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Xd ) and write T µT = (µT 1 , µ2 ), Σ= Σ11 Σ21 Σ12 Σ22 . Then X1 ∼ Ek (µ1 , Σ11 , ψ) and X2 ∼ Ed−k (µ2 , Σ22 , ψ). • Conditional distributions. Assuming that Σ is nonsingular, X2 |X1 = x1 ∼ Ed−k (µ2,1 , Σ22,1 , ψ), where −1 µ2,1 = µ2 + Σ21 Σ−1 11 (x1 − µ1 ) and Σ22,1 = Σ22 − Σ21 Σ11 Σ12 . Typically ψ is a different characteristic generator than the original ψ. • Quadratic forms. If Σ is nonsingular, then d D 2 = (X − µ)T Σ−1 (X − µ) = R2 . The variable D is called the Mahalanobis distance. • Convolutions.

The notation X ∼ Nd (µ, Σ) is used to denote that X has a d-dimensional normal distribution with mean vector µ and covariance matrix Σ. (2) X ∼ Nd (µ, Σ) if and only if its characteristic function is given by 1 φX (t) = E(exp{itT X}) = exp itT µ − tT Σt . 2 (3) A random vector X with E(X) = µ and Cov(X) = Σ, such that |Σ| > 0, satisfies X ∼ Nd (µ, Σ) if and only if it has the density fX (x) = 1 (2π)d |Σ| − exp (x − µ)T Σ−1 (x − µ) 2 Next we list some useful properties of the multivariate normal distribution.

Instead of maximizing the likelihood function we can maximize the log-likelihood function given by ln L(γ, β; Y1, . . , YNu ) = −Nu ln β − Nu 1 +1 γ ln 1 + i=1 γ Yi . β Maximizing the log-likelihood numerically gives estimates γ and β. 60. 4) works, see Figure 14. 4 Estimation of Value-at-Risk and Expected shortfall Recall that Value-at-Risk at confidence level p for a risk X with distribution function F is, by definition, the quantile qp (F ). Hence, the POT method gives the Value-at-Risk estimator VaRp,POT n (1 − p) Nu β =u+ γ −b γ −1 .

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