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1799): √ k ˆP n k − 2 →d N 0 22 +1 +1 2 2 − 1 log 2 n→ 2 For the estimator based on an exponential regression model, we have that √ k ˆ RMA n k − 2 →d N 0 a2 under some conditions on U x and if we suppose that k n → (see Matthys and Beirlant, 2003), where a = and 2 1 1 2 0 1 − u + u log u 1−u with k/n → 0 2 du equals the variance of K U with U uniformly distributed on (0,1) and K U = 1 log u + 1+ 2 dilog u where dilog u = u 1 log t dt 1−t u≥0 denotes the dilogarithm function. For the POT ML estimator, Nu ˆ uMLP − →d N 0 1 + 2 for Nu → provided > −1/2, under the assumption that the excesses exactly follow a GPD (Smith, 1987).

9 shows the comparison of r2 and r3 for different c and p. From this figure one may conclude that in most cases the values c = 0 4 and c = 0 5 correspond to the best values of u1 and u2 , respectively. The value c = 0 4 gives a cautious decision in the sense that r2 is the same irrespective of distribution and r3 is not maximized. In all cases, the Weibull distribution has the largest r2 and r3 . Together with the previous conclusion, this implies that the confidence intervals for this distribution are worse than for Pareto and Fréchet distributions.

3 Detection of tail heaviness and dependence Before a serious analysis of the data is carried out, it is necessary to detect heavy tails in the data. , 1997; Markovich and Krieger, 2006b) can be applied. Here, we consider several simple procedures that may help us to detect heavy tails and the dependence structure of the data. We illustrate by means of real data how these methods are applied to analyze traffic measurements. 1 Rough tests of tail heaviness Here, we consider several methods in order to check whether measurements X n = X1 X2 Xn are derived from a heavy-tailed DF F x = P X1 ≤ x or not.

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