Estimating asymptotic dependence functionals in multivariate regularly varying models

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 functional