Parameter Estimation for Exponentially Tempered Power Law Distributions

Abstract

Tail estimates are developed for power law probability distributions with exponential tempering, using a conditional maximum likelihood approach based on the upper-order statistics. Tempered power law distributions are intermediate between heavy power-law tails and Laplace or exponential tails, and are sometimes called “semi-heavy” tailed distributions. The estimation method is demonstrated on simulated data from a tempered stable distribution, and for several data sets from geophysics and finance that show a power law probability tail with some tempering.

Keywords:

• Exponential tempering
• Heavy tails
• Tail estimation
• Tempered stable

Mathematics Subject Classification:

• Primary 62F12
• Secondary 60E07

Acknowledgments

Mark M. Meerschaert was partially supported by NSF grants DMS-102548, DMS-0803360, and NIH grant R01-EB012079.

This work was initiated while Professor Shao was visiting the Department of Statistics and Probability at Michigan State University.