Confidence intervals for the difference between the coefficients of variation of Weibull distributions for analyzing wind speed dispersion

Abstract

Wind energy is an important renewable energy source for generating electricity that has the potential to replace fossil fuels. Herein, we propose confidence intervals for the difference between the coefficients of variation of Weibull distributions constructed using the concepts of the generalized confidence interval (GCI), Bayesian methods, the method of variance estimates recovery (MOVER) based on Hendricks and Robey’s confidence interval, a percentile bootstrap method, and a bootstrap method with standard errors. To analyze their performances, their coverage probabilities and expected lengths were evaluated via Monte Carlo simulation. The simulation results indicate that the coverage probabilities of GCI were greater than or sometimes close to the nominal confidence level. However, when the Weibull shape parameter was small, the Bayesian- highest posterior density interval was preferable. All of the proposed confidence intervals were applied to wind speed data measured at 90-meter wind energy potential stations at various regions in Thailand.

Introduction

Nowadays, Thailand is increasingly using alternative energy sources to oil and coal owing to the government pushing forward a policy concerning alternative energy development and efficiency. The alternative energy development in Thailand has primarily relied on energy production in domestic with emphasis on solar, wind, hydro energy, biomass, biogas, municipal Solid waste (MSW), geothermal power and biofuels including ethanol and biodiesel in term of electricity, heat and biofuels (https://www.dede.go.th/ewt_news.php?nid=47340). Wind energy is an important natural energy source for use in power generation because it has many advantages: it is clean and environmentally friendly, it reduces the level of carbon dioxide emissions that contribute toward global warming, and it offers unlimited renewable energy. In recent years, power generation from wind energy in Thailand has limitation about potential of wind speeds such as wind power density, wind speed, and the height of wind turbine (Chancham et al., 2009). In addition, there are problems about where to install wind turbines because wind energy is related to the topography of the installation area and wind strength variation throughout the year. Potential areas to harvest wind energy in Thailand are coastal areas and the northeastern region of Thailand. Since the coefficient of variation can be used to analyze areas with high wind speed variability, we are interested in analyzing wind datasets from the aforementioned areas in terms of the difference between coefficients of variation of Weibull distributions. The approach in this study could be useful in finding areas that are the most suitable for placing wind turbine systems to produce electricity.

In statistics, the Weibull distribution, which is named after Swedish mathematician Waloddi Weibull (Weibull, 1951), is a continuous probability distribution commonly used to analyze wind speed data. It has also been widely applied in engineering, industry, and weather forecasting. For example, Chaichana et al. (2015) studied the potential of wind energy for high-tech agricultural projects as part of the Royal Initiative at Chiang Mai province, Thailand. Moreover, Pang, Forster & Troutt (2001) estimated the parameters for a Weibull distribution when applied to wind speed data obtained from an observatory in Hong Kong. Previously, many researchers have estimated the parameters and functions of parameters for Weibull distributions. For instance, Green et al. (1994) used the Markov chain Monte Carlo (MCMC) method to estimate three parameters of a Weibull distribution. Colosimo & Ho (1999) derived confidence intervals for the Weibull mean lifetime based on censored reliability datasets. Krishnamoorthy, Lin & Xia (2009) constructed confidence intervals for the mean of Weibull distribution based on the generalized variable approach and compared this with Wald confidence intervals. In another study, Krishnamoorthy & Lin (2010) presented the confidence interval for stress–strength reliability involving two Weibull distributions. Ibrahim (2010) compared the performance of Bayesian approaches using Jeffreys’ prior and an extension of Jeffreys’ prior with maximum likelihood estimation to estimate the parameters of a Weibull distribution. Kundu & Howlader (2010) used Bayesian inference to estimate the scale parameter of an inverted Weibull distribution based on Type-II censored data. Yalcinkaya & Birgoren (2017) compared a confidence interval of the lower percentiles in a Weibull distribution of small samples based on maximum likelihood estimation and the Bayesian-Weibull method. Saraiva & Suzuki (2017) compared methods to estimate the parameters of a Weibull distribution, which are a maximum likelihood method and a Bayesian method with the Metropolis–Hastings algorithm.

The coefficient of variation is used for measuring variations in data as well as comparing the degree of variation between several datasets in which the measurement units are different. It has been used in many fields. Faber & Korn (1991) used the coefficient of variation to measure variation in the mean synaptic response of nerves in the central nervous system. Banik & Kibria (2011) used the coefficient of variation to analyze data on psychiatric disorders, while Yosboonruang, Niwitpong & Niwitpong (2019) used it to examine variation in rainfall data in Thailand. In this study, we focus on the coefficient of variation. Many researchers have provided confidence intervals for estimating the single and function of coefficients of variation. For example in a normal distribution: a normal coefficient of variation (Vangel, 1996), common coefficient of variation (Tian, 2005), coefficient of variation (Mahmoudvand & Hassani, 2009) and (Donner & Zou, 2010). Moreover, Pang et al. (2005) proposed confidence intervals for estimating the coefficient of variation of a three-parameter Weibull distribution via a simulation-based Bayesian approach. Niwitpong (2013) provided confidence intervals for the coefficient of variation of a lognormal distribution with a restricted parameter space. Buntao & Niwitpong (2013) constructed confidence intervals for the difference between the coefficients of variation of lognormal distributions. Sangnawakij, Niwitpong & Niwitpong (2015) presented confidence intervals for the ratio of coefficients of variation of gamma distributions. Yosboonruang, Niwitpong & Niwitpong (2020) proposed confidence intervals for the difference between the coefficients of variation of delta-lognormal distributions. La-ongkaew, Niwitpong & Niwitpong (2009) constructed confidence interval for single coefficient of variation of Weibull distribution. It could be seen that not many researches mentioned above focused on coefficient of variation or difference between the coefficients of variation of Weibull distributions. Consequently, we extended this study to construct confidence intervals for the difference between the coefficients of variation of Weibull distributions.

Herein, we propose new confidence intervals for the difference between the coefficients of variation of Weibull distributions using the generalized confidence interval (GCI) method, Bayesian methods, the method of variance estimates recovery (MOVER) method, a percentile bootstrap method, and a bootstrap interval with standard errors, the derivations of which are given in this article. As an empirical application, we use wind speed data measured at 90-meter wind energy potential stations in Thailand. Finally, we discuss the results of our study and draw conclusions on them.

Methods

Let a random variable X follow the 2-Parameter Weibull distribution. The probability density function of X is given by

$${\displaystyle f\left(x;a,k\right)=\frac{k}{a}{\left(\frac{x}{a}\right)}^{k-1}exp\left[-{\left(\frac{x}{a}\right)}^k\right],x>0,}$$ (1)

where the positive constant a is called the scale parameter and the positive constant k is called the shape parameter. The mean and the variance of X are $\text{E}\left(X\right)=a\Gamma \left(1+\frac{1}{k}\right)$ and $\text{Var}\left(X\right)=a^2\left[\Gamma \left(1+\frac{2}{k}\right)-{\left(\Gamma \left(1+\frac{1}{k}\right)\right)}^2\right]$, respectively. Therefore, the coefficient of variation of X can be written as

$${\displaystyle \text{CV}\left(X\right)=\lambda =\sqrt{\frac{\Gamma \left(1+\frac{2}{k}\right)}{{\left(\Gamma \left(1+\frac{1}{k}\right)\right)}^2}-1}.}$$ (2)

Suppose that X =(X₁, X₂, …, X_n) be a random sample of size n from Weibull distribution with scale parameter a and shape parameter k, denoted as Weibull(a, k). To estimate the parameters a and k, the maximum likelihood estimation is applied. The maximum likelihood estimators (MLEs) can be obtained from Cohen (1965), they are as follows. The MLE $\hat{k}$ of k is solution of the following equation

$${\displaystyle \frac{1}{\hat{k}}-\frac{\sum _{i=1}^n\left[x_i^{\hat{k}}ln\left(x_i\right)\right]}{\sum \left(\underset{i}{\overset{\hat{k}}{x}}\right)}+\frac{1}{n}\sum _{i=1}^{n}ln\left(x_i\right)=0}$$ (3)

and the MLE $\hat{a}$ of a is given by

$${\displaystyle \hat{a}={\left[\sum _{i=1}^n{x}_i^{\hat{k}}∕n\right]}^{\frac{1}{\hat{k}}}.}$$ (4)

For solving the Eq.(3), the Newton–Raphson iterative method can be applied. We obtained a starting value for the Newton–Raphson iterative method. Let x₁, x₂,…, x_n be a sample of observations from a Weibull(a, k) distribution. Let z_i = ln(x_i), i =1 , 2, …, n. Menon (1963) showed that the estimator

$${\displaystyle {\hat{k}}_u=\left(\frac{\pi }{\sqrt{6}}\right)\left(\sqrt{\frac{n-1}{s^2}}\right)}$$ (5)

is asymptotically unbiased estimator, which is the starting value.

The following algorithm is used to compute the MLEs:

Algorithm 1

1. Generate x₁, x₂,…, x_n, let z_i = ln(x_i).

2. Compute $s_1={\sum }_{i=1}^n{z}_i$, $\overline{z}=s_1∕n$ and $s^2={\sum }_{i=1}^n{\left(z_i-\overline{z}\right)}^2$.

3. Compute ${\hat{k}}_u$ = $\left(\frac{\pi }{\sqrt{6}}\right)\left(\sqrt{\frac{n-1}{s^2}}\right)$.

4. For l = 1 to number of iterations.

5. Compute $w_i=x_i^{{\hat{k}}_u}$, $s_2={\sum }_{i=1}^n\left(w_i\right)$, $s_3={\sum }_{i=1}^n\left(w_i{z}_i\right)$ and $s_4={\sum }_{i=1}^n\left(w_i{z}_i^2\right)$.

6. Compute $F=\left(\frac{1}{{\hat{k}}_u}\right)+\overline{z}-\left(\frac{s_3}{s_2}\right)$.

7. Compute $f=\left(\frac{1}{{\hat{k}}_u^2}\right)+\left[\left(s_2{s}_4\right)-\left(s_3^2\right)\right]∕\left(s_2^2\right)$.

8. Compute ${\hat{k}}_u={\hat{k}}_u$ + $\left(\frac{F}{f}\right)$ .

9. If (F ≤ error tolerance) return $\hat{k}={\hat{k}}_u$.

10. End l loop.

11. Compute $\hat{a}$ from Eq. (4).

Let X₁, X₂, …, X_n and Y₁, Y₂, …, Y_m be random samples from two independent Weibull distributions with scale parameters a_X, a_Y and shape parameters k_X, k_Y, respectively. Moreover, let λ_X be the coefficient of variation of X. Similarly, let λ_Y be the coefficient of variation of Y. Thus, the difference between the coefficients of variation is defined by

$${\displaystyle \delta ={\lambda }_X-{\lambda }_Y=\sqrt{\frac{\Gamma \left(1+\frac{2}{k_X}\right)}{{\left(\Gamma \left(1+\frac{1}{k_X}\right)\right)}^2}-1}-\sqrt{\frac{\Gamma \left(1+\frac{2}{k_Y}\right)}{{\left(\Gamma \left(1+\frac{1}{k_Y}\right)\right)}^2}-1}.}$$ (6)

And the estimator of δ is given by

$${\displaystyle \hat{\delta }={\hat{\lambda }}_X-{\hat{\lambda }}_Y=\sqrt{\frac{\Gamma \left(1+\frac{2}{{\hat{k}}_X}\right)}{{\left(\Gamma \left(1+\frac{1}{{\hat{k}}_X}\right)\right)}^2}-1}-\sqrt{\frac{\Gamma \left(1+\frac{2}{{\hat{k}}_Y}\right)}{{\left(\Gamma \left(1+\frac{1}{{\hat{k}}_Y}\right)\right)}^2}-1}.}$$ (7)

The Generalized Confidence Interval

The generalized confidence interval (GCI) was introduced by Weerahandi (1993). The most important concept of GCI is the generalized pivotal quantity (GPQ) defined as

Definition Suppose that X = (X₁, X₂, …, X_n) be a random variables from distribution, which depends on a parameter of interest φ, and a nuisance parameter γ. Furthermore, suppose that x =(x₁, x₂, …, x_n) be the observed value of X. To obtain a GCI for φ, a GPQ R(X; x, φ, γ) is also required to satisfy the following two conditions.

(i) For a fixed x, the distribution of R(X; x, φ, γ) is free of unknown parameters.

(ii) The observed value of R(X; x, φ, γ) at X = x, denoted as r(X; x, φ, γ) does not depend on nuisance parameters.

Let R_φ(α∕2) and R_φ(1 − α∕2) are the 100(α/2)-th and 100(1 − α/2)-th percentile of R_φ(X; x), respectively. The 100 (1 − α)% two-sided confidence interval based on GCI for parameter of interest is given by [R_φ(α∕2), R_φ(1 − α∕2)]. Note that 1 − α is the probability that population parameter will be in the interval. The 100 (1 − α)% confidence interval will include the true value of the population parameter with probability 1 − α, i.e., if α = 0.05, that probability is about 0.95 that the 95% confidence interval will include the true population parameter.

For Weibull distribution, Thoman, Bain & Antle (1969) showed that the distributions of $\frac{\hat{k}}{k}$ and $\hat{k}ln\left(\frac{\hat{a}}{a}\right)$ do not depend on a and k, where $\hat{a}$ and $\hat{k}$ are the MLEs of a and k, respectively. Let $\widehat{a^{\ast }}$ and $\widehat{k^{\ast }}$ be the MLEs based on a sample of size n from a Weibull(1, 1). Then $\frac{\hat{k}}{k}\sim \widehat{k^{\ast }}$ and $\hat{k}ln\left(\frac{\hat{a}}{a}\right)\sim \widehat{k^{\ast }}ln\left(\widehat{a^{\ast }}\right)$. Both distributions do not depend on any parameters. Hence, they are pivotal quantity.

Krishnamoorthy, Mukherjee & Guo (2007) presented the GPQs of scale and shape parameters from Weibull distribution. Let ${\hat{a}}_0$ and ${\hat{k}}_0$ be the observed values of $\hat{a}$ and $\hat{k}$. The GPQs of the parameters are given by

$${\displaystyle R_k=\frac{k}{\hat{k}}{\hat{k}}_0=\frac{{\hat{k}}_0}{{\hat{k}}^{\ast }}}$$ (8)

and

$${\displaystyle R_a={\left(\frac{a}{\hat{a}}\right)}^{\frac{\hat{k}}{{\hat{k}}_0}}{\hat{a}}_0={\left(\frac{1}{{\hat{a}}^{\ast }}\right)}^{\frac{{\hat{k}}^{\ast }}{{\hat{k}}_0}}{\hat{a}}_0.}$$ (9)

From the random variables X and Y, the GPQs for k_X and k_Y are given by

$${\displaystyle R_{k_X}=\frac{{\hat{k}}_{X_0}}{{\hat{k}}_X^{\ast }}}$$ (10)

and

$${\displaystyle R_{k_Y}=\frac{{\hat{k}}_{Y_0}}{{\hat{k}}_Y^{\ast }}.}$$ (11)

Thus, the GPQ for δ is defined by

$${\displaystyle R_{\delta }=R_{{\lambda }_X}-R_{{\lambda }_Y}=\sqrt{\frac{\Gamma \left(1+\frac{2}{R_{k_X}}\right)}{{\left(\Gamma \left(1+\frac{1}{R_{k_X}}\right)\right)}^2}-1}-\sqrt{\frac{\Gamma \left(1+\frac{2}{R_{k_Y}}\right)}{{\left(\Gamma \left(1+\frac{1}{R_{k_Y}}\right)\right)}^2}-1}.}$$ (12)

Therefore, the 100 (1 − α)% two-sided confidence interval for the difference between the coefficients of variation of Weibull distributions based on GCI is

$${\displaystyle C{I}_{\delta \left(gci\right)}=\left[L_{\delta \left(gci\right)},U_{\delta \left(gci\right)}\right]=\left[R_{\delta }\left(\alpha ∕2\right),R_{\delta }\left(1-\alpha ∕2\right)\right],}$$ (13)

where R_δ(α∕2) denotes the 100(α/2)-th percentile of R_δ.

The following algorithm is used to construct confidence interval based on GCI for the difference between the coefficients of variation of Weibull distributions:

Algorithm 2

1. Compute ${\hat{a}}_X$, ${\hat{a}}_Y$, ${\hat{k}}_X$ and ${\hat{k}}_Y$ from Algorithm 1.

2. Generate $X_1^{\ast }$, $X_2^{\ast }$, …, $X_n^{\ast }$ and $Y_1^{\ast }$, $Y_2^{\ast }$, …, $Y_m^{\ast }$ from Weibull(1, 1).

3. Compute ${\hat{a}}_X^{\ast }$, ${\hat{a}}_Y^{\ast }$, ${\hat{k}}_X^{\ast }$ and ${\hat{k}}_Y^{\ast }$ from Algorithm 1.

4. Compute R_kX and R_kY from Eqs. (10) and (11).

5. Compute R_δ from Eq. (12).

6. Repeat steps 1-5 for q times, where q is the number of generalized computation.

7. Compute 95% confidence interval based on GCI, as given in Eq. (13).

The bootstrap confidence intervals

The bootstrap method was introduced by Efron & Tibshirani (1993). This is a resampling method to determine precision measures for statistical estimation.

Let ${\mathbf{x}}_i=\left(x_1,x_2,\dots ,x_n\right)$ and ${\mathbf{y}}_i=\left(y_1,y_2,\dots ,y_m\right)$ be random samples of size n and m from Weibull distributions, and let ${\mathbf{x}}_i^{\ast b}=\left(x_1^{\ast b},x_2^{\ast b},\dots ,x_n^{\ast b}\right)$ and ${\mathbf{y}}_i^{\ast b}=\left(y_1^{\ast b},y_2^{\ast b},\dots ,y_m^{\ast b}\right)$ be bootstrapped samples drawn with replacement from the original data, using the same sample sizes. After resampling B bootstrap samples, the difference between the coefficients of variation are calculated in each bootstrap sample, as follows: ${\delta }^{\ast b}={\lambda }_X^{\ast b}-{\lambda }_Y^{\ast b},$b =1 , 2, …, B.

The percentile bootstrap confidence interval

The percentile bootstrap confidence interval is based on the percentile of the distribution of the bootstrapped replications. The value of the bootstrap statistic, δ^∗b are ordered from the smallest to the largest.

Therefore, the 100 (1 − α)% two-sided confidence interval for the difference between the coefficients of variation of Weibull distributions based on percentile bootstrap is given by

$${\displaystyle C{I}_{\delta \left(pb\right)}=\left[L_{\delta \left(pb\right)},U_{\delta \left(pb\right)}\right]=\left[{\delta }^{\ast b}\left(\alpha ∕2\right),{\delta }^{\ast b}\left(1-\alpha ∕2\right)\right],}$$ (14)

where δ^∗b(α∕2) denotes the 100(α/2)-th percentile of δ^∗b.

The following algorithm is used to construct confidence interval based on percentile bootstrap for the difference between the coefficients of variation of Weibull distributions:

Algorithm 3

1. Generate X₁, X₂, …, X_n from Weibull(a_X, k_X) and Y₁, Y₂, …, Y_m from Weibull(a_Y, k_Y).

2. For b = 1.

3. Resampling bootstrap samples $X_1^{\ast },X_2^{\ast },\dots ,X_n^{\ast }$ from X₁, X₂, …, X_n and compute ${\lambda }_X^{\ast b}$.

4. Resampling bootstrap samples $Y_1^{\ast },Y_2^{\ast },\dots ,Y_m^{\ast }$ from Y₁, Y₂, …, Y_m and compute ${\lambda }_Y^{\ast b}$.

5. Compute ${\delta }^{\ast b}={\lambda }_X^{\ast b}-{\lambda }_Y^{\ast b}$.

6. Repeat steps 3-5 for B times, where B is the number of bootstrap sample.

7. Sort δ^∗b from the smallest to the largest.

8. Compute 95% confidence interval based on percentile bootstrap, as given in Eq. (14).

The bootstrap confidence interval with standard errors

From the B bootstrap statistic denoted as δ^∗b, b =1 , 2, …, B, we can calculate the standard error of a statistic. They can be estimated using the standard deviation of the bootstrap distribution.

Therefore, the 100 (1 − α)% two-sided confidence interval for the difference between the coefficients of variation of Weibull distributions based on bootstrap confidence interval with standard errors can be written as

$${\displaystyle C{I}_{\delta \left(bs\right)}=\left[L_{\delta \left(bs\right)},U_{\delta \left(bs\right)}\right]=\left[\hat{\delta }-Z_{\left(\alpha ∕2\right)}SE,\hat{\delta }+Z_{\left(\alpha ∕2\right)}SE\right],}$$ (15)

where SE is the standard error of a statistic.

The following algorithm is used to construct confidence interval based on bootstrap confidence interval with standard errors for the difference between the coefficients of variation of Weibull distributions:

Algorithm 4

1. Generate X₁, X₂, …, X_n from Weibull(a_X, k_X) and Y₁, Y₂, …, Y_m from Weibull(a_Y, k_Y).

2. For b = 1.

3. Resampling bootstrap samples $X_1^{\ast },X_2^{\ast },\dots ,X_n^{\ast }$ from X₁, X₂, …, X_n and compute ${\lambda }_X^{\ast b}$.

4. Resampling bootstrap samples $Y_1^{\ast },Y_2^{\ast },\dots ,Y_m^{\ast }$ from Y₁, Y₂, …, Y_m and compute ${\lambda }_Y^{\ast b}$.

5. Compute ${\delta }^{\ast b}={\lambda }_X^{\ast b}-{\lambda }_Y^{\ast b}$.

6. Repeat steps 3-5 for B times, where B is the number of bootstrap sample.

7. Compute SE.

8. Compute 95% confidence interval based on bootstrap confidence interval with standard errors, as given in Eq. (15).

The method of variance estimates recovery

According to Donner & Zou (2010), this approach can be used to construct a confidence interval for a function of two parameters, λ_X − λ_Y, defined as

$${\displaystyle C{I}_m=\left[L_m,U_m\right],}$$ (16)

where the lower limit and upper limit for ${\hat{\lambda }}_X$ - ${\hat{\lambda }}_Y$ are given by

$${\displaystyle L_m=\left({\hat{\lambda }}_X-{\hat{\lambda }}_Y\right)-\sqrt{{\left({\hat{\lambda }}_X-l_X\right)}^2+{\left(u_Y-{\hat{\lambda }}_Y\right)}^2}}$$ (17)

$${\displaystyle U_m=\left({\hat{\lambda }}_X-{\hat{\lambda }}_Y\right)+\sqrt{{\left(u_X-{\hat{\lambda }}_X\right)}^2+{\left({\hat{\lambda }}_Y-l_Y\right)}^2}.}$$ (18)

Hendricks & Robey (1936) presented confidence intervals for λ_X and λ_Y defined as

$${\displaystyle \left(l_{X.HR},u_{X.HR}\right)=\left({\hat{\lambda }}_X-t_{\left(\alpha ∕2,n-1\right)}\frac{{\hat{\lambda }}_X}{\sqrt{2n}},{\hat{\lambda }}_X+t_{\left(\alpha ∕2,n-1\right)}\frac{{\hat{\lambda }}_X}{\sqrt{2n}}\right)}$$ (19)

and

$${\displaystyle \left(l_{Y.HR},u_{Y.HR}\right)=\left({\hat{\lambda }}_Y-t_{\left(\alpha ∕2,m-1\right)}\frac{{\hat{\lambda }}_Y}{\sqrt{2m}},{\hat{\lambda }}_Y+t_{\left(\alpha ∕2,m-1\right)}\frac{{\hat{\lambda }}_Y}{\sqrt{2m}}\right),}$$ (20)

where t_{(α∕2,n−1)} and t_{(α∕2,m−1)} denote the 100(α/2)-th percentile of t-distribution with n − 1 and m − 1 degrees of freedom, respectively. To construct a confidence interval for the difference between the coefficients of variation of Weibull distributions based on MOVER with Hendricks and Robey’s confidence interval, we substitute l_X.HR, u_X.HR, l_Y.HR and u_Y.HR for l_X, u_X, l_Y and u_Y, respectively.

Therefore, the 100 (1 − α)% two-sided confidence interval for the difference between the coefficients of variation of Weibull distributions based on MOVER with Hendricks and Robey’s confidence interval becomes

$${\displaystyle C{I}_{\delta \left(m.HR\right)}=\left[L_{\delta \left(m.HR\right)},U_{\delta \left(m.HR\right)}\right],}$$ (21)

where

$${\displaystyle L_{\delta \left(m.HR\right)}=\left({\hat{\lambda }}_X-{\hat{\lambda }}_Y\right)-\sqrt{{\left({\hat{\lambda }}_X-l_{X.HR}\right)}^2+{\left(u_{Y.HR}-{\hat{\lambda }}_Y\right)}^2}}$$ (22)

$${\displaystyle U_{\delta \left(m.HR\right)}=\left({\hat{\lambda }}_X-{\hat{\lambda }}_Y\right)+\sqrt{{\left(u_{X.HR}-{\hat{\lambda }}_X\right)}^2+{\left({\hat{\lambda }}_Y-l_{Y.HR}\right)}^2}.}$$ (23)

The following algorithm is used to construct confidence interval based on MOVER for the difference between the coefficients of variation of Weibull distributions:

Algorithm 5

1. Generate X₁, X₂, …, X_n from Weibull(a_X, k_X) and Y₁, Y₂, …, Y_m from Weibull(a_Y, k_Y).

2. Compute the intervals for λ_X and λ_Y from Eqs. (19) and (20).

3. Compute 95% confidence interval based on MOVER, as given in Eq. (21).

The Bayesian confidence intervals

Here, we derive Bayesian estimates of the parameters of a Weibull distribution. Once again, for random variable X = (X₁, X₂, …, X_n) from a Weibull distribution, we transform scale parameter $a^{\prime }={\left(\frac{1}{a}\right)}^k$, and so we can provide its probability density function in another form as follows:

$${\displaystyle f\left(x;a^{\prime },k\right)=a^{\prime }k{x}^{k-1}exp\left[-a^{\prime }{x}^k\right],x>0.}$$ (24)

It is assumed that the prior distributions for the scale and shape parameters are gamma with hyperparameters (v₁, v₂, z₁, z₂):

$${\displaystyle \pi \left(a^{\prime }\right)\sim gamma\left(v_1,z_1\right)}$$ (25)

and

$${\displaystyle \pi \left(k\right)\sim gamma\left(v_2,z_2\right).}$$ (26)

Hence, using Bayes’ theorem, the joint posterior density function of a′ and k is given by

$${\displaystyle \pi \left(a^{\prime },k|x\right)\propto L\left(a^{\prime },k|x\right)\pi \left(a^{\prime }\right)\pi \left(k\right),}$$ (27)

where L(a′, k|x) is a likelihood function.

However, the posterior distributions cannot be computed, and so we used the MCMC method with Gibbs sampling to provide them for the parameters. We used the Gibbs sampling procedure to generate samples from the joint posterior distribution, as given in Eq. (27). The conditional posterior distributions of parameters are

$${\displaystyle \pi \left(a^{\prime }|k,x\right)\sim gamma\left(v_2+n,z_2+\sum x^k\right)}$$ (28)

and

$${\displaystyle \pi \left(k|a^{\prime },x\right)\propto k^{v_1+n-1}exp\left[-v_{1}k-a^{\prime }\sum x^k\right].}$$ (29)

For Eq. (29), we used the Metropolis-Hasting (MH) algorithm to update shape parameter k. Moreover, the random walk Metropolis (RWM) and Gibbs sampling methods are applied as follows (Geman & Geman, 1984):

Algorithm 6 The Gibbs algorithm

1. Take the initial value of parameter (a′⁽⁰⁾, k⁽⁰⁾).

2. Generate a′^(t) ∼ gamma(v₂ + n, z₂ + ∑x^k(t−1)).

3. Using Random walk Metropolis (RWM) algorithm for calculate k^(t).

4. Repeat step 2–3 for T times, where T is the number of replications of MCMC.

5. Burn in 1000 samples and compute the parameter of interest.

Algorithm 7 The random walk Metropolis (RWM)

1. Start with (a′^(t), k^(t−1)).

2. Generate $\epsilon \sim N\left(0,{\sigma }_k^2\right)$.

3. Compute k^∗ = k^(t−1) + ε.

4. Compute $A_k=\frac{L\left(k^{\ast },a^{\prime }|x\right)\pi \left(k^{\ast }\right)}{L\left(k,a^{\prime }|x\right)\pi \left(k\right)}$.

5. Generate u ∼ U(0, 1).

6. If u ≤ min(1, A_k) set k^(t) = k^∗ and If u > min(1, A_k) set k^(t) = k^(t−1).

The Bayesian-MCMC

Once again, let X and Y be random variables from Weibull distributions. First, we used Algorithms 6 and 7 to calculate the difference between the coefficients of variation denoted as δ^t, t =1 , 2, …, T, based on the Bayesian-MCMC.

Therefore, the 100 (1 − α)% two-sided confidence interval for the difference between the coefficients of variation of Weibull distributions based on the Bayesian-MCMC is given by

$${\displaystyle C{I}_{\delta \left(MCMC\right)}=\left[L_{\delta \left(MCMC\right)},U_{\delta \left(MCMC\right)}\right]=\left[{\delta }^t\left(\alpha ∕2\right),{\delta }^t\left(1-\alpha ∕2\right)\right],}$$ (30)

where ${\delta }^t\left(\alpha ∕2\right)$ denotes the 100(α/2)-th percentile of δ^t.

The following algorithm is used to construct confidence interval based on Bayesian-MCMC for the difference between the coefficients of variation of Weibull distributions:

Algorithm 8

1. Calculate the initial value of parameter (a′⁽⁰⁾, k⁽⁰⁾).

2. Calculate δ^t from Algorithms 6 and 7.

3. Compute 95% confidence interval based on Bayesian-MCMC, as given in Eq. (30).

The Bayesian-Highest Posterior Density (HPD) Interval

Here, the assumption is that the density of every point inside the interval is greater than that of every point outside the interval, and it is also the shortest interval (Box & Tiao, 2011). In this article, the HPD interval was computed by using the R package with HDInterval.

Therefore, the 100 (1 − α)% two-sided confidence interval for the difference between the coefficients of variation of Weibull distributions based on the Bayesian-HPD interval can be written as

$${\displaystyle C{I}_{\delta \left(HPD\right)}=\left[L_{\delta \left(HPD\right)},U_{\delta \left(HPD\right)}\right].}$$ (31)

The following algorithm is used to construct confidence interval based on Bayesian-HPD for the difference between the coefficients of variation of Weibull distributions:

Algorithm 9

1. Calculate the initial value of parameter (a′⁽⁰⁾, k⁽⁰⁾).

2. Calculate δ^t from Algorithms 6 and 7.

3. Compute 95% confidence interval based on Bayesian-HPD interval, as given in Eq. (31).

Results

A simulation study was conducted to compare the performances of the six confidence intervals for the difference between the coefficients of variation of Weibull distributions: GCI, the Bayesian-MCMC method, the Bayesian-HPD interval, MOVER based on Hendricks and Robey’s confidence interval, the percentile bootstrap method, and the bootstrap confidence interval with standard errors. The R program was used to estimate the coverage probabilities and expected lengths of the proposed confidence intervals using the following parameter settings:

Scale parameter a_X = a_Y = 0.5 and 2.

Shape parameter k_X = 1 and k_Y = 0.5, 1, 2, 2.5, 4 and 9, with the differences between the coefficients of variation of −1.2360, 0, 0.4772, 0.5720, 0.7194 and 0.8671, respectively.

Sample sizes (n, m) = (10,10), (10,20), (20,20), (30,30), (30,50), (50,50), (50,100) and (100,100).

Hyperparameters v₁ = v₂ = z₁ = z₂ = 0.1.

The number of replications for each situation was 5,000, along with 2,500 pivotal quantities for GCI. Moreover, we used 500 bootstrap samples for the bootstrap methods and we generated 20,000 realizations of MCMC using the Gibbs and RWM algorithms with a burn-in of 1000. The following performance indicators were used to determine the best-performing method: a coverage probability of greater than or equal to 0.95 (the nominal confidence level) and the shortest expected length.

The following algorithm is used to estimate the coverage probability and expected length:

Algorithm 10

1. Set M, q, B, T, n, m, a_X, k_X, a_Y and k_Y.

2. Generate X₁, X₂, …, X_n from Weibull(a_X, k_X) and Y₁, Y₂, …, Y_m from Weibull(a_Y, k_Y), respectively.

3. Use Algorithm 2 to construct generalized confidence interval (CI_δ(gci)).

4. Use Algorithm 3 to construct percentile bootstrap confidence interval (CI_δ(pb)).

5. Use Algorithm 4 to construct bootstrap confidence interval with standard errors (CI_δ(bs)).

6. Use Algorithm 5 to construct MOVER based on Hendricks and Robey’s confidence interval (CI_δ(m.HR)).

7. Use Algorithm 8 to construct Baysian-MCMC confidence interval (CI_δ(MCMC)).

8. Use Algorithm 9 to construct Baysian-HPD confidence interval (CI_δ(HPD)).

9. If (L ≤ δ ≤ U), then set P = 1, else set P = 0.

10. Compute (U − L).

11. Repeat steps 2-10 for M times.

12. Compute mean of P for the coverage probability.

13. Compute mean of (U − L) for the expected length.

For a = 0.5 (Table 1 and Fig. 1), the simulation results indicate that the coverage probabilities of CI_δ(gci) were greater than or sometimes close to the nominal confidence level of 0.95 in almost all cases. CI_δ(pb) yielded coverage probabilities greater than 0.95 when k_Y = 1 for (n, m) = (10, 10) and (20, 20). For the Bayesian methods, the coverage probabilities of CI_δ(MCMC) and CI_δ(HPD) were greater than or sometimes close to the nominal confidence level, while the expected lengths of CI_δ(HPD) for (n, m) = (10, 10) and k_Y = 0.5, 2, 4; (n, m) = (20, 20) and k_Y = 0.5; (n, m) = (10, 20), (30, 30), (30, 50), (50, 50), (50, 100) and k_Y = 0.5, 1; and (n, m) = (100, 100) and k_Y = 1 were the shortest. Moreover, CI_δ(bs) and CI_δ(m.HR) yielded coverage probabilities under 0.95 for all cases.

Table 1. Comparison results of the 95% two-sided confidence intervals for the difference between the coefficients of variation of Weibull distributions for a = 0.5.

(n, m)	k	Coverage probability (Expected length)
		CIδ(gci)	CIδ(pb)	CIδ(bs)	CIδ(m.HR)	CIδ(MCMC)	CIδ(HPD)
(10,10)	0.5	0.9568	0.6324	0.6956	0.8430	0.9392	0.9514
		(8.1452)	(1.6874)	(1.7114)	(2.3678)	(5.4003)	(4.6368)
	1	0.9494	0.9614	0.9032	0.9610	0.9388	0.9706
		(2.3498)	(1.1568)	(1.1587)	(1.3535)	(1.8406)	(1.7986)
	2	0.9510	0.9046	0.8860	0.9410	0.9454	0.9578
		(1.7089)	(0.9219)	(0.9248)	(1.0787)	(1.3696)	(1.3018)
	2.5	0.9492	0.8710	0.8610	0.9152	0.9370	0.9400
		(1.6386)	(0.8863)	(0.8894)	(1.0359)	(1.3144)	(1.2340)
	4	0.9580	0.8604	0.8526	0.9026	0.9360	0.9510
		(1.5526)	(0.8366)	(0.8404)	(0.9827)	(1.2451)	(1.1469)
	9	0.9510	0.8374	0.8334	0.8854	0.9120	0.8764
		(1.5173)	(0.8076)	(0.8105)	(0.9515)	(1.2218)	(1.1060)
(10,20)	0.5	0.9564	0.7056	0.7924	0.8598	0.9476	0.9664
		(3.6785)	(1.6034)	(1.6216)	(1.7373)	(3.1102)	( 2.9756)
	1	0.9504	0.9378	0.8948	0.9344	0.9416	0.9616
		(1.8251)	(1.0235)	(1.0258)	(1.1510)	(1.5018)	(1.4525)
	2	0.9536	0.8620	0.8536	0.9018	0.9406	0.9362
		(1.5737)	(0.8642)	(0.8664)	(1.0037)	(1.2677)	(1.1743)
	2.5	0.9543	0.8480	0.8448	0.8952	0.9382	0.9300
		(1.5418)	(0.8407)	(0.8432)	(0.9810)	(1.2400)	(1.1376)
	4	0.9488	0.8380	0.8312	0.8836	0.9386	0.9154
		(1.5419)	(0.8264)	(0.8302)	(0.9668)	(1.2290)	(1.1124)
	9	0.9482	0.8262	0.8250	0.8786	0.9204	0.8794
		(1.5300)	(0.8106)	(0.8144)	(0.9544)	(1.2208)	(1.0958)
(20,20)	0.5	0.9524	0.6292	0.7778	0.8282	0.9480	0.9536
		(3.2987)	(1.4974)	(1.5220)	(1.5855)	(2.8839)	(2.6795)
	1	0.9530	0.9564	0.9170	0.9450	0.9444	0.9656
		(1.2350)	(0.8758)	(0.8814)	(0.9149)	(1.1264)	(1.1166)
	2	0.9522	0.9024	0.9010	0.9330	0.9434	0.9486
		(0.9234)	(0.6836)	(0.6889)	(0.7236)	(0.8471)	(0.8260)
	2.5	0.9526	0.8900	0.8918	0.9276	0.9458	0.9470
		(0.8915)	(0.6566)	(0.6615)	(0.6969)	(0.8178)	(0.7917)
	4	0.9534	0.8730	0.8870	0.9166	0.9430	0.9344
		(0.8589)	(0.6348)	(0.6403)	(0.6666)	(0.7866)	(0.7532)
	9	0.9522	0.8652	0.8720	0.9114	0.9194	0.8866
		(0.8395)	(0.6149)	(0.6225)	(0.6479)	(0.7729)	(0.7340)
(30,30)	0.5	0.9506	0.6428	0.8124	0.8228	0.9448	0.9501
		(2.3298)	(1.3805)	(1.4109)	(1.2717)	(2.1510)	(2.0452)
	1	0.9522	0.9426	0.9244	0.9382	0.9480	0.9618
		(0.9229)	(0.7415)	(0.7481)	(0.7346)	(0.8728)	(0.8678)
	2	0.9510	0.8972	0.9082	0.9298	0.9462	0.9482
		(0.6990)	(0.5770)	(0.5819)	(0.5820)	(0.6621)	(0.6506)
	2.5	0.9482	0.8918	0.9064	0.9280	0.9452	0.9476
		(0.6787)	(0.5581)	(0.5629)	(0.5625)	(0.6430)	(0.6287)
	4	0.9532	0.8776	0.9036	0.9212	0.9476	0.9394
		(0.6526)	(0.5364)	(0.5429)	(0.5370)	(0.6171)	(0.5991)
	9	0.9446	0.8602	0.8814	0.9036	0.9080	0.8784
		(0.6365)	(0.5187)	(0.5246)	(0.5207)	(0.6040)	(0.5835)
(30,50)	0.5	0.9574	0.6856	0.8760	0.8416	0.9530	0.9634
		(1.7122)	(1.2865)	(1.3262)	(1.0347)	(1.6309)	(1.5927)
	1	0.9400	0.9356	0.9210	0.9240	0.9360	0.9500
		(0.8011)	(0.6768)	(0.6822)	(0.6526)	(0.7652)	(0.7591)
	2	0.9494	0.8848	0.8996	0.9226	0.9452	0.9448
		(0.6690)	(0.5518)	(0.5568)	(0.5556)	(0.6354)	(0.6203)
	2.5	0.9462	0.8812	0.8948	0.9160	0.9452	0.9418
		(0.6581)	(0.5399)	(0.5455)	(0.5440)	(0.6246)	(0.6074)
	4	0.9450	0.8696	0.8880	0.9098	0.9414	0.9382
		(0.6447)	(0.5297)	(0.5359)	(0.5289)	(0.6095)	(0.5897)
	9	0.9498	0.8608	0.8878	0.9056	0.9214	0.8936
		(0.6306)	(0.5144)	(0.5208)	(0.5170)	(0.5981)	(0.5769)
(50,50)	0.5	0.9520	0.6502	0.8596	0.8202	0.9474	0.9504
		(1.6289)	(1.2294)	(1.2703)	(0.9739)	(1.5586)	(1.5103)
	1	0.9474	0.9368	0.9330	0.9278	0.9456	0.9552
		(0.6699)	(0.6106)	(0.6165)	(0.5626)	(0.6485)	(0.6460)
	2	0.9456	0.9058	0.9208	0.9304	0.9460	0.9480
		(0.5171)	(0.4724)	(0.4766)	(0.4489)	(0.5010)	(0.4951)
	2.5	0.9512	0.8900	0.9178	0.9264	0.9448	0.9450
		(0.4992)	(0.4545)	(0.4586)	(0.4319)	(0.4840)	(0.4770)
	4	0.9558	0.8844	0.9154	0.9228	0.9482	0.9412
		(0.4796)	(0.4384)	(0.4432)	(0.4119)	(0.4640)	(0.4552)
	9	0.9458	0.8744	0.9082	0.9148	0.9132	0.8908
		(0.4695)	(0.4246)	(0.4299)	(0.4008)	(0.4564)	(0.4463)
(50,100)	0.5	0.9532	0.7166	0.9172	0.8386	0.9518	0.9580
		(1.1384)	(1.1032)	(1.1514)	(0.7422)	(1.1119)	(1.0987)
	1	0.9584	0.9364	0.9448	0.9346	0.9538	0.9610
		(0.5672)	(0.5404)	(0.5448)	(0.4843)	(0.5522)	(0.5486)
	2	0.9480	0.8898	0.9176	0.9244	0.9460	0.9484
		(0.4897)	(0.4464)	(0.4505)	(0.4230)	(0.4759)	(0.4684)
	2.5	0.9518	0.8820	0.9098	0.9116	0.9496	0.9442
		(0.4814)	(0.4380)	(0.4425)	(0.4145)	(0.4668)	(0.4581)
	4	0.9494	0.8734	0.9082	0.9164	0.9466	0.9418
		(0.4700)	(0.4309)	(0.4361)	(0.4041)	(0.4567)	(0.4471)
	9	0.9486	0.8704	0.9006	0.9074	–	–
		(0.4648)	(0.4206)	(0.4253)	(0.3974)	–	–
(100,100)	0.5	0.9488	0.6768	0.9102	0.8198	0.9468	0.9472
		(1.0705)	(1.0617)	(1.1132)	(0.6829)	(1.0486)	(1.0308)
	1	0.9496	0.9362	0.9548	0.9266	0.9496	0.9524
		(0.4523)	(0.4660)	(0.4692)	(0.3949)	(0.4453)	(0.4440)
	2	0.9434	0.9076	0.9346	0.9256	0.9444	0.9460
		(0.3514)	(0.3558)	(0.3584)	(0.3152)	(0.3461)	(0.3437)
	2.5	0.9514	0.9060	0.9386	0.9286	0.9486	0.9492
		(0.3394)	(0.3435)	(0.3462)	(0.3032)	(0.3341)	(0.3313)
	4	0.9532	0.9004	0.9390	0.9270	0.9516	0.9484
		(0.3269)	(0.3344)	(0.3376)	(0.2896)	(0.3214)	(0.3179)
	9	0.9514	0.8954	0.9336	0.9224	–	–
		(0.3188)	(0.3223)	(0.3250)	(0.2809)	–

Figure 1. Graphs to compare the performance of the methods with varying difference between CVs in terms of (A) coverage probability and (B) expected length.

From the simulation results for a = 2 (Table 2 and Fig. 2), CI_δ(gci) performed well in terms of coverage probability in almost all cases, while those of CI_δ(pb), CI_δ(bs) and CI_δ(m.HR) were the same as for a = 0.5.CI_δ(pb) yielded coverage probabilities of under 0.95 in almost all cases except when (n, m) = (10, 10), (20, 20) and k_Y = 0.5. For the Bayesian methods, the coverage probabilities of CI_δ(HPD) were greater than 0.95 and its expected lengths were shortest for (n, m) = (10, 20), (30, 50), (50, 100) and k_Y = 0.5, 1; (n, m) = (20, 20) and k_Y = 2; (n, m) = (30, 30) and k_Y = 1, 2, 2.5; (n, m) = (50, 50) and k_Y = 0.5, 1, 2.5; and (n, m) = (100, 100) for all k_Y except for k_Y = 4 and 9.

Table 2. Comparison results of the 95% two-sided confidence intervals for the difference between the coefficients of variation of Weibull distributions for a = 2.

(n, m)	k	Coverage probability (Expected length)
		CIδ(gci)	CIδ(pb)	CIδ(bs)	CIδ(m.HR)	CIδ(MCMC)	CIδ(HPD)
(10,10)	0.5	0.9512	0.6242	0.6852	0.8266	0.9368	0.9466
		(7.9720)	(1.6763)	(1.6970)	(2.3414)	(5.2853)	(4.5489)
	1	0.9514	0.9630	0.9034	0.9632	0.9338	0.9674
		(2.3568)	(1.1613)	(1.1622)	(1.3561)	(1.8368)	(1.7938)
	2	0.9430	0.8992	0.8744	0.9330	0.9300	0.9482
		(1.7036)	(0.9226)	(0.9261)	(1.0758)	(1.3570)	(1.2881)
	2.5	0.9476	0.8730	0.8622	0.9128	0.9348	0.9396
		(1.6354)	(0.8877)	(0.8907)	(1.0342)	(1.3011)	(1.2200)
	4	0.9474	0.8532	0.8490	0.8958	0.9294	0.9290
		(1.5569)	(0.8388)	(0.8411)	(0.9825)	(1.2356)	(1.1332)
	9	0.9472	0.8552	0.8830	0.9058	0.9402	0.9388
		(0.6355)	(0.5167)	(0.5229)	(0.5159)	(0.5988)	(0.5770)
(10,20)	0.5	0.9500	0.7112	0.8002	0.8624	0.9398	0.9596
		(3.6607)	(1.5948)	(1.1618)	(1.7329)	(3.0961)	(2.9601)
	1	0.9502	0.9366	0.9000	0.9358	0.9394	0.9576
		(1.8261)	(1.0249)	(1.0273)	(1.1529)	(1.4955)	(1.451)
	2	0.9512	0.8626	0.8586	0.9088	0.9380	0.9370
		(1.5731)	(0.8627)	(0.8655)	(1.0039)	(1.2633)	(1.1696)
	2.5	0.9470	0.8464	0.8390	0.8936	0.9330	0.9260
		(1.5644)	(0.8484)	(0.8518)	(0.9886)	(1.2476)	(1.1432)
	4	0.9604	0.8326	0.8300	0.8782	0.9324	0.9186
		(1.5208)	(0.8193)	(0.8219)	(0.9596)	(1.2102)	(1.0956)
	9	0.9544	0.8356	0.8326	0.8840	0.9382	0.9204
		(1.5123)	(0.8095)	(0.8134)	(0.9496)	(1.1960)	(1.0712)
(20,20)	0.5	0.9538	0.6378	0.7808	0.8274	0.9440	0.9484
		(3.2994)	(1.5021)	(1.5258)	(1.5847)	(2.8817)	(2.6754)
	1	0.9478	0.9510	0.9156	0.9420	0.9402	0.9598
		(1.2330)	(0.8735)	(0.8781)	(0.9134)	(1.1231)	(1.1132)
	2	0.9514	0.9008	0.9038	0.9346	0.9420	0.9526
		(0.9275)	(0.6855)	(0.6899)	(0.7258)	(0.8490)	(0.8271)
	2.5	0.9512	0.8846	0.8936	0.9220	0.9414	0.9460
		(0.8922)	(0.6606)	(0.6657)	(0.6971)	(0.8163)	(0.7900)
	4	0.9506	0.8738	0.8844	0.9186	0.9430	0.9468
		(0.8582)	(0.6286)	(0.6334)	(0.6660)	(0.7847)	(0.7507)
	9	0.9486	0.8564	0.8692	0.9058	0.9370	0.9314
		(0.8336)	(0.6079)	(0.6133)	(0.6444)	(0.7584)	(0.7186)
(30,30)	0.5	0.9540	0.6384	0.8140	0.8240	0.9494	0.9486
		(2.3309)	(1.3775)	(1.4077)	(1.2727)	(2.1542)	(2.0485)
	1	0.9482	0.9486	0.9278	0.9346	0.9458	0.9600
		(0.9231)	(0.7463)	(0.7532)	(0.7347)	(0.8719)	(0.8667)
	2	0.9570	0.9056	0.9134	0.9316	0.9508	0.9520
		(0.7005)	(0.5752)	(0.5798)	(0.5827)	(0.6629)	(0.6512)
	2.5	0.9568	0.8922	0.9074	0.9314	0.9530	0.9536
		(0.6806)	(0.5584)	(0.5636)	(0.5635)	(0.6437)	(0.6293)
	4	0.9484	0.8700	0.8902	0.9090	0.9404	0.9376
		(0.6537)	(0.5371)	(0.5435)	(0.5374)	(0.6176)	(0.5992)
	9	0.9528	0.8602	0.8876	0.9068	0.9436	0.9378
		(0.6341)	(0.5199)	(0.5262)	(0.5194)	(0.5916)	(0.5747)
(30,50)	0.5	0.9532	0.6688	0.8768	0.8408	0.9496	0.9568
		(1.7003)	(1.2713)	(1.3092)	(1.0311)	(1.6218)	(1.5842)
	1	0.9540	0.9416	0.9306	0.9352	0.9500	0.9576
		(0.8051)	(0.6830)	(0.6899)	(0.6549)	(0.7671)	(0.7609)
	2	0.9504	0.8884	0.9036	0.9200	0.9464	0.9444
		(0.6717)	(0.5544)	(0.5595)	(0.5569)	(0.6366)	(0.6212)
	2.5	0.9512	0.8738	0.8988	0.9168	0.9462	0.9412
		(0.6576)	(0.5405)	(0.5464)	(0.5436)	(0.6227)	(0.6055)
	4	0.9528	0.8776	0.8914	0.9124	0.9474	0.9436
		(0.6435)	(0.5277)	(0.5341)	(0.5288)	(0.6088)	(0.5888)
	9	0.9472	0.8552	0.8830	0.9058	0.9402	0.9388
		(0.6355)	(0.5167)	(0.5229)	(0.5159)	(0.5988)	(0.5770)
(50,50)	0.5	0.9532	0.6506	0.8538	0.8144	0.9484	0.9532
		(1.6366)	(1.2485)	(0.9732)	(0.9766)	(1.5637)	(1.5147)
	1	0.9498	0.9378	0.9380	0.9338	0.9470	0.9564
		(0.6698)	(0.6120)	(0.6178)	(0.5627)	(0.6484)	(0.6458)
	2	0.9512	0.9012	0.9158	0.9288	0.9476	0.9486
		(0.5152)	(0.4690)	(0.4729)	(0.4476)	(0.4990)	(0.4931)
	2.5	0.9498	0.8882	0.9198	0.9300	0.9478	0.9500
		(0.4994)	(0.4542)	(0.4588)	(0.4320)	(0.4834)	(0.4764)
	4	0.9484	0.8852	0.9108	0.9212	0.9464	0.9436
		(0.4791)	(0.4367)	(0.4416)	(0.4116)	(0.4637)	(0.4549)
	9	0.9496	0.8812	0.9072	0.9142	0.9446	0.9430
		(0.4689)	(0.4254)	(0.4299)	(0.4005)	(0.4520)	(0.4416)
(50,100)	0.5	0.9546	0.7092	0.9182	0.8404	0.9522	0.9566
		(1.1315)	(1.1025)	(1.1526)	(0.7399)	(1.1044)	(1.0913)
	1	0.9506	0.9314	0.9384	0.9272	0.9508	0.9562
		(0.5698)	(0.5402)	(0.5444)	(0.4860)	(0.5548)	(0.5511)
	2	0.9450	0.8936	0.9102	0.9134	0.9414	0.9390
		(0.4902)	(0.4502)	(0.4552)	(0.4232)	(0.4749)	(0.4670)
	2.5	0.9530	0.8856	0.9162	0.9254	0.9524	0.9490
		(0.4802)	(0.4380)	(0.4432)	(0.4141)	(0.4659)	(0.4572)
	4	0.9512	0.8882	0.9138	0.9198	0.9470	0.9454
		(0.4730)	(0.4317)	(0.4369)	(0.4052)	(0.4579)	(0.4482)
	9	0.9504	0.8750	0.9034	0.9094	–	–
		(0.4658)	(0.4256)	(0.4319)	(0.3979)	–	–
(100,100)	0.5	0.9468	0.6818	0.9048	0.8158	0.9496	0.9468
		(1.0733)	(1.0585)	(1.1081)	(0.6835)	(1.0506)	(1.0325)
	1	0.9498	0.9374	0.9524	0.9300	0.9496	0.9512
		(0.4518)	(0.4644)	(0.4673)	(0.3944)	(0.4446)	(0.4432)
	2	0.9552	0.9142	0.9438	0.9364	0.9538	0.9544
		(0.3505)	(0.3527)	(0.3548)	(0.3145)	(0.3449)	(0.3424)
	2.5	0.9530	0.9043	0.9424	0.9294	0.9504	0.9504
		(0.3393)	(0.3447)	(0.3475)	(0.3032)	(0.3340)	(0.3310)
	4	0.9472	0.8916	0.9320	0.9166	0.9446	0.9428
		(0.3259)	(0.3323)	(0.3351)	(0.2889)	(0.3206)	(0.3170)
	9	0.9484	0.8944	0.9304	0.9170	–	–
		(0.3189)	(0.3253)	(0.3286)	(0.2810)	–

Figure 2. Graphs to compare the performance of the methods with varying sample size in terms of (A) coverage probability and (B) expected length.

Empirical Application of the Derived Confidence Intervals

Example 1

Wind speed data set were collected at 90-meter wind energy potential stations in Trad and Chonburi provinces in 2016 by the Department of Alternative Energy Development and Efficiency, Ministry of Energy (https://www.dede.go.th/ewt_news.php?nid=47706). These are coastal provinces in eastern Thailand that have been identified as potential areas for harvesting wind energy. Figure 3 shows Q-Q plots of the Weibull distributions of the data from the two provinces. Furthermore, the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) values reported in Table 3 indicate that these two datasets fit Weibull distributions because the latter have the smallest values for both criteria.

Figure 3. The Weibull Q-Q plot for the wind speed data of two provinces, Thailand: (A) Trad Province (B) Chonburi Province.

Table 3. The AIC and BIC values of the wind speed data of Trad province and Chonburi province.

Method	Trad		Chonburi
	AIC	BIC	AIC	BIC
Weibull	37.0272	37.9970	22.8054	23.6011
Exponential	37.2675	37.7524	56.2471	56.6450
Gamma	37.5735	38.5433	24.0780	24.8737
Log-normal	39.1594	40.1292	24.3327	25.1285

The statistical summary of the estimates of the parameters for these datasets is (n, m) = (12,11), ${\hat{a}}_X=1.7621$, ${\hat{a}}_Y=4.5839$, ${\hat{k}}_X=1.4832$, ${\hat{k}}_Y=9.0794$, ${\hat{\lambda }}_X=0.6860$ and ${\hat{\lambda }}_Y=0.1317$, while the actual difference between the coefficients of variation of the two Weibull distributions $\hat{\delta }=0.5543$. The confidence intervals based on the proposed methods are given in Table 4. The results show that the Bayesian-HPD interval performed well in terms of the coverage probability and expected length when the sample sizes are small, which supports the simulation results. Finally, Fig. 4 shows a trace plot of the generated δ value.

Table 4. The 95% confidence intervals for the difference between the coefficients of variation of the wind speed data of Trad and Chonburi provinces, Thailand.

Method	Confidence intervals for δ
	Lower	Upper	Length
GCI	0.3316	1.1013	0.7697
MOVER	0.2398	0.8689	0.6291
PB	0.2081	0.8036	0.5155
BS	0.2531	0.8555	0.6024
MCMC	0.2912	1.0095	0.7183
HPD	0.2600	0.9216	0.6616

Figure 4. Plot of generated δ of example 1 vs. iteration of the MCMC algorithm.

Example 2

Data on wind speeds measured at 90-meter wind energy potential stations in the southern and northeastern regions of Thailand were collected in April–May, 2019 by the Department of Alternative Energy Development and Efficiency, Ministry of Energy (https://www.dede.go.th/more_news.php?cid=501). The summary statistics for the southern region are n = 12, ${\hat{a}}_X=2.5334$, ${\hat{k}}_X=2.3956$ and ${\hat{\lambda }}_X=0.4441$ and the northeastern region are m = 20, ${\hat{a}}_Y=4.4608$, ${\hat{k}}_Y=3.8457$ and ${\hat{\lambda }}_Y=0.2906$. The actual difference between the coefficients of variation of the Weibull distributions of these datasets $\hat{\delta }=0.1538$. Weibull Q-Q plots of these data are shown in Fig. 5 and the correctness of fitting the data to Weibull distributions in terms of the smallest AIC and BIC values are reported in Table 5. The 95% confidence intervals for δ by applying the six methods (Table 6) indicate that the Bayesian-HPD interval performed well in terms of the coverage probability and expected length when the difference between the coefficients of variation was small, which is once again consistent with the simulation study results. Finally, the trace plot of the generated δ value are presented in Fig. 6.

Figure 5. The Weibull Q-Q plot for the wind speed data of two Thailand’s regions: (A) Southern (B) Northeastern.

Table 5. The AIC and BIC values of the wind speed data of the southern and northeastern regions of Thailand.

Method	Southern		Northeastern
	AIC	BIC	AIC	BIC
Weibull	37.9176	38.8871	68.3758	70.3673
Exponential	45.4623	45.9472	97.6711	98.6669
Gamma	39.2535	75.5790	71.5482	73.5397
Log-normal	41.5439	42.5137	73.5875	40.2233

Table 6. The 95% confidence intervals for the difference between the coefficients of variation of the wind speed data of the southern and northeastern regions of Thailand.

Method	Confidence intervals for δ
	Lower	Upper	Length
GCI	−0.0270	0.4807	0.5077
MOVER	−0.0678	0.3755	0.4435
PB	−0.0450	0.3763	0.4213
BS	−0.0553	0.3630	0.4183
MCMC	−0.0241	0.4727	0.4768
HPD	−0.0445	0.4267	0.4712

Figure 6. Plot of generated δ of example 2 vs. iteration of the MCMC algorithm.

Discussion

It can be seen from the results of the study that GCI performed well in almost all cases since its coverage probability was greater than or close to the nominal confidence level. For the Bayesian methods, Bayesian-MCMC and Bayesian-HPD interval produced similar results and performed well when k_Y (the difference between the coefficients of variation) was small, zero, or negative. For large sample sizes, the Bayesian methods were not suitable when k_Y was large. Moreover, the expected length of the Bayesian-HPD interval was always shorter than Bayesian-MCMC. In relation to the expected lengths, they tended to decrease when the sample sizes and/or k_Y increased.

Conclusions

This paper proposed six methods for deriving the confidence interval for the difference between the coefficients of variation of Weibull distributions: the GCI method, the Bayesian-MCMC method, the Bayesian-HPD interval, MOVER based on Hendricks and Robey’s confidence interval, the percentile bootstrap method, and the bootstrap confidence interval with standard errors. The performances of these methods were compared using the coverage probability and the expected length. The results of simulation studies showed that the GCI method and the Bayesian-HPD method were the best in different scenarios. The Bayesian-HPD was preferable when shape parameter k_Y was small. The percentile bootstrap method can be used when the sample sizes are small and k_Y = 1. However, MOVER based on Hendricks and Robey’s confidence interval and the bootstrap confidence interval with standard errors are not recommended since these methods yielded coverage probabilities under the nominal confidence level of 0.95 for almost all cases.

Supplemental Information

Supplemental Information 2. The wind speeds measured at 90-meter wind energy potential stations in the southern and northeastern regions of Thailand.

Collected April-May 2019.

Funding Statement

This research was funded by King Mongkut’s University of Technology North Bangkok. Contract no. KMUTNB-PHD-62-04 The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.