Parameter Estimation of Stable Distributions

ABSTRACT

In finance, economics, statistical physics, signal processing, telecommunications, etc., we frequently meet data sets with outliers that transport important information. α-stable distributions are found more suitable in modeling these kind of data. But the lack of simple and effective methods of estimating their parameters limited their applications to wider variety of fields. In this article we develop an unbiased estimator for the stable index α. With the structure of U-statistic, it inherits all the good statistical properties from U-statistics. A consistent estimator of its asymptotic variance is provided. The asymptotic normality of the given estimator holds when using the estimated variance for standardization. Simulation studies are performed. The results support our theory.

Keywords:

• Asymptotic normality
• Simulation
• Stable distribution
• Stable index
• U-statistics

Mathematics Subject Classification:

• 62E07
• 62F12

Acknowledgment

The author would like to thank an anonymous referee for constructive suggestions that led to a great improvement to the expression of this article.