We propose a testing and estimation methodology for univariate and bivariate symmatric $α$-stable distributions using a modified version of the Greenwood statistic. Originally designed for positive-valued random variables, the Greenwood statistic, and its modified version tailored for symmetric distributions, have been predominantly applied to univariate random samples. In this paper, we extend the modified Greenwood statistic to a bivariate setting and examine its probabilistic properties within the class of $α$-stable distributions, with a focus on the sub-Gaussian case. Additionally, we introduce a novel testing approach that considers two variations of the modified Greenwood statistic as test statistics for the bivariate case. In the univariate setting, we adapt the proposed testing methodology for estimating the stability index. The simulation studies presented demonstrate that our proposed methodology outperforms classical approaches previously used in this context and serves as an effective tool for distinguishing between Gaussian and $α$-stable distributions with a stability index close to 2. The theoretical and simulation results are further illustrated with practical data examples.

翻译：本文提出一种利用修正Greenwood统计量对单变量及双变量对称$α-$稳定分布进行检验与估计的方法。原始Greenwood统计量针对正随机变量设计，而面向对称分布定制的修正版本主要应用于单变量随机样本。本文将修正Greenwood统计量拓展至双变量场景，并在$α-$稳定分布族中重点分析其亚高斯情形下的概率性质。此外，我们引入一种新型检验方法，将修正Greenwood统计量的两种变体作为双变量情形的检验统计量。在单变量设定下，我们对该检验方法进行适配以估计稳定性指数。仿真研究表明，所提方法优于此前在该领域中使用的经典方法，并能有效区分高斯分布与稳定性指数接近2的$α-$稳定分布。最终通过实际数据案例验证了理论结果与仿真结论。