Simulation comparison of modified confidence intervals based on robust estimators for coefficient of variation: skewed distributions case with real applications

Abstract

In this article, we propose confidence intervals for the population coefficient of variation based on some robust estimators such as trimmed mean, winsorized mean, Hodges-Lehmann estimator and trimean. The proposed confidence intervals and their bootstrap versions were compared with the existing confidence intervals for the population coefficient of variation. The performances of the proposed confidence intervals were evaluated via a Monte-Carlo simulation study by considering the coverage probability, average width, standard deviation of widths, and coefficient of variation of widths as comparison criteria. The proposed confidence intervals performed well in terms of coverage probability on symmetrical distributions. In the case of skewed distributions, they were closer to the nominal confidence level and had narrower widths than the others. As a result, we proposed to use confidence intervals based on the Hodges-Lehmann estimator and winsorized mean for small sample sizes ($n=10\text{and}15$) and larger sample sizes for skewed distributions, respectively. The real-life datasets were analyzed to support the simulation results and confirm the practical applications of the proposed confidence intervals.

Keywords:

• Average width
• bootstrap
• coefficient of variation
• coverage probability
• Hodges-Lehmann estimator
• trimean
• trimmed mean
• winsorized mean

1. Introduction

The coefficient of variation (CV) is a statistical tool that is utilized extensively across various study subfields to measure the dispersion for a variable of interest. It is a unit-free measure, therefore, it can be used to compare the homogeneity of multiple variables scaled differently. This is a significant advantage of the CV over the variance or standard deviation, since when the latter are used to report variation, all variables must have the same means and be expressed in the same units. In such cases, it would be correct to use the population CV since it is not dependent on the measuring unit. The population CV is equal to ratio of the population standard deviation (σ) to the population mean $\left(\mu ,\mu \ne 0\right).$ The greater the CV value, the greater the dispersion, and the lower the value, the lower the risk. The lowest value indicates the highest level of safety (Banik & Kibria, 2011). It is commonly used in many fields, such as engineering, atmospheric, medical sciences and environmental data.

Confidence intervals or hypothesis testing can be used to inference about the CV of a population whose distribution is unknown. While many researchers have studied constructing confidence intervals based on normality assumption for the population CV (Miller, 1991; Panichkitkosolkul, 2015), others proposed confidence intervals based on the skewed distributions (Curto & Pinto, 2009). Thangjai, Niwitpong, and Niwitpong (2020) obtained novel approaches to constructing confidence intervals for the common CV of several normal populations. Amiri and Zwanzig (2010) stated that the shape of the distribution should be known to make inferences about the CV.

Most of the methods in the literature are generally based on parametric models. In addition, bootstrap methods can be considered alternative methods. On the other hand, extremely skewed distributions are common in real-life applications, as well. Niwitpong (2013) presented the confidence intervals for the population CV under log-normal distribution and two theorems were proved for the approximated coverage probability and the expected length of confidence interval. Sangnawakij, Niwitpong, and Niwitpong (2015) proposed confidence intervals for the ratio of the CVs of Gamma distributions. They used the method of variance of estimates recovery with the methods of Score and Wald intervals and compared the coverage probability and interval width via a Monte Carlo simulation. On the other hand, the literature provides confidence intervals for the CV ratio and the difference between two lognormal values (Nam & Kwon, 2017; Niwitpong, Niwitpong, & Thangjai, 2019). Consulin et al. (2018) evaluated the performance of some CV estimators via Monte Carlo simulation and they highly recommended the use of the CV as a measure of relative dispersion through Ranked set sampling. Abu-Shawiesh, Akyüz, and Kibria (2019) developed three confidence intervals based on variance for the CV of both symmetric and skewed distributions. They observed that the proposed augmented-large-sample (AA&K-ALS) confidence interval performed well in terms of coverage probability in all cases, while large-sample (A&A-LS) and adjusted degrees of freedom (AA&K-ADJ) confidence intervals had much lower coverage probability than the nominal level for skewed distributions. Forkman (2009) conducted statistically inferences for the estimation of CV in normal distribution. This article discussed inference for the coefficient of variation when there was reason to believe that the data is normally distributed, but not lognormally distributed. La-Ongkaew, Niwitpong, and Niwitpong (2022) proposed the confidence intervals for the ratio of coefficients of variation of two Weibull distributions and these intervals have been used on wind speed data, whereas Puggard, Niwitpong, and Niwitpong (2022) compared some interval methods to estimate the population CV of several Birnbaum–Saunders distributions. Chankham, Niwitpong, and Niwitpong (2022) analyzed an air pollution indicator using confidence intervals for the CV of inverse Gaussian distribution.

Recently, sampling plans based on CV have received increasing attention in the literature from many authors due to their importance in measuring product quality. Rao, Aslam, Sherwani, Shehzad, and Jun (2022) obtained a generalized multiple dependent state (GMDS) sampling plan based on population CV and compared with the existing sampling plans. The study’s results indicate that GMDS sampling plan based on population CV provides the desired level of protection with a smaller sample size requirement, making it more cost-effective than existing plans. Albatineh, Kibria, Wilcox, and Zogheib (2014) evaluated the performance of some confidence intervals for the population CV using ranked set sampling and simulated data that are generated fromnormal, skew normal, Gamma, log-normal, and Weibull distributions. The simulation results indicate that the confidence intervals based on ranked set sampling outperformed in terms of coverage probability and interval width. On the other hand, Shahzad et al. (2023) developed some new estimators for the estimation of CV under double stratified random sampling. It was expressed that the traditional estimators of the population CV are based on conventional moments; and also, these are highly affected by the presence of extreme values. The proposed method has shown that estimators can be used in the presence of extreme observations.

Evaluating all the results together, we take into account the CV parameter, which is widely used in describing variation within a data set, is more informative than others among scale parameters, and is preferred over variance or standard deviation in many fields (Banik & Kibria, 2011).

When both standard deviation and mean of the population are unknown, the CV may be considered to obtain the interval estimation of the CV. Studies on the confidence interval for CV in such cases are limited. Considering all these, this study aims to obtain new confidence intervals for the population CV by modifying the robust confidence interval proposed by Abu-Shawiesh, Banik, and Kibria (2011). The performance of confidence intervals and their bootstrap versions is compared with some known confidence intervals in terms of coverage probability (CP), average width (AW), standard deviation of width (SDW), and coefficient of variation of width (CVW).

This study is organised as follows: In Section 2, the statistical methodology for deriving the confidence intervals for the population CV is given. That section also includes a review for some existing confidence intervals for a population CV and an explanation for a robust confidence interval for the population standard deviation ($\sigma$). It also includes a derivation of the modified confidence intervals for the population CV and the bootstrap versions of the proposed confidence intervals. In Section 3, simulations are used to evaluate the performance of the proposed confidence intervals in terms of CP, AW, SDW, and CVW. Section 4 illustrates the use of these confidence intervals using three real-life dataset examples. Finally, Section 5 reports our conclusions.

2. Statistical methodology

Consider the independent and identically distributed random sample X₁, X₂, …, X_n from a distribution with a finite mean μ and variance σ². We want to determine the (1 − α)100% confidence interval for the population CV under the probability of type I error (α). The population CV can be estimated by $\widehat{\mathrm{\text{CV}}},$ the sample CV, defined as $\widehat{\mathit{\text{CV}}}=S/\overline{X}$ where S and $\overline{X}$ are sample standard deviation and sample mean, respectively.

2.1. Some selected confidence intervals for the population coefficient of variation

In this section, some confidence intervals for the population CV that are important and useful are reviewed.

2.1.1. Hendricks and Robey confidence interval (denoted by HR)

Hendricks and Robey (1936) proposed a confidence interval as in Eq. (2.1)(2.1) $${\mathit{\text{CI}}}_{\mathit{\text{HR}}}=\left(\widehat{\mathit{\text{CV}}}-t_{\left(n-1,\alpha /2\right)}{S}_{\widehat{\mathit{\text{CV}}}},\widehat{\mathit{\text{CV}}}+t_{\left(n-1,\alpha /2\right)}{S}_{\widehat{\mathit{\text{CV}}}}\right)$$(2.1) . (2.1) $${\mathit{\text{CI}}}_{\mathit{\text{HR}}}=\left(\widehat{\mathit{\text{CV}}}-t_{\left(n-1,\alpha /2\right)}{S}_{\widehat{\mathit{\text{CV}}}},\widehat{\mathit{\text{CV}}}+t_{\left(n-1,\alpha /2\right)}{S}_{\widehat{\mathit{\text{CV}}}}\right)$$(2.1) where $t_{(n-1,\alpha /2)}$ is $100\left(1-\alpha /2\right)\mathit{\text{th}}$ percentile of the student t-distribution with degrees of freedom (n-1) and $S_{\widehat{\mathit{\text{CV}}}}=\widehat{\mathit{\text{CV}}}/\sqrt{2n}$ is the standard error of $\widehat{\mathit{\text{CV}}}.$ This confidence interval is very sensitive to minor violations of the assumption of normality.

2.1.2. Miller confidence interval (denoted by Mill)

Miller’s method has confidence interval limits (CI_Mill) given as follows (Miller, 1991): (2.2) $${\mathit{\text{CI}}}_{\mathit{\text{Mill}}}=(\widehat{\mathit{\text{CV}}}-z_{1-\alpha /2}\sqrt{\frac{{\widehat{\mathit{\text{CV}}}}^2}{n-1}\left(\frac{1}{2}+{\widehat{\mathit{\text{CV}}}}^2\right)},\widehat{\mathit{\text{CV}}}+z_{1-\alpha /2}\sqrt{\frac{{\widehat{\mathit{\text{CV}}}}^2}{n-1}\left(\frac{1}{2}+{\widehat{\mathit{\text{CV}}}}^2\right)})$$(2.2) where $z_{(1-\alpha /2)}$ is the $100\left(1-\alpha /2\right)\mathrm{t}\mathrm{h}$ percentile of the standard normal distribution.

2.1.3. Gulhar, Kibria, Albatineh, and Ahmed confidence interval (denoted by GKAA)

Gulhar, Kibria, and Albatineh (2012) proposed a confidence interval for population CV as follows: (2.3) $${\mathit{\text{CI}}}_{\mathit{\text{GKA}}\&A}=\left(\frac{\sqrt{n-1}\widehat{\mathit{\text{CV}}}}{\sqrt{{\mathrm{\chi }}_{v,1-\alpha /2}^2}},\frac{\sqrt{n-1}\widehat{\mathit{\text{CV}}}}{\sqrt{{\mathrm{\chi }}_{v,\alpha /2}^2}}\right)$$(2.3) where ${\mathrm{\chi }}_{v,1-\alpha /2}^2$ and ${\mathrm{\chi }}_{v,\alpha /2}^2$ are, respectively, the $100\left(1-\alpha /2\right)\mathit{\text{th}}$ and $100\left(\alpha /2\right)\mathrm{t}\mathrm{h}$ percentiles of the Chi-square distribution with degrees of freedom $v=n-1.$

2.1.4. Bootstrap percentile confidence interval (denoted by percentile bootstrap)

This method is based on the percentile of the distribution of bootstrap replications. Doğan (2017) stated that all bootstrap methods give similar confidence interval limit values for normally distributed data. On the other hand, these methods for skewed distributions produce different confidence intervals. The lower and upper limits of the confidence interval are obtained as follows:

1. Calculate estimator CV based on a random sample $\{x_1,x_2,\dots ,x_n\}.$

2. Obtain a bootstrap sample $x^{\mathrm{*}b}=\{x_1^{\mathrm{*}b},x_2^{\mathrm{*}b},\dots ,x_{}^{\mathrm{*}b}$} of size n by simple random sampling with replacement.

3. Calculate ${\mathit{\text{CV}}}^{\mathrm{*}b}$ based on the bootstrap sample.

4. Repeat steps (2) to (3) b = 2000 times.

5. Sort the estimators ${\widehat{\mathit{\text{CV}}}}^{\mathrm{*}b}\mathrm{ }$ in ascending order.

6. The lower and upper values of the bootstrap confidence interval for the population CV are the $\left(\alpha /2\right)\mathrm{t}\mathrm{h}$ and $(1-\alpha /2)\mathrm{\text{th}}$ quantiles of the estimator, ${{\mathit{\text{CV}}}^{\mathrm{*}b}}_{[\alpha /2]}\mathrm{ }$ and ${{\mathit{\text{CV}}}^{\mathrm{*}b}}_{[1-\alpha /2]},$ respectively.

2.1.5. Bootstrap-t confidence interval (denoted by parametric bootstrap)

The CV of bootstrap samples is as: (2.4) $${\widehat{\mathit{\text{CV}}}}^j=\frac{S^j}{{\overline{X}}^j}$$(2.4) where ${\overline{X}}^j={\sum }_{i=1}^n{X}_i^j/n$ and $S^j={\sum }_{i=1}^n{\left(X_i^j-{\overline{X}}^j\right)}^2/\left(n-1\right).$ Accordingly, the mean of CVs of bootstrap samples and standard error of statistics $\widehat{\mathit{\text{CV}}}$ are defined as follows: (2.5) $$\overline{\widehat{{\mathit{\text{CV}}}^{\mathrm{*}}}}=\frac{\sum _{j=1}^b{\widehat{\mathit{\text{CV}}}}^j}{b},$$(2.5) (2.6) $$S_{\widehat{\mathit{\text{CV}}}}=\sqrt{\frac{\sum _{j=1}^b{\left({\widehat{\mathit{\text{CV}}}}^j-\overline{\widehat{{\mathit{\text{CV}}}^{\mathrm{*}}}}\right)}^2}{b}}.$$(2.6)

Thus, the bootstrap-t confidence interval for population CV is defined as follows: (2.7) $$P\left(\widehat{\mathit{\text{CV}}}-T^{\left(\left(\alpha /2\right)b\right)}{S}_{\widehat{\mathit{\text{CV}}}}<\mathit{\text{CV}}<\widehat{\mathit{\text{CV}}}+T^{\left(\left(1-\alpha /2\right)b\right)}{S}_{\widehat{\mathit{\text{CV}}}}\right)=1-\alpha \mathrm{ }$$(2.7) where $T^j=\frac{{\widehat{\mathit{\text{CV}}}}^j-\overline{\widehat{{\mathit{\text{CV}}}^{\mathrm{*}}}}\mathrm{ }}{S_{\widehat{\mathit{\text{CV}}}}},$ j = 1,2,…,b (Nam & Kwon, 2017; Santiago & Smith, 2013).

2.2. Robust confidence interval for the population standard deviation

The estimator Q_n for a random sample X₁, X₂,…, X_n with distribution function F can be expressed as follows (Abu-Shawiesh et al., 2011): (2.8) $$Q_n=2.2219{\left(\left|x_i-x_j\right|\right)}_{\left(g\right)},i<j;i=1,2,\dots ,n;j=1,2,\dots ,n$$(2.8) where $g=\left(\begin{array}{c}h\\ 2\end{array}\right)\approx \left(\raisebox{1ex}{$\left(\begin{array}{c}n\\ 2\end{array}\right)$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\right)$ and $\left[\frac{n}{2}\right]+1.$ Q_n is the gth order statistic of the $\left(\begin{array}{c}n\\ 2\end{array}\right)$ interpoint distance. Let X₁, X₂,…, X_n and n be a random sample and sample size, respectively. x_i and x_j are values of random variable X_k, where k = 1,2,…,n, i < j; i = 1,2,…,n; j = 1,2,…n. d_n is unbiasing factor. Its values for larger values of sample size are obtained as follows (Abu-Shawiesh et al., 2011): (2.9) $$d_n=\{\begin{array}{cc}\frac{n}{n+1.4},& \mathit{\text{for odd value of}}\mathrm{ }n\\ \frac{n}{n+3.8},& \mathit{\text{for even value of}}\mathrm{ }n\end{array}$$(2.9)

Abu-Shawiesh et al. (2011) proposed a robust confidence interval for the population standard deviation (σ) based on the estimator Q_n. This confidence interval is given as in Eq. (2.10)(2.10) $$P\left(\frac{1.28\sqrt{n}\times d_n{Q}_{\mathrm{n}}}{z_{\alpha /2}+1.28\sqrt{n}}\le \sigma \le \frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{1-\alpha /2}+1.28\sqrt{n}}\right)=1-\alpha$$(2.10) . (2.10) $$P\left(\frac{1.28\sqrt{n}\times d_n{Q}_{\mathrm{n}}}{z_{\alpha /2}+1.28\sqrt{n}}\le \sigma \le \frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{1-\alpha /2}+1.28\sqrt{n}}\right)=1-\alpha$$(2.10)

2.3. Modified confidence intervals for the population coefficient of variation

There are many studies in the literature based on modifications of confidence intervals (Abu-Shawiesh et al., 2011, 2019; Banik, Albatineh, Abu-Shawiesh, & Golam Kibria, 2014; Gulhar et al., 2012). It is known that the sample mean estimator $\overline{X}$ does not show robust statistical properties in estimating the non-normal population mean, and the coverage probabilities of the confidence intervals have much lower values than the nominal confidence interval. The population mean is very sensitive to outliers and deviation from the normality assumption. In such cases, it would be a good choice to use robust location estimators for estimation of population mean. We propose alternatives for interval estimation of the population CV that is based on robust estimators preferred instead of the sample mean for skewed distributions.

Let’s divide both sides of the confidence interval in Eq. (2.10)(2.10) $$P\left(\frac{1.28\sqrt{n}\times d_n{Q}_{\mathrm{n}}}{z_{\alpha /2}+1.28\sqrt{n}}\le \sigma \le \frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{1-\alpha /2}+1.28\sqrt{n}}\right)=1-\alpha$$(2.10) by µ for $\mu \ne 0.$ We can write as: (2.11) $$P\left(\frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{\alpha /2}+1.28\sqrt{n}}.\frac{1}{\mu }\le \frac{\sigma }{\mu }\le \frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{1-\frac{\alpha }{2}}+1.28\sqrt{n}}.\frac{1}{\mu }\right)=1-\alpha .$$(2.11)

It is obtained a confidence interval for the population CV as follows: (2.12) $$P\left(\frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{\alpha /2}+1.28\sqrt{n}}.\frac{1}{\mu }\le \mathit{\text{CV}}\le \frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{1-\frac{\alpha }{2}}+1.28\sqrt{n}}.\frac{1}{\mu }\right)=1-\alpha$$(2.12) where $z_{\alpha /2}$ and $z_{1-\alpha /2}$ are the $(\alpha /2)\mathit{\text{th}}$ and $(1-\alpha /2)\mathit{\text{th}}$ percentiles of standard normal distribution, respectively.

Since the population mean is unknown in Eq. (2.12)(2.12) $$P\left(\frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{\alpha /2}+1.28\sqrt{n}}.\frac{1}{\mu }\le \mathit{\text{CV}}\le \frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{1-\frac{\alpha }{2}}+1.28\sqrt{n}}.\frac{1}{\mu }\right)=1-\alpha$$(2.12) , we consider using as an estimate the trimmed mean, winsorized mean, Hodges-Lehmann estimator, and trimean instead of the population mean, respectively.

2.3.1. Confidence interval based on the trimmed mean (denoted by interval I)

Since μ is not known in Eq. (2.12)(2.12) $$P\left(\frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{\alpha /2}+1.28\sqrt{n}}.\frac{1}{\mu }\le \mathit{\text{CV}}\le \frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{1-\frac{\alpha }{2}}+1.28\sqrt{n}}.\frac{1}{\mu }\right)=1-\alpha$$(2.12) , it can be replaced with the robust estimator of μ which is $\widehat{\mu }$=${\widehat{\mu }}_{\mathit{\text{tm}}}.$ (2.13) $$P\left(\frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{\alpha /2}+1.28\sqrt{n}}.\frac{1}{{\widehat{\mu }}_{\mathit{\text{tm}}}}\le {\mathit{\text{CV}}}_{\mathit{\text{tm}}}\le \frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{1-\alpha /2}+1.28\sqrt{n}}.\frac{1}{{\widehat{\mu }}_{\mathit{\text{tm}}}}\right)=1-\alpha$$(2.13) where tm is trimmed mean. The random sample of size n is $\{x_1,x_2,\dots ,x_n\},$ and ith order statistics is shown with X_(i). The equation of the trimmed mean changes depending on whether the data is symmetrical or skewed. For positively skewed distributions, it is as follows: (2.14) $${\widehat{\mu }}_{\mathit{\text{tm}}}=\frac{1}{n-u_n}\sum _{i=1}^{n-u_n}{X}_{\left(i\right)}.$$(2.14)

For the symmetrical distribution, it is as: (2.15) $${\widehat{\mu }}_{\mathit{\text{tm}}}=\frac{1}{n-{2l}_n}\sum _{i=\mathit{\text{ln}}+1}^{n-u_n}{X}_{\left(i\right)}$$(2.15) where l_n and u_n are the number of terms to be removed from the low and high end of the consecutive data, respectively.

2.3.2. Confidence interval based on the winsorized mean (denoted by interval II)

We consider winsorized mean (wm) as a robust estimator of the population mean. (2.16) $$P\left(\frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{\alpha /2}+1.28\sqrt{n}}.\frac{1}{{\widehat{\mu }}_{\mathit{\text{wm}}}}\le {\mathit{\text{CV}}}_{\mathit{\text{wm}}}\le \frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{1-\alpha /2}+1.28\sqrt{n}}.\frac{1}{{\widehat{\mu }}_{\mathit{\text{wm}}}}\right)=1-\alpha$$(2.16)

Winsorized mean is defined for positively skewed distributions as follows: (2.17) $${\widehat{\mu }}_{\mathit{\text{wm}}}=\frac{1}{n}\left\{\sum _{i=1}^{n-u_n}{X}_{\left(i\right)}+u_n{X}_{\left(n-u_n\right)}\right\}$$(2.17) $u_n=\left[\rho n+0.5\right],$ and $\rho \mathrm{ }$ replacement percentage [.] expression indicates the largest integer function. When replacement is made on both ends of the consecutive data, it is obtained as: (2.18) $${\widehat{\mu }}_{\mathit{\text{wm}}}=\frac{1}{n}\left\{\sum _{i=l_{n+}1}^{n-u_n}{X}_{\left(i\right)}+l_n{X}_{\left(l_n+1\right)}+u_n{X}_{(n-u_n)}\right\}$$(2.18) where l_n and u_n are the number of terms to be replaced from the low and high end of the consecutive data, respectively.

2.3.3. Confidence interval based on the Hodges-Lehmann estimator (denoted by interval III)

(2.19) $$P\left(\frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{\alpha /2}+1.28\sqrt{n}}.\frac{1}{{\widehat{\mu }}_{\mathit{\text{HL}}}}\le {\mathit{\text{CV}}}_{\mathit{\text{tr}}}\le \frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{1-\alpha /2}+1.28\sqrt{n}}.\frac{1}{{\widehat{\mu }}_{\mathit{\text{HL}}}}\right)=1-\alpha$$(2.19)

where HL refers to Hodges-Lehmann and ${\widehat{\mu }}_{\mathit{\text{HL}}}=\mathit{\text{median}}\left(\frac{x_i+x_j}{2},1\le i<j\le n\right).$ This estimator is obtained from Wilcoxon signed-rank statistic and based on the median of all pairwise differences (Rosenkranz, 2010).

2.3.4. Confidence interval based on the trimean (denoted by interval IV)

(2.20) $$P\left(\frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{\alpha /2}+1.28\sqrt{n}}.\frac{1}{{\widehat{\mu }}_{\mathit{\text{tr}}}}\le {\mathit{\text{CV}}}_{\mathit{\text{tr}}}\le \frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{1-\alpha /2}+1.28\sqrt{n}}.\frac{1}{{\widehat{\mu }}_{\mathit{\text{tr}}}}\right)=1-\alpha$$(2.20)

where ${\widehat{\mu }}_{\mathit{\text{tr}}}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\times \left(\mathit{\text{md}}+\frac{Q_1+Q_3}{2}\right)$ is sample trimean (tr). Also, md, Q₁, and Q₃ are median, first quantile, and third quantile, respectively. Trimean is resistant to extreme values and is preferred to mean and median for extremely skewed distributions (Bonett, 2006).

2.4. Bootstrap versions of proposed confidence intervals

In some studies, it has been suggested to use the critical value obtained by the bootstrap method instead of the table value in the confidence intervals (Abu-Shawiesh et al., 2011; Banik et al., 2014; Gulhar et al., 2012). Even if it is known that the sampling distribution of a statistic is approximately normally distributed, this may not apply to real-life problems. Thus, we also propose to examine confidence intervals based on the critical value obtained from bootstrap samples. The lower and upper limits are as follows: (2.21) $$\mathit{\text{lower}}:\frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{\alpha /2}^{\mathrm{*}}+1.28\sqrt{n}}.\frac{1}{\mu }$$(2.21) (2.22) $$\mathit{\text{upper}}:\frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{1-\alpha /2}^{\mathrm{*}}+1.28\sqrt{n}}.\frac{1}{\mu }$$(2.22) where $z_{\alpha /2}^{\mathrm{*}}$ and $z_{1-\alpha /2}^{\mathrm{*}}$ are the $(\alpha /2)\mathrm{\text{th}}$ and $(1-\alpha /2)\mathrm{\text{th}}$ quantiles of statistics $z_j^{\mathrm{*}},j=1,2,\dots ,b.$ (2.23) $$z_j^{\mathrm{*}}=\frac{{\widehat{\mathit{\text{CV}}}}^j-\overline{{\widehat{\mathit{\text{CV}}}}^{\mathrm{*}}}}{{\widehat{\sigma }}_{\widehat{\mathit{\text{CV}}}}}$$(2.23) where ${\widehat{\mathit{\text{CV}}}}^j$ is the sample CV of the bootstrap samples, $\overline{{\widehat{\mathit{\text{CV}}}}^{\mathrm{*}}}=\frac{{\sum }_{j=1}^b{\widehat{\mathit{\text{CV}}}}^j}{b}$ and ${\widehat{\sigma }}_{\widehat{\mathit{\text{CV}}}}=\sqrt{\frac{{\sum }_{j=1}^b{\left({\widehat{\mathit{\text{CV}}}}^j-\overline{{\widehat{\mathit{\text{CV}}}}^{\mathrm{*}}}\right)}^2}{b}}.$ In this study, we denoted the bootstrap confidence intervals as bootstrap intervals I-IV.

3. Performance comparison

In this section, the simulation study was performed to compare the performance of the confidence intervals. We examined the proposed confidence intervals and their bootstrap versions to estimate the population CV. Also, these confidence intervals were compared with bootstrap percentile, parametric bootstrap, and some known confidence intervals in terms of CP and AW. In order to compare the performance of the various intervals, other criteria were the SDW and CVW. The reason for choosing the HR, Mill, and GKAA confidence intervals was to use intervals that studied both the normal and non-normal distribution. Since a theoretical comparison was not possible, a simulation study was conducted to compare the performance of the intervals. All simulations were performed using codes written in MATLAB. We note that trimming and replacement ratios are prefered as in Bonett (2006). The simulation study was designed as follows:

• Sample sizes n = 10, 15, 20, 30, 50, 100,

• Probability of type I error α = 0.05,

• The number of bootstrap and simulation replications are b = 2000 and s = 5000, respectively,

• Trimming and replacement ratios $0.5/\sqrt{n-4},$

• Normal (10,2) with skewness = 0,

• Beta (2,2) with skewness = 0,

• Beta (2,4) with skewness= 1.32,

• Gamma (6.25,2) with skewness= 0.8,

• Gamma (4,1) with skewness= 1,

• Chi-square (2) with skewness = 2,

• Lognormal (0,1) with skewness = 6.18.

The CP, AW, SDW, and CVW were measured for each case. AWs were obtained by dividing the total differences between the lower (l) and upper limits (u) found for each replication by the number of replications. The CPs were determined as the proportion of cases where the population CV was between the lower and upper interval limit. It is always desirable that the CP be as close as possible to the nominal level and the AW be as narrow as possible. SDW and CVW were obtained as the standard deviation of the widths and the ratio of SDW to AW, respectively. Thus, they are obtained by using the following four formulas: (3.1) $$\mathit{\text{CP}}=\frac{P\left(l\le \mathit{\text{CV}}\le u\right)}{s},$$(3.1) (3.2) $$\mathit{\text{AW}}=\frac{\sum _{i=1}^s\left(u_i-l_i\right)}{s},$$(3.2) (3.3) $$\mathit{\text{SDW}}=\sqrt{\frac{\sum _{i=1}^s{\left(W_i-\mathit{\text{AW}}\right)}^2}{s-1}},$$(3.3) (3.4) $$\mathit{\text{CVW}}=\frac{\mathit{\text{SDW}}}{\mathit{\text{AW}}}.$$(3.4)

If we need to rephrase, it can be said that s is number of simulation replications, l and u are the lower and upper limits of confidence interval, and W_i, i = 1,2,…,s are interval widths.

The lower and upper limits of proposed confidence intervals in MATLAB are based on the following algorithm:

• Step-1: The sample sizes n = 10, 15, 20, 30, 50, and 100 are generated from the known distributions.

• Step-2: The trimming/winsorizing percentages based on these samples are obtained (percent= $0.5/\sqrt{n-4}$). If the distribution is symmetrical, trimming/winsorizing is performed from both ends of the data. Otherwise (which is right-skewed distribution), it is obtained only from the top end.

• Step-3: The samples are sorted in ascending order and trimmed mean, winsorized mean, HL estimator, and trimean are calculated.

• Step-4: The lower and upper limits of confidence intervals in Eqs. (2.12)(2.12) $$P\left(\frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{\alpha /2}+1.28\sqrt{n}}.\frac{1}{\mu }\le \mathit{\text{CV}}\le \frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{1-\frac{\alpha }{2}}+1.28\sqrt{n}}.\frac{1}{\mu }\right)=1-\alpha$$(2.12) , (2.15)(2.15) $${\widehat{\mu }}_{\mathit{\text{tm}}}=\frac{1}{n-{2l}_n}\sum _{i=\mathit{\text{ln}}+1}^{n-u_n}{X}_{\left(i\right)}$$(2.15) , (2.18)(2.18) $${\widehat{\mu }}_{\mathit{\text{wm}}}=\frac{1}{n}\left\{\sum _{i=l_{n+}1}^{n-u_n}{X}_{\left(i\right)}+l_n{X}_{\left(l_n+1\right)}+u_n{X}_{(n-u_n)}\right\}$$(2.18) and (2.19)(2.19) $$P\left(\frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{\alpha /2}+1.28\sqrt{n}}.\frac{1}{{\widehat{\mu }}_{\mathit{\text{HL}}}}\le {\mathit{\text{CV}}}_{\mathit{\text{tr}}}\le \frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{1-\alpha /2}+1.28\sqrt{n}}.\frac{1}{{\widehat{\mu }}_{\mathit{\text{HL}}}}\right)=1-\alpha$$(2.19) are obtained.

• Step-5: Bootstrap samples of size n are generated by simple random sampling with replacement following Step 3.

• Step-6: The CV is obtained based on the bootstrap samples.

• Step-7: Steps (5) to (6) are repeated 2000 times.

• Step-8: The z critical value is obtained from the bootstrap samples as in Eq. (22).

• Step-9: The limits of bootstrap confidence intervals are get using bootstrapped critical values in Eqs. (2.13)(2.13) $$P\left(\frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{\alpha /2}+1.28\sqrt{n}}.\frac{1}{{\widehat{\mu }}_{\mathit{\text{tm}}}}\le {\mathit{\text{CV}}}_{\mathit{\text{tm}}}\le \frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{1-\alpha /2}+1.28\sqrt{n}}.\frac{1}{{\widehat{\mu }}_{\mathit{\text{tm}}}}\right)=1-\alpha$$(2.13) , (2.16)(2.16) $$P\left(\frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{\alpha /2}+1.28\sqrt{n}}.\frac{1}{{\widehat{\mu }}_{\mathit{\text{wm}}}}\le {\mathit{\text{CV}}}_{\mathit{\text{wm}}}\le \frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{1-\alpha /2}+1.28\sqrt{n}}.\frac{1}{{\widehat{\mu }}_{\mathit{\text{wm}}}}\right)=1-\alpha$$(2.16) , (2.19)(2.19) $$P\left(\frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{\alpha /2}+1.28\sqrt{n}}.\frac{1}{{\widehat{\mu }}_{\mathit{\text{HL}}}}\le {\mathit{\text{CV}}}_{\mathit{\text{tr}}}\le \frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{1-\alpha /2}+1.28\sqrt{n}}.\frac{1}{{\widehat{\mu }}_{\mathit{\text{HL}}}}\right)=1-\alpha$$(2.19) , and (2.20)(2.20) $$P\left(\frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{\alpha /2}+1.28\sqrt{n}}.\frac{1}{{\widehat{\mu }}_{\mathit{\text{tr}}}}\le {\mathit{\text{CV}}}_{\mathit{\text{tr}}}\le \frac{1.28\sqrt{n}\times d_n{Q}_n}{z_{1-\alpha /2}+1.28\sqrt{n}}.\frac{1}{{\widehat{\mu }}_{\mathit{\text{tr}}}}\right)=1-\alpha$$(2.20) .

• Step 10: The values of CP, AW, SDW, and CVW in Eq. (3.1)–(3.4) are calculated based on these limits.

Simulation results based on random samples generated from Normal, Beta and some skewed distributions are presented in Tables 1–7.

Table 1. Simulated CP, AW, SDW and CVW values of confidence intervals for N (10,2) and CV = 0.2.

	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW
	n = 10				n = 15				n = 20				n = 30				n = 50				n = 100
HR	0.9040	0.1973	0.0493	0.2498	0.9176	0.1539	0.0305	0.1982	0.9344	0.1314	0.0219	0.1665	0.9336	0.1051	0.0144	0.1365	0.9342	0.0800	0.0084	0.1050	0.9430	0.0562	0.0040	0.0720
Mill	0.8920	0.1881	0.0507	0.2693	0.9110	0.1517	0.0324	0.2132	0.9332	0.1315	0.0236	0.1795	0.9352	0.1066	0.0157	0.1469	0.9394	0.0819	0.0093	0.1129	0.9498	0.0577	0.0045	0.0774
GKAA	0.9416	0.2218	0.0554	0.2498	0.9426	0.1661	0.0329	0.1982	0.9432	0.1390	0.0231	0.1665	0.9398	0.1091	0.0149	0.1365	0.9404	0.0817	0.0086	0.1050	0.9460	0.0569	0.0041	0.0720
Percentile bootstrap	0.7756	0.1559	0.0551	0.3536	0.8310	0.1316	0.0394	0.2993	0.8732	0.1181	0.0319	0.2698	0.8912	0.0983	0.0225	0.2294	0.9418	0.0779	0.0146	0.1868	0.9440	0.0561	0.0075	0.1343
Parametric bootstrap	0.8144	0.1470	0.0579	0.3941	0.8554	0.1283	0.0420	0.3273	0.8902	0.1169	0.0344	0.2940	0.9030	0.0985	0.0244	0.2478	0.9496	0.0787	0.0158	0.2008	0.9490	0.0568	0.0082	0.1445
Interval I	0.8850	0.2566	0.0787	0.3068	0.8938	0.1883	0.0466	0.2475	0.9120	0.1557	0.0311	0.2000	0.9210	0.1220	0.0194	0.1591	0.9374	0.0905	0.0109	0.1207	0.9422	0.0627	0.0051	0.0812
Interval II	0.8830	0.2568	0.0792	0.3086	0.8908	0.1883	0.0466	0.2476	0.9118	0.1557	0.0312	0.2001	0.9192	0.1220	0.0194	0.1592	0.9390	0.0905	0.0109	0.1207	0.9428	0.0627	0.0051	0.0812
Interval III	0.8874	0.2567	0.0787	0.3066	0.8916	0.1883	0.0466	0.2475	0.9136	0.1557	0.0311	0.1999	0.9196	0.1220	0.0194	0.1592	0.9378	0.0905	0.0109	0.1208	0.9416	0.0627	0.0051	0.0813
Interval IV	0.8828	0.2567	0.0790	0.3075	0.8924	0.1884	0.0468	0.2483	0.9110	0.1558	0.0312	0.2005	0.9194	0.1220	0.0195	0.1595	0.9368	0.0905	0.0110	0.1213	0.9406	0.0627	0.0051	0.0816
Bootstrap interval I	0.8798	0.2769	0.0904	0.3266	0.8926	0.1921	0.0486	0.2531	0.9136	0.1566	0.0320	0.2041	0.9192	0.1216	0.0198	0.1626	0.9460	0.0900	0.0112	0.1239	0.9406	0.0624	0.0052	0.0835
Bootstrap interval II	0.8790	0.2772	0.0914	0.3299	0.8906	0.1922	0.0487	0.2534	0.9112	0.1567	0.0320	0.2045	0.9176	0.1216	0.0198	0.1629	0.9470	0.0900	0.0111	0.1238	0.9406	0.0624	0.0052	0.0835
Bootstrap interval III	0.8796	0.2770	0.0903	0.3261	0.8896	0.1921	0.0486	0.2531	0.9126	0.1567	0.0320	0.2041	0.9170	0.1216	0.0198	0.1628	0.9456	0.0900	0.0112	0.1240	0.9488	0.0624	0.0052	0.0836
Bootstrap interval IV	0.8784	0.2771	0.0908	0.3278	0.8906	0.1923	0.0489	0.2543	0.9116	0.1567	0.0321	0.2048	0.9176	0.1216	0.0198	0.1631	0.9464	0.0900	0.0112	0.1244	0.9488	0.0625	0.0052	0.0839

n: sample size, CP: coverage probability, AW: average width, SDW: standard deviation of width, CVW: coefficient of variation of width

Table 2. Simulated CP, AW, SDW and CVW values of confidence intervals for Beta (2,2) and CV = 0.4472.

	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW
	n = 10				n = 15				n = 20				n = 30				n = 50				n = 100
HR	0.9324	0.4544	0.1069	0.2352	0.9392	0.3521	0.0661	0.1878	0.9412	0.2956	0.0476	0.1610	0.9500	0.2379	0.0309	0.1297	0.9500	0.1797	0.0180	0.1001	0.9534	0.1257	0.0088	0.0700
Mill	0.9304	0.5023	0.1533	0.3052	0.9442	0.4001	0.0974	0.2434	0.9496	0.3392	0.0707	0.2083	0.9640	0.2768	0.0463	0.1673	0.9686	0.2103	0.0271	0.1289	0.9752	0.1459	0.0133	0.0911
GKAA	0.9462	0.5111	0.1202	0.2352	0.9462	0.3799	0.0713	0.1878	0.9484	0.3126	0.0503	0.1610	0.9466	0.2469	0.0320	0.1297	0.9476	0.1837	0.0184	0.1001	0.9502	0.1270	0.0089	0.0700
Percentile bootstrap	0.8568	0.3885	0.1186	0.3053	0.8954	0.3169	0.0771	0.2432	0.9034	0.2724	0.0563	0.2067	0.9302	0.2240	0.0367	0.1640	0.9376	0.1727	0.0217	0.1257	0.9434	0.1223	0.0107	0.0875
Parametric bootstrap	0.8814	0.3908	0.1343	0.3437	0.9104	0.3236	0.0857	0.2648	0.9154	0.2793	0.0621	0.2224	0.9360	0.2296	0.0399	0.1735	0.9448	0.1765	0.0231	0.1311	0.9468	0.1243	0.0113	0.0912
Interval I	0.8740	0.6038	0.1765	0.2923	0.8828	0.4437	0.1037	0.2336	0.9032	0.3638	0.0692	0.1901	0.8974	0.2870	0.0423	0.1474	0.9334	0.2126	0.0237	0.1113	0.9410	0.1477	0.0111	0.0755
Interval II	0.8568	0.6079	0.1863	0.3065	0.8818	0.4439	0.1041	0.2345	0.8982	0.3640	0.0700	0.1922	0.8944	0.2871	0.0425	0.1481	0.9358	0.2126	0.0236	0.1110	0.9428	0.1477	0.0111	0.0751
Interval III	0.8814	0.6031	0.1748	0.2898	0.8804	0.4436	0.1033	0.2329	0.9036	0.3639	0.0693	0.1905	0.8976	0.2871	0.0427	0.1486	0.9320	0.2127	0.0239	0.1122	0.9484	0.1477	0.0113	0.0763
Interval IV	0.8680	0.6059	0.1809	0.2987	0.8668	0.4452	0.1081	0.2428	0.8884	0.3650	0.0724	0.1983	0.8800	0.2878	0.0450	0.1563	0.9380	0.2129	0.0251	0.1179	0.9450	0.1478	0.0120	0.0812
Bootstrap interval I	0.8838	0.6091	0.1749	0.2871	0.8876	0.4368	0.0994	0.2275	0.9092	0.3576	0.0671	0.1876	0.9032	0.2829	0.0417	0.1474	0.9378	0.2106	0.0237	0.1125	0.9450	0.1470	0.0114	0.0773
Bootstrap interval II	0.8654	0.6139	0.1891	0.3081	0.8880	0.4370	0.1000	0.2289	0.9066	0.3579	0.0681	0.1903	0.9010	0.2831	0.0420	0.1483	0.9400	0.2106	0.0236	0.1122	0.9466	0.1470	0.0113	0.0769
Bootstrap interval III	0.8900	0.6082	0.1720	0.2827	0.8868	0.4366	0.0991	0.2270	0.9102	0.3577	0.0673	0.1881	0.9016	0.2830	0.0421	0.1486	0.9460	0.2107	0.0239	0.1133	0.9414	0.1470	0.0115	0.0782
Bootstrap interval IV	0.8760	0.6116	0.1809	0.2957	0.8758	0.4383	0.1040	0.2373	0.8938	0.3588	0.0704	0.1961	0.8852	0.2837	0.0444	0.1565	0.9438	0.2109	0.0251	0.1190	0.9424	0.1471	0.0122	0.0829

n: sample size, CP: coverage probability, AW: average width, SDW: standard deviation of width, CVW: coefficient of variation of width

Table 3. Simulated CP, AW, SDW, and CVW values of confidence intervals for Beta (2,4) and CV = 0.5345.

	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW
	n = 10				n = 15				n = 20				n = 30				n = 50				n = 100
HR	0.9340	0.5340	0.1202	0.2252	0.9466	0.4165	0.0750	0.1800	0.9512	0.3526	0.0542	0.1537	0.9522	0.2824	0.0355	0.1258	0.9596	0.2544	0.0203	0.0798	0.9576	0.1498	0.0101	0.0674
Mill	0.9362	0.6232	0.1921	0.3083	0.9572	0.5007	0.1242	0.2480	0.9644	0.4290	0.0905	0.2111	0.9720	0.3477	0.0600	0.1725	0.9796	0.2658	0.0344	0.1294	0.9816	0.1867	0.0171	0.0916
GKAA	0.9588	0.6006	0.1352	0.2252	0.9616	0.4494	0.0809	0.1800	0.9596	0.3730	0.0573	0.1537	0.9562	0.2930	0.0368	0.1258	0.9636	0.2592	0.0207	0.0799	0.9580	0.1514	0.0102	0.0674
Percentile bootstrap	0.8646	0.4399	0.1180	0.2682	0.8950	0.3585	0.0764	0.2132	0.9098	0.3100	0.0562	0.1811	0.9256	0.2538	0.0370	0.1459	0.9402	0.2561	0.0213	0.0832	0.9404	0.1389	0.0111	0.0799
Parametric bootstrap	0.8908	0.4418	0.1335	0.3022	0.9178	0.3657	0.0842	0.2303	0.9270	0.3173	0.0612	0.1930	0.9376	0.2602	0.0398	0.1528	0.9464	0.2505	0.0227	0.0906	0.9444	0.1413	0.0118	0.0835
Interval I	0.8154	0.8522	0.2650	0.3110	0.9130	0.5101	0.1158	0.2269	0.9138	0.4933	0.0936	0.1896	0.9273	0.3658	0.0515	0.1407	0.9416	0.2408	0.0268	0.1113	0.9488	0.1387	0.0120	0.0865
Interval II	0.8666	0.7460	0.2109	0.2827	0.9376	0.4603	0.0973	0.2115	0.9302	0.4392	0.0759	0.1727	0.9574	0.3353	0.0438	0.1306	0.9438	0.2376	0.0232	0.0976	0.9416	0.1296	0.0109	0.0841
Interval III	0.9370	0.6976	0.1894	0.2715	0.9358	0.4484	0.0940	0.2096	0.9500	0.4257	0.0725	0.1702	0.9604	0.3327	0.0438	0.1318	0.9408	0.2378	0.0236	0.0992	0.9420	0.1316	0.0112	0.0851
Interval IV	0.8950	0.7149	0.2038	0.2851	0.9298	0.4590	0.1016	0.2214	0.9234	0.4353	0.0785	0.1803	0.9250	0.3404	0.0484	0.1421	0.9484	0.2433	0.0264	0.1085	0.9402	0.1354	0.0126	0.0931
Bootstrap interval I	0.8318	0.8579	0.2561	0.2985	0.9226	0.5015	0.1088	0.2170	0.9146	0.4848	0.0892	0.1840	0.9280	0.3605	0.0499	0.1385	0.9406	0.2342	0.0265	0.1132	0.9468	0.1379	0.0122	0.0885
Bootstrap interval II	0.8816	0.7514	0.2042	0.2718	0.9448	0.4525	0.0910	0.2011	0.9382	0.4316	0.0722	0.1672	0.9630	0.3304	0.0424	0.1283	0.9462	0.2332	0.0230	0.0978	0.9438	0.1288	0.0111	0.0862
Bootstrap interval III	0.9318	0.7027	0.1825	0.2597	0.9366	0.4411	0.0888	0.2013	0.9538	0.4184	0.0692	0.1654	0.9642	0.3279	0.0426	0.1299	0.9450	0.2354	0.0234	0.0994	0.9434	0.1308	0.0114	0.0872
Bootstrap interval IV	0.9064	0.7210	0.2014	0.2793	0.9304	0.4516	0.0968	0.2143	0.9314	0.4278	0.0753	0.1760	0.9316	0.3355	0.0472	0.1406	0.9418	0.2409	0.0262	0.1088	0.9442	0.1345	0.0128	0.0952

n: sample size, CP: coverage probability, AW: average width, SDW: standard deviation of width, CVW: coefficient of variation of width

Table 4. Simulated CP, AW, SDW and CVW values of confidence intervals for Gamma (4,1) and CV = 0.5.

	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW
	n = 10				n = 15				n = 20				n = 30				n = 50				n = 100
HR	0.9098	0.4853	0.1181	0.2434	0.9156	0.3808	0.0757	0.1987	0.9176	0.3234	0.0560	0.1731	0.9126	0.2601	0.0379	0.1456	0.9172	0.2291	0.0222	0.0970	0.9222	0.1497	0.0110	0.0735
Mill	0.9118	0.5491	0.1811	0.3298	0.9294	0.4446	0.1195	0.2687	0.9354	0.3826	0.0893	0.2334	0.9412	0.3119	0.0614	0.1969	0.9520	0.2408	0.0360	0.1495	0.9622	0.1701	0.0179	0.1052
GKAA	0.9456	0.5458	0.1328	0.2434	0.9442	0.4109	0.0816	0.1987	0.9394	0.3421	0.0592	0.1731	0.9278	0.2699	0.0393	0.1456	0.9248	0.2335	0.0227	0.0972	0.9254	0.1612	0.0111	0.0689
Percentile bootstrap	0.8002	0.3823	0.1102	0.2884	0.8350	0.3192	0.0815	0.2554	0.8606	0.2841	0.0724	0.2548	0.8718	0.2400	0.0587	0.2444	0.8908	0.2323	0.0436	0.1876	0.9184	0.1532	0.0303	0.1978
Parametric bootstrap	0.8380	0.3672	0.1106	0.3011	0.8620	0.3128	0.0787	0.2516	0.8794	0.2807	0.0687	0.2446	0.8806	0.2390	0.0566	0.2369	0.8988	0.2332	0.0445	0.1908	0.9232	0.1551	0.0328	0.2115
Interval I	0.8254	0.7168	0.2163	0.3018	0.8968	0.4970	0.1119	0.2252	0.9254	0.4149	0.0776	0.1869	0.9388	0.3096	0.0451	0.1458	0.9462	0.2272	0.0234	0.1029	0.9466	0.1524	0.0104	0.0682
Interval II	0.9114	0.6408	0.1791	0.2794	0.9478	0.4542	0.0963	0.2120	0.9550	0.3754	0.0651	0.1734	0.9480	0.2862	0.0394	0.1377	0.9448	0.2120	0.0208	0.0981	0.9432	0.1449	0.0096	0.0663
Interval III	0.9358	0.6034	0.1625	0.2693	0.9504	0.4452	0.0934	0.2098	0.9542	0.3659	0.0624	0.1706	0.9492	0.2862	0.0393	0.1374	0.9532	0.2142	0.0210	0.0980	0.9488	0.1483	0.0099	0.0668
Interval IV	0.9248	0.6187	0.1718	0.2776	0.9394	0.4561	0.0995	0.2181	0.9492	0.3746	0.0666	0.1779	0.9438	0.2932	0.0426	0.1454	0.9562	0.2193	0.0228	0.1039	0.9576	0.1518	0.0108	0.0711
Bootstrap interval I	0.8242	0.7507	0.2628	0.3501	0.8994	0.5017	0.1132	0.2257	0.9267	0.4144	0.0779	0.1881	0.9404	0.3079	0.0455	0.1479	0.9664	0.2257	0.0239	0.1059	0.9432	0.1514	0.0108	0.0713
Bootstrap interval II	0.9208	0.6709	0.2189	0.3263	0.9540	0.4582	0.0966	0.2107	0.9572	0.3749	0.0655	0.1746	0.9518	0.2846	0.0396	0.1391	0.9430	0.2106	0.0211	0.1001	0.9456	0.1439	0.0099	0.0688
Bootstrap interval III	0.9462	0.6313	0.1961	0.3107	0.9538	0.4497	0.0962	0.2139	0.9570	0.3655	0.0632	0.1730	0.9508	0.2846	0.0400	0.1406	0.9512	0.2128	0.0216	0.1015	0.9432	0.1473	0.0103	0.0699
Bootstrap interval IV	0.9306	0.6487	0.2172	0.3349	0.9432	0.4608	0.1031	0.2237	0.9528	0.3742	0.0677	0.1808	0.9456	0.2916	0.0434	0.1487	0.9564	0.2178	0.0234	0.1074	0.9544	0.1508	0.0112	0.0743

n: sample size, CP: coverage probability, AW: average width, SDW: standard deviation of width, CVW: coefficient of variation of width

Table 5. Simulated CP, AW, SDW and CVW values of confidence intervals for Gamma (6.25, 2) and CV = 0.4.

	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW
	n = 10				n = 15				n = 20				n = 30				n = 50				n = 100
HR	0.9056	0.3892	0.0958	0.2461	0.9194	0.3056	0.0598	0.1956	0.9216	0.2593	0.0442	0.1707	0.9238	0.2087	0.0289	0.1387	0.9298	0.1593	0.0169	0.1062	0.9294	0.1118	0.0085	0.0760
Mill	0.8992	0.4125	0.1276	0.3094	0.9250	0.3343	0.0817	0.2443	0.9342	0.3175	0.0615	0.1937	0.9402	0.2345	0.0405	0.1728	0.9516	0.1806	0.0239	0.1322	0.9572	0.1276	0.0121	0.0948
GKAA	0.9472	0.4377	0.1077	0.2461	0.9480	0.3298	0.0645	0.1956	0.9414	0.2742	0.0468	0.1707	0.9366	0.2165	0.0300	0.1387	0.9390	0.1628	0.0173	0.1062	0.9342	0.1130	0.0086	0.0761
Percentile bootstrap	0.8050	0.3052	0.0886	0.2905	0.8304	0.2544	0.0664	0.2610	0.9386	0.2260	0.0551	0.2438	0.8818	0.1896	0.0435	0.2295	0.9024	0.1590	0.0331	0.2082	0.9238	0.1122	0.0209	0.1862
Parametric bootstrap	0.8176	0.2882	0.0877	0.3043	0.8518	0.2471	0.0649	0.2626	0.8528	0.2223	0.0539	0.2424	0.8894	0.1885	0.0432	0.2293	0.9084	0.1595	0.0340	0.2131	0.9262	0.1134	0.0226	0.1992
Interval I	0.8314	0.5663	0.1733	0.3061	0.8780	0.3966	0.0953	0.2402	0.9234	0.3329	0.0652	0.1958	0.9184	0.2512	0.0379	0.1509	0.9446	0.1501	0.0204	0.1360	0.9590	0.1104	0.0093	0.0842
Interval II	0.9006	0.5177	0.1483	0.2864	0.9252	0.3694	0.0845	0.2286	0.9404	0.3075	0.0565	0.1837	0.9512	0.2362	0.0340	0.1440	0.9488	0.1444	0.0186	0.1288	0.9468	0.1096	0.0088	0.0802
Interval III	0.9206	0.4931	0.1376	0.2791	0.9276	0.3624	0.0820	0.2263	0.9402	0.3005	0.0545	0.1815	0.9508	0.2352	0.0337	0.1434	0.9526	0.1449	0.0186	0.1283	0.9496	0.1112	0.0089	0.0800
Interval IV	0.9086	0.5018	0.1432	0.2855	0.9174	0.3685	0.0857	0.2326	0.9436	0.3050	0.0570	0.1869	0.9446	0.2388	0.0354	0.1484	0.9512	0.1476	0.0195	0.1321	0.9534	0.1130	0.0093	0.0823
Bootstrap interval I	0.8226	0.5958	0.2452	0.4115	0.8790	0.4030	0.0973	0.2415	0.9352	0.3344	0.0663	0.1982	0.9202	0.2507	0.0384	0.1533	0.9460	0.1531	0.0207	0.1352	0.9586	0.1108	0.0096	0.0866
Bootstrap interval II	0.9004	0.5450	0.2102	0.3857	0.9236	0.3752	0.0859	0.2288	0.9430	0.3088	0.0575	0.1862	0.9536	0.2356	0.0344	0.1459	0.9492	0.1525	0.0188	0.1224	0.9426	0.1100	0.0090	0.0803
Bootstrap interval III	0.9252	0.5190	0.1925	0.3710	0.9262	0.3684	0.0847	0.2299	0.9454	0.3019	0.0559	0.1850	0.9520	0.2342	0.0344	0.1464	0.9492	0.1540	0.0190	0.1233	0.9474	0.1106	0.0092	0.0831
Bootstrap interval IV	0.9116	0.5285	0.2090	0.3955	0.9170	0.3747	0.0889	0.2374	0.9384	0.3067	0.0585	0.1908	0.9450	0.2383	0.0362	0.1519	0.9484	0.1567	0.0199	0.1269	0.9510	0.1124	0.0096	0.0854

n: sample size, CP: coverage probability, AW: average width, SDW: standard deviation of width, CVW: coefficient of variation of width

Table 6. Simulated CP, AW, SDW and CVW values of confidence intervals for chi-square (2) and CV = 1.

	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW
	n = 10				n = 15				n = 20				n = 30				n = 50				n = 100
HR	0.8810	0.9364	0.2377	0.2539	0.8810	0.7406	0.1611	0.2175	0.8744	0.6356	0.1233	0.1940	0.8696	0.5132	0.0839	0.1635	0.8674	0.3935	0.0515	0.1309	0.8626	0.2780	0.0261	0.0940
Mill	0.9292	1.4766	0.6408	0.4340	0.9438	1.2118	0.4605	0.3800	0.9522	1.0594	0.3571	0.3371	0.9590	0.8677	0.2454	0.2828	0.9692	0.6709	0.1504	0.2241	0.9812	0.4785	0.0761	0.1590
GKAA	0.9466	1.0531	0.2673	0.2539	0.9308	0.7991	0.1738	0.2175	0.9142	0.6722	0.1304	0.1940	0.9174	0.5325	0.0871	0.1635	0.8810	0.4022	0.0527	0.1309	0.8756	0.2811	0.0264	0.0940
Percentile bootstrap	0.8094	0.8000	0.2267	0.2834	0.8318	0.6748	0.1900	0.2816	0.8388	0.6034	0.1695	0.2810	0.8600	0.5130	0.1446	0.2819	0.9156	0.4194	0.1161	0.2769	0.9240	0.3223	0.0885	0.2746
Parametric bootstrap	0.8424	0.8383	0.2435	0.2905	0.8600	0.7025	0.1893	0.2695	0.8616	0.6243	0.1632	0.2614	0.8698	0.5261	0.1358	0.2580	0.9136	0.4294	0.1122	0.2613	0.9266	0.3303	0.0910	0.2755
Interval I	0.8628	1.4251	0.3955	0.2775	0.9560	0.8762	0.1687	0.1925	0.9158	0.7592	0.1159	0.1527	0.9376	0.5267	0.0570	0.1082	0.9456	0.3724	0.0286	0.0767	0.9484	0.2388	0.0121	0.0508
Interval II	0.9200	1.1357	0.2925	0.2576	0.9244	0.7270	0.1393	0.1917	0.9000	0.6140	0.0912	0.1485	0.9354	0.4448	0.0508	0.1141	0.9444	0.3199	0.0270	0.0845	0.9456	0.2130	0.0121	0.0569
Interval III	0.9278	1.0159	0.2580	0.2539	0.9252	0.7253	0.1320	0.1820	0.9398	0.6048	0.0867	0.1434	0.9336	0.4688	0.0488	0.1040	0.9400	0.3472	0.0253	0.0729	0.9444	0.2399	0.0113	0.0471
Interval IV	0.9004	1.1165	0.2855	0.2557	0.9330	0.7947	0.1519	0.1912	0.9214	0.6613	0.0972	0.1470	0.9346	0.5145	0.0581	0.1129	0.9448	0.3806	0.0314	0.0825	0.9460	0.2631	0.0142	0.0542
Bootstrap interval I	0.8860	1.3369	0.3351	0.2507	0.9556	0.8355	0.1585	0.1897	0.9256	0.7312	0.1152	0.1576	0.9264	0.5125	0.0602	0.1175	0.9410	0.3646	0.0316	0.0866	0.9472	0.2353	0.0138	0.0587
Bootstrap interval II	0.9164	1.0680	0.2576	0.2412	0.9100	0.6933	0.1321	0.1905	0.9336	0.5918	0.0938	0.1586	0.9374	0.4328	0.0535	0.1235	0.9438	0.3132	0.0291	0.0928	0.9488	0.2099	0.0132	0.0627
Bootstrap interval III	0.9298	0.9549	0.2256	0.2363	0.9242	0.6930	0.1317	0.1900	0.9318	0.5830	0.0897	0.1539	0.9328	0.4562	0.0530	0.1162	0.9442	0.3400	0.0286	0.0843	0.9462	0.2364	0.0133	0.0564
Bootstrap interval IV	0.9242	1.0491	0.2464	0.2349	0.9244	0.7585	0.1467	0.1934	0.9302	0.6370	0.0972	0.1526	0.9356	0.5005	0.0605	0.1208	0.9414	0.3727	0.0339	0.0910	0.9400	0.2592	0.0161	0.0620

n: sample size, CP: coverage probability, AW: average width, SDW: standard deviation of width, CVW: coefficient of variation of width

Table 7. Simulated CP, AW, SDW and CVW values of confidence intervals for Lognormal (0,1) and CV = 1.3108.

	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW	CP	AW	SDW	CVW
	n = 10				n = 15				n = 20				n = 30				n = 50				n = 100
HR	0.5614	0.9829	0.3238	0.3294	0.5376	0.8169	0.2627	0.3216	0.5938	0.7099	0.2149	0.3028	0.5920	0.5928	0.1728	0.2916	0.6516	0.4740	0.1264	0.2667	0.7130	0.3408	0.0776	0.2278
Mill	0.7100	1.6530	1.0019	0.6062	0.7188	1.4908	0.9363	0.6280	0.7122	1.3308	0.7957	0.5979	0.7216	1.1600	0.7093	0.6114	0.7312	0.9622	0.5400	0.5613	0.7264	0.7018	0.3578	0.5098
GKAA	0.7756	1.1054	0.3641	0.3294	0.7866	0.8814	0.2835	0.3216	0.7984	0.7509	0.2273	0.3028	0.8052	0.6151	0.1793	0.2916	0.8998	0.4844	0.1292	0.2667	0.9254	0.3445	0.0785	0.2278
Percentile bootstrap	0.8924	0.8211	0.3244	0.3951	0.8922	0.7697	0.3485	0.4527	0.9012	0.7209	0.3413	0.4735	0.9020	0.6762	0.3525	0.5213	0.9122	0.6322	0.3492	0.5524	0.9298	0.5635	0.3300	0.5857
Parametric bootstrap	0.8918	0.8090	0.3079	0.3807	0.8942	0.7510	0.3190	0.4248	0.9140	0.6945	0.3047	0.4387	0.9002	0.6404	0.3069	0.4792	0.9184	0.5933	0.3009	0.5072	0.9224	0.5385	0.2991	0.5554
Interval I	0.9096	0.9290	0.3490	0.2840	0.9062	0.7671	0.1528	0.1991	0.9220	0.6622	0.1065	0.1608	0.9122	0.4559	0.0554	0.1215	0.9300	0.3229	0.0286	0.0886	0.9344	0.2046	0.0134	0.0657
Interval II	0.9144	0.9992	0.2666	0.2668	0.9110	0.6503	0.1306	0.2040	0.9326	0.5398	0.0876	0.1624	0.9300	0.3832	0.0521	0.1360	0.9348	0.2741	0.0282	0.1030	0.9360	0.1793	0.0141	0.0789
Interval III	0.9238	0.8995	0.2401	0.2669	0.9238	0.6421	0.1220	0.1870	0.9312	0.5431	0.0808	0.1488	0.9302	0.4187	0.0469	0.1119	0.9356	0.3121	0.0248	0.0794	0.9340	0.2152	0.0114	0.0530
Interval IV	0.9172	0.9910	0.2600	0.2623	0.9138	0.7115	0.1369	0.1924	0.9308	0.5912	0.0899	0.1521	0.9304	0.4574	0.0541	0.1184	0.9336	0.3210	0.0293	0.0860	0.9366	0.2052	0.0133	0.0564
Bootstrap interval I	0.9044	0.7920	0.3691	0.3066	0.9014	0.7515	0.1701	0.2263	0.9206	0.6535	0.1365	0.2089	0.9266	0.4490	0.0789	0.1757	0.9310	0.3168	0.0433	0.1366	0.9372	0.1994	0.0203	0.1020
Bootstrap interval II	0.9182	0.9800	0.2879	0.2938	0.9118	0.6262	0.1409	0.2250	0.9340	0.5332	0.1158	0.2171	0.9340	0.3773	0.0696	0.1844	0.9338	0.2691	0.0391	0.1452	0.9488	0.1747	0.0190	0.1088
Bootstrap interval III	0.9288	0.8812	0.2539	0.2881	0.9252	0.6307	0.1520	0.2369	0.9342	0.5365	0.1106	0.2062	0.9330	0.4126	0.0710	0.1722	0.9342	0.3061	0.0405	0.1322	0.9462	0.2098	0.0201	0.0958
Bootstrap interval IV	0.9238	0.9716	0.2814	0.2897	0.9274	0.6990	0.1648	0.2357	0.9356	0.5835	0.1184	0.2029	0.9314	0.4505	0.0781	0.1735	0.9314	0.3345	0.0445	0.1329	0.9400	0.2293	0.0223	0.0971

n: sample size, CP: coverage probability, AW: average width, SDW: standard deviation of width, CVW: coefficient of variation of width

Firstly, we compare the performance of the confidence intervals for the some symmetric distributions. The simulation results for random samples generated from both N (10,2) and Beta (2,2) are given in Tables 1 and 2. The results show that the proposed confidence intervals are quite close to the nominal confidence level (1-α) except for sample sizes n = 10, 15, and $\alpha =0.05.$ The percentile and parametric bootstrap confidence intervals have the narrowest widths for all sample sizes. We noticed that bootstrap intervals I–IV that were obtained with bootstrapped critical value are narrower than intervals I–IV. On the other hand, we note that proposed confidence intervals give higher AWs than the others for symmetrical distributions. The existing confidence intervals are not close to nominal level except GKAA for n = 10, 15 and they aren’t close to nominal level except HR, Mill, and GKAA for n = 20, 30 under the normal distribution. The SDWs of the proposed confidence intervals are larger because of these results. It is also evident from the CV of the widths. Interval IV has the highest AWs in symmetrical distributions (Tables 1 and 2).

In the study, we preferred some skewed distributions in the range skewness coefficient 0.8 − 6.18 to evaluate the performance of the confidence intervals. Because we also want to see the effect of the severity of the skewness on the confidence intervals. We reported them in Tables 3–7.

Table 3–6 shows the results for moderate skewness. It can say the CPs of all proposed confidence intervals and bootstrap versions are closer to the nominal confidence level than HR, Mill, GKAA, percentile and parametric bootstrap confidence intervals for sample sizes n = 20, 30, 50, and 100. Also, they have narrower widths for similar CPs. Bootstrap interval III performed well in small sample sizes (n = 10,15) in terms of CP and its AWs were narrower than Mill confidence interval for n = 20, 30. It is determined that bootstrap interval II that the confidence interval based on the winsorize mean gives the narrowest widths with sample sizes $n\ge 50\mathrm{ }$ for moderate skewness. However, it was obtained to be narrowest except n = 10, 15 under distributions where the skewness coefficient value was two or more (Tables 3–6).

In Table 7, we present the performance of confidence intervals for the Lognormal (0,1) distribution that is highly skewed. It was obtained that all proposed confidence intervals are closer to the nominal confidence level than the others for all sample sizes and they have narrower AWs except n = 10,15. Also, bootstrap interval II has the narrowest AWs according to HR, Mill, GKAA, percentile, and parametric bootstrap confidence intervals for n = 20, 30, 50, and 100. On the other hand, bootstrap interval III performed better for small sample sizes n = 10,15. We obtained that their SDWs were smaller than the others. CVW values also support these results. Interval I which is based on trimmed mean has AWs higher than intervals II-IV in skewed distributions (Tables 3–7).

4. Applications using real data

In this section, three real-life data sets will be analyzed to illustrate the application of all considered confidence intervals CIs for the population CV. These data have different sample sizes and different degrees of skewness.

4.1. Dataset I: survival times

The data were obtained from Lawless (2011). They include the survival times in weeks for 20 male rats exposed to high levels of radiation. Descriptive statistics and normality test results are as in Table 8.

Table 8. Descriptive statistics and goodness of fit test results for the survival time.

Statistics	Values
Sample size	20
Qn	48.8818
Mean	113.4500
$0.5/\sqrt{n-4}\mathrm{ }$ trimmed mean	114.6667
$0.5/\sqrt{n-4}\mathrm{ }$ winsorized mean	115.3500
HL estimator	114.5000
TR estimator	117.0000
Median	119.0000
Variance	1280.892
Standard deviation	35.78955
Minimum	40.00
Maximum	165.00
Range	125.00
Interquartile range	64.00
Skewness	−0.355
Kurtosis	−0.788
CV	0.3155
Shapiro-Wilk (SW)	0.961 (p-value = 0.571)

The histogram and Q-Q plot of survival times are shown in Figure 1. The goodness of fit test for normality has a p-value greater than 0.05 (SW test statistic = 0.961, p-value = 0.571). We can conclude that the data are normally distributed. Thus, the widths based on trimmed and winsorized means are obtained using both ends (trimming/replacement). The calculated 95% confidence intervals are reported in Table 9.

Figure 1. Histogram and normal Q-Q plot of the survival time.

Table 9. The 95% confidence intervals for the CV of the survival time.

	Limits	Width
HR	0.2111–0.4199	0.2088
Mill	0.2056–0.4253	0.2197
GKAA	0.2399–0.4608	0.2209
Percentile bootstrap	0.2144–0.4020	0.1876
Parametric bootstrap	0.2187–0.4123	0.1936
Interval I	0.2669–0.5447	0.2783
Interval II	0.2653–0.5415	0.2762
Interval III	0.2672–0.5455	0.2780
Interval IV	0.2615–0.5339	0.2723
Bootstrap interval I	0.2643–0.5373	0.2734
Bootstrap interval II	0.2627–0.5341	0.2714
Bootstrap interval III	0.2647–0.5381	0.2730
Bootstrap interval IV	0.2590–0.5266	0.2675

Table 9 includes the confidence interval limits for the CV of the survival time which is 0.3155. All confidence intervals cover this value. The percentile and parametric bootstrap confidence intervals differ slightly in terms of width and they give the narrowest interval widths. Interval I has the widest widths. This result supports the simulation outputs for symmetrical distributions (Table 9).

4.2. Dataset II: mosquito survival rates

The real-life data used in the second application are on mosquito survival rates in wet climates compiled by Johnson and McFarland (1993). Descriptive statistics and goodness of fit test results for dataset II are given in Table 10.

Table 10. Descriptive statistics and goodness of fit test results for the mosquito survival rate.

Statistics	Values
Sample size	8
Qn	0.0755
Mean	0.1283
$0.5/\sqrt{n-4}\mathrm{ }$ trimmed mean	0.0325
$0.5/\sqrt{n-4}\mathrm{ }$ winsorized mean	0.0469
HL estimator	0.0638
TR estimator	0.0690
Median	0.0375
Variance	0.0360
Std. Deviation	0.1910
Minimum	0.0100
Maximum	0.5390
Range	0.5290
Interquartile range	0.1810
Skewness	1.8340
Kurtosis	2.7980
CV	1.4897
Shapiro-Wilk test statistic (SW)	0.704 (p-value = 0.002)

According to Figure 2 and Table 10, we can say that the hypothesis data come from a normal distribution that is rejected at the 0.05 level (SW test statistic= 0.704, p-value < 0.05). All confidence interval limits and widths for dataset II are as in Table 11.

Figure 2. Histogram and normal Q-Q plot of the mosquito survival rate.

Table 11. The 95% confidence intervals for the CV of mosquito survival rate.

	Limits	Width
HR	0.6473–2.3321	1.6848
Mill	0.4131–2.5662	2.1531
GKAA	0.9849–3.0319	2.0470
Percentile bootstrap	0.6947–2.1494	1.4547
Parametric bootstrap	0.6796–2.2997	1.6201
Interval I	1.0224–3.4361	2.4137
Interval II	0.7089–2.3824	1.6735
Interval III	0.5212–1.7517	1.1369
Interval IV	0.4816–1.6184	1.2305
Bootstrap interval I	0.9653–3.1737	2.2084
Bootstrap interval II	0.6693–2.2004	1.5312
Bootstrap interval III	0.4921–1.6180	1.0402
Bootstrap interval IV	0.4547–1.4948	1.1258

In Table 11, it was obtained that bootstrap interval III has the narrowest width. This result is consistent with the simulation result obtained with small sample sizes.

4.3. Dataset III: urinary tract infections

Data were obtained from the study by Santiago and Smith (2013). They are on the duration of male patient urinary tract infections in days. The histogram and Q-Q plot were shown in Figure 3. Also, descriptive statistics and goodness of fit test results for the urinary tract infection are obtained as in Table 12.

Figure 3. Histogram and normal Q-Q plot of the urinary tract infection.

Table 12. Descriptive statistics and goodness of fit test results for the urinary tract infection.

Statistics	Values
Sample size	54
Qn	0.1543
Mean	0.2103
$0.5/\sqrt{n-4}\mathrm{ }$ trimmed mean	0.1668
$0.5/\sqrt{n-4}\mathrm{ }$ winsorized mean	0.1940
HL estimator	0.1757
TR estimator	0.1580
Median	0.1424
Variance	0.0449
Std. Deviation	0.2119
Minimum	0.0035
Maximum	1.0889
Range	1.0854
Interquartile range	0.2431
Skewness	1.9340
Kurtosis	4.7580
CV	1.0079
Shapiro-Wilk test statistic(SW)	0.807 (p-value = 0.000)

According to the histogram, Q-Q plot, and goodness of fit test of urinary tract infection, the data have a non-normal distribution (Figure 3). The limits of 95% confidence intervals and widths for the CV of urinary tract infection are reported in Table 13. We can see that all confidence intervals include the CV value. Bootstrap interval II has a narrower width than the others. This result supports the simulation results obtained for large sample sizes (Table 13).

Table 13. The 95% confidence intervals for the CV of the urinary tract infection.

	Limits	Width
HR	0.8133–1.2024	0.3890
Mill	0.6738–1.3419	0.6681
GKAA	0.8472–1.2442	0.3970
Percentile bootstrap	0.7777–1.2123	0.4346
Parametric bootstrap	0.7838–1.2319	0.4481
Interval I	0.7154–1.0920	0.3766
Interval II	0.6849–1.0086	0.3237
Interval III	0.6790–1.0364	0.3574
Interval IV	0.7551–1.1526	0.3975
Bootstrap interval I	0.7147–1.0762	0.3615
Bootstrap interval II	0.6943–1.0080	0.3137
Bootstrap interval III	0.6784–1.0214	0.3431
Bootstrap interval IV	0.7544–1.1360	0.3816

5. Some concluding remarks

Finding the mean and variance of the parameter estimates is a fundamental step in establishing the confidence interval (CI) for a parameter of interest. A different method will be used to generate the confidence interval when determining the precise mean and variance is challenging. Although there are many studies for the population CV under the assumption of normal distribution, there are insufficient studies to obtain confidence intervals for CV in skewed distributions and in cases where the population standard deviation is unknown. We proposed modified confidence intervals based on robust estimators for the population CV. Also, we examined the bootstrap versions of these confidence intervals that were obtained from bootstrapped critical values. These intervals were compared to some existing confidence intervals. Simulation results showed that the proposed confidence intervals and bootstrap versions were close to the nominal confidence level but performed poorly in terms of AW, SDW, and CVW on symmetrical distributions. Interval I had the weakest overall performance for the normal distribution. For moderate and high skewness, their CPs are closer to the nominal confidence level and they have narrower widths than the others in large sample sizes $n=20,30,50,100.$ In addition, bootstrap interval II has the narrowest average widths. The SDWs and CVWs for this interval are smaller than the others. On the other hand, the proposed confidence intervals did not perform well for small sample sizes ($n=10,15$) except for interval III and bootstrap interval III. Consequently, we propose bootstrap interval III for small sample ($n=10,15$) sizes and bootstrap interval II for larger sample sizes under skewed distributions. Three real data sets were analyzed to illustrate the findings of the study and the simulation results were verified. Furthermore, the proposed confidence intervals are easy to compute and can be recommended for practitioners in various fields of industry, engineering, medical and physical sciences.

Authors’ contributions

All authors contributed equally and significantly in writing of this article. All authors have read and agreed the last version of the paper.

Disclosure statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability statement

The information that assists the conclusions of the present research are included in the paper’s application section.

Additional information

Funding

The authors weren’t financially supported by this research.