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：缓和平稳分布常用于金融应用（如期权定价），其中平稳分布的尾部往往过于厚重。鉴于概率密度函数缺乏显式形式，估计依赖于数值算法，这些算法通常耗时较长。本文比较了多种参数化估计方法，包括最大似然估计法及不同的广义矩估计方法。我们研究了大样本性质，并推导了估计量的一致性、渐近正态性及渐近有效性结果。此外，我们通过模拟研究分析了有限样本性质，以经验偏差、精度和渐近置信区间覆盖率为衡量指标，并比较了计算成本。研究涵盖了缓和平稳分布的相关子类，如经典缓和平稳分布与缓和平稳从属过程。同时，我们探讨了通过缓和平稳从属过程驱动布朗运动所衍生的正态缓和平稳分布。在金融应用方面，通过对资产指数对数收益率与能源现货价格的实证分析，我们阐明了缓和平稳模型的优势。