Interval Estimation of the Ratio of Medians of the Bivariate Log-Normal Distribution

The ratio of medians serves as a key metric for comparing two positively skewed populations and is especially appropriate for log-normal data, where the median more accurately represents the central tendency than the mean. This study introduces two Asymptotic confidence interval (CI) procedures for the ratio of medians of two dependent (correlated) log-normal distributions. The proposed methods utilize the Delta method and the Logarithmic transformation method and are applicable to both equal and unequal sample sizes. Their performance is assessed through extensive Monte Carlo simulation studies across a broad range of parameter settings, including varying standard deviations, correlation coefficients, and sample sizes. Moreover, a Percentile Bootstrap CI is included as a benchmark to evaluate the performance of the proposed methods for small sample sizes of n=10,20 and n=25. The methods are evaluated based on coverage probability (CP) and average width (AW). Simulation results indicate that both approaches generally achieve satisfactory CPs, while the Asymptotic approach tends to produce narrower CIs, and the Logarithmic transformation approach often provides CPs closer to the nominal confidence level, particularly for highly variable data. Two real-world datasets from Thailand, comprising PM2.5 concentration data and rainfall measurements, are used to demonstrate the practical applicability of the suggested techniques. These applications demonstrate the use of the proposed methods for interval estimation of the ratio of medians in different environmental datasets. The findings provide useful guidance for interval estimation of the ratio of medians in correlated log-normal populations.