# Nonlinear functional analysis vol.3: Variational methods and by E. Zeidler, L.F. Boron By E. Zeidler, L.F. Boron

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Additional info for Nonlinear functional analysis vol.3: Variational methods and optimization

Example text

8) in case F = G1,α or G2,α but considers no asymptotics. Viewing F as an element of Qi (δ), we can establish asymptotics for UMVU estimators of a, b and of resulting estimators of extreme quantiles F −1 (1 − qn ) with qn → 0 and k = k(n) → ∞ as n → ∞. 6 that the error of the resulting estimator of F −1 (1 − qn ) is of the order −γ(i) −1/2 Op (qn k (log2 (nqn /k) + 1)1/2 ), where γ(i) = 1/α, −1/α, 0 if F ∈ Qi (δ), i = 1, 2, 3. This demonstrates the superiority of the estimators to the ones proposed by Dekkers and de Haan  and Dekkers et al.

22) explains the idea behind the peaksover-threshold method (POT), widely used for instance by hydrologists to model large floods having a GPD (cf. Todorovic and Zelenhasic , Hosking and Wallis ). 1 below, and independently by Pickands . 21), if we drop the assumption F ∈ D(G) and merely consider a sequence an t + bn , n ∈ N, of thresholds satisfying a certain regularity condition. 1 in Rychlik  . 5. Suppose that there exist an > 0, bn ∈ R with 1 − F (bn) −→n→∞ 0 and (1 − F (bn+1 ))/(1 − F (bn )) −→n→∞ 1 such that for any s ≥ 0, 1− 1 − F (an s + bn ) −→n→∞ L(s) 1 − F (bn ) for some continuous df L.

Be iid rv. Then we have for an EVD G and norming constants an > 0, bn ∈ R, Zn:n − bn −→D G an Zn−i+1:n − bn ⇐⇒ an i≤k −→D G(k) for any k ∈ N, where the distribution G(k) /Bk has Lebesgue density g (k) (x1 , . . , xk ) = G(xk ) ′ i≤k G (xi )/G(xi ) for x1 > · · · > xk and zero elsewhere. 3. Let ξ1 , ξ2 , . . be a sequence of independent and standard ex(k) (k) ponential rv. 12. j≤i ξj ))i≤k . 2. We have to show the only-if part of the assertion. Consider without loss of generality Zi = F −1 (Ui ), where U1 , U2 , .