Confidence intervals for rainfall dispersions using the ratio of two coefficients of variation of lognormal distributions with excess zeros.

Rainfall fluctuation is directly affected by the Earth's climate change. It can be described using the coefficient of variation (CV). Similarly, the ratio of CVs can be used to compare the rainfall variation between two regions. The ratio of CVs has been widely used in statistical inference in a number of applications. Meanwhile, the confidence interval constructed with this statistic is also of interest. In this paper, confidence intervals for the ratio of two independent CVs of lognormal distributions with excess zeros using the fiducial generalized confidence interval (FGCI), Bayesian methods based on the left-invariant Jeffreys, Jeffreys rule, and uniform priors, and the Wald and Fieller log-likelihood methods are proposed. The results of a simulation study reveal that the highest posterior density (HPD) Bayesian using the Jeffreys rule prior method performed the best in terms of the coverage probability and the average length for almost all cases of small sample size and a large sample size together with a large variance and a small proportion of non-zero values. The performance of the statistic is demonstrated on two rainfall datasets from the central and southern regions in Thailand.

Introduction

The Earth’s climate is changing due to increased greenhouse gas emission from human activities, and climate change has resulted in dramatic weather events such as heatwaves, heavy rainfall, droughts, etc. Thailand is a country situated in the southeastern region of Asia that is affected by the southwest and northeast monsoons at different times of the year [1]. The country is divided into five regions: North, Northeast, Central, East, and South. Over the past three decades, Thailand has suffered from increased temperatures and fluctuating rainfall. In this study, rainfall is of interest because too much rain causes flooding and too little causes droughts at different times of the year in each area of the country. Rainfall fluctuation can be described using the CV, meaning that the ratio of CVs can be used to compare the rainfall variation between two regions. The lognormal distribution with excess zeros, a mixed distribution of discrete and continuous random variables, has been applied in many studies involving observations with zero values (such as rainfall data) [2-6]. Observations with zero values conform to a binomial distribution whereas the positive values with skewness follow a lognormal distribution. Moreover, the lognormal distribution with excess zeros has been widely used in fields such as fishery surveys [7-9], climatology [10, 11], and medicine [12, 13].

The CV is the ratio of the standard deviation to the mean that is more widely used than the standard deviation when comparing the dispersion in two or more datasets. There have been many applications of the CV to compare the dispersion in datasets. For example, Verrill and Johnson [14] compared the CVs of the xylan percentage for laboratory quality control data in the United States Department of Agriculture forest products laboratory, while Gulhar et al. [15] established confidence intervals for health-related datasets, including birth weight and cigarette smoking. Nam and Kwon [16] applied CVs to estrogen metabolites in blood and urine measurement data. Marek [17] used CVs for mining and ore geology to classify bituminous coal deposits in the Upper Silesian Coal Basin in Poland.

In statistical inference, the confidence intervals for the CV and the function of CVs have attracted the interest of several researchers. Wong and Wu [18] recommended a small sample asymptotic method to construct the confidence intervals for the CV of small sample sizes following normal, gamma, or Weibull distributions. Mahmoudvand and Hassani [19] proposed two confidence intervals for the CV for a normal distribution based on an asymptotically unbiased estimator for the CV that worked well, especially for small sample sizes. Hayter [20] applied confidence intervals (a directional bound and a non-directional upper bound) for the CV of a normal distribution applied to win probabilities. For the ratio of CVs, Nam and Kwon [16] introduced confidence intervals based on the Wald-type, Fieller-type, and log methods, and the method of variance estimate recovery (MOVER) of lognormal distributions. Hasan and Krishnamoorthy [21] suggested MOVER and the fiducial method to establish confidence intervals for small sample sizes from lognormal distributions. Wong and Jiang [22] proposed the Bartlett-corrected likelihood ratio method to construct confidence intervals for lognormal distributions. In addition, Sangnawakij and Niwitpong [23] used the score and Wald methods to construct confidence intervals for the CV and function of CVs with bounded parameter spaces in two gamma distributions for which both proposed methods were evidently suitable. They also constructed confidence intervals using MOVER, the generalized confidence interval (GCI), and the asymptotic confidence interval for CV and the difference between CVs with two parameters exponential distributions of which the GCI was satisfied [24]. Moreover, for a lognormal distribution with excess zeros, Buntao and Niwitpong [25] presented the generalized pivotal approach (GPA) and the closed-form method for obtaining the confidence intervals for the difference between CVs and reported that GPA performed well. Later, the same authors established the confidence intervals for the ratio of CVs under the GPA and MOVER methods, the results revealing that GPA was the most accurate [26]. Recently, Yosboonruang et al. [27] compared GCI and a modified Fletcher method in the confidence interval construction for CV (GCI was the most suitable). After that, they suggested FGCI, comparable with MOVER, to establish the confidence intervals for the CV of a lognormal distribution with excess zeros with three parameters [28]. Moreover, Yosboonruang et al. [29] reported a Bayesian confidence interval compared with FGCI, the results indicating that the Bayesian method performed the best.

In this paper, the ratio of CVs from two lognormal distributions with excess zeros is proposed. Therefore, the concept of FGCI [28], the Bayesian method [29], Wald- and Fieller-type methods [16] were extended to establish confidence intervals for the ratio of two independent CVs from lognormal distributions with excess zeros. The proposed methods for constructing the confidence intervals are presented in the next section. Subsequently, the results of a simulation study are reported to assess the coverage probabilities and average lengths for comparing the proposed methods. Next, application of the proposed methods to rainfall datasets from two regions in Thailand is demonstrated. Last, the paper is brought to a close with a discussion and conclusion.

Materials and methods

Let , for i = 1, 2, j = 1, 2, …, n, be a semi-continuous random sample that conforms a lognormal distribution with excess zeros with the probability of zero values δ, mean μ, and variance , denoted by . The zero observations have a binomial distribution, while the non-zero observations follow a lognormal distribution. The numbers of zero and non-zero observations are defined as n and n, respectively, where n = n + n. This leads to the distribution function of X: where δ = P(x = 0), n ∼ B(n, δ) [6], and is a lognormal cumulative distribution function [11], so ln X follows a normal distribution with mean μ and variance for X > 0. Thus, the probability density function of X can be expressed as such that if x = 0, then I0 [x] = 1 and I(0,∞) [x] = 0, and if x > 0, then I(0,∞) [x] = 1. According to Aitchison [30], the population mean and variance of X are and , respectively, where δ = 1 − δ. Thus, the CV of X can be defined as The aim here is to construct the confidence interval for the ratio of the CVs: In accordance with a lognormal distribution with excess zeros, the maximum likelihood estimators (MLEs) of parameters δ, δ, and μ are , , and , respectively. For , the unbiased estimator is . The approaches used to construct the confidence intervals are in the following subsections.

The fiducial generalized confidence interval

Fiducial inference was initially suggested by Fisher [31], whereupon Hannig [32] and Li, Zhou, and Tian [33] introduced the generalized fiducial quantities (GFQs) for δ and , which are respectively and where .

Moreover, there have been several studies using FGCI to construct confidence intervals [28, 29, 32, 34–37]. From Eqs (5) and (6), the GFQ for η is defined as The approach in this study points toward constructing the confidence interval for the ratio of CVs. Thus, the GFQ for ϕ is in the form Therefore, the 100(1 − α)% confidence interval for ϕ is where R(α/2) and R(1 − α/2) are the (α/2)-th and (1 − α/2)-th percentiles of R, respectively.

Algorithm 1

for k = 1 : M do

Generate datasets x, for i = 1, 2, j = 1, 2, …, n, from the lognormal distributions with excess zeros;

Calculate and ;

for l = 1 : m do

Generate Beta (n, n + 1) and Beta (n + 1, n);

Calculate , , and R;

end for

Calculate the (α/2)-th and (1 − α/2)-th percentiles of R

end for

Bayesian methods

The probability density function of a lognormal distribution with excess zeros (Eq (2)) has unknown parameters δ, μ and . The joint likelihood function is defined as Since we are interested in the ratio of the CVs, then the Fisher information matrix of the unknown parameters computed by the second-order derivative of the log-likelihood function can be expressed as

Subsequently, the equitailed confidence intervals and the HPD intervals are constructed for the left-invariant Jeffreys, the Jeffreys rule, and the uniform priors.

The left-invariant Jeffreys prior

Because the lognormal distribution with excess zeros is a combination of binomial and lognormal distributions, the Jeffreys priors for δ and are computed under these distributions. According to Jeffreys [38] and Ghosh et al. [39], the invariant Jeffreys prior is obtained using a Fisher information matrix (I(θ)), which is given as .

Because the left-invariant Jeffreys prior is non-informative, the Jeffreys invariant prior for a binomial proportion (δ) is given by This leads to the posterior distribution of δ as which is a beta distribution with parameters n + 1/2 and n + 1/2, denoted by δ ∣ n ∼ Beta (n + 1/2, n + 1/2). Similarly for the lognormal distribution, the left-invariant Jeffreys prior for , which is [39]. Subsequently, by combining the prior distributions of δ and , we obtain for a lognormal distribution with excess zeros. Accordingly, the joint posterior density function for a lognormal distribution with excess zeros is written as where and . Therefore, the posterior distribution of δ is a beta distribution with parameters n + 1/2 and n + 1/2, denoted by δ∣x ∼ Beta (n + 1/2, n + 1/2). Similarly, the posterior distribution of is an inverse gamma distribution with parameters (n − 1)/2 and , denoted by .

The Jeffreys rule prior

The Jeffreys rule prior, which was previously referred to as the square root of the determinant of the Fisher information for a binomial proportion, is and for from a lognormal distribution is [40]. According to the CV of the lognormal distribution with excess zeros as in Eq (3), the parameters are independent, then the Jeffreys rule prior for is . Therefore, the joint posterior density function is defined as where and . Subsequently, the posterior densities of δ and follow a beta distribution, Beta (n + 1/2, n + 3/2), and an inverse gamma distribution, , respectively.

The uniform prior

For the uniform prior, the prior probability is a constant function whereby all possible values are equally likely to be a priori [41, 42]. Accordingly, for binomial and lognormal distributions, the uniform priors of δ and are proportional to 1 [43, 44], which implies that the uniform prior for a lognormal distribution with excess zeros is . The joint posterior distribution for a lognormal distribution with excess zeros is given by where and . Thus, the posterior distribution for δ follows a beta distribution, Beta (n + 1, n + 1), and that of is an inverse gamma distribution, .

The posterior distributions of δ and can be replaced by following Eq (4), and then the equitailed confidence intervals and HPD intervals are constructed by imposing Algorithm 2.

Algorithm 2

for k = 1 : M do

Generate datasets x, for i = 1, 2, j = 1, 2, …, n, from the lognormal distributions with excess zeros;

Calculate and ;

for l = 1 : m do

Generate the posterior densities of δ ∣ x,

• left-invariant Jeffreys prior: δ∣x ∼ Beta (n + 1/2, n + 1/2),

• Jeffreys rule prior: δ∣x ∼ Beta (n + 1/2, n + 3/2),

• uniform prior: δ∣x ∼ Beta (n + 1, n + 1);

Generate the posterior densities of ,

• left-invariant Jeffreys prior: ,

• Jeffreys rule prior: ,

• uniform prior: ;

Calculate ϕ from Eq (4) for the left-invariant Jeffreys, the Jeffreys rule, and the uniform priors;

end for

Construct HPD intervals and equitailed confidence intervals for ϕ based on the left-invariant Jeffreys, the Jeffreys rule, and the uniform priors

end for

The Wald log-likelihood method

According to Eq (10), the log-likelihood function is From Eq (4), the parameter of interest is ϕ. Subsequently, the log-likelihood function is reparameterized in terms of ϕ by substituting η1 = ϕη2, , and into Eq (17) as follows

Theorem 1. Let , where i = 1, 2, j = 1, 2, …, n. Let , for X > 0. Likewise, let , where δ = 1 − δ for i = 1, 2, be the ratio of CVs of lognormal distributions with excess zeros. The unrestricted MLEs of δ, μ, and are , , and , respectively. Consequently, and . Therefore, the asymptotic variance of is

Proof. Following Nam and Kwon [16], since the MLE of is , where for i = 1, 2, then the log-likelihood function for reparameterization from Eq (17) can be written as The asymptotic variance of is obtained using the Fisher information which is also written as By the second-order partial derivative, the Fisher information elements are and the other elements are zeros. By the left-hand block of the matrix , I11, the asymptotic variance of is where η1 = ϕη2 and for i = 1, 2.

Following the MLEs of the parameters , , , for i = 1, 2, and , the variance estimate for is The asymptotically standard normal distribution is Therefore, the 100(1 − α)% two-sided confidence interval for ϕ based on the Wald log-likelihood method is where z1− is the (1 − α/2)-th percentile of the standard normal distribution.

The Fieller log-likelihood method

Following Eq (17) and since , the log-likelihood function can be written as

Theorem 2. Let , where i = 1, 2, j = 1, 2, …, n, and let , for X > 0. The CV of a lognormal distribution with excess zeros is , where δ = 1 − δ for i = 1, 2. Let , , and be the unrestricted MLEs of δ, μ, and , respectively. Likewise, . Therefore, the asymptotic variance of are for i = 1, 2.

Proof. The MLEs of the parameters are obtained from the first-order derivative of Eq (23) as , , and . Thus, , for i = 1, 2. Similarly to Theorem 1, the elements of the Fisher information matrix are for m = n = 1, 2, i = 1, 2, I for m = n = 3, 4, 5, 6 follows from Theorem 1 when and I = 0 for m, n = 1, 2, …, 6 and m ≠ n. The asymptotic variances of and are and

Since δ = 1 − δ and the MLEs of δ and are and , respectively, then the estimated variance of is where , for i = 1, 2. According to Fieller [45], the statistic Therefore, the 100(1 − α)% two-sided confidence interval for ϕ based on the Fieller log-likelihood method is where z1− is the (1 − α/2)-th percentile of a standard normal distribution.

Note that: The variance of the estimator for ratio of CVs from Theorem 1 and the variance of the estimator for CV from Theorem 2 are equal to the variances which are reported by Nam and Kwon [16] when δ1,1, δ2,1 = 1.

Algorithm 3

for k = 1 : M do

Generate datasets x, for i = 1, 2, j = 1, 2, …, n, from the lognormal distributions with excess zeros;

Calculate and ;

Calculate from Eq (20) and from Eq (25);

Construct two-sided confidence intervals for ϕ based on the Wald log-likelihood and Fieller log-likelihood methods

end for

Results and discussion

Simulation studies

Simulation studies were conducted to compare the performances of the methods used to construct the confidence intervals using FGCI, the Bayesian methods (the left-invariant Jeffreys, the Jeffreys rule, and the uniform priors), and the Wald and Fieller log-likelihood methods. The optimal method was the one with a coverage probability equal to or greater than the nominal confidence level of 0.95 together with the shortest average length. Following Wu and Hsieh [46], cases that were expected to have non-zero values of less than 10 were not considered. Sample sizes (n), δ, and were set as reported in Table 1. For all of the simulations, 15,000 runs were generated and 5,000 replicates were defined for the FGCI and Bayesian methods via Monte Carlo simulation using RStudio version 1.1.463.

Table 1

The coverage probabilities and average lengths of 95% two-sided confidence intervals for the ratio of CVs of lognormal distributions with excess zeros.

n1: n2	δ1: δ2	σ12:σ22	Coverage Probabilities (Average Lengths)
			FGCI	Equitailed			HPD			Wald	Fieller
				B-LIJ	B-JR	B-U	B-LIJ	B-JR	B-U
25:25	0.5:0.5	0.5:0.5	0.9654	0.9835	0.9799	0.9859	0.9872	0.9844	0.9893	0.9061	0.9516
			(1.3234)	(1.4643)	(1.3937)	(1.5297)	(1.3608)	(1.3037)	(1.4109)	(0.7803)	(0.8304)
		0.5:1.0	0.9586	0.9719	0.9671	0.9760	0.9725	0.9679	0.9760	0.9330	0.9475
			(1.1076)	(1.1805)	(1.1220)	(1.2369)	(1.1129)	(1.0636)	(1.1579)	(0.7528)	(0.9368)
		1.0:2.0	0.9535	0.9573	0.9489	0.9640	0.9555	0.9496	0.9628	0.9407	0.8281
			(1.7466)	(1.7645)	(1.6379)	(1.9158)	(1.4734)	(1.3975)	(1.5569)	(1.0605)	(1.1056)
		2.0:2.0	0.9514	0.9559	0.9477	0.9636	0.9573	0.9514	0.9627	0.9127	0.8597
			(10.9684)	(11.3842)	(9.5735)	(14.3922)	(6.2440)	(5.5766)	(7.1173)	(2.5565)	(9.6299)
	0.8:0.8	0.5:0.5	0.9561	0.9675	0.9631	0.9717	0.9711	0.9667	0.9754	0.9215	0.9579
			(1.1465)	(1.2155)	(1.1818)	(1.2381)	(1.1582)	(1.1287)	(1.1775)	(0.8839)	(0.9510)
		0.5:1.0	0.9555	0.9618	0.9566	0.9665	0.9623	0.9580	0.9679	0.9319	0.9654
			(0.8710)	(0.9016)	(0.8747)	(0.9254)	(0.8672)	(0.8431)	(0.8887)	(0.7100)	(0.8447)
		1.0:2.0	0.9538	0.9529	0.9477	0.9584	0.9495	0.9455	0.9545	0.9360	0.9438
			(1.1656)	(1.1758)	(1.1340)	(1.2198)	(1.0753)	(1.0432)	(1.1082)	(0.8974)	(1.8105)
		2.0:2.0	0.9501	0.9503	0.9445	0.9554	0.9511	0.9463	0.9552	0.9147	0.9723
			(4.1302)	(4.2305)	(3.9862)	(4.5007)	(3.2591)	(3.1220)	(3.4102)	(2.1424)	(3.9999)
25:50	0.5:0.5	0.5:0.5	0.9631	0.9804	0.9769	0.9821	0.9776	0.9725	0.9805	0.8478	0.8945
			(1.1354)	(1.2463)	(1.1768)	(1.3098)	(1.1396)	(1.0852)	(1.1826)	(0.6463)	(0.6637)
		0.5:1.0	0.9561	0.9735	0.9692	0.9765	0.9714	0.9653	0.9746	0.8943	0.9739
			(0.9068)	(0.9743)	(0.9210)	(1.0214)	(0.9043)	(0.8615)	(0.9387)	(0.5858)	(0.6421)
		1.0:2.0	0.9513	0.9600	0.9535	0.9651	0.9583	0.9517	0.9632	0.9133	0.9937
			(1.4721)	(1.5075)	(1.3869)	(1.6444)	(1.2475)	(1.1733)	(1.3248)	(0.8232)	(1.1562)
		2.0:2.0	0.9548	0.9577	0.9499	0.9635	0.9543	0.9473	0.9605	0.8630	0.9895
			(10.3410)	(10.0671)	(8.3466)	(12.8534)	(5.5641)	(4.9297)	(6.4182)	(2.1428)	(2.7247)
	0.8:0.8	0.5:0.5	0.9553	0.9683	0.9653	0.9718	0.9706	0.9674	0.9773	0.9011	0.9323
			(0.9915)	(1.0529)	(1.0218)	(1.0811)	(0.9942)	(0.9679)	(1.0177)	(0.7378)	(0.7615)
		0.5:1.0	0.9549	0.9660	0.9621	0.9697	0.9650	0.9602	0.9719	0.9099	0.9621
			(0.7237)	(0.7571)	(0.7351)	(0.7791)	(0.7208)	(0.7016)	(0.7395)	(0.5652)	(0.6086)
		1.0:2.0	0.9503	0.9548	0.9513	0.9586	0.9581	0.9543	0.9619	0.9227	0.9943
			(0.9815)	(0.9901)	(0.9555)	(1.0304)	(0.8970)	(0.8709)	(0.9267)	(0.7072)	(0.8692)
		2.0:2.0	0.9500	0.9530	0.9481	0.9577	0.9581	0.9534	0.9623	0.8895	0.9829
			(3.7856)	(3.7697)	(3.5480)	(4.0469)	(2.8749)	(2.7497)	(3.0279)	(1.7976)	(2.1417)
25:100	0.5:0.5	0.5:0.5	0.9592	0.9803	0.9756	0.9830	0.9736	0.9659	0.9778	0.7939	0.8203
			(1.0311)	(1.1495)	(1.0758)	(1.2122)	(1.0394)	(0.9820)	(1.0809)	(0.5785)	(0.5853)
		0.5:1.0	0.9543	0.9745	0.9689	0.9788	0.9685	0.9582	0.9719	0.8447	0.9081
			(0.7945)	(0.8644)	(0.8109)	(0.9090)	(0.7896)	(0.7474)	(0.8201)	(0.4895)	(0.5095)
		1.0:2.0	0.9526	0.9637	0.9580	0.9684	0.9611	0.9515	0.9649	0.8653	0.9762
			(1.3343)	(1.3555)	(1.2358)	(1.4897)	(1.0972)	(1.0246)	(1.1716)	(0.6752)	(0.7689)
		2.0:2.0	0.9516	0.9553	0.9462	0.9624	0.9495	0.9401	0.9568	0.8298	0.9202
			(9.5088)	(10.2704)	(8.2102)	(13.2450)	(5.3998)	(4.7107)	(6.2762)	(1.9127)	(2.1279)
	0.8:0.8	0.5:0.5	0.9548	0.9670	0.9643	0.9713	0.9675	0.9619	0.9740	0.8671	0.8899
			(0.9126)	(0.9673)	(0.9359)	(0.9988)	(0.9077)	(0.8810)	(0.9333)	(0.6615)	(0.6709)
		0.5:1.0	0.9554	0.9627	0.9597	0.9654	0.9625	0.9575	0.9697	0.8796	0.9219
			(0.6407)	(0.6715)	(0.6502)	(0.6935)	(0.6332)	(0.6151)	(0.6517)	(0.4826)	(0.4985)
		1.0:2.0	0.9507	0.9555	0.9520	0.9595	0.9584	0.9555	0.9647	0.8897	0.9595
			(0.8574)	(0.8692)	(0.8360)	(0.9081)	(0.7766)	(0.7520)	(0.8048)	(0.5893)	(0.6416)
		2.0:2.0	0.9525	0.9512	0.9471	0.9551	0.9550	0.9503	0.9611	0.8657	0.9255
			(3.5906)	(3.5590)	(3.3357)	(3.8322)	(2.6723)	(2.5466)	(2.8233)	(1.6200)	(1.7423)
50:50	0.2:0.2	0.5:0.5	0.9675	0.9885	0.9869	0.9902	0.9904	0.9889	0.9913	0.8934	0.9381
			(1.4029)	(1.5611)	(1.4427)	(1.6879)	(1.4160)	(1.3269)	(1.5007)	(0.6473)	(0.6791)
		0.5:1.0	0.9625	0.9778	0.9743	0.9803	0.9755	0.9729	0.9767	0.9166	0.9160
			(1.2525)	(1.3672)	(1.2661)	(1.4770)	(1.2576)	(1.1810)	(1.3302)	(0.7180)	(1.0312)
		1.0:2.0	0.9524	0.9617	0.9527	0.9697	0.9631	0.9555	0.9689	0.9392	0.7470
			(2.4409)	(2.4928)	(2.1804)	(2.9603)	(1.8501)	(1.7018)	(2.0355)	(1.1185)	(1.7014)
		2.0:2.0	0.9533	0.9613	0.9512	0.9697	0.9571	0.9500	0.9647	0.9063	0.8193
			(40.7663)	(36.0997)	(21.8269)	(65.5736)	(11.6103)	(9.0939)	(15.9990)	(2.8913)	(5.2774)
	0.5:0.5	0.5:0.5	0.9598	0.9823	0.9807	0.9835	0.9835	0.9827	0.9849	0.9105	0.9327
			(0.7841)	(0.8854)	(0.8689)	(0.8957)	(0.8595)	(0.8443)	(0.8689)	(0.5544)	(0.5713)
		0.5:1.0	0.9549	0.9679	0.9648	0.9701	0.9689	0.9671	0.9715	0.9281	0.9475
			(0.6816)	(0.7365)	(0.7209)	(0.7477)	(0.7210)	(0.7062)	(0.7316)	(0.5435)	(0.6019)
		1.0:2.0	0.9508	0.9583	0.9545	0.9616	0.9589	0.9546	0.9619	0.9427	0.9540
			(0.9225)	(0.9453)	(0.9228)	(0.9676)	(0.8966)	(0.8774)	(0.9154)	(0.7578)	(1.0524)
		2.0:2.0	0.9500	0.9528	0.9483	0.9561	0.9556	0.9521	0.9587	0.9277	0.9967
			(2.7610)	(2.8132)	(2.7145)	(2.9196)	(2.3746)	(2.3092)	(2.4431)	(1.7547)	(2.8522)
	0.8:0.8	0.5:0.5	0.9563	0.9671	0.9643	0.9697	0.9677	0.9658	0.9701	0.9279	0.9477
			(0.7258)	(0.7740)	(0.7646)	(0.7788)	(0.7565)	(0.7477)	(0.7610)	(0.6165)	(0.6389)
		0.5:1.0	0.9532	0.9570	0.9547	0.9606	0.9585	0.9565	0.9615	0.9365	0.9583
			(0.5606)	(0.5818)	(0.5743)	(0.5881)	(0.5715)	(0.5644)	(0.5777)	(0.4986)	(0.5417)
		1.0:2.0	0.9529	0.9500	0.9467	0.9524	0.9503	0.9482	0.9537	0.9443	0.9637
			(0.7013)	(0.7099)	(0.7000)	(0.7197)	(0.6869)	(0.6779)	(0.6960)	(0.6267)	(0.7872)
		2.0:2.0	0.9501	0.9502	0.9473	0.9525	0.9549	0.9532	0.9572	0.9347	0.9973
			(1.8696)	(1.8701)	(1.8372)	(1.9029)	(1.7035)	(1.6779)	(1.7292)	(1.4471)	(1.7632)
50:100	0.2:0.2	0.5:0.5	0.9658	0.9882	0.9855	0.9891	0.9873	0.9830	0.9879	0.8433	0.8855
			(1.2156)	(1.3690)	(1.2439)	(1.4802)	(1.2081)	(1.1179)	(1.2694)	(0.5533)	(0.5645)
		0.5:1.0	0.9630	0.9821	0.9784	0.9839	0.9817	0.9752	0.9824	0.9113	0.9712
			(1.0392)	(1.1399)	(1.0410)	(1.2270)	(1.0282)	(0.9547)	(1.0788)	(0.5707)	(0.6263)
		1.0:2.0	0.9549	0.9655	0.9599	0.9703	0.9662	0.9575	0.9695	0.9301	0.9753
			(2.0945)	(2.2044)	(1.8812)	(2.6662)	(1.6118)	(1.4499)	(1.7906)	(0.8727)	(0.5866)
		2.0:2.0	0.9538	0.9571	0.9485	0.9642	0.9548	0.9444	0.9610	0.8528	0.9797
			(67.8516)	(61.5960)	(34.1736)	(164.6780)	(13.8966)	(10.3774)	(22.4465)	(2.3299)	(3.0886)
	0.5:0.5	0.5:0.5	0.9635	0.9821	0.9806	0.9829	0.9805	0.9774	0.9819	0.8793	0.9045
			(0.6718)	(0.7570)	(0.7399)	(0.7661)	(0.7309)	(0.7154)	(0.7383)	(0.4717)	(0.4779)
		0.5:1.0	0.9559	0.9739	0.9723	0.9752	0.9715	0.9679	0.9737	0.9097	0.9511
			(0.5556)	(0.6052)	(0.5918)	(0.6130)	(0.5882)	(0.5758)	(0.5951)	(0.4297)	(0.4496)
		1.0:2.0	0.9505	0.9586	0.9567	0.9611	0.9617	0.9573	0.9643	0.9262	0.9882
			(0.7576)	(0.7811)	(0.7609)	(0.7979)	(0.7341)	(0.7172)	(0.7473)	(0.5904)	(0.6832)
		2.0:2.0	0.9503	0.9529	0.9502	0.9563	0.9583	0.9540	0.9609	0.9050	0.9792
			(2.5209)	(2.5261)	(2.4280)	(2.6227)	(2.0993)	(2.0352)	(2.1614)	(1.4996)	(1.6864)
	0.8:0.8	0.5:0.5	0.9535	0.9659	0.9639	0.9668	0.9657	0.9630	0.9684	0.9121	0.9310
			(0.6263)	(0.6669)	(0.6586)	(0.6734)	(0.6496)	(0.6418)	(0.6555)	(0.5218)	(0.5300)
		0.5:1.0	0.9529	0.9593	0.9575	0.9616	0.9575	0.9550	0.9609	0.9172	0.9489
			(0.4624)	(0.4817)	(0.4756)	(0.4872)	(0.4710)	(0.4654)	(0.4763)	(0.3999)	(0.4148)
		1.0:2.0	0.9531	0.9518	0.9495	0.9543	0.9517	0.9489	0.9546	0.9286	0.9765
			(0.5747)	(0.5844)	(0.5767)	(0.5925)	(0.5615)	(0.5547)	(0.5688)	(0.4967)	(0.5467)
		2.0:2.0	0.9529	0.9518	0.9485	0.9531	0.9599	0.9575	0.9618	0.9165	0.9705
			(1.6544)	(1.6684)	(1.6388)	(1.7006)	(1.4996)	(1.4773)	(1.5243)	(1.2334)	(1.3380)
100:100	0.2:0.2	0.5:0.5	0.9637	0.9885	0.9875	0.9888	0.9908	0.9904	0.9917	0.8920	0.9182
			(0.7637)	(0.8899)	(0.8673)	(0.9033)	(0.8618)	(0.8411)	(0.8735)	(0.4694)	(0.4801)
		0.5:1.0	0.9616	0.9755	0.9732	0.9781	0.9785	0.9772	0.9804	0.9227	0.9277
			(0.7215)	(0.8006)	(0.7793)	(0.8150)	(0.7834)	(0.7633)	(0.7971)	(0.5276)	(0.5839)
		1.0:2.0	0.9513	0.9607	0.9556	0.9641	0.9615	0.9579	0.9639	0.9433	0.9315
			(1.0450)	(1.0894)	(1.0542)	(1.1220)	(1.0187)	(0.9904)	(1.0444)	(0.8018)	(1.3385)
		2.0:2.0	0.9522	0.9589	0.9543	0.9627	0.9599	0.9571	0.9631	0.9270	0.9835
			(3.4830)	(3.5748)	(3.3877)	(3.7837)	(2.8333)	(2.7227)	(2.9520)	(1.8358)	(2.4826)
	0.5:0.5	0.5:0.5	0.9562	0.9829	0.9820	0.9836	0.9831	0.9820	0.9835	0.9056	0.9177
			(0.5149)	(0.5854)	(0.5805)	(0.5878)	(0.5765)	(0.5719)	(0.5788)	(0.3961)	(0.4021)
		0.5:1.0	0.9543	0.9725	0.9710	0.9739	0.9706	0.9701	0.9723	0.9307	0.9420
			(0.4555)	(0.4926)	(0.4879)	(0.4958)	(0.4873)	(0.4826)	(0.4904)	(0.3895)	(0.4093)
		1.0:2.0	0.9513	0.9558	0.9546	0.9588	0.9581	0.9555	0.9603	0.9457	0.9545
			(0.5917)	(0.6083)	(0.6020)	(0.6142)	(0.5957)	(0.5897)	(0.6013)	(0.5379)	(0.6305)
		2.0:2.0	0.9523	0.9529	0.9510	0.9550	0.9590	0.9585	0.9622	0.9387	0.9918
			(1.4645)	(1.4979)	(1.4795)	(1.5164)	(1.4009)	(1.3850)	(1.4158)	(1.1982)	(1.3712)
	0.8:0.8	0.5:0.5	0.9520	0.9681	0.9674	0.9699	0.9682	0.9667	0.9688	0.9345	0.9434
			(0.4892)	(0.5242)	(0.5212)	(0.5255)	(0.5174)	(0.5146)	(0.5188)	(0.4333)	(0.4410)
		0.5:1.0	0.9507	0.9599	0.9582	0.9603	0.9601	0.9579	0.9611	0.9379	0.9499
			(0.3813)	(0.3973)	(0.3948)	(0.3992)	(0.3932)	(0.3908)	(0.3951)	(0.3531)	(0.3677)
		1.0:2.0	0.9512	0.9498	0.9492	0.9518	0.9501	0.9487	0.9522	0.9437	0.9623
			(0.4669)	(0.4715)	(0.4685)	(0.4746)	(0.4642)	(0.4612)	(0.4671)	(0.4406)	(0.4900)
		2.0:2.0	0.9510	0.9505	0.9485	0.9524	0.9554	0.9527	0.9563	0.9415	0.9786
			(1.1197)	(1.1283)	(1.1201)	(1.1361)	(1.0821)	(1.0748)	(1.0892)	(0.9968)	(1.0932)

Table 1 and Figs 1–3 present the coverage probabilities and average lengths of the confidence intervals for the various methods. The results show that the coverage probabilities of FGCI were consistently close to the nominal confidence level of 0.95 for all cases. The coverage probabilities of the Bayesian method using the uniform prior (B-U) for both the equitailed confidence interval and HPD interval were greater than or close to the nominal confidence level of 0.95 for all cases. The Bayesian methods using the left-invariant Jeffreys (B-LIJ) and the Jeffreys rule (B-JR) priors based on equitailed confidence intervals and HPD intervals attained coverage probabilities greater than or close to the nominal confidence level of 0.95 in almost every case. However, those attained by the Wald log-likelihood method were less than the nominal confidence level 0.95 for all cases whereas those produced by the Fieller log-likelihood method were greater than or close to the nominal confidence level 0.95 in some cases.

Fig 1

Line graphs for comparing the coverage probabilities (CP) and average lengths (AL) of all methods in cases of the different sample sizes.

Fig 3

Line graphs for comparing the coverage probabilities (CP) and average lengths (AL) of all methods in cases of the different variances.

The average lengths of B-JR based on the HPD interval was the shortest for most of the cases when the sample size was small (n1 and/or n2 = 25). For large sample sizes (n1, n2 = 50, 100), B-JR based on the HPD interval had mainly narrow average lengths for the cases of δ1,1, δ2,1 = 0.2 for all variances and δ1,1, δ2,1 = 0.5, 0.8 together with , , while those of FGCI were the shortest for cases with small variance(s) ( and/or ).

An empirical example

As previously mentioned, the CV can be used to measure the dispersion in a dataset, especially in cases like rainfall data that conform to a lognormal distribution with excess zeros. Therefore, daily rainfall data from the central and southern regions (Chumphon province) in August 2017, collected by the Central and Southern Region Irrigation Hydrology Center were used to construct confidence intervals for evaluating the proposed methods. The datasets were shown in Tables 2 and 3. The data with zero values implied a binomial distribution whereas the positive observations for both regions were skewed, as shown in Fig 4, and so log transformation of the positive values was applied. It is imperative to check the distribution of the data, and so a minimum Akaike information criteria (AIC) analysis was conducted. The AIC values of the rainfall data from the central and southern regions for normal, lognormal, and Cauchy distributions were 1491.7130, 1251.7700, and 1390.5430 and 952.6130, 840.1782, and 919.0321, respectively. Thus, the lognormal distribution was suitable for both datasets. This was further confirmed using normal Q-Q plots (Fig 5). The summary statistics for the rainfall data from the central and southern regions were n1 = 390, , , , and n2 = 248, , , , , respectively. The ratio of CVs between two regions was ϕ = 1.2066, and the 95% confidence intervals for ϕ are reported in Table 4.

Table 2

Daily rainfall data from central region on August, 2017.

Date	Stations
	N.67	C.2	C.13	C.30	Ct.4	Ct.5A	Ct.7	Ct.9	Ct.2A	S.9	S.13	S.28	T.7
1	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	6.4	0.0	0.0
2	0.0	0.0	0.0	9.2	4.2	0.0	0.0	0.0	17.0	0.0	9.6	2.2	0.0
3	0.0	0.0	0.0	1.1	2.6	0.0	0.0	0.0	0.0	0.0	1.5	0.0	0.0
4	8.2	0.0	0.0	0.0	0.6	0.0	28.5	6.1	1.4	26.2	0.0	0.0	0.0
5	3.9	4.4	36.1	0.0	0.4	0.0	10.0	3.0	0.0	56.3	15.2	8.6	0.0
6	0.0	0.0	2.2	0.0	0.0	10.5	0.0	0.0	1.8	4.1	3.2	4.5	2.9
7	0.0	0.0	0.4	13.5	0.0	19.2	1.0	10.2	4.1	1.5	2.5	1.3	27.9
8	4.0	19.1	0.0	0.0	0.0	7.8	0.0	0.0	3.1	7.8	0.0	0.0	0.0
9	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
10	11.0	0.0	17.4	0.0	3.0	0.0	0.0	0.0	0.0	8.0	0.0	0.0	0.0
11	0.0	0.0	0.0	3.8	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
12	0.0	0.0	9.5	0.0	18.8	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
13	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	20.0
14	1.3	0.9	0.0	0.0	0.0	0.0	13.5	10.8	0.0	0.0	0.0	0.0	1.1
15	44.2	9.3	10.8	3.4	70.0	0.0	8.3	3.0	52.0	9.3	0.0	0.0	0.0
16	12.0	6.4	26.6	22.2	54.5	54.8	11.5	25.4	9.2	12.3	5.7	2.2	12.4
17	2.6	0.0	9.5	12.5	15.3	65.6	18.5	2.0	9.4	0.0	2.6	12.8	16.2
18	1.0	11.8	12.3	21.6	12.9	14.3	4.0	46.7	0.8	6.2	18.8	6.8	2.2
19	0.0	0.6	44.8	5.1	0.0	0.0	36.5	17.0	14.7	2.1	0.0	26.7	9.5
20	67.8	17.9	3.8	7.6	0.2	17.2	5.5	8.0	9.9	0.0	20.5	0.5	40.0
21	0.0	0.8	0.0	15.4	0.2	6.4	0.0	14.2	0.0	0.0	0.0	0.0	0.0
22	15.0	0.0	0.0	8.5	3.1	0.0	2.2	5.4	0.0	0.0	0.0	0.0	0.0
23	0.0	0.0	0.0	13.9	0.0	8.3	1.5	0.0	0.0	10.0	0.0	0.0	10.5
24	38.5	2.3	24.8	6.5	8.0	5.4	0.0	0.8	30.0	6.2	6.4	0.0	0.0
25	13.0	7.6	0.0	9.3	10.0	10.2	2.0	3.4	0.0	0.0	3.3	0.0	0.0
26	0.0	5.1	0.0	1.9	0.3	7.4	0.0	44.7	0.0	13.6	26.8	26.3	3.9
27	24.7	55.7	12.7	18.2	30.0	0.0	35.0	22.1	13.4	13.0	62.5	4.6	89.9
28	11.0	14.7	0.0	0.0	53.6	10.2	14.0	1.0	7.0	10.6	0.0	2.5	4.9
29	4.8	2.2	0.0	0.0	4.3	15.4	0.0	0.0	21.1	0.0	3.1	0.0	0.0
30	4.4	24.8	0.0	0.0	13.6	0.0	8.0	16.5	19.1	7.3	2.4	5.2	41.9
31	0.0	0.0	0.0	12.7	1.3	0.0	2.0	3.8	0.0	0.0	0.0	0.0	7.2

Source: Central region irrigation hydrology center (http://hydro-5.rid.go.th/)

Table 3

Daily rainfall data from southern region on August, 2017.

Date	Stations
	100191	100251	100261	100271	100281	100291	100301	100311
1	2.0	0.0	4.7	13.6	2.3	2.3	14.7	11.3
2	0.0	0.0	4.8	2.2	3.9	3.9	3.2	0.0
3	6.0	0.0	3.4	4.1	0.0	0.0	8.0	2.0
4	5.0	0.0	2.9	0.0	0.0	0.0	40.8	5.0
5	0.0	0.0	5.3	0.0	0.0	0.0	0.0	0.0
6	0.0	0.0	2.6	0.0	0.0	0.0	13.9	2.1
7	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
8	0.0	0.0	30.0	0.0	0.0	0.0	7.2	1.3
9	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
10	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
11	0.0	0.0	0.0	0.0	0.0	0.0	8.6	0.0
12	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
13	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
14	63.0	10.5	3.6	6.1	42.2	42.2	18.6	10.6
15	0.0	7.0	16.1	22.6	18.3	18.3	1.6	0.0
16	3.0	0.0	17.0	7.4	5.2	5.2	1.8	0.0
17	25.0	5.0	9.0	21.6	17.3	17.3	32.7	27.0
18	16.0	6.0	7.5	16.2	11.3	11.3	20.2	12.8
19	2.0	0.4	7.0	14.0	3.7	3.7	5.0	1.5
20	9.0	0.0	3.1	6.3	8.4	8.4	11.2	1.7
21	0.0	0.0	10.6	3.6	0.0	0.0	0.0	0.0
22	0.0	4.0	6.0	12.2	4.4	4.4	0.3	0.0
23	6.0	0.7	16.9	25.7	0.0	0.0	6.6	1.5
24	0.0	0.8	0.9	24.4	0.0	0.0	0.0	2.1
25	8.0	0.4	19.1	27.1	0.0	0.0	8.0	6.0
26	6.0	0.0	9.2	26.4	0.0	0.0	9.2	3.0
27	1.0	0.5	6.2	13.2	0.0	0.0	3.9	1.2
28	8.0	1.0	27.8	18.7	4.5	4.5	10.0	8.5
29	0.0	1.2	0.0	0.0	0.0	0.0	0.0	1.2
30	0.0	0.0	0.0	0.0	0.0	0.0	1.4	0.0
31	0.0	0.0	12.3	0.0	0.0	0.0	0.0	0.0

Source: Southern region irrigation hydrology center (http://hydro-8.com/main/Submenu/3-RAIN/3-RAIN-02.html)

Fig 4

Density of rainfall data sets from central and southern regions in Thailand.

Fig 5

Normal Q-Q plot of log-transformed for rainfall data sets from central and southern regions in Thailand.

Table 4

The 95% two-sided confidence intervals for the ratio of CVs of daily rainfall data between central and southern region in August, 2017.

Methods	Confidence intervals
	Lower	Upper	Length
FGCI	1.0540	1.8070	0.7530
B-LIJ (Equitailed)	1.0416	1.8182	0.7766
B-JR (Equitailed)	1.0514	1.7957	0.7443
B-U (Equitailed)	1.0420	1.8296	0.7876
B-LIJ (HPD)	0.9905	1.7545	0.7640
B-JR (HPD)	1.0119	1.7532	0.7412
B-U (HPD)	1.0297	1.8060	0.7762
Wald	1.0316	1.7332	0.7016
Fieller	1.0659	1.7831	0.7172

The lower and upper bounds from the results indicate that the dispersion of rainfall in the central region was more than the southern region. This is because the southern region has abundant precipitation throughout the year due to being located on the peninsula surrounded by the Andaman Sea and the Gulf of Thailand. The central region is located on the plains that cause irregular precipitation, thus the dispersion of the rainfall data is larger than in the southern region.

Conclusion

FGCI, Bayesian methods based on the left-invariant Jeffreys, Jeffreys rule, and uniform priors, and the Wald and Fieller log-likelihood methods were used to construct the confidence intervals for the ratio of CVs of lognormal distributions with excess zeros. Coverage probabilities and the average lengths were used to evaluate the performance of the proposed methods.

The simulation results indicate that the coverage probabilities for all cases of the FGCI and Bayesian methods using the uniform prior and almost all cases of the Bayesian method using the left-invariant Jeffreys and Jeffreys rule priors were close to or greater than the target. However, when considering the average lengths, the Bayesian method using the Jeffreys rule prior based on the HPD interval produced the shortest ones in cases of small sample sizes and a large sample size together with a small expected number of non-zero observations and a large variance, while FGCI was optimal for the other cases. Therefore, the HPD Bayesian method using the Jeffreys rule prior and the FGCI method are suitable for constructing confidence intervals for the ratio of CVs of lognormal distributions with excess zeros.

Nam and Kwon [16] introduced the Wald-type and Fieller-type methods for the ratio of CVs of lognormal distributions that were appropriate for medium sample sizes. In the present study, this method was extended for a lognormal distribution with zero-inflated observations. However, the coverage probabilities of the Wald log-likelihood method were less than the target for all cases whereas those of the Fieller log-likelihood method were greater than the target for a few cases when the probability of non-zero values was more than half and for a large variance. Moreover, the average lengths of these methods were wider than the FGCI and Bayesian methods. Hence, the Wald and Fieller log-likelihood methods are not recommended for constructing confidence intervals for the ratio of CVs of lognormal distributions with excess zeros.

Furthermore, the confidence intervals evaluation in the empirical study is coincidental with the simulation results.

19 Jan 2022

A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'.

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PLOS ONE

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Reviewer #1: No

Reviewer #2: Partly

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2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: Yes

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Reviewer #1: Yes

Reviewer #2: Yes

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Reviewer #1: Yes

Reviewer #2: Yes

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5. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: The authors consider Fiducial, Bayesian, Frequentist interval estimations for rainfall dispersions using the ratio of two coefficients of variation of lognormal distributions with excess zeros. The topic and application are interesting. I have a few comments given below.

1. The paper considered Fiducial, Bayesian, and Frequentist methods. Why is the title about Bayesian interval estimations?

2. Besides the implication from simulations, how would one choose prior for Bayesian methods particularly for this rainfall application? Empirical Bayes might be another option.

3. Is it reasonable to assume that non-zero $X$ follows lognormal? Nonparametric fiducial approach recently drew some attention.

4. Real data application, I am not convinced by the statement that the HPD Bayesian using the Jeffreys rule prior is the best.

The shortest interval does not mean the best as the ground truth is not known, right?

Reviewer #2: In this paper, some approaches have been proposed for the ratio of coefficients of variation in the log-normal distribution with excess zeros. The paper can e accepted for publication in Plos One. However, the English language must be improved.

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Reviewer #1: No

Reviewer #2: No

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6 Feb 2022

Response to the Academic Editor and Reviewers

Journal: PLOS ONE

Manuscript ID: PONE-D-21-38266

Title name: Bayesian interval estimations for rainfall dispersions using the ratio of two coefficients of variation of lognormal distributions with excess zeros

Authors: Noppadon Yosboonruang, Sa-Aat Niwitpong, Suparat Niwitpong

Dear Academic Editor and Reviewers,

We are very grateful for your constructive comments to our manuscript. The manuscript was revised in accordance with your valuable suggestions. Below are our point-by-point replies to your comments.

Lists of correction:

Reviewer 1

1. The paper considered Fiducial, Bayesian, and Frequentist methods. Why is the title about Bayesian interval estimations?

Response:

We defined the topic as Bayesian interval estimations because it is a method that can apply to various problems of interest to the researchers. The prior distribution can determine as appropriate with the distribution of data. In addition, the results obtained from this study indicate that Bayesian is the best method when compared with the other methods. To avoid doubt, we changed the title to “Confidence intervals for rainfall dispersions using the ratio of two coefficients of variation of lognormal distributions with excess zeros”.

2. Besides the implication from simulations, how would one choose prior for Bayesian methods particularly for this rainfall application? Empirical Bayes might be another option.

Response:

According to testing the distribution of rainfall data, there was a lognormal distribution with excess zeros. Therefore, we choose the prior suitability for Bayesian by considering the values of a random variable of their posterior distributions that correspond to those for a lognormal distribution with excess zeros, see, for example, Maneerat et al. (2021a, 2021b) and Yosboonruang et al. (2021). We will find new confidence intervals for your valuable comments using the Empirical Bayes in the next study.

3. Is it reasonable to assume that non-zero follows lognormal? Nonparametric fiducial approach recently drew some attention.

Response:

Yes, the reason to assume the distribution of non-zero follows lognormal either from the normal Q-Q plot or the minimum AIC and BIC. For example, rainfall data has a right-skewed distribution that may correspond with various distributions; therefore, consider the distribution of such data from the normal Q-Q plot or the minimum AIC and BIC, based on an empirical example in the manuscript. The following papers show that the delta-lognormal distribution fits the rainfall data, see for example:

Yosboonruang et al. (2021) used the minimum AIC and the lowest BIC to test the fitting of the distributions to the daily rainfall data. Then, they confirmed it with the normal Q-Q plots to show the distributions of such data.

Maneerat et al. (2021b) checked a fit of distribution of rainfall data by a minimum AIC and showed the distribution of the datasets by histogram plots and normal Q-Q plots.

We will study the value of your suggestion for the nonparametric fiducial approach the next time.

4. Real data application, I am not convinced by the statement that the HPD Bayesian using the Jeffreys rule prior is the best. The shortest interval does not mean the best as the ground truth is not known, right?

Response:

Thank you for your recommendation, we agree with your opinion. Therefore, we deleted the sentence “The results reveal that the Bayesian method using the Jeffreys rule prior based on the HPD interval had the shortest interval length from the simulation results for large sample sizes.” in the summary of an empirical example in a revised manuscript. Likewise, we adjusted the last sentence in the conclusions section on page 13/20 in a revised manuscript as “Furthermore, the confidence intervals evaluation in the empirical study is coincidental with the simulation results.”.

References

Maneerat P, Niwitpong S, Niwitpong S. Bayesian confidence intervals for a single mean and the difference between two means of delta-lognormal distributions. Commun Stat-Simul C. 2021a;50(10):2906–2934.

Maneerat P, Niwitpong S, Niwitpong S. Simultaneous confidence intervals for all pairwise comparisons of the means of delta-lognormal distributions with application to rainfall data. PLoS ONE. 2021b;16(7):e0253935.

Yosboonruang N, Niwitpong S, Niwitpong S. Simultaneous confidence intervals for all pairwise differences between the coefficients of variation of rainfall series in Thailand. PeerJ. 2021;9:e11651.

Reviewer 2

In this paper, some approaches have been proposed for the ratio of coefficients of variation in the log-normal distribution with excess zeros. The paper can be accepted for publication in PLOS ONE. However, the English language must be improved.

Response:

Thank you for reviewing the manuscript. The manuscript has been improved by native.

Best regards,

Noppadon Yosboonruang, Sa-Aat Niwitpong, and Suparat Niwitpong

The authors

10 Mar 2022

Confidence intervals for rainfall dispersions using the ratio of two coefficients of variation of lognormal distributions with excess zeros

PONE-D-21-38266R1

Dear Dr. Niwitpong,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication.

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If your institution or institutions have a press office, please notify them about your upcoming paper to help maximize its impact. If they’ll be preparing press materials, please inform our press team as soon as possible -- no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org.

Kind regards,

Bryan C Daniels

Academic Editor

PLOS ONE

Additional Editor Comments (optional):

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #1: All comments have been addressed

Reviewer #2: All comments have been addressed

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2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Partly

Reviewer #2: Partly

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: Yes

**********

4. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: Yes

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #2: Yes

**********

6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: All comments from the last round were addressed. No further comments given in this round.

Reviewer #2: (No Response)

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7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

Reviewer #2: No

14 Mar 2022

PONE-D-21-38266R1

Confidence intervals for rainfall dispersions using the ratio of two coefficients of variation of lognormal distributions with excess zeros

Dear Dr. Niwitpong:

I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department.

If your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact onepress@plos.org.

If we can help with anything else, please email us at plosone@plos.org.

Thank you for submitting your work to PLOS ONE and supporting open access.

Kind regards,

PLOS ONE Editorial Office Staff

on behalf of

Dr. Bryan C Daniels

Academic Editor

PLOS ONE