Minimum-Distance Estimator for Stable Exponent

Abstract

Assume that X ₁, X ₂,…, X _n is a sequence of i.i.d. random variables with α-stable distribution (α ∈ (0,2], the stable exponent, is the unknown parameter). We construct minimum distance estimators for α by minimizing the Kolmogorov distance or the Cramér–von-Mises distance between the empirical distribution function G _n , and a class of distributions defined based on the sum-preserving property of stable random variables. The minimum distance estimators can also be obtained by minimizing a U-statistic estimate of an empirical distribution function involving the stable exponent. They share the same invariance property with the maximum likelihood estimates. In this article, we prove the strong consistency of the minimum distance estimators. We prove the asymptotic normality of our estimators. Simulation study shows that the new estimators are competitive to the existing ones and perform very closely even to the maximum likelihood estimator.

Keywords:

• Cramér–von-Mises distance
• Kolmogorov-distance
• Minimum-distance estimate
• Stable distribution
• Stable exponent
• U-statistics

Mathematics Subject Classification:

• Primary 62E07
• Secondary 62F12

Acknowledgments

The author would like to thank the anonymous referee and the Associate Editor for constructive suggestions which led to the improvement of an earlier version of this article. The research of the author is supported in part by the Natural Sciences and Engineering Research Council of Canada.