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：调和稳定分布常用于金融应用（例如期权定价），在这些应用中，稳定分布的尾部可能过于厚重。由于概率密度函数没有显式形式，其估计依赖于数值算法（如快速傅里叶变换），而这些算法通常耗时较长。我们比较了几种参数估计方法，例如极大似然法和不同的广义矩方法。我们研究了这些估计量的大样本性质，并推导了相合性、渐近正态性和渐近效率结果。此外，我们进行了模拟研究，以分析基于经验偏差和精确度度量的有限样本性质，并比较了计算成本。我们涵盖了调和稳定分布的相关子类，例如经典调和稳定分布和调和稳定子序器。此外，我们讨论了通过布朗运动与调和稳定子序器的子序化而得到的正态调和稳定分布。我们将其应用于资产指数对数收益率和能源现货价格的金融案例，展示了调和稳定模型的优势。