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Bayesian inference for quantiles of the log-normal distribution.

Aldo Gardini1, Carlo Trivisano1, Enrico Fabrizi2

  • 1Dipartimento di Scienze Statistiche'P. Fortunati', Università di Bologna, Bologna, Italy.

Biometrical Journal. Biometrische Zeitschrift
|July 23, 2020
PubMed
Summary

This study introduces a new Bayesian method for estimating quantiles in log-normal distributions, crucial for environmental data. The proposed approach ensures reliable estimates, particularly in small sample sizes.

Keywords:
Bessel functionsenvironmental monitoringgeneralized inverse Gaussiansmall samples

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Area of Science:

  • Statistics
  • Environmental Science
  • Bayesian Inference

Background:

  • Log-normal distribution is widely used for positive right-skewed data in environmental applications.
  • Traditional Bayesian quantile estimation can result in posterior distributions without finite moments.
  • Prior choices significantly impact the properties of posterior distributions for quantiles.

Purpose of the Study:

  • To develop a robust Bayesian method for estimating quantiles of the log-normal distribution.
  • To address the issue of non-finite posterior moments arising from common prior choices.
  • To improve quantile estimation, especially in small sample scenarios.

Main Methods:

  • Utilized a Bayesian perspective for quantile estimation of the log-normal distribution.
  • Proposed the generalized inverse Gaussian distribution as a prior for the variance.
  • Investigated the impact of prior parameter choices on posterior properties.

Main Results:

  • Demonstrated that the generalized inverse Gaussian prior, with a specific parameter restriction, ensures the existence of posterior moments.
  • Showcased that carefully chosen prior parameters yield quantile estimators with good frequentist properties in small samples.
  • The proposed Bayesian method outperformed existing frequentist approaches in small sample simulations.

Conclusions:

  • The generalized inverse Gaussian prior offers a superior alternative for Bayesian quantile estimation in log-normal models.
  • The developed methodology provides reliable and accurate quantile estimates, especially valuable in environmental and occupational health studies with limited data.
  • This Bayesian approach enhances the precision and stability of quantile estimation compared to traditional methods.