Oracle Automatic Storage Management for 10g and 11g

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Ud )du1 . . dud −∞ and then f is called the density of F . The components of X are independent if and only if d F (x) = Fi (xi ) i=1 or equivalently if and only if the joint density f (if the density exists) satisfies d f (x) = fi (xi ). i=1 Recall that the distribution of a random vector X is completely determined by its characteristic function given by φX (t) = E(exp{i tT X}), t ∈ Rd . 1 The multivariate normal distribution with mean µ and covariance matrix Σ has the density (with |Σ| being the absolute value of the determinant of Σ) f (x) = 1 (2π)d |Σ| exp 1 − (x − µ)T Σ−1 (x − µ) , 2 Its characteristic function is given by 1 φX (t) = exp i tT µ − tT Σt , 2 43 t ∈ Rd .

Since this is true for every ε ∈ (0, 1), choosing ε arbitrarily small gives lim x→∞ P(X + Y > x) = 1. P(X > x) Hence, P(X > x, X + Y > x) x→∞ P(X + Y > x) P(X > x) = 1. = lim x→∞ P(X + Y > x) lim P(X > x | X + Y > x) = lim x→∞ We have found that if the insurance company suffers a large loss, it is likely that this is due to a large loss in the fire insurance line only. 32 6 Hill estimation Suppose we have an iid sample of positive random variables X1 , . . , Xn from an unknown distribution function F with a regularly varying right tail.

Tail probability F (x), for x large, based on the sample points X1 , . . , Xn and (H) the the Hill estimate αk,n . Notice that F (x) = F x Xk,n Xk,n ≈ x Xk,n −α F (Xk,n ) (H) ≈ x Xk,n −αk,n (H) k Fn (Xk,n ) ≈ n 34 x Xk,n −b αk,n . This argument can be made more rigorously. Hence, the following estimator seems reasonable: (H) k F (x) = n x Xk,n −b αk,n . This leads to an estimator of the quantile qp (F ) = F ← (p). qp (F ) = inf x ∈ R : F (x) ≤ 1 − p (H) k = inf x ∈ R : n n (1 − p) = k x Xk,n −b αk,n ≤1−p (H) −1/b αk,n Xk,n .

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