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：温控稳定分布在金融应用中（例如期权定价）频繁使用，在这些场景中，稳定分布的尾部可能过于厚重。鉴于概率密度函数的非显式形式，参数估计依赖于通常耗时的数值算法。我们比较了几种参数估计方法，例如最大似然法和不同形式的广义矩方法。我们研究了大样本性质，并推导了估计量的一致性、渐近正态性和渐近效率结果。此外，我们通过模拟研究分析了由经验偏差、精度和渐近置信区间覆盖概率衡量的有限样本性质，并比较了计算成本。我们涵盖了温控稳定分布的相关子类，例如经典温控稳定分布和温控稳定子序器。此外，我们讨论了通过用温控稳定子序器对布朗运动进行子序化而得到的正态温控稳定分布。我们对资产指数对数收益率和能源现货价格的金融应用说明了温控稳定模型的优势。