Diverging moments and parameter estimation

Heavy-tailed distributions are enjoying increased popularity and are becoming more readily applicable as the arsenal of analytical and numerical tools grows. They play key roles in modeling approaches in networking, finance, and hydrology, to name but a few areas. The tail parameter [alpha] is of central importance, because it governs both the existence of moments of positive order and the thickness of the tails of the distribution. Some of the best-known tail estimators, such as those of Koutrouvelis and Hill, are either parametric or show a lack of robustness or accuracy. This article develops a shift- and scale-invariant nonparametric estimator for both, upper and lower bounds for orders with finite moments. The estimator builds on the equivalence between tail behavior and the regularity of the characteristic function at the origin and achieves its goal by deriving a simplified wavelet analysis that is particularly suited to characteristic functions. KEY WORDS: Characteristic functions; Diverging moments; Heavy-tailed distributions; Wavelet transform.