Extreme value theory-based confidence intervals for the parameters of a symmetric Lévy-stable distribution

The Lévy-stable distribution is a very flexible and suitable model for heavy-tailed data that is characterized by four parameters. The most important one, the shape parameter, lies between 0 and 2. Then, we have the location, scale and skewness parameters which respectively belong to the real line, the set of non-negative numbers and the interval [-1,1]. The shape parameter, also called characteristic exponent or stability index, was thoroughly studied as the extreme value index of a heavy-tailed distribution (see, e.g., Hill, 1975). Peng (2001) has proposed, via the extreme value theory (EVT) approach, an asymptotically normal estimator for the location parameter. By adopting the same methodology, Meraghni and Necir (2007) provided a consistent estimator of the scale parameter with an asymptotic normality property. In this paper, we exploit the asymptotic normality of the EVT-based estimators in the case of a symmetric Lévy-stable distribution to construct confidence intervals for the respective parameters. In a simulation study with the same dataset, we evaluate the accuracy of these intervals through lengths and coverage probabilities. The results show that the proposed approach yields accurate confidence intervals, particularly for values of the stability index that are less than or equal to 1.5.