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Parameter estimation of stable distributions. (English) Zbl 1084.62019
Summary: In finance, economics, statistical physics, signal processing, telecommunications, etc., we frequently meet data sets with outliers that transport important information. \(\alpha\)-stable distributions are found more suitable in modeling this kind of data. But the lack of simple and effective methods of estimating their parameters limited their applications to wider varieties of fields. We develop an unbiased estimator for the stability index \(\alpha\). With the structure of U-statistics, it inherits all 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.

MSC:
62F12 Asymptotic properties of parametric estimators
60E07 Infinitely divisible distributions; stable distributions
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