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 as the fast Fourier transform 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 and precision 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：由于概率密度函数的非显性形式,估计依赖数字算法作为快速的Fourier变异,通常耗费时间。我们比较了几种参数估计方法,如最大可能性方法和不同的瞬间通用方法。我们研究大量抽样特性,并为我们的估测员取得一致性、无症状的正常性、无症状效率结果。此外,我们进行模拟研究,分析根据经验偏差和精确度衡量的有限抽样属性,比较计算成本。我们涵盖典型的温和稳定分布和相关稳定分布的亚类,如典型的温和稳定分布和温和稳定次协调员。此外,我们讨论正常的温和稳定分布,这是通过将布朗运动与温和稳定的次协调员相调和产生的。我们在记录资产指数收益和能源现价方面的财务应用说明了温和稳定模型的好处。</s>