Page Header

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

Wansiri Khooriphan, Sa-Aat Niwitpong, Suparat Niwitpong


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.


[1] J. Aitchison, “On the distribution of a positive random variable having a discrete probability mass at the origin,” Journal of the American Statistical Association, vol. 50, no. 271, pp. 901– 908, Sep. 1955.

[2] J. Aitchison and J. A. C. Brown, The lognormal distribution: With special reference to its uses in economics London. UK: Cambridge University Press, 1963.

[3] N. Yosboonruang, S. A. Niwitpong, and S. Niwitpong, “Measuring the dispersion of rainfall using Bayesian confidence intervals for coefficient of variation of delta-lognormal distribution: A study from Thailand,” PeerJ, vol. 7, 2019, Art. no. e7344.

[4] P. Maneerat and S. A. Niwitpong, “Estimating the average daily rainfall in Thailand using confidence intervals for the common mean of several delta-lognormal distributions,” PeerJ, vol. 9, 2021, Art. no. e10758.

[5] K. Krishnamoorthy and X. Wang, “Fiducial confidence limits and prediction limits for a gamma distribution: Censored and uncensored cases,” Environmetrics, vol. 27, no. 8, pp. 479– 493, 2016.

[6] P. Ren, G. Liu, and X. Pu, “Simultaneous confidence intervals for mean differences of multiple zeroinflated gamma distributions with applications to precipitation,” Communications in Statistics - Simulation and Computation, 2021, doi: 10.1080/03610918.2021.1966466.

[7] K. Muralidharan and B. K. Kale, “Modified gamma distributions with singularity at zero,” Communications in Statistics - Simulation and Computation, vol. 31, no. 1, pp. 143–158, 2002.

[8] J. B. Lecomte, H. P. Benot, S. Ancelet, M. P. Etienne, L. Bel, and E. Parent, “Compound Poissongamma vs. delta-gamma to handle zero-inflated continuous data under a variable sampling volume,” Methods in Ecology and Evolution, vol. 4, no. 12, pp. 1159–1166, 2013.

[9] G. Casella and R. L. Berger, Statistical Inference, 2nd ed. Massachusetts: Cengage Learning, 2001

[10] K. Krishnamoorthy, T. Mathew, and S. Mukherjee, “Normal-based methods for a gamma distribution,” Technometrics, vol. 50, no. 1, pp. 69–78, 2008.

[11] X. Li, X. Zhou, and L. Tian, “Interval estimation for the mean of lognormal data with excess zeros,” Statistics Probability Letters, vol. 83, no. 11, pp. 2447–2453, 2013.

[12] P. Maneerat, S. A. Niwitpong, and S. Niwitpong, “Bayesian confidence intervals for the difference between variances of deltalognormal distributions,” Biometrical Journal, vol. 62, no. 7, pp. 1769– 1790, 2020.

[13] W. M. Bolstad and J. M. Curran, Introduction to Bayesian Statistics, 3rd ed. New Jersey: Wiley, 2016.

[14] T. A. Kalkur and A. Rao, “Bayes estimator for coefficient of variation and inverse coefficient of variation for the normal distribution,” International Journal of Statistics and Systems, vol. 12, no. 4, pp. 721–732, 2017.

[15] P. Sangnawakij, S. A. Niwitpong, and S. Niwitpong, “Confidence intervals for the ratio of coefficients of variation of the gamma distributions,” in Lecture Notes in Computer Science. Cham: Springer, 2015.

[16] M. M. A. Ananda, O. Dag, and S. Weerahandi, “Heteroscedastic two-way ANOVA under constraints,” Communications in Statistics - Theory and Methods, pp. 1–16, 2022, doi: 10.1080/ 03610926.2022.2059682.

Full Text: PDF

DOI: 10.14416/j.asep.2022.12.001


  • There are currently no refbacks.