Since rainfall data often contain zero observations, the ratio of the variances of delta-gamma distributions can be used to compare the rainfall dispersion between two rainfall datasets. To this end, we constructed the confidence interval for the ratio of the variances of two delta-gamma distributions by using the fiducial quantity method, Bayesian credible intervals based on the Jeffreys, uniform, or normal-gamma-beta priors, and highest posterior density (HPD) intervals based on the Jeffreys, uniform, or normal-gamma-beta priors. The performances of the proposed confidence interval methods were evaluated in terms of their coverage probabilities and average lengths via Monte Carlo simulation. Our findings show that the HPD intervals based on Jeffreys prior and the normal-gamma-beta prior are both suitable for datasets with a small and large probability of containing zeros, respectively. Rainfall data from Phrae province, Thailand, are used to illustrate the practicability of the proposed methods with real data.

The gamma distribution is commonly used to model environmental data. However, rainfall data often contain zero observations, which violates the assumption that all observations must be positive in a gamma distribution, and so a gamma model with excess zeros treated as a binary random variable is required. Rainfall dispersion is important and interesting, the confidence intervals for the variance of a gamma distribution with excess zeros help to examine rainfall intensity, which may be high or low risk. Herein, we propose confidence intervals for the variance of a gamma distribution with excess zeros by using fiducial quantities and parametric bootstrapping, as well as Bayesian credible intervals and highest posterior density intervals based on the Jeffreys’, uniform, or normal-gamma-beta prior. The performances of the proposed confidence interval were evaluated by establishing their coverage probabilities and average lengths via Monte Carlo simulations. The fiducial quantity confidence interval performed the best for a small probability of the sample containing zero observations (δ) whereas the Bayesian credible interval based on the normal-gamma-beta prior performed the best for large δ. Rainfall data from the Kiew Lom Dam in Lampang province, Thailand, are used to illustrate the efficacies of the proposed methods in practice.

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.

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