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The asymptotics of L-statistics for non i.i.d. variables with heavy tails

Barczyk A., Janssen A., Pauly M.

Probability and Mathematical Statistics

2011

Vol. 31, Fasc. 2

285--299

The purpose of this paper is to study the asymptotic behaviour of linear combinations of order statistics (L-statistics)... [formula] for variables with heavy tails. The order statisticsXi:kn correspond to a non i.i.d. triangular array (Xi,n)1≤i≤kn of infinitesimal and rowwise independent random variables. We give sufficient conditions for the convergence of L-statistics to non-normal limit laws and it is shown that only the extremes contribute to the limit distribution, whereas the middle parts vanish. As an example we consider the case, where the extremal partial sums belong to the domain of attraction of a stable law.We also study L-statistics with scores defined by ci,n := J(i/(n + 1)) with a regularly varying function J, a case which has often been treated in the literature.

Asymptotic distribution of unbiased linear estimators in the presence of heavy-tailed stochastic regressors and residuals

Samorodnitsky G., Rachey S. T., Kurz-Kim J.-R., Stoyanov S.

Probability and Mathematical Statistics

2007

Vol. 27, Fasc. 2

275--302

Under the symmetric а-stable distributional assumption for the disturbances, Blattberg and Sargent [3] consider unbiased line- ar estimators for a regression model with non-stochastic regressors. We study both the rate of convergence to the true value and the asymptotic distribution of the normalized error of the linear unbiased estimators. By doing this, we allow the regressors to be stochastic and disturbances to be heavy-tailed with either finite or infinite variances, where the tail-thickness parameters of the regressors and disturbances may be different.

Computing the portfolio conditional value-at-risk in the α-stable case

Stoyanov S., Samorodnitsky G., Rachev S. T., Ortobelli S.

Probability and Mathematical Statistics

2006

Vol. 26, Fasc. 1

1--22

The class of α-stable distributions is an attractive probabilistic model of asset returns distribution in the field of finance. When dealing with real issues, such as optimal portfolio selection, it is important that we can compute the Conditional Value-at-Risk (CVaR) accurately. The CVaR is also known as the expected tail loss (ETL) proposed in literature as a coherent risk measure. In our paper we propose an integral expression for the calculation of the CVaR of a stable law. We compare the current approach to some existing method and we demonstrate how to relate the derived result to some common multivariate distributional assumptions.