A Bayesian Approach to Confidence Interval Estimation for the Ratio of Variances of Two Log-Normal Populations: Application to Particulate Matter (PM2.5) Concentration Data

Abstract

Abstract: In Thailand, Particulate Matter (PM2.5) pollution is a significant environmental and public health issue. It comes from a variety of sources, including industrial emissions, urban traffic, and seasonal biomass burning in different regions. Effective monitoring, risk assessment, and policymaking require an understanding of PM2.5 variability in addition to average concentration levels. This work examines three different types of priors — the independence Jeffreys prior, the uniform prior, and the Jeffreys-rule prior — to investigate Bayesian credible intervals (BCIs) for the ratio of variances of two independent log-normal distributions. Through extensive simulations, the performance of these BCIs was evaluated under different sample sizes (10 to 200) and variance ratios (one set at 0.1 and the remaining ones varying from 0.1 to 2.0). A balanced and an unbalanced design effect analysis was conducted. The BCI based on the uniform prior is a conservative alternative, according to the simulation results, since it consistently produces wide intervals while achieving coverage probabilities (CPs) near the nominal 0.95 level. The most effective interval estimates, with comparatively narrow intervals and reliable CPs, are provided by the Jeffreys-rule prior, especially for moderate to large sample sizes. However, in some cases, particularly with large sample sizes and unbalanced designs, where CPs fall below the nominal level, the BCI based on the independence Jeffreys prior performs worse than expected. When large, balanced sample conditions are satisfied, all approaches converge toward nominal CPs, with the Jeffreys-rule prior providing the most accurate estimates. The proposed procedures were applied to analyze daily PM2.5 mass concentration data from two ecologically distinct areas of Thailand: Si Phum, Chiang Mai (rural) and Phaya Thai, Bangkok (urban), to demonstrate their practical utility. The results indicate that PM2.5 concentrations in Bangkok are more stable, while the rural site in Si Phum exhibits greater variability, likely due to seasonal agricultural activities. These findings underscore the importance of location-specific statistical modeling for environmental monitoring and provide guidance for targeted public health interventions and air quality management strategies.