximum likelihood estimators and Kendall's tau approximation. This new copula can be significantly different from the standard t copula in terms of risk measures such as tail dependence, value at risk and expected shortfall.
Estimation of a Conditional Copula and Association Measures
2011
This paper is concerned with studying the dependence structure between two random variables Y 1 and Y 2 conditionally upon a covariate X. The dependence structure is modelled via a copula function, which depends on the given value of the covariate in a general way. Gijbels et al. (Comput. Statist. Data Anal., 55, 2011, 1919) suggested two non-parametric estimators of the 'conditional' copula and investigated their numerical performances. In this paper we establish the asymptotic properties of the proposed estimators as well as conditional association measures derived from them. Practical recommendations for their use are then discussed.
Copulas: A new technique to model dependence in petroleum decision making
Journal of Petroleum Science and …, 2007
A key step in valuing petroleum investment opportunities is to construct a model that portrays the uncertainty inherent in the investment decision. In almost all such cases, several of the random variables that are relevant for the model are correlated. Properly accounting for and modelling these correlations is essential in deriving reliable valuations for decision support.The Envelope method and the Iman–Conover method are popular approaches to model dependency in the petroleum industry. Although these models work well in many cases, there are situations where they fail to accurately account for important characteristics of the correlations.In many cases the structure of the dependence between two random variables is important. The approaches typically used to model dependence in oil and gas evaluations often fail to address the dependence structure. The copulas technique, which is well known in financial risk management and insurance applications, has proven to be a superior tool for modelling dependency structures. Yet, to our knowledge, it has rarely been used in petroleum applications.A copula is a statistical concept that relates random variables. It is a function that links the marginal distributions to the joint distribution. A copula can model the dependence structure given any type of marginal distribution, which is not possible with other correlation measures. This is illustrated by the fact that the copula approach is able to separate the marginal distribution from the correlation.The objective of this paper is to illustrate the potential benefits of using copulas to model dependencies in oil and gas applications with a particular focus on the reserves problem. This paper introduces the copulas method and then compares and contrasts it with some of the more commonly used approaches to model dependence. We then show that the traditional methods have problems in accurately revealing the dependence structure in the tails of the variable distributions. Finally, we illustrate how the dependence structure can be captured and modelled using the copulas approach.
Copulas: an Approach How to Model the Dependence Structure of Random Vectors
Copulas enabling to characterize the joint distributions of random vectors bymeans of the corresponding one-dimensional marginal distributions are presented anddiscussed. Some properties of copulas and several construction methods, especially when apartial knowledge is available, are included. Possible applications are indicated.
Dependence measures for perturbations of copulas
Fuzzy Sets and Systems, 2017
In this paper (which is a substantially extended version of a conference paper from AGOP 2015 [10]), we investigate the effects of specific class of perturbations of bivariate copulas on several measures of dependence (Spearman's rho, Blomqvist's beta, Gini's gamma, Kendall's tau), and tail dependence along both diagonal sections. It is demonstrated that the influence of the perturbation parameter on the values of the first three of the above coefficients of dependence is linear, while on the last one it is quadratic. Interesting numerical analyses for several important classes of Archimedean copulas are presented. It is also demonstrated that the considered perturbations do not change the coefficients of tail dependencies along the main diagonal but linearly reduce their values along the second diagonal. An interesting possible application for analyzing dependencies along the second diagonal of copulas represent insurance data, where censoring introduces a negative dependence between the investigated components of the claims. As a by-product, we present a new class of perturbations of copulas that linearly reduce the more popular coefficients of tail dependencies along the main diagonal, while preserving their values along the second diagonal. Subsequently using suitable elements of both above classes of perturbations, any original copula can be transformed to a resulting one, having coefficients of tail dependencies along both diagonals linearly reduced (with any couple of preselected linear proportions from [0, 1]).