Abstract
The inverse of the coefficient of variation (ICV), otherwise known as the signal to-noise ratio, is the ratio of the population standard deviation to the population mean. It has often been used in the fields of finance and image processing, among others. In this study, various methods were applied to estimate the confidence intervals (CIs) for the difference between and the ratio of the ICVs of two delta-gamma distributions. The fiducial quantity method, Bayesian CI estimates based on the Jeffreys, uniform, or normal-gamma-beta (NGB) prior, and highest posterior density (HPD) intervals based on the Jeffreys, uniform, or NGB priors were used in this endeavor. A Monte Carlo simulation study was conducted to assess the performances of the proposed CI estimation methods in terms of their coverage probabilities and average lengths. The results indicate that the HPD interval based on the NGB prior or the Jeffreys prior performed well for a small probability of the samples containing zero observations (\(\delta\)) whereas the fiducial quantity method performed well for large values of \(\delta\). Furthermore, we demonstrate the practicability of the proposed methods using rainfall data from Lampang province, Thailand.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1134%2FS1995080223110227/MediaObjects/12202_2024_7491_Fig1_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1134%2FS1995080223110227/MediaObjects/12202_2024_7491_Fig2_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1134%2FS1995080223110227/MediaObjects/12202_2024_7491_Fig3_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1134%2FS1995080223110227/MediaObjects/12202_2024_7491_Fig4_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1134%2FS1995080223110227/MediaObjects/12202_2024_7491_Fig5_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1134%2FS1995080223110227/MediaObjects/12202_2024_7491_Fig6_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1134%2FS1995080223110227/MediaObjects/12202_2024_7491_Fig7_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1134%2FS1995080223110227/MediaObjects/12202_2024_7491_Fig8_HTML.png)
REFERENCES
G. Casella and R. L. Berger, Statistical Inference, 2nd ed. (Cengage Learning, US, 2001).
T. Holgersson, P. Karlsson, and R. Mansoor, ‘‘Estimating mean-standard deviation ratios of financial data,’’ J. Appl. Stat. 39, 657–671 (2013).
A. N. Albatineh, B. M. G. Kibria, and B. Zogheib, ‘‘Asymptotic sampling distribution of inverse coefficient of variation and its applications,’’ J. Adv. Stat. Probab. 2, 15–20 (2014).
A. N. Albatineh, I. Boubakari, and B. M. G. Kibria, ‘‘New confidence interval estimator of the signal to noise ratio based on asymptotic sampling distribution,’’ Commun. Stat. Theory Methods 46, 574–590 (2017).
F. George and B. M. G. Kibria, ‘‘Confidence intervals for estimating the population signal to noise ratio: A simulation study,’’ J. Appl. Stat. 39, 1225–1240 (2012).
K. K. Sharma and H. Krishna, ‘‘Asymptotic sampling distribution of inverse coefficient of variation and its applications,’’ IEEE Trans. Reliab. 43, 630–633 (1994).
S. Niwitpong, ‘‘Confidence intervals for functions of signal-to-noise ratios of normal distributions,’’ Studies Comput. Intell. 760, 255–265 (2018).
L. Saothayanun and W. Thangjai, ‘‘Confidence intervals for the signal to noise ratio of two-parameter exponential distribution,’’ IEEE Trans. Reliab. 43, 630–633 (2018).
W. Thangjai and S. Niwitpong, ‘‘Confidence intervals for the signal-to-noise ratio and difference of signal-to-noise ratios of gamma distributions,’’ Adv. Appl. Math. Sci. 18, 503–520 (2019).
W. Thangjai and S. Niwitpong, ‘‘Confidence intervals for the signal-to-noise ratio and difference of signal-to-noise ratios of lognormal distribution,’’ Stats 2, 164–173 (2019).
W. Thangjai and S. Niwitpong, ‘‘Confidence intervals for common signal-to-noise ratio of several log-normal distributions,’’ Iran. J. Sci. Technol. Trans., A: Sci. 44, 99–107 (2020).
W. Thangjai and S. Niwitpong, ‘‘Confidence intervals for difference of signal-to-noise ratios of two-parameter exponential distributions,’’ Int. J. Stat. Appl. Math. 5 (3), 47–54 (2020).
X. Wang, C. Zou, L. Yi, J. Wang, and X. Li, ‘‘Fiducial inference for gamma distributions: Two-sample problems,’’ Commun. Stat. – Simul. Comput. 50, 811–821 (2019).
J. Aitchison, ‘‘On the distribution of a positive random variable having a discrete probability mass at the origin,’’ J. Am. Stat. Assoc. 50 (271), 901 (1955).
J. Aitchison and J. A. C. Brown, The Lognormal Distribution: With Special Reference to its Uses in Economics (Cambridge University Press, London, UK, 1963).
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 7, e7344 (2019).
P. Maneerat, S. A. Niwitpong, and S. Niwitpong, ‘‘Bayesian approach to construct confidence intervals for comparing the rainfall dispersion in Thailand,’’ PeerJ 8, e8502 (2020).
P. Maneerat, S. A. Niwitpong, and S. Niwitpong, ‘‘Bayesian confidence intervals for the difference between variances of delta-lognormal distributions,’’ Biometr. J. 62, 1769–1790 (2020).
P. Maneerat, S. A. Niwitpong, and S. Niwitpong, ‘‘Estimating the average daily rainfall in Thailand using confidence intervals for the common mean of several delta-lognormal distributions,’’ PeerJ 9, e10758 (2021).
P. Maneerat, S. A. Niwitpong, and S. Niwitpong, ‘‘Bayesian confidence intervals for variance of delta-lognorma distribution with an application to rainfall dispersion,’’ Stat. Interface 14, 229–241 (2021).
Q. Zhang, J. Xu, J. Zhao, H. Liang, and X. Li, ‘‘Simultaneous confidence intervals for ratios of means of zero-inflated log-normal populations,’’ J. Stat. Comput. Simul. 92, 1113–1132 (2022).
P. Ren, G. Lui, and X. Pu, ‘‘Simultaneous confidence intervals for mean differences of multiple zero-inflated gamma distributions with applications to precipitation,’’ Commun. Stat. – Simul. Comput. 52, 4705 (2021).
K. Muralidharan and B. K. Kale, ‘‘Modified gamma distributions with singularity at zero,’’ Commun. Stat. – Simul. Comput. 31, 143–158 (2002).
J. B. Lecomte, H. P. Benot, S. Ancelet, M. P. Etienne, L. Bel, and E. Parent, ‘‘Compound Poisson-gamma vs. delta-gamma to handle zero-inflated continuous data under a variable sampling volume,’’ Methods Ecol. Evol. 4, 1159–1166 (2013).
T. Kaewprasert, S. A. Niwitpong, and S. Niwitpong, ‘‘Bayesian estimation for the mean of delta-gamma distributions with application to rainfall data in Thailand,’’ PeerJ 10, 1–27 (2022).
W. Khooriphan, S. A. Niwitpong, and S. Niwitpong, ‘‘Bayesian estimation of rainfall dispersion in Thailand using gamma distribution with excess zeros,’’ PeerJ 10, e14023 (2022).
P. Sangnawakij and S. A. Niwitpong, ‘‘Confidence intervals for functions of coefficients of variation with bounded parameter spaces in two gamma distributions,’’ Songklanakarin J. Sci. Technol. 39, 27–39 (2017).
K. Krishnamoorthy and X. Wang, ‘‘Fiducial confidence limits and prediction limits for a gamma distribution: Censored and uncensored cases,’’ Environmetrics 27, 479–493 (2016).
X. Li, X. Zhou, and L. Tian, ‘‘Interval estimation for the mean of lognormal data with excess zeros,’’ Stat. Probab. Lett. 83, 2447–2453 (2013).
W. M. Bolstad and J. M. Curran, Introduction to Bayesian Statistics, 3rd ed. (Wiley, Hoboken, 2016).
H. Jeffreys, Theory of Probability (Oxford Univ. Press, UK, 1961).
T. A. Kalkur and A. Rao, ‘‘Bayes estimator for coefficient of variation and inverse coefficient of variation for the normal distribution,’’ Int. J. Stat. Syst. 12, 721–732 (2017).
Funding
The first author would like to express gratitude to the Thai Scientific Achievement Scholarship (SAST) for financial assistance. This research was supported by the National Science, Research, and Innovation Fund (NSRF) and King Mongkut’s University of Technology North Bangkok (Grant no. KMUTNB-FF-66-44).
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The authors of this work declare that they have no conflicts of interest.
Additional information
Publisher’s Note.
Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
(Submitted by M. UЁ nver)
Rights and permissions
About this article
Cite this article
Khooriphan, W., Niwitpong, SA. & Niwitpong, S. Confidence Intervals of the Inverse of Coefficient of Variation of Delta-Gamma Distribution. Lobachevskii J Math 44, 4739–4762 (2023). https://doi.org/10.1134/S1995080223110227
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080223110227