The log-normal distribution is very popular for modeling positive right-skewed data and represents a common distributional assumption in many environmental applications. Here we consider the estimation of quantiles of this distribution from a Bayesian perspective. We show that the prior on the variance of the log of the variable is relevant for the properties of the posterior distribution of quantiles. Popular choices for this prior, such as the inverse gamma, lead to posteriors without finite moments. We propose the generalized inverse Gaussian and show that a restriction on the choice of one of its parameters guarantees the existence of posterior moments up to a prespecified order. In small samples, a careful choice of the prior parameters leads to point and interval estimators of the quantiles with good frequentist properties, outperforming those currently suggested by the frequentist literature. Finally, two real examples from environmental monitoring and occupational health frameworks highlight the improvements of our methodology, especially in a small sample situation.

中文翻译：

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对数正态分布分位数的贝叶斯推理

对数正态分布在对正右偏数据建模时非常流行，并且代表了许多环境应用中的常见分布假设。在这里，我们从贝叶斯角度考虑该分布的分位数估计。我们证明变量对数方差的先验与分位数后验分布的性质相关。此先验的流行选择，例如逆伽玛，导致后验没有有限矩。我们提出了广义逆高斯分布，并证明对其参数之一的选择的限制保证了后验矩的存在达到预先指定的阶数。在小样本中，仔细选择先验参数会导致分位数的点和区间估计器具有良好的频率特性，优于当前频率特性文献中建议的那些。最后，来自环境监测和职业健康框架的两个真实例子突出了我们方法的改进，特别是在小样本情况下。