Some comments on the estimation of a dependence index in bivariate extreme value statistics

Modeling of censored bivariate extremal events

“…The main inspiration for our estimator of small tail probability (denoted by p n below) comes from the recent contribution Beirlant et al (2011). We retain the notation and the framework previously introduced, and denote further τ = ρ/M (1) n (k) and τ 1 = τ I{τ 1 > τ 2 } with ρ a consistent estimator of ρ, and thus p n is given by…”

“…Our framework of bivariate randomly censoring is easily explained if we consider two independent bivariate random vectors (X, Y ) and ( X, Y ). Then the random vector (X, Y ) is componentwise randomly censored by ( X, Y ), and we will establish our estimators based on samples from (X * , Y * ) and (δ (1) , δ (2) ) de ned by X * = min(X, X), Y * = min(Y, Y ), δ (1) = I{X ≤ X}, δ (2) = I{Y ≤ Y }, (1.4) with I{·} the indicator function.…”

“…, n be two independent and identically distributed samples from independent parents (X, Y ) and ( X, Y ) with unit Fréchet distributed margins. Then by (1.4) the samples (1) and δ (2) , respectively. De ne…”

“…Figure 9 draws sample paths of our estimators η (0) n and η (1) n for the coe cient of tail dependence of (Loss, ALAE) with w = 0.5, which shows that η (1) n is more stable than H (c) n and H (c) n . Next, we estimate the tail probability P(Loss > 200000, ALAE > 100000) by our tail probability estimators (3.2) plugged in the estimator (2.2) of the coe cient of tail dependence with w = 2/3.…”

Hidden regular variation and the rank transform

“…We calculate the coefficient of tail dependence η of Ledford and Tawn (1996), (1997) using the Hill estimator for the shape parameter of the distribution of componentwise minima taken after rank transformation to standard Pareto margins. (See also Beirlant and Vandewalle (2002), Drees et al (2004), andPeng (1999).) The Hill and altHill plots used to inform our choice of k for this estimation are shown in Figure 1.…”