The lack of closed representations for the density functions of the α-stable distributions, when considering Bayesian inference using Markov Chain Monte Carlo methods, has historically lead to the use of bivariate probability density functions [Buckle. Bayesian inference for stable distributions. J Am Stat Assoc. 1995;90:605–613] and Fast Fourier Transforms of their characteristic functions [Lombardi. Bayesian inference for α-stable distributions: a random walk MCMC approach. Comput Stat Data Anal. 2007;51(5):2688–2700]. We present a novel approach using a full power series representation for the probability density functions. The Bayesian estimation analysis is provided for two different parameterization systems for one-dimensional stable distributions. We provide an algorithm that makes use only of the power series representation. Three goodness-of-fit tests, based on the empirical distribution functions, and two types of loss functions with their respective decision rules to minimize the Bayesian risk, are included. A simulation study and two empirical applications are also presented.

在考虑使用马尔可夫链蒙特卡罗方法的贝叶斯推理时，α 稳定分布的密度函数缺乏封闭表示，历史上导致使用二元概率密度函数 [Buckle. 稳定分布的贝叶斯推理。J Am Stat 协会 1995;90:605-613] 和其特征函数的快速傅立叶变换 [Lombardi. α 的贝叶斯推理-稳定分布：随机游走 MCMC 方法。计算统计数据分析。2007;51(5):2688–2700]。我们提出了一种使用概率密度函数的全幂级数表示的新方法。为一维稳定分布的两个不同参数化系统提供了贝叶斯估计分析。我们提供了一种仅使用幂级数表示的算法。包括三个基于经验分布函数的拟合优度测试，以及两种类型的损失函数，它们具有各自的决策规则以最小化贝叶斯风险。还介绍了模拟研究和两个经验应用。