# Statistical Inference on the Ratio of Delta-Lognormal Coefficients of Variation

### Abstract

The coefficient of variation is useful to measure and compare the dispersion of the data when different units are used in different datasets. This article aims to propose new confidence intervals for the ratio of two independent coefficients of variation with delta-lognormal distribution. The proposed methods include the concept of the generalized confidence interval and the method of variance estimate recovery. They are applied with three methods, variance stabilizing transformation, Wilson score method, and Jeffreys method. The performance of the confidence intervals was assessed by the coverage probabilities and the expected lengths via the Monte Carlo simulation. The outcomes of the simulation study showed that the generalized confidence interval is appropriate to construct the confidence interval for the ratio of delta-lognormal coefficients of variation. Two rainfall datasets from Nakhon Ratchasima, Thailand are used to demonstrate the proposed confidence intervals.

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### References

[2] R. Ananthakrishnan and M. K. Soman, “Statistical distribution of daily rainfall and its association with the coefficient of variation of rainfall series,” International Journal of Climatology, vol. 9, pp. 485–500, 1989.

[3] K. Shimizu, “A bivariate mixed lognormal distribution with an analysis of rainfall data,” American Meteor Society, vol. 32, pp. 161–171, 1993.

[4] W. J. Owen and T. A. DeRouen, “Estimation of the mean for lognormal data containing zeroes and left-censored values, with applications to the measurement of worker exposure to air contaminants,” Biometrics, vol. 36, pp. 707–719, 1980.

[5] S. J. Ganocy, “Calculation of the mean and variance of lognormal data which contains leftcensored observations,” in Proceedings MidWest SAS Users Group, 1995, pp. 220–225.

[6] L. Tian and J. Wu, “Confidence intervals for the mean of lognormal data with excess zeros,” Biometrical Journal, vol. 48, pp. 149–156, 2006.

[7] X. H. Zhou and W. Tu, “Confidence intervals for the mean of diagnostic test charge data containing zeros,” Biometrics, vol. 56, pp. 1118–1125, 2000.

[8] X. Li, X. Zhou, and L. Tian, “Interval estimation for the mean of lognormal data with excess zeros,” Statistics and Probability Letters, vol. 83, pp. 2447– 2453, 2013.

[9] D. Fletcher, “Confidence intervals for the mean of the delta-lognormal distribution,” Environmental and Ecological Statistics, vol. 15, pp. 175–189, 2008.

[10] W. H. Wu and H. N. Hsieh, “Generalized confidence interval estimation for the mean of delta-lognormal distribution: An application to New Zealand trawl survey data,” Journal of Applied Statistics, vol. 41, pp. 1471–1485, 2014.

[11] A. C. M. Wong and J. Wu, “Small sample asymptotic inference for the coefficient of variation: normal and nonnormal models,” Journal of Statistical Planning and Inference, vol. 104, pp. 73–82, 2002.

[12] L. Tian, “Inferences on the mean of zero-inflated lognormal data: the generalized variable approach,” Statistics in Medicine, vol. 24, pp. 3223–3232, 2005.

[13] A. Donner and G. Y. Zou, “Closed-form confidence intervals for functions of the normal mean and standard deviation,” Statistical Methods in Medical Research, vol. 21, pp. 347–359, 2012.

[14] A. Wongkhao, S.-A. Niwitpong, and S. Niwitpong, “Confidence intervals for the ratio of two independent coefficients of variation of normal distribution,” Far East Journal of Mathematical Sciences, vol. 98, pp. 741–757, 2015.

[15] A. J. Hayter, “Confidence bounds on the coefficient of variation of a normal distribution with applications to win-probabilities,” Journal of Statistical Computation and Simulation, vol. 85, pp. 3778– 3791, 2015.

[16] P. Sangnawakij and S.-A. Niwitpong, “Confidence intervals for coefficients of variation in two-parameter exponential distributions,” Communications in Statistics - Simulation and Computation, vol. 46, pp. 6618–6630, 2017.

[17] S.-A. Niwitpong, “Confidence intervals for coefficient of variation of lognormal distribution with restricted parameter space,” Applied Mathematical Sciences, vol. 7, pp. 3805–3810, 2013.

[18] N. Yosboonruang, S.-A. Niwitpong, and S. Niwitpong, “Confidence intervals for the coefficient of variation of the delta-lognormal distribution,” in Econometrics for Financial Applications - Studies in Computational Intelligence, Cham, Switzerland: Springer Nature, 2018, pp. 327–337.

[19] N. Yosboonruang, S. Niwitpong, and S. Niwitpong, “Confidence intervals for coefficient of variation of three parameters delta-lognormal distribution,” in Structural Changes and their Econometric Modeling - Studies in Computational Intelligence, Cham, Switzerland: Springer Nature, 2019, pp. 352– 363.

[20] S. Verrill and R. A. Johnson, “Confidence bounds and hypothesis tests for normal distribution coefficients of variation,” Communications in Statistics - Theory and Methods, vol. 36, pp. 2187– 2206, 2007.

[21] N. Buntao and S. Niwitpong, “Confidence intervals for the ratio of coefficients of variation of deltalognormal distribution,” Applied Mathematical Sciences, vol. 7, pp. 3811–3818, 2013.

[22] J. Nam and D. Kwon, “Inference on the ratio of two coefficients of variation of two lognormal distributions,” Communications in Statistics - Theory and Methods, vol. 46, pp. 8575–8587, 2016.

[23] M. S. Hasan and K. Krishnamoorthy, “Improved confidence intervals for the ratio of coefficients of variation of two lognormal distributions,” Journal of Statistical Theory and Applications, vol. 16, pp. 345–353, 2017.

[24] J. Aitchison, “On the distribution of a positive random variable having a discrete probability and mass at the origin,” Journal of the American Statistical Association, vol. 50, pp. 901–908, 1955.

[25] W. K. de la Mare, “Estimating confidence intervals for fish stock abundance estimates from trawl surveys,” CCAMLR Science, vol. 1, pp. 203–207, 1994.

[26] K. W. Tsui and S. Weerahandi, “Generalized p-values in significance testing of hypotheses in the presence of nuisance parameters,” Journal of the American Statistical Association, vol. 84, pp. 602–607, 1989.

[27] K. Krishnamoorthy and T. Mathew, “Inferences on the means of lognormal distributions using generalized p-values and generalized confidence intervals,” Journal of Statistical Planning and Inference, vol. 115, pp. 103–121, 2003.

[28] E. B. Wilson, “Probable inference, the law of succession, and statistical inference,” Journal of the American Statistical Association, vol. 22. pp. 209–212, 1927.

[29] L. D. Brown, T. T. Cai, and A. DasGupta, “Interval estimation for a binomial proportion,” Statistical Science, vol. 16, pp. 101–133, 2001.

[30] G. Y. Zou and A. Donner, “Construction of confidence limits about effect measures: A general approach,” Statistics in Medicine, vol. 27, pp. 1693– 1702, 2008.

[31] G. Y. Zou, W. Huang, and X. Zhang, “A note on confidence interval estimation for a linear function of binomial proportions,” Computational Statistics and Data Analysis, vol. 53, pp. 1080– 1085, 2009.

[32] A. DasGupta, Asymptotic Theory of Statistics and Probability. Berlin, Germany: Springer, 2008.

[33] Y. Guan, “Variance stabilizing transformations of poisson, binomial and negative binomial distributions,” Statistics and Probability Letters, vol. 79, pp. 1621–1629, 2009.

[34] A. Donner and G. Y. Zou, “Estimating simultaneous confidence intervals for multiple contrasts of proportions by the method of variance estimates recovery,” Statistics in Biopharmaceutical Research, vol. 3, pp. 320–335, 2011.

[35] J. O. Berger, Statistical Decision Theory and Bayesian Analysis. Berlin, Germany: Springer- Verlag, 1985.

[36] G. Blom, “Transformations of the binomial, negative binomial, poisson and χ2 distributions,” Biometrika, vol. 41, pp. 302–316, 1954.