Lithuanian Mathematical Journal

, Volume 52, Issue 3, pp 259–281 | Cite as

Estimating asymptotic dependence functionals in multivariate regularly varying models

  • Georg Mainik


This paper deals with semiparametric estimation of the asymptotic portfolio risk factor γ ξ introduced in [G. Mainik and L. Rüschendorf, On optimal portfolio diversification with respect to extreme risks, Finance Stoch., 14:593–623, 2010] for multivariate regularly varying random vectors in \( \mathbb{R}_{+}^d \). The functional γ ξ depends on the spectral measure Ψ, the tail index α, and the vector ξ of portfolio weights. The representation of γ ξ is extended to characterize the portfolio loss asymptotics for random vectors in ℝ d . The earlier results on uniform strong consistency and uniform asymptotic normality of the estimates of γ ξ are extended to the general setting, and the regularity assumptions are significantly weakened. Uniform consistency and asymptotic normality are also proved for the estimators of the functional \( \gamma_\xi^{{{1} \left/ {\alpha } \right.}} \) that characterizes the asymptotic behavior of the portfolio loss quantiles. The techniques developed here can also be applied to other dependence functionals.


tail dependence multivariate regular variation portfolio risk functional CLT functional SLLN 


primary 60 F17; secondary 60 F05 60 F15 60 H12 60 G70 


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Copyright information

© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  1. 1.RiskLab, Department of MathematicsETH ZürichZürichSwitzerland

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