Tempered stable distributions are frequently used in financial applications (e.g., for option pricing) in which the tails of stable distributions would be too heavy. Given the non-explicit form of the probability density function, estimation relies on numerical algorithms which typically are time-consuming. We compare several parametric estimation methods such as the maximum likelihood method and different generalized method of moment approaches. We study large sample properties and derive consistency, asymptotic normality, and asymptotic efficiency results for our estimators. Additionally, we conduct simulation studies to analyze finite sample properties measured by the empirical bias, precision, and asymptotic confidence interval coverage rates and compare computational costs. We cover relevant subclasses of tempered stable distributions such as the classical tempered stable distribution and the tempered stable subordinator. Moreover, we discuss the normal tempered stable distribution which arises by subordinating a Brownian motion with a tempered stable subordinator. Our financial applications to log returns of asset indices and to energy spot prices illustrate the benefits of tempered stable models.

翻译：修正稳定分布常被用于金融应用（如期权定价）中，在这些场景下标准稳定分布的尾部可能过重。鉴于其概率密度函数不存在显式形式，参数估计依赖于数值算法，而这些算法通常耗时较长。我们比较了多种参数估计方法，包括极大似然估计法和不同的广义矩方法。我们研究了大样本性质，推导了估计量的一致性、渐近正态性和渐近效率结果。此外，我们通过模拟研究分析了基于经验偏差、精度和渐近置信区间覆盖率的有限样本性质，并比较了计算成本。我们涵盖了修正稳定分布的相关子类，如经典修正稳定分布和修正稳定子序。同时，我们讨论了通过布朗运动与修正稳定子序的复合得到的正态修正稳定分布。我们对资产指数对数收益率和能源现货价格的金融应用实例展示了修正稳定模型的优势。