Brahim Brahimi

Copula Representation of Bivariate L-Moments: A New Estimation Method for Multiparameter 2-Dimensional Copula Models

Recently, Serfling and Xiao (2007) extended the L-moment theory (Hosking, 1990) to the multivariate setting. In the present paper, we focus on the two-dimension random vectors to establish a link between the bivariate L-moments (BLM) and the underlying bivariate copula functions. This connection provides a new estimate of dependence parameters of bivariate statistical data. Consistency and asymptotic normality of the proposed estimator are established. Extensive simulation study is carried out to compare estimators based on the BLM, the maximum likelihood, the minimum distance and rank approximate Z-estimation. The obtained results show that, when the sample size increases, BLM-based estimation performs better as far as the bias and computation time are concerned. Moreover, the root mean squared error (RMSE) is quite reasonable and less sensitive in general to outliers than those of the above cited methods. Further, we expect that the BLM method is an easy-to-use tool for the estimation of multiparameter copula models.

Gaussian approximation to the extreme value index estimator of a heavy-tailed distribution under random censoring

We make use of the empirical process theory to approximate the adapted Hill estimator, for censored data, in terms of Gaussian processes. Then, we derive its asymptotic normality, only under the usual second-order condition of regular variation. Our methodology allows us to relax the assumptionsmade in Einmahl et al. (2008) on the heavy-tailed distribution functions and the sample fraction of upper order statistics.

A Bias-Reduced Estimator for the Mean of a Heavy-Tailed Distribution with an Infinite Second Moment

We use bias-reduced estimators of high quantiles, of heavy-tailed distributions, to introduce a new estimator of the mean in the case of infinite second moment. The asymptotic normality of the proposed estimator is established and checked, in a simulation study, by four of the most popular goodness-of-fit tests for different sample sizes. Moreover, we compare, in terms of bias and mean squared error, our estimator with Pengs estimator (Peng, 2001) and we evaluate the accuracy of some resulting confidence intervals.

Erratum to: ‘Statistical estimate of the proportional hazard premium of loss’

The asymptotic variance of the risk premium estimator, proposed by Necir et al. (2007), is revised, by using the right asymptotic approximation of the uniform empirical quantile process.

Optimal number of upper order statistics used in estimation for the coefficient of tail dependence

Beirlant et al . (2011) introduced a bias-reduced estimator for the coeffcient of tail dependence and for bivariate tail probability in bivariate extreme value statistics. In this paper, we are interested in the problem of choice of the number of extreme order statistics of bivariate observations exceeding high thresholds, we want to optimize the estimators on the choice we expose different methods for determining this number. The effciency of our methods is illustrated on a simulation study and by an application to real data. Keywords: Coeffcient of tail dependence; Bias reduction; Extended Pareto distribution; Tail probability; Copula; Hill estimator; Moment estimator