Diverging Moments, Wavelets, and Nonparametric Tail Estimation

• Activity Number: 395
• Type: Contributed
• Date/Time: Wednesday, August 10, 2005 : 10:30 AM to 12:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Rudolf Riedi*+ and Paulo Goncalves
• Companies: Rice University and INRIA
• Keywords: non-parametric ; tail estimation ; wavelets ; heavy-tails ; multifractals ; stable parameter

Abstract

Heavy-tailed distributions become of increasing importance in various applications as the arsenal of analytical and numerical tools grows. Examples of interest include the stable and, more generally, heavy-tailed distributions for which moments beyond a critical order diverge. Applications include networking, quantitative finance, and turbulence. In practice, however, standard estimators of moments typically will report a finite value, even for diverging moments. This talk provides a nonparametric, wavelet-based estimator of the critical order of moments and compares it to standard estimators of the tail parameter of stable and Pareto distributions. The approach taken exploits the tight connection between the tail and the regularity of the characteristic function at the origin with the idea to estimate the latter using wavelets. While regularity estimation via wavelets may be cumbersome and numerically expensive in general, it simplifies drastically here due to particular properties of the wavelet transform of characteristic functions. These properties will be established in the talk as well. If time permits, applications to multifractal model identification will be discussed.