Confidence Interval for the Difference Between Variances of Delta-Gamma Distribution

Abstract

Since environmental data are often right-skewed, the gamma distribution is commonly used to model them. However, rainfall data often contain zero observations, so the delta-gamma model is a better fit in these circumstances. Since the variance of delta-gamma distributions is a useful measure of rainfall dispersion, we focused on the difference between the variances of two delta-gamma populations for comparison of the precipitation in two areas in Thailand. We constructed the confidence interval for the difference between the variances of delta-gamma distributions by using various Bayesian and highest posterior density (HPD) methods based on the Jeffrey’s, uniform, or normal-gamma-beta priors and compared with the fiducial quantity (FQ) approach. The performances of the proposed confidence interval methods were evaluated by examining their coverage probabilities and average lengths via a Monte Carlo simulation study. The results indicate that for a small probability of zero observations (δ), the confidence intervals based on FQ and HPD with either the Jeffrey’s or uniform priors are suitable whereas for large δ, the HPD with the normal-gamma-beta prior is recommended. Rainfall data from Lamphun province, Thailand, are used to illustrate the practical efficacies of the proposed methods.