Archimedean Copula Models Research Articles

A Semiparametric Estimation of Copula Models Based on the Method of Moments

Using the classical estimation method of moments, we propose a new semiparametric estimation procedure for multi-parameter copula models. Consistency and asymptotic normality of the obtained estimators are established. By considering an Archimedean copula model, an extensive simulation study, comparing these estimators with the pseudo maximum likelihood, rho-inversion and tau-inversion ones, is carried out. We show that, with regards to the other methods, the moment based estimation is quick and simple to use with reasonable bias and root mean squared error.

A goodness-of-fit test for Archimedean copula models in the presence of right censoring

A goodness-of-fit testing procedure for Archimedean copula (AC) models is developed based on right-censored data. The proposed approach extends an existing method, which is suitable for the Clayton model, to general AC models. Asymptotic properties of the proposed test statistics under the true model assumption are derived. Simulation analysis shows that the proposed test has reasonable performance. Finally, two real data examples are analyzed for illustrative purposes.

Archimedean copula estimation and model selection via [formula omitted]-norm symmetric distribution

Based on the relationship between Archimedean copulas and l 1 -norm symmetric distributions, we propose a method to not only estimate the copula parameter but also select the copula model through the observation data in this paper. The strong consistency of the estimator is proved, and a Radial Information Criteria (RIC) is provided to select the appropriate Archimedean copula model fitting the data best. It can be extended to the multivariate cases conveniently because the selection is achieved by using the one-dimensional radial distribution to capture the dependence structure for multivariate data. The Monte Carlo simulation experiments illustrate that the proposed approach works well in parameter estimation and model selection for both bivariate and multivariate cases. An application in modelling the dependence structure of real stock indices is carried out with good performance as well.

Correlation smile matching for collateralized debt obligation tranches with α-stable distributions and fitted Archimedean copula models

As an extension of the standard Gaussian copula model to price collateralized debt obligation (CDO) tranche swaps we present a generalization of a one-factor copula model based on stable distributions. For special parameter values these distributions coincide with Gaussian or Cauchy distributions, but changing the parameters allows a continuous deformation away from the Gaussian copula. All these factor copulas are embedded in a framework of stochastic correlations. We furthermore generalize the linear dependence in the usual factor approach to a more general Archimedean copula dependence between the individual trigger variable and the common latent factor. Our analysis is carried out on a non-homogeneous correlation structure of the underlying portfolio. CDO tranche market premia, even throughout the correlation crisis in May 2005, can be reproduced by certain models. From a numerical perspective, all these models are simple, since calculations can be reduced to one-dimensional numerical integrals.

Archimedean copula model selection under dependent truncation

One-sided truncated survival data arise when a pair of time-to-event variables (X, Y) is observed only when X<Y. Existing methods of analysis rely on the assumption of quasi-independence between X and Y. Recently, Lakhal-Chaieb et al. (Biometrika 2006; 93:655-669) modeled potential dependency between these random variables via a semi-survival Archimedean copula. In this paper, we present a model selection procedure to rank a set of semi-survival Archimedean copula families according to their ability to fit a given data set subject to dependent truncation. The proposed procedure is based on a truncated version of Kendall's tau (J. Multivariate Anal. 1996; 56:60-74). The performance of the proposal is illustrated through simulations and three real data sets.

Frailty models and copulas: similarities and differences

Copulas and frailty models are important tools to model bivariate survival data. Equivalence between Archimedean copula models and shared frailty models, e.g. between the Clayton-Oakes copula model and the shared gamma frailty model, has often been claimed in the literature. In this note we show that, in both the models, there is indeed a well-known equivalence between the copula functions; the modeling of the marginal survival functions, however, is quite different. The latter fact leads to different joint survival functions.

Some properties of the Kendall distribution in bivariate Archimedean copula models under censoring

Suppose that ( T 1 , T 2 ) can be modelled by an Archimedean copula model and it is subject to dependent or independent right censoring. In this paper, we present some distributional results for the random variable V = S ( T 1 , T 2 ) under different censoring patterns (singly or doubly censored). The results are expected to be useful in setting up both the model fitting and checking procedures for Archimedean copula models for censored bivariate data. As an application of the theoretical results we obtained, a simple moment estimator of the dependence parameter in Archimedean copula models is proposed. Simulation studies have shown that the proposed estimator parameter works well and the estimator is used in analyzing a medical data set.

Mixture Cure Survival Models With Dependent Censoring

SummaryThe paper is motivated by cure detection among the prostate cancer patients in the National Institutes of Health surveillance epidemiology and end results programme, wherein the main end point (e.g. deaths from prostate cancer) and the censoring causes (e.g. deaths from heart diseases) may be dependent. Although many researchers have studied the mixture survival model to analyse survival data with non-negligible cure fractions, none has studied the mixture cure model in the presence of dependent censoring. To account for such dependence, we propose a more general cure model that allows for dependent censoring. We derive the cure models from the perspective of competing risks and model the dependence between the censoring time and the survival time by using a class of Archimedean copula models. Within this framework, we consider the parameter estimation, the cure detection and the two-sample comparison of latency distributions in the presence of dependent censoring when a proportion of patients is deemed cured. Large sample results by using martingale theory are obtained. We examine the finite sample performance of the proposed methods via simulation and apply them to analyse the surveillance epidemiology and end results prostate cancer data.

On the Tail Behavior of Sums of Dependent Risks

The tail behavior of sums of dependent risks was considered by Wüthrich (2003) and by Alink et al. (2004, 2005) in the case where the variables are exchangeable and connected through an Archimedean copula model. It is shown here how their result can be extended to a broader class of dependence structures using multivariate extreme-value theory. An explicit form is given for the asymptotic probability of extremal events, and the behavior of the latter is studied as a function of the indices of regular variation of both the copula and the common distribution of the risks.

On the Tail Behavior of Sums of Dependent Risks

The tail behavior of sums of dependent risks was considered by Wüthrich (2003) and by Alink et al. (2004, 2005) in the case where the variables are exchangeable and connected through an Archimedean copula model. It is shown here how their result can be extended to a broader class of dependence structures using multivariate extreme-value theory. An explicit form is given for the asymptotic probability of extremal events, and the behavior of the latter is studied as a function of the indices of regular variation of both the copula and the common distribution of the risks.

Copula model generated by Dabrowska's association measure

SUMMARY We propose a new archimedean copula model for bivariate survival data that is motivated by Dabrowska's (1988) measure of association. The model can represent negatively correlated or moderately positively correlated data but not highly positively correlated data. Local and global measures of association are calculated. A generalisation is presented.

Compound Poisson approximations for individual models with dependent risks

This paper shows how compound Poisson distributions can be used to approximate the distribution of the total claim amount in the context of single- or multi-class individual risk models where dependence between the contracts arises through mixtures. Some of these models are generated by Archimedean copulas, and others are seen to fall under the purview of a general multi-class shock model whose structure is both intuitive and easily tractable. A numerical study is used to illustrate the quality of the approximation as a function of the heterogeneity and the dependence in the portfolio. A theoretical result is also provided which helps to explain the effect of dependence on the total claim amount when the contracts are linked through an Archimedean copula model.

Multivariate survival distributions

When two or more observed survival times depend, via a proportional hazards model, on the same unobserved variable, called in this context a frailty, this common dependence induces an association between the observed times. This paper focuses on the class of multivariate survival distributions generated by such models. These turn out to be a subclass of the archimedean copula models described among others by Genest and Mackay (1986). A special case, important in survival analysis, is Gumbells (1960) Type B distribution of extreme values, which is obtained from a positive stable frailty distribution. We discuss characterizations of the frailty distribution via measures of association and methods for parametric and nonparametric inference from data possibly subject to censoring. We briefly contrast this situation with that obtained when repeated events can be observed on the same subject, leading to ordered failure time data.