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Journal of Financial Econometrics Vol. 2, No. 3, pp. 349-369
Journal of Financial Econometrics, Vol. 2, No. 3, © Oxford University Press 2004; all rights reserved.

Which Extreme Values Are Really Extreme?

Jesús Gonzalo
     Universidad Carlos III de Madrid

José Olmo
     Universidad Carlos III de Madrid

Address correspondence to Jesús Gonzalo, Department of Economics, Universidad Carlos III de Madrid, 28903, Getafe, Madrid, Spain, or e-mail: jesus.gonzalo{at}uc3m.es.

We define the extreme values of any random sample of size n from a distribution function F as the observations exceeding a threshold and following a type of generalized Pareto distribution (GPD) involving the tail index of F. The threshold is the order statistic that minimizes a Kolmogorov-Smirnov statistic between the empirical distribution of the corresponding largest observations and the corresponding GPD. To formalize the definition we use a semiparametric bootstrap to test the corresponding GPD approximation. Finally, we use our methodology to estimate the tail index and value at risk (VaR) of some financial indexes of major stock markets.

KEYWORDS: bootstrap, extreme values, goodness-of-fit test, Hill estimator, Pickands theorem, VaR

Received March 5, 2003; revised February 19, 2004; accepted April 21, 2004


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