Confidence Intervals Based on the Difference of Medians for Independent Log-Normal Distributions

Abstract

In this paper, we study the inferences of the difference of medians for two independent log-normal distributions. These methods include traditional methods such as the parametric bootstrap approach, the normal approximation approach, the method of variance estimates recovery approach, and the generalized confidence interval approach. The simultaneous confidence intervals for the difference in the median for more than two independent log-normal distributions are also discussed. Our simulation studies evaluate the performances of the proposed confidence intervals in terms of coverage probabilities and average lengths. We find that the parametric bootstrap approach would be a suitable choice for smaller sample sizes for the two independent distributions and multiple independent distributions. However, the method of variance estimates recovery and normal approximation approaches are alternative competitors for constructing simultaneous confidence intervals, especially when the populations have large variance. We also include two practical applications demonstrating the use of the techniques on observed data, where one data set works for the PM2.5 mass concentrations in Bangkapi and Dindaeng in Thailand and the other data came from the study of nitrogen-bound bovine serum albumin produced by three groups of diabetic mice. Both applications show that the confidence intervals from the parametric bootstrap approach have the smallest length.

1. Introduction

Random variables that are inherently positive occur in many real-life applications. In biological and medical research, the real data always has the properties of being strictly positive, right-skewed and possibly having heterogeneous variances. A frequently used approach for analyzing this data is to log-transform the observations and assume normality and then apply standard methods and interpret the back-transformed confidence intervals. This approach leads to the confidence intervals for the mean values, variance and coefficient of covariance based on the single treatment, and this has been demonstrated by several scholars (see Land [1], Angus [2], Zhou and Gao [3], Krishnammoorthy and Mathew [4], Olsson [5] and Harvey and Van der Merwe [6]). Another option is to study the comparison of two independent log-normal distributions in terms of the differences or ratios of their means. Zhou et al. [7] studied the parametric z-score approach and nonparametric bootstrap approach to compare the means of two log-normal samples when the variances of the log-transformed data are unequal. Zhou and Tu [8] considered constructing confidence intervals for the ratio in the means of two independent populations and proposed a maximum likelihood-based method with a two-stage bootstrap approach. Wu et al. [9] proposed two likelihood-based approaches for inference about the ratio of means of two independent log-normal distributions. Chen and Zhou [10] explored methods of constructing confidence intervals for estimating the ratio or difference of two log-normal means and discussed how to conduct hypothesis tests. Gupta and Li [11] presented a score test for testing the equality of the means of two independent log-normal populations. Jafari and Abdollahnezhad [12] proposed a novel approach for testing the equality of two log-normal populations by using the computational approach test, which does not require explicit knowledge of the sampling distribution of the test statistics.

As we know, the median survival time is reported more than the mean survival time in medical studies. The log-normal distribution is positively skewed and has extremely low measurements, and the median would be a better estimator rather than the mean. The studying of the median of the log-normal distribution proposed by Zellner [13] demonstrated the Bayesian and non-Bayesian estimators of the median of the log-normal distribution. To compare two samples, Krishnamoorthy and Mathew [4] proposed several different methods to compute the confidence intervals for those parameters and evaluated them concerning their coverage probability. Chen and Zhou [10] studied the methods of likelihood profiles and generalized pivotal quantities. After that, Rao and D’Cunha [14] studied the Bayesian approach for constructing the confidence interval of the median of a log-normal distribution, which has a short average length compared with the MLE approach. Recently, Singhasomboon et al. [15] proposed several methods, including normal approximation (NA), the method of variance estimates recovery (MOVER) and the generalized confidence interval (GCI) approaches, to construct the confidence interval of the ratio of the medians for two independent log-normal distributions. However, the inference based on the difference of the medians for two independent log-normal distributions has not been illustrated until now. Moreover, in experiments involving multiple treatments, various sets of multiple comparisons among the treatments and corresponding simultaneous confidence intervals (SCIs) have been considered. For example, multiple comparisons between different processes are studied in the industrial field, and multiple comparisons of different treatment regimens are considered in the clinical trials of medical products. In controlled studies that include several experimental treatments, multiple comparisons among these treatments are common. Tukey’s method and Dunnett’s method are the common methods for pairwise comparisons under the normal assumption.

In this paper, we will take the challenge of illustrating different approaches to construct the confidence intervals based on the difference in the median for two or more independent log-normal distributions. The rest of the paper is organized as follows. The proposed problems are presented in Section 2. Several methods to construct the confidence intervals of the difference of the medians of two log-normal distributions are demonstrated in Section 3. Simultaneous confidence intervals for the difference of medians of several log-normal distributions are studied in Section 4. The simulation studies will be presented in Section 5. Applications are given to illustrate the proposed methods of constructing confidence intervals in Section 6. Some conclusions are given in Section 7.

2. The Proposed Problem

Let X be a random variable having a log-normal distribution, and $\mu$ and ${\sigma }^2$ denote the mean and variance of $ln\left(X\right)$, respectively, so that $Z=LN\left(X\right)\sim N(\mu ;{\sigma }^2)$. Assume we have $X_{ij}$, $i=1,\cdots ,k,j=1,\cdots ,n_i$ be random samples of a size $n_i$ from independent log-normal distributions $LN({\mu }_i,{\sigma }_i^2)$. Define $Z_{ij}=ln\left(X_{ij}\right)$, and we know that the unbiased estimators of ${\mu }_i$ and ${\sigma }_i^2$ are

$$\begin{array}{c}\hfill \overline{Z_i}=\frac{1}{n_1}\sum _{j=1}^{n_i}{Z}_{ij},S_i^2=\frac{1}{n_i-1}\sum _{j=1}^{n_i}{(Z_{ij}-\overline{Z_i})}^2,\end{array}$$

(1)

where $\overline{Z_i}\sim N({\mu }_i,\frac{{\sigma }_i^2}{n_i})$, $\frac{(n_i-1)S_i^2}{{\sigma }_i^2}\sim {\chi }_{n_i-1}^2$ and $\overline{Z_i}$ and $S_i^2$ are independent. It is well known that the medians of two log-normal distributions can be obtained by $M_1=e^{{\mu }_1}$ and $M_2=e^{{\mu }_2}$. For simplicity, we denote the interested parameter to be $\eta =M_1-M_2$. By definition of the plug-in estimate, the unbiased point estimator of $\eta$ is

$$\widehat{\eta }=e^{\overline{Z_1}}-e^{\overline{Z_2}}.$$

(2)

In the rest of this paper, we consider the four different approaches—the parametric bootstrap (PB) approach, NA approach, FGCI approach and MOVER approach—to construct the confidence interval for the differences of the medians for two samples, such as $X_1$ and $X_2$ at the beginning. Then, we extend the proposed methods into multiple samples to, for example, construct the SCIs for k independent samples. All the related formulas to construct the confidence intervals are studied, and the performances are investigated.

3. Confidence Intervals of the Differences of the Medians for Two Samples

In the following, we will study the PB approach, NA approach, Fiducial generalized confidence interval (FGCI) approach and MOVER approach to construct the confidence interval of $\eta$. For continuity, we denote $i=1,2$ in the rest of this section.

3.1. Parametric Bootstrap Approach

The PB approach has been successfully used in testing problems which involve nuisance parameters (see Krishnamoorthy et al. [16], Tian et al. [17], Krishnamoorthy and Lu [18], Sadooghi-Alvandi and Jafari [19] and Tian and Yang [20]). In the following, we will use the idea of the PB approach to study the confidence interval of $\eta$.

In fact, we know that $Y_i=e^{{\overline{Z}}_i}\sim LN({\mu }_i,\frac{{\sigma }_i^2}{n_i})$ and

$$\begin{array}{ccc}\hfill E(Y_1-Y_2)& =& e^{{\mu }_1+\frac{{\sigma }_1^2}{n_1}}-e^{{\mu }_2+\frac{{\sigma }_2^2}{n_2}},\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill Var(Y_1-Y_2)& =& e^{2{\mu }_1+\frac{{\sigma }_1^2}{n_1}}\left(e^{\frac{{\sigma }_1^2}{n_1}}-1\right)+e^{2{\mu }_2+\frac{{\sigma }_2^2}{n_2}}\left(e^{\frac{{\sigma }_2^2}{n_2}}-1\right).\hfill \end{array}$$

(4)

Since ${\mu }_i$ and ${\sigma }_i^2$ are unknown parameters, then the unbiased estimator of $Var(Y_1-Y_2)$ is

$${\widehat{v}}_{12}=e^{2{\overline{Z}}_1+\frac{S_1^2}{n_1}}\left(e^{\frac{S_1^2}{n_1}}-1\right)+e^{2{\overline{Z}}_2+\frac{S_2^2}{n_2}}\left(e^{\frac{S_2^2}{n_2}}-1\right).$$

(5)

Thus, the confidence interval of $\eta$ can be constructed by the following random quantity:

$$T=\left|\frac{\widehat{\eta }-\eta }{\sqrt{{\widehat{v}}_{12}}}\right|.$$

(6)

Therefore, the approximate $100(1-\alpha )\%$ confidence interval of $\eta$ is

$$\left(\widehat{\eta }-T_{\alpha }\sqrt{{\widehat{v}}_{12}},\widehat{\eta }+T_{\alpha }\sqrt{{\widehat{v}}_{12}}\right),$$

(7)

where $T_{\alpha }$ denotes the approximate $(1-\alpha )$th quantile of the distribution of T.

3.2. Fiducial Generalized Confidence Intervals Approach

Weerahandi [21] introduced the concept of a generalized pivotal quantity (GPQ) for a scalar parameter. Hannig et al. [22] identified an important subclass of GPQs called FGPQs. Subsequently, the generalized fiducial inference has attracted a great amount of attention due to its advantage of handling the inference problems under certain complex situations (see Hannig and Lee [23] and Cisewski and Hannig [24]). In the following, we employ the FGCI method to study the inference of $\eta$.

Let ${\tilde{Z}}_i$ denote the observed value of ${\overline{Z}}_i$ and ${\tilde{S}}_i$ denote the observed value of ${\overline{S}}_i$. The FGPQ of $\eta$ is

$$R_{\eta }=e^{R_{{\mu }_1}}-e^{R_{{\mu }_2}},$$

(8)

where

$$R_{{\mu }_i}={\tilde{Z}}_i-\frac{W_i}{\sqrt{n_i}}\frac{{\tilde{S}}_i\sqrt{n_i-1}}{V_i},$$

(9)

and $W_i\sim N(0,1)$, $V_i^2\sim {\chi }_{n_i-1}^2$.

Therefore, the $100(1-\alpha )\%$ two-sided generalized confidence interval for $\eta$, based on the FGCI approach, is given as follows:

$$(R_{\eta }(\alpha /2),R_{\eta }(1-\alpha /2)),$$

(10)

where $R_{\eta }\left(\alpha \right)$ is the $\alpha$th percentile of $R_{\eta }$.

3.3. Normal Approximation Approach

In the NA approach, the main statistical tool used to obtain an asymptotically normal distribution of an estimator is the Delta method. Consider an estimator $g(v_1,v_2)=e^{v_1}-e^{v_2}$ to be a function of two other basic statistics, and in our case, $v_1={\overline{Z}}_1$ and $v_2={\overline{Z}}_2$. We know $v_1$ and $v_2$ have simple forms and are jointly asymptotically normal. By using the procedure of the Delta method, the asymptotic distribution of the estimator based on the Taylor series of $g(v_1,v_2)$ at ${\mu }_1$ and ${\mu }_2$ is given as follows:

$$g(v_1,v_2)=g({\mu }_1,{\mu }_2)+\frac{\partial g({\mu }_1,{\mu }_2)}{\partial v_1}(v_1-{\mu }_1)+\frac{\partial g({\mu }_1,{\mu }_2)}{\partial v_2}(v_2-{\mu }_2)+Reminder.$$

(11)

After some calculations, we obtain

$$g(v_1,v_2)=e^{{\mu }_1}-e^{{\mu }_2}+e^{{\mu }_1}(v_1-{\mu }_1)+e^{{\mu }_2}(v_2-{\mu }_2),$$

(12)

and the asymptotic mean and variance of the NA estimator are given by

$$\begin{array}{c}\hfill E\left[g(v_1,v_2)\right]=e^{{\mu }_1}-e^{{\mu }_2},\end{array}$$

(13)

$$\begin{array}{c}\hfill Var\left(g(v_1,v_2)\right)=\frac{e^{2{\mu }_1}{\sigma }_1^2}{n_1}+\frac{e^{2{\mu }_2}{\sigma }_2^2}{n_2}.\end{array}$$

(14)

Therefore, as $n_1,n_2\to \infty$, we have

$$\widehat{\eta }\to N\left(e^{{\mu }_1}-e^{{\mu }_2},\frac{e^{2{\mu }_1}{\sigma }_1^2}{n_1}+\frac{e^{2{\mu }_2}{\sigma }_2^2}{n_2}\right).$$

Thus, the $100(1-\alpha )\%$ two-sides approximation confidence interval for $\eta$ based on the NA approach is

$$\left(\widehat{\eta }-z_{\alpha /2}\sqrt{\frac{e^{2{\overline{Z}}_1}{S}_1^2}{n_1}+\frac{e^{2{\overline{Z}}_2}{S}_2^2}{n_2}},\widehat{\eta }+z_{\alpha /2}\sqrt{\frac{e^{2{\overline{Z}}_1}{S}_1^2}{n_1}+\frac{e^{2{\overline{Z}}_2}{S}_2^2}{n_2}}\right),$$

where $z_{\alpha /2}$ is the $\alpha /2$th quartile value from $N(0,1)$.

3.4. The Confidence Interval Based on the MOVER Approach

The MOVER approach, which was first proposed by Graybill and Wang [25], is a method for finding a confidence interval for a linear combination of parameters based on the individual confidence intervals of the parameters. Zou and Donner [26] justified the validity of the MOVER confidence interval for any set of parameters, including the location parameters, and concluded that the MOVER approach did not require any specific underlying distribution for the parameter. In the following, we construct the confidence interval of $\eta$ based on the MOVER approach.

Let $L_i$ and $U_i$ be the lower and upper limiters of the confidence interval of ${\mu }_i$, which can be obtained as follows:

$$\begin{array}{c}\hfill L_i={\overline{Z}}_i-z_{\alpha /2}\sqrt{\frac{S_i^2}{n_i}},U_i={\overline{Z}}_i+z_{\alpha /2}\sqrt{\frac{S_i^2}{n_i}}.\end{array}$$

(15)

The lower and upper limits of the confidence interval of $M_i$ are $e^{L_i}$ and $e^{U_i}$, respectively. According to Zou and Donner [26], the lower and upper limits of the $100(1-\alpha )\%$ confidence interval for $\eta$ based on the MOVER approach are given by

$$\begin{array}{c}\hfill L=e^{{\overline{Z}}_1}-e^{{\overline{Z}}_2}-\sqrt{{\left(e^{{\overline{Z}}_1}-e^{{\overline{Z}}_1-z_{\alpha /2}\sqrt{S_1^2/n_1}}\right)}^2+{\left(e^{{\overline{Z}}_2}-e^{{\overline{Z}}_2+z_{\alpha /2}\sqrt{S_2^2/n_2}}\right)}^2},\end{array}$$

(16)

$$\begin{array}{c}\hfill U=e^{{\overline{Z}}_1}-e^{{\overline{Z}}_2}+\sqrt{{\left(e^{{\overline{Z}}_1}-e^{{\overline{Z}}_1+z_{\alpha /2}\sqrt{S_1^2/n_1}}\right)}^2+{\left(e^{{\overline{Z}}_2}-e^{{\overline{Z}}_2-z_{\alpha /2}\sqrt{S_2^2/n_2}}\right)}^2},\end{array}$$

(17)

where $z_{\alpha /2}$ is the $\alpha /2$th quartile value from $N(0,1)$.

4. Simultaneous Confidence Intervals for the Difference of the Medians of Several Log-Normal Distributions

In this section, we study the PB, FGCI, NA and MOVER approaches to obtain the SCIs of the differences of the medians for several log-normal distributions. In the following section, we denote $i\ne j=1,2,\cdots ,k$.

4.1. Parametric Bootstrap Simultaneous Confidence Intervals

The parametric bootstrapping method has been used to estimate confidence intervals, especially for studies with multiple samples. Sadooghi-Alvandi and Malekzadeh [27] proposed the PB approach to construct SCIs for the ratios of the means of several log-normal distributions. Li et al. [28] used the PB and FG methods to build SCIs for the differences in two-parameter exponential means. Thangjai et al. [29] used the GCI, MOVER and PB approaches to construct SCIs for the differences in the means of several two-parameter exponential distributions.

Note that the unbiased estimator of $M_i$ is

$${\widehat{M}}_i=e^{{\overline{Z}}_i}.$$

(18)

and

$$\begin{array}{c}\hfill Var({\widehat{M}}_i-{\widehat{M}}_j)=e^{2{\mu }_i+\frac{{\sigma }_i^2}{n_i}}\left(e^{\frac{{\sigma }_i^2}{n_i}}-1\right)+e^{2{\mu }_j+\frac{{\sigma }_j^2}{n_j}}\left(e^{\frac{{\sigma }_j^2}{n_j}}-1\right),\end{array}$$

(19)

Then, the unbiased estimator of $Var({\widehat{M}}_i-{\widehat{M}}_j)$ is

$$V_{ij}=e^{2{\overline{Z}}_i+\frac{S_i^2}{n_i}}\left(e^{\frac{S_i^2}{n_i}}-1\right)+e^{2{\overline{Z}}_j+\frac{S_j^2}{n_j}}\left(e^{\frac{S_j^2}{n_j}}-1\right).$$

(20)

We define the following:

$$T_n=\underset{i\ne j}{max}\left|\frac{({\widehat{M}}_i-{\widehat{M}}_j)-(M_i-M_j)}{\sqrt{V_{ij}}}\right|,$$

(21)

In addition, the approximate $100(1-\alpha )\%$ two-sided SCIs for $({\widehat{M}}_i-{\widehat{M}}_j)$ are

$$\begin{array}{c}\hfill (M_i-M_j)\in ({\widehat{M}}_i-{\widehat{M}}_j)\pm q_{\alpha }\sqrt{V_{ij}},\end{array}$$

where $q_{\alpha }$ denotes the approximate $(1-\alpha )$th quantile of the distribution of $T_n$.

Note that the distribution of $T_n$ depends on the nuisance parameters ${\sigma }_i^2$ but does not depend on the values of ${\mu }_i$. Therefore, without loss of generality, all ${\mu }_i$ values are assumed to be zero. Let

$$e^{{\overline{Z}}_{i}B}\sim LN\left(0,\frac{S_i^2}{n_i}\right),S_i^{2B}\sim \frac{{\tilde{S}}_i^2}{n_i-1}{\chi }_{n_i-1}^2,$$

(22)

where ${\tilde{S}}_i$ is the observed value of $S_i$. Based on Equation (21), we define the PB analogues of $T_n$ as follows:

$$T_n^B=\underset{i\ne j}{max}\left|\frac{e^{{\overline{Z}}_{i}B}-e^{{\overline{Z}}_{j}B}}{\sqrt{V_{ij}^B}}\right|,$$

(23)

where

$$V_{ij}^B=e^{2{\overline{Z}}_{i}B+\frac{S_i^{2B}}{n_i}}\left(e^{\frac{S_i^{2B}}{n_i}}-1\right)+e^{2{\overline{Z}}_{j}B+\frac{S_j^{2B}}{n_j}}\left(e^{\frac{S_j^{2B}}{n_j}}-1\right).$$

(24)

Therefore, the $100(1-\alpha )\%$ two-sided PB SCIs for $(M_i-M_j)$ are

$$\begin{array}{c}\hfill SC{I}_{ij\left(PB\right)}=\left(({\widehat{M}}_i-{\widehat{M}}_j)-q_{\alpha }^B\sqrt{V_{ij}},({\widehat{M}}_i-{\widehat{M}}_j)+q_{1-\alpha }^B\sqrt{V_{ij}}\right),\end{array}$$

(25)

where $q_{\alpha }^B$ and $q_{1-\alpha }^B$ denote the approximate $\alpha$th and $(1-\alpha )$th quantiles of the distribution of $T_n^B$, respectively.

4.2. Simultaneous Fiducial Generalized Confidence Intervals

Scholars have used the FGCI method to study the confidence intervals of multiple samples, mainly including Thangjai and Niwitpong [30] and Hannig et al. [31]. In the following, we study the FGCI method to obtain the SCIs for the differences in the medians of several log-normal distributions. The fiducial pivotal variable for $M_i$ follows immediately as

$$R_{M_i}=e^{R_{{\mu }_i}},$$

(26)

where $R_{{\mu }_i}$ is defined by Equation (9).

Consequently, we have

$$\begin{array}{c}\hfill R_{M_{ij}}=R_{M_i}-R_{M_j}=exp\left({\tilde{Z}}_i-\frac{W_i}{\sqrt{n_i}}\frac{{\tilde{S}}_i\sqrt{n_i-1}}{V_i}\right)-exp\left({\tilde{Z}}_j-\frac{W_j}{\sqrt{n_j}}\frac{{\tilde{S}}_j\sqrt{n_j-1}}{V_j}\right).\end{array}$$

(27)

We define

$$D=\underset{i\ne j}{max}\left|\frac{({\widehat{M}}_i-{\widehat{M}}_j)-R_{M_{ij}}}{\sqrt{V_{ij}}}\right|,$$

(28)

where $V_{ij}$ is given in Equation (20).

Therefore, the $100(1-\alpha )\%$ SCIs for $(M_i-M_j)$ in the FGCI approach are given by

$$SC{I}_{ij\left(FGCI\right)}=(L_{ij\left(FGCI\right)},U_{ij\left(FGCI\right)})=\left(({\widehat{M}}_i-{\widehat{M}}_j)-D_{\alpha }\sqrt{V_{ij}},({\widehat{M}}_i-{\widehat{M}}_j)+D_{1-\alpha }\sqrt{V_{ij}}\right),$$

(29)

where $D_{\alpha }$ and $D_{1-\alpha }$ denote the $\alpha$th and $(1-\alpha )$th percentile of D, respectively.

4.3. Normal Approximation Approach Simultaneous Confidence Intervals

Using the same arguments as above, we consider the estimator $g(v_i,v_j)=e^{v_i}-e^{v_j}$ to be a function of two other basic statistics $v_i={\overline{Z}}_i$ and $v_j={\overline{Z}}_j$. Similarly, the asymptotic mean and variance of the NA estimator are given by

$$\begin{array}{c}\hfill E\left[g(v_i,v_j)\right]=e^{{\mu }_i}-e^{{\mu }_j},Var\left(g(v_i,v_j)\right)=\frac{e^{2{\mu }_i}{\sigma }_i^2}{n_i}+\frac{e^{2{\mu }_j}{\sigma }_j^2}{n_j},\end{array}$$

(30)

As $n_i,n_j\to \infty$, we have

$$\widehat{M_i}-\widehat{M_j}\to N\left(e^{{\mu }_i}-e^{{\mu }_j},\frac{e^{2{\mu }_i}{\sigma }_i^2}{n_i}+\frac{e^{2{\mu }_j}{\sigma }_j^2}{n_j}\right).$$

Therefore, the $100(1-\alpha )\%$ two-sides approximation SCIs for $(M_i-M_j)$ based on the NA approach are

$$SC{I}_{ij\left(NA\right)}=\left(\widehat{M_i}-\widehat{M_j}-z_{\alpha /2}\sqrt{\frac{e^{2{\overline{Z}}_i}{S}_i^2}{n_i}+\frac{e^{2{\overline{Z}}_2}{S}_j^2}{n_j}},\widehat{M_i}-\widehat{M_j}+z_{\alpha /2}\sqrt{\frac{e^{2{\overline{Z}}_i}{S}_i^2}{n_i}+\frac{e^{2{\overline{Z}}_j}{S}_j^2}{n_j}}\right),$$

(31)

where $z_{\alpha /2}$ is the $\alpha /2$th quartile value for $N(0,1)$.

4.4. Method of Variance Estimates Recovery Simultaneous Confidence Intervals

Thangjai et al. [32] proposed the MOVER method to study the SCIs for all differences in the means of the normal distributions with unknown coefficients of variation, followed by Thangjai and Niwitpong [33], who studied the SCIs for the mean difference of a log-normal distribution. In a similar procedure, we consider the MOVER method to obtain the SCIs for the differences in the medians of several log-normal distributions.

Let $l_i$ and $u_i$ be the lower and upper limiters of the confidence interval of ${\mu }_i$, respectively, which can be obtained as follows:

$$\begin{array}{c}\hfill l_i=\widehat{M_i}-z_{\alpha }\sqrt{var\left(\widehat{M_i}\right)},u_i=\widehat{M_i}+z_{\alpha }\sqrt{var\left(\widehat{M_i}\right)},\end{array}$$

(32)

where $z_{\alpha }$ is the $\alpha /2$th quantile value for $N(0,1)$.

The $100(1-\alpha )\%$ two-sided MOVER SCIs for $M_i-M_j$ are

$$SC{I}_{ij\left(MOVER\right)}=(L_{ij},U_{ij}),$$

(33)

where

$$L_{ij}=\widehat{M_i}-\widehat{M_j}-\sqrt{{(\widehat{M_i}-l_i)}^2+{(u_j-\widehat{M_j})}^2},U_{ij}=\widehat{M_i}-\widehat{M_j}+\sqrt{{(\widehat{M_i}-l_i)}^2+{(u_j-\widehat{M_j})}^2},$$

(34)

and $\widehat{M_i}$ and $\widehat{M_j}$ are defined as in Equation (18).

5. Simulation

To evaluate the performances of the proposed confidence interval constructed by different approaches, we carry out simulation studies in this section. The coverage probability (CP), average length (AL) and associated standard deviation (SD) were calculated based on R software. For each of the generated sets, we used the R code with $N=1000$ runs to compute the confidence intervals. The percentage of these 1000 confidence intervals that includes the assumed $\eta$ is an estimator of the CP. The values of the AL and associated SD were estimated similarly. All the results were calculated based on R software.

For the case where $k=2$, we considered the parameters with the following cases of equal means and unequal means. Furthermore, we also studied the case of smaller values of ${\sigma }_i^2$. The detailed information is listed as follows:

I:

${\mu }_1={\mu }_2=0$, (A) ${\sigma }_1^2={\sigma }_2^2=1$ and (B) ${\sigma }_1^2=1,{\sigma }_2^2=2$;

II:

${\mu }_1=0,{\mu }_2=1$, (C) ${\sigma }_1^2={\sigma }_2^2=1$ and (D) ${\sigma }_1^2=1,{\sigma }_2^2=2$;

III:

Smaller values of ${\sigma }_i^2$, where (E) ${\sigma }_1^2=0.1$ and ${\sigma }_2^2=0.2$.

The corresponding results are displayed in the

Table 1. CP and AL of 95% confidence intervals for $\eta$ under ${\mu }_1={\mu }_2=0$.

		CP				AL(SD)
${\mathit{\sigma }}_{\mathbf{1}}^{\mathbf{2}}/{\mathit{\sigma }}_{\mathbf{2}}^{\mathbf{2}}$	$({\mathbf{n}}_{\mathbf{1}},{\mathbf{n}}_{\mathbf{2}})$	PB	NA	FGCI	MOVER	PB	NA	FGCI	MOVER
$1/1$	(10, 10)	0.9492	0.9481	0.9479	0.9491	1.7316(0.5742)	1.8372(0.5395)	1.9497(0.5729)	2.0636(0.6294)
	(10, 20)	0.9492	0.9439	0.9459	0.9475	1.5200(0.4656)	1.5481(0.4194)	1.6320(0.4729)	1.7291(0.5194)
	(10, 50)	0.9491	0.9449	0.9489	0.9497	1.4090(0.5187)	1.4131(0.4382)	1.4335(0.4699)	1.4790(0.4938)
	(10, 100)	0.9497	0.9584	0.9409	0.9468	1.3261(0.6026)	1.3584(0.4833)	1.3589(0.4595)	1.3991(0.5139)
	(20, 20)	0.9500	0.9487	0.9439	0.9514	1.1616(0.2451)	1.2708(0.2409)	1.2980(0.2657)	1.3459(0.2829)
	(20, 50)	0.9493	0.9484	0.9479	0.9532	0.9976(0.2072)	1.0507(0.2409)	1.0650(0.2270)	1.1033(0.2241)
	(20, 100)	0.9491	0.9578	0.9729	0.9503	0.9723(0.2293)	0.9786(0.2249)	0.9745(0.2189)	1.0012(0.2382)
	(50, 20)	0.9495	0.9475	0.9369	0.9518	1.0233(0.2229)	1.0384(0.1946)	1.0568(0.2209)	1.0887(0.2233)
	(50, 100)	0.9497	0.9639	0.9449	0.9536	0.6660(0.0832)	0.6884(0.0840)	0.6749(0.0835)	0.6979(0.0839)
	(100, 100)	0.9515	0.9717	0.9400	0.9547	0.5318(0.0462)	0.5569(0.0492)	0.5442(0.0514)	0.5658(0.0503)
$1/2$	(10, 10)	0.9494	0.9419	0.9451	0.9507	2.2579(0.9409)	2.2689(0.7691)	2.4670(0.8666)	2.7321(1.0954)
	(10, 20)	0.9497	0.9481	0.9491	0.9563	1.7230(0.4946)	1.8440(0.5149)	1.9620(0.5442)	2.0518(0.5975)
	(10, 50)	0.9498	0.9472	0.9481	0.9557	1.4406(0.4524)	1.5184(0.4408)	1.5510(0.4367)	1.6204(0.4700)
	(10, 100)	0.9497	0.9549	0.9421	0.9561	1.4023(0.4807)	1.4099(0.4867)	1.4171(0.4684)	1.4899(0.5052)
	(20, 20)	0.9492	0.9425	0.9431	0.9519	1.4427(0.3835)	1.5893(0.4123)	1.6271(0.4073)	1.7016(0.4498)
	(20, 50)	0.9495	0.9486	0.9471	0.9495	1.1166(0.2170)	1.2200(0.2211)	1.2166(0.2357)	1.2799(0.2447)
	(20, 100)	0.9496	0.9585	0.9520	0.9593	1.0339(0.2175)	1.0539(0.2152)	1.0553(0.2182)	1.0949(0.2129)
	(50, 20)	0.9496	0.9482	0.9401	0.9491	1.0132(0.2160)	1.3960(0.4178)	1.3935(0.4427)	1.5048(0.4406)
	(50, 100)	0.9495	0.9606	0.9421	0.9588	0.7213(0.0862)	0.7958(0.0994)	0.7815(0.0971)	0.8074(0.0959)
	(100, 100)	0.9512	0.9555	0.9411	0.9586	0.6639(0.0743)	0.6871(0.0796)	0.6763(0.0789)	0.6953(0.0784)

,

Table 2. CP and AL of 95% confidence intervals for $\eta$ under ${\mu }_1=0$ and ${\mu }_2=1$.

		CP				AL(SD)
${\mathit{\sigma }}_{\mathbf{1}}^{\mathbf{2}}/{\mathit{\sigma }}_{\mathbf{2}}^{\mathbf{2}}$	$({\mathbf{n}}_{\mathbf{1}},{\mathbf{n}}_{\mathbf{2}})$	PB	NA	FGCI	MOVER	PB	NA	FGCI	MOVER
$1/1$	(10, 10)	0.9491	0.9438	0.9321	0.9500	3.6918(1.4588)	3.7243(1.2638)	3.8488(1.2896)	3.9868(1.3905)
	(10, 20)	0.9497	0.9461	0.9450	0.9514	2.5455(0.6103)	2.7828(0.6764)	2.8719(0.6975)	2.9739(0.7017)
	(10, 50)	0.9492	0.9456	0.9470	0.9462	1.9346(0.4167)	1.9996(0.3917)	2.0778(0.4225)	2.1445(0.4550)
	(10, 100)	0.9492	0.9533	0.9469	0.9513	1.6452(0.4381)	1.6558(0.3787)	1.7422(0.4330)	1.7735(0.4420)
	(20, 20)	0.9497	0.9432	0.9429	0.9476	2.5507(0.6719)	2.5663(0.5985)	2.5861(0.5999)	2.6476(0.6369)
	(20, 50)	0.9493	0.9468	0.9430	0.9495	1.6891(0.2622)	1.7693(0.2606)	1.7704(0.2716)	1.8409(0.2857)
	(20, 100)	0.9494	0.9536	0.9489	0.9526	1.3062(0.1907)	1.3985(0.1909)	1.4029(0.2038)	1.4445(0.2171)
	(50, 20)	0.9497	0.9491	0.9419	0.9501	2.3814(0.6929)	2.4207(0.6465)	2.4268(0.6346)	2.5208(0.6563)
	(50, 100)	0.9496	0.9518	0.9348	0.9574	1.1637(0.1221)	1.2098(0.1297)	1.1802(0.1253)	1.2211(0.1230)
	(100, 100)	0.9496	0.9645	0.9409	0.9536	1.1021(0.1167)	1.1334(0.1219)	1.1131(0.1255)	1.1440(0.1197)
$1/2$	(10, 10)	0.9499	0.9292	0.9339	0.9518	5.1381(2.5064)	5.1467(2.2107)	5.2742(2.4259)	5.9315(2.6442)
	(10, 20)	0.9496	0.9347	0.9319	0.9541	3.7284(1.3176)	3.7299(1.1684)	3.7620(1.1219)	4.1265(1.3107)
	(10, 50)	0.9493	0.9444	0.9469	0.9546	2.4400(0.5010)	2.5191(0.5075)	2.6070(0.5483)	2.7469(0.5764)
	(10, 100)	0.9491	0.9567	0.9419	0.9547	1.8929(0.4245)	2.0057(0.3985)	2.0614(0.4377)	2.1102(0.4322)
	(20, 20)	0.9497	0.9448	0.9309	0.9522	3.5669(1.3025)	3.5670(1.1508)	3.6458(1.2554)	3.8154(1.1987)
	(20, 50)	0.9495	0.9455	0.9421	0.9543	2.2339(0.4296)	2.3474(0.4610)	2.3233(0.4655)	2.4163(0.4489)
	(20, 100)	0.9494	0.9514	0.9500	0.9536	1.5475(0.2185)	1.7557(0.2435)	1.7733(0.2509)	1.8342(0.2605)
	(50, 20)	0.9498	0.9482	0.9479	0.9482	3.4640(1.1473)	3.4739(1.1556)	3.4702(1.1461)	3.7706(1.2801)
	(50, 100)	0.9494	0.9533	0.9489	0.9464	1.1423(0.2070)	1.1602(0.2174)	1.1837(0.2324)	1.2485(0.2347)
	(100, 100)	0.9497	0.9536	0.9419	0.9567	1.5436(0.2196)	1.5681(0.2243)	1.5520(0.2322)	1.5905(0.2364)

and

Table 3. CP and AL of 95% confidence intervals for $\eta$ under ${\sigma }_1^2=0.1$ and ${\sigma }_2^2=0.2$.

		CP				AL(SD)
${\mathit{\sigma }}_{\mathbf{1}}^{\mathbf{2}}/{\mathit{\sigma }}_{\mathbf{2}}^{\mathbf{2}}$	$({\mathbf{n}}_{\mathbf{1}},{\mathbf{n}}_{\mathbf{2}})$	PB	NA	FGCI	MOVER	PB	NA	FGCI	MOVER
$1/3$	(10, 10)	0.9491	0.9283	0.9429	0.9365	11.1077(3.2546)	11.7098(2.9978)	11.4597(3.0595)	11.7612(3.0190)
	(10, 20)	0.9594	0.9423	0.9369	0.9472	7.7195(1.4732)	7.7784(1.4440)	7.7880(1.4708)	7.9277(1.4324)
	(10, 50)	0.9597	0.9516	0.9458	0.9567	4.7684(0.5352)	5.0529(0.5882)	5.0449(0.5952)	5.0724(0.5459)
	(10, 100)	0.9497	0.9567	0.9439	0.9527	3.6086(0.2946)	3.6730(0.2979)	3.6192(0.3093)	3.6690(0.2884)
	(20, 20)	0.9598	0.9505	0.9428	0.9511	7.7516(1.7286)	7.9261(1.4955)	7.9128(1.4532)	7.8549(1.4618)
	(20, 50)	0.9494	0.9478	0.9448	0.9456	4.8537(0.5515)	4.9782(0.5624)	4.8973(0.5858)	5.0544(0.6003)
	(20, 100)	0.9496	0.9576	0.9439	0.9567	3.3816(0.2716)	3.6025(0.2887)	3.5042(0.2939)	3.5867(0.2979)
	(50, 20)	0.9493	0.9439	0.9469	0.9446	7.7641(1.5155)	7.8552(1.5629)	7.7985(1.5316)	7.8857(1.4983)
	(50, 100)	0.9499	0.9527	0.9459	0.9528	3.3907(0.2842)	3.5642(0.2941)	3.4584(0.3007)	3.5491(0.2972)
	(100, 100)	0.9495	0.9649	0.9470	0.9617	3.4410(0.2857)	3.5277(0.2844)	3.4533(0.3043)	3.5498(0.2878)
$0.1/0.3$	(10, 10)	0.9491	0.9396	0.9429	0.9456	0.8360(0.1927)	0.8545(0.1826)	0.9076(0.1862)	0.8766(0.1922)
	(10, 20)	0.9596	0.9388	0.9449	0.9556	0.6847(0.1100)	0.6879(0.1061)	0.7068(0.1118)	0.6924(0.1130)
	(10, 50)	0.9496	0.9355	0.9339	0.9418	0.5358(0.0980)	0.5420(0.0899)	0.5660(0.0991)	0.5484(0.0926)
	(10, 100)	0.9495	0.9496	0.9499	0.9415	0.4859(0.0981)	0.4907(0.0961)	0.5083(0.1007)	0.4888(0.0953)
	(20, 20)	0.9593	0.9524	0.9419	0.9447	0.6044(0.0902)	0.6055(0.0935)	0.6190(0.0969)	0.6149(0.0946)
	(20, 50)	0.9594	0.9585	0.9560	0.9537	0.4247(0.0437)	0.4523(0.0459)	0.4536(0.0505)	0.4576(0.0478)
	(20, 100)	0.9494	0.9515	0.9389	0.9627	0.3842(0.0453)	0.3857(0.0429)	0.3846(0.0463)	0.3897(0.0435)
	(50, 20)	0.9565	0.9498	0.9459	0.9546	0.5592(0.0969)	0.5647(0.0964)	0.5647(0.0958)	0.5615(0.0943)
	(50, 100)	0.9494	0.9536	0.9450	0.9573	0.2925(0.0198)	0.3056(0.0200)	0.2992(0.0218)	0.3071(0.0205)
	(100, 100)	0.9498	0.9656	0.9409	0.9552	0.2654(0.0179)	0.2729(0.0178)	0.2668(0.0200)	0.2726(0.0177)

with sample sizes of $(n_1,n_2)$ ranging from 10 to 100 for all the different cases.

For the case where $k\ge 3$, we assumed that $k=3$. We studied the equal means and equal variance. For the equal means, we also considered the small and large variances and employed a similar idea for the equal variance:

I:

Equal means, where ${\mu }_1={\mu }_2={\mu }_3=0$, (A) ${\sigma }_1^2=0.2,{\sigma }_2^2=0.4,{\sigma }_3^2=0.6$, (B) ${\sigma }_1^2=1.0$, ${\sigma }_2^2=3.0$ and ${\sigma }_3^2=5.0$.

II:

Equal variance, where ${\sigma }_1^2={\sigma }_2^2={\sigma }_3^2=1$, (C) ${\mu }_1=0.3,{\mu }_2=0.5,{\mu }_3=0.7$, and (D) ${\mu }_1=1.0$, ${\mu }_2=1.5$ and ${\mu }_2=2.0$.

The corresponding results are displayed in the

Table 4. CP and AL of 95% SCIs for different methods under $({\mu }_1,{\mu }_2,{\mu }_3)=(0,0,0)$.

		CP				AL(SD)
${\mathit{\sigma }}_{\mathbf{1}}^{\mathbf{2}}/{\mathit{\sigma }}_{\mathbf{2}}^{\mathbf{2}}/{\mathit{\sigma }}_{\mathbf{3}}^{\mathbf{2}}$	$({\mathbf{n}}_{\mathbf{1}},{\mathbf{n}}_{\mathbf{2}},{\mathbf{n}}_{\mathbf{3}})$	PB	NA	FGCI	MOVER	PB	NA	FGCI	MOVER
(0.2/0.4/0.6)	(10, 10, 10)	0.9420	0.9477	0.9470	0.9507	1.5356(0.4007)	1.1123(0.2747)	2.0186(0.6390)	1.1769(0.3125)
	(15, 15, 15)	0.9500	0.9473	0.9570	0.9493	1.2692(0.2779)	0.9025(0.1852)	1.5005(0.3468)	0.9282(0.1863)
	(15, 15, 20)	0.9593	0.9437	0.9683	0.9450	1.1991(0.2256)	0.8550(0.1611)	1.3852(0.2897)	0.8688(0.1643)
	(20, 20, 20)	0.9623	0.9547	0.9673	0.9573	1.1166(0.2017)	0.7925(0.1406)	1.2718(0.2483)	0.8024(0.1434)
	(20, 25, 20)	0.9633	0.9517	0.9687	0.9503	1.0861(0.1838)	0.7602(0.1287)	1.2335(0.2282)	0.7713(0.1350)
	(25, 25, 25)	0.9687	0.9487	0.9753	0.9547	1.0084(0.1570)	0.7056(0.1074)	1.1248(0.1946)	0.7124(0.1107)
	(30, 30, 30)	0.9697	0.9450	0.9733	0.9473	0.9398(0.1363)	0.6397(0.0892)	1.0135(0.1523)	0.6460(0.0921)
	(40, 40, 40)	0.9823	0.9490	0.9783	0.9450	0.8320(0.1034)	0.5518(0.0658)	0.8893(0.1144)	0.5574(0.0687)
	(50, 50, 50)	0.9830	0.9453	0.9850	0.9533	0.7585(0.0831)	0.4973(0.0554)	0.8007(0.0938)	0.4961(0.0550)
	(100, 100, 100)	0.9933	0.9440	0.9890	0.9523	0.5699(0.0432)	0.3484(0.0269)	0.5854(0.0467)	0.3500(0.0258)
(1.0/3.0/5.0)	(15, 15, 15)	0.9433	0.9567	0.9640	0.9737	4.3151(2.3953)	2.8497(1.3731)	8.4039(5.0095)	3.5100(1.8688)
	(15, 15, 20)	0.9483	0.9600	0.9757	0.9703	3.7087(1.6799)	2.5770(1.1063)	6.9593(3.9353)	3.0154(1.3755)
	(20, 20, 20)	0.9533	0.9607	0.9713	0.9687	3.5634(1.6259)	2.3260(0.8973)	6.0509(2.9523)	2.7507(1.2300)
	(25, 25, 25)	0.9540	0.9563	0.9777	0.9693	3.1887(1.3638)	2.0840(0.7418)	4.9558(2.1291)	2.4056(0.9488)
	(30, 25, 25)	0.9617	0.9557	0.9780	0.9667	3.3162(1.3572)	2.0206(0.7256)	4.9573(2.1523)	2.3297(0.9164)
	(30, 30, 30)	0.9620	0.9533	0.9787	0.9687	2.8943(1.0616)	1.8695(0.6506)	4.2183(1.6384)	2.0990(0.7339)
	(30, 40, 30)	0.9637	0.9540	0.9847	0.9597	2.7805(1.0220)	1.7917(0.5870)	3.9711(1.5026)	1.9507(0.7055)
	(40, 40, 40)	0.9740	0.9570	0.9837	0.9683	2.5214(0.7560)	1.5919(0.4392)	3.3735(1.0453)	1.7023(0.4682)
	(50, 50, 50)	0.9777	0.9573	0.9873	0.9600	2.2499(0.5736)	1.4225(0.3646)	2.9951(0.8788)	1.4917(0.3919)
	(100, 100, 100)	0.9890	0.9553	0.9883	0.9563	1.6622(0.3058)	0.9720(0.1696)	1.9071(0.3587)	1.0116(0.1837)

and

Table 5. CP and AL of 95% SCIs for different methods under $({\sigma }_1^2,{\sigma }_2^2,{\sigma }_3^2)=(1.0,2.0,3.0)$.

		CP				AL(SD)
${\mathit{\sigma }}_{\mathbf{1}}^{\mathbf{2}}/{\mathit{\sigma }}_{\mathbf{2}}^{\mathbf{2}}/{\mathit{\sigma }}_{\mathbf{3}}^{\mathbf{2}}$	$({\mathbf{n}}_{\mathbf{1}},{\mathbf{n}}_{\mathbf{2}},{\mathbf{n}}_{\mathbf{3}})$	PB	NA	FGCI	MOVER	PB	NA	FGCI	MOVER
0.3/0.5/0.7	(20, 20, 20)	0.9447	0.959	0.9670	0.9577	4.6570(1.7122)	3.3280(1.0976)	7.2362(2.9444)	3.6462(1.2777)
	(25, 25, 25)	0.9460	0.9520	0.9677	0.9580	4.1106(1.2885)	2.8925(0.8107)	6.0044(2.1589)	3.1610(0.9568)
	(30, 30, 30)	0.9607	0.9507	0.9730	0.9563	3.8176(1.0980)	2.6427(0.7326)	5.2922(1.6064)	2.7976(0.8127)
	(40, 40, 40)	0.9683	0.9540	0.9810	0.9573	3.3625(0.8476)	2.2759(0.5410)	4.3404(1.1284)	2.3659(0.5503)
	(40, 50, 40)	0.9657	0.9493	0.9830	0.9560	3.2662(0.7798)	2.2071(0.5009)	4.2387(1.1023)	2.3004(0.5482)
	(50, 50, 50)	0.9683	0.9500	0.9867	0.9550	3.0407(0.6803)	2.0276(0.4250)	3.8598(0.8637)	2.0672(0.4394)
	(50, 50, 70)	0.9810	0.9507	0.9893	0.9537	2.7494(0.5047)	1.8351(0.3201)	3.3988(0.6500)	1.9110(0.3437)
	(50, 70, 70)	0.9807	0.9557	0.9890	0.9553	2.6034(0.4711)	1.7435(0.3005)	3.2162(0.5819)	1.7867(0.3147)
	(70, 70, 70)	0.9787	0.9477	0.9860	0.9580	2.6345(0.4668)	1.6984(0.3004)	3.1862(0.5793)	1.7417(0.3019)
	(100, 100, 100)	0.9893	0.9560	0.9893	0.9530	2.2829(0.3493)	1.4046(0.2039)	2.6377(0.4167)	1.4264(0.2069)
1.0/1.5/2.0	(20, 20, 20)	0.9423	0.9463	0.9563	0.9510	14.9760(5.9846)	10.7241(3.9954)	24.0100(10.3319)	12.0315(5.0424)
	(25, 25, 25)	0.9510	0.9480	0.9580	0.9570	13.3986(4.7372)	9.4666(3.1494)	20.0042(7.5117)	10.2049(3.5172)
	(30, 30, 30)	0.9583	0.9513	0.9660	0.9547	12.3695(3.9432)	8.7083(2.5695)	17.6625(5.9396)	9.0229(2.7079)
	(40, 40, 40)	0.9573	0.9477	0.9693	0.9590	10.6545(2.9137)	7.2922(1.9126)	14.3658(4.1688)	7.7888(2.1221)
	(40, 50, 40)	0.9627	0.9507	0.9727	0.9543	10.5144(2.8711)	7.1298(1.8948)	14.1034(4.0772)	7.5692(2.0211)
	(50, 50, 50)	0.9620	0.9463	0.9793	0.9563	9.7127(2.3405)	6.4738(1.4645)	12.6974(3.0742)	6.7811(1.5474)
	(50, 50, 70)	0.9737	0.9433	0.9830	0.9547	8.6573(1.8020)	5.7387(1.1445)	10.9302(2.2641)	5.9907(1.2123)
	(50, 70, 70)	0.9753	0.9510	0.9887	0.9470	8.3458(1.6541)	5.5265(1.0617)	10.5696(2.2639)	5.6480(1.1066)
	(70, 70, 70)	0.9737	0.9457	0.9853	0.9597	8.5508(1.6556)	5.4250(1.0474)	10.5478(2.1923)	5.6042(1.0852)
	(100, 100, 100)	0.9877	0.9540	0.9897	0.9523	7.3910(1.2372)	4.5635(0.6974)	8.6206(1.4487)	4.6114(0.7459)

, with the sample sizes of $(n_1,n_2,n_3)$ ranging from 10 to 100.

From the results in Table 1, Table 2 and Table 3, we found that the PB and MOVER approaches had better CPs when the sample sizes were small, and the NA approach performed better for the larger sample sizes. As the sample size increased, the ALs of the confidence interval decreased. The one with smaller values for the mean and variance appeared to have a shorter AL, and the PB approach had the smallest AL compared with the other approaches. Meanwhile, we know that the AL of the confidence interval decreased and the CP of the confidence interval increased when one sample size was fixed and another sample size increased for all approaches. Moreover, the CPs for all the approaches were close to the nominal level when the samples had equal means or equal variances. However, the powers of these approaches were different under the case of unequal means or unequal variances, especially for the smaller sample sizes. For example, with $n_1=(10,20)$ and $n_2=(10,20)$, the CPs calculated by the PB and MOVER approaches were closest to the nominal level, followed by the NA and FGCI approaches when the two samples had equal means. However, this phenomenon changed a little when the two samples had equal means, especially for the NA approach, where the CPs were not close to the nominal level.

We can see from Table 4 and Table 5 that the values of ALs and SDs of the confidence interval decreased as the sample size increased for all methods, and the NA method had the smallest values for the AL, followed by the MOVER and PB methods. The CPs of the NA approach were close to the nominal confidence level for the case of equal means, especially for the small variances. However, for the unequal means, the CPs of the NA approach were close to the nominal confidence level for most cases. Furthermore, the CPs obtained by the PB approach were close to the nominal confidence level in the small sample sizes. When the sample size of $n_i$ gradually increased, the interval calculated by the NA and MOVER approaches may lead to over-coverage, which may be caused by the fact that the ALs calculated by these two approaches were wider than those of the other two methods. We also notice that the MOVER approach always provided the longest interval length, while the PB approach provided the shortest interval length.

6. Application

The log-normal distribution was positively skewed, and for the skewed distributions, the median was a better estimator than the mean, which was affected by the extreme values. The PB approach, NA approach, FGCI approach and MOVER approach were studied for constructing the confidence intervals of the population means and medians. In this section, we illustrate all the proposed approaches for how to obtain the confidence intervals of the difference of two medians of two or more than two independent log-normal distributions in the real problems. The first data set works on the Air Quality Index (e.g., PM2.5) in the cities of Bangkapi and Dindaeng in Thailand. The second data set works on the nitrogen-bound bovine serum albumin produced by three groups of diabetic mice: normal, alloxan diabetic and alloxan diabetic treated with insulin.

6.1. Air Quality Index Data Set

The Air Quality Index is based on measurements of particulate matter (PM), ozone, nitrogen dioxide, sulfur dioxide and carbon monoxide emissions. Most of the stations on the map are monitoring both PM2.5 and PM10 data. Air pollution, especially PM from vehicles, has a major impact on people’s health, namely those who live in urban areas and close to traffic. Singhasomboon et al. [15] studied the PM2.5 mass concentrations in Bangkapi and Dindaeng in Thailand. In this example, we will use these data sets to study the difference of the medians for PM2.5 in theses cities. Singhasomboon et al. [15]) showed that the QQ plots for these two data sets are positively skewed and the logarithmical transformed data are approximately normally distributed. Using the “fitdistr” function in the R program, we obtained the estimators of the corresponding parameters in the log-normal distributions. Furthermore, the Kolmogorov–Smirnov (K-S) test (Hassani and Silva [34]), the Anderson–Darling (A-D) goodness-of-fit test (Anderson [35]) as well as the p-value (pval) are reported in

Table 6. Estimators of the log-normal distribution for the PM2.5 mass concentration data.

City	n	$\widehat{\mathsf{\mu }}$	${\widehat{\mathsf{\sigma }}}^2$	K-S(pval)	A-D(pval)
Bangkapi	90	2.9286	0.3110	0.0509(0.9737)	0.3127(0.9281)
Dindaeng	90	3.0805	0.2390	0.0769(0.6613)	0.5048(0.7414)

. The K-S statistic for the data from PM2.5 in Bangkapi was 0.0509, and the corresponding p-value was 0.9737. The K-S statistic for the data from PM2.5 in Dindaeng was 0.0769, and the corresponding p-value was 0.6613. Therefore, these data were reasonably fitted for the log-normal distributions. The fitting curves of the probability densities from these data sets are also displayed in Figure 1.

Table 7. The 95% confidence intervals for $\eta$ based on the proposed approaches.

Approach	Confidence Interval	Length
PB	(−5.9736, −0.2595)	5.7141
FGCI	(−5.9479, −0.2119)	5.7360
NA	(−6.1624, 0.0292)	6.1916
MOVER	(−6.1653, −0.0476)	6.1177

gives the 95% two-sided confidence intervals for $\eta$ based on the proposed CI construction approaches. These results show that the PB approach had a smaller length than the MOVER approach. Except for the NA approach, we can conclude that the median of the PM2.5 in Bangkapi is smaller than that in Dindaeng, which is consistent with the study results by Singhasomboon et al. [15].

6.2. The Nitrogen-Bound Bovine Serum Albumin Data

In the following, we consider another data set that has unequal and small sample sizes, which was studied by Schaarschmidt [36]. The data set can be found in the handbook written by Hand et al. [37], consisting of 57 observations of nitrogen-bound bovine serum albumin produced by three groups of diabetic mice: normal, alloxan diabetic and alloxan diabetic treated with insulin. Using the “fitdistr” function in the R program, we obtained the estimators of the corresponding parameters in the log-normal distributions. The estimators of the corresponding parameters of the log-normal distribution for each group are shown in

Table 8. Estimators of the log-normal distribution for the data of three groups.

Group	n	$\widehat{\mathsf{\mu }}$	${\widehat{\mathsf{\sigma }}}^2$	K-S(pval)	A-D(pval)
Normal	20	4.859	0.927	0.1846(0.5031)	0.4839(0.7612)
Alloxan	18	4.867	0.850	0.1012(0.9927)	0.2375(0.9766)
AlloxanInsulin	19	4.397	0.696	0.0999(0.9913)	0.1671(0.9971)

. Similarly, the Kolmogorov–Smirnov (K-S) test, the Anderson–Darling (A-D) goodness-of-fit tests as well as the p-value (pval) are reported in Table 8. The K-S statistic for the data in the normal group was 0.1846, with a corresponding p-value of 0.5031. The K-S statistic for the data in the alloxan group was 0.1012, with a corresponding p-value of 0.9927. Finally, the K-S statistic for the data in the alloxan with insulin group was 0.0999, with a corresponding p-value 0.9913. Therefore, these data from the three groups are reasonable fits for the log-normal distributions. The corresponding fitting curves of the probability densities from these data sets are displayed in Figure 2.

The 95% confidence intervals and corresponding lengths for the differences of the pairwise comparisons for the treated groups and control group based on the PB, NA, FGCI and MOVER methods are summarized in

Table 9. The 95% confidence intervals for the differences of the medians for 3 groups.

Comparision	PB	NA	FGCI	MOVER
Normal-Alloxan	(−73.8085, 71.7379)	(−78.6276, 76.5570)	(−70.5716, 79.6785)	(−83.0634, 80.2609)
Alloxan-AlloxanInsulin	(−11.1186, 108.5659)	(−14.4459, 111.8933)	(−14.622, 119.1691)	(−9.5601, 122.2428)
AlloxanInsulin-Normal	(−107.4132, 12.0364)	(−110.0271, 14.6504)	(−119.8781, 11.2019)	(−119.9627, 10.0617)

and

Table 10. The lengths of 95% confidence intervals for the differences of the medians for 3 groups.

Comparision	PB	NA	FGCI	MOVER
Normal-Alloxan	145.5465	155.1847	150.2501	163.3244
Alloxan-AlloxanInsulin	119.6846	126.3392	133.7911	131.8028
AlloxanInsulin-Normal	119.4496	124.6776	131.0799	130.0243

.

From Table 9 and Table 10, we can conclude that there was no change between the two treated groups and the control group according to the medians. Furthermore, we can find that the confidence intervals from the PB approach had the smallest length, which is consistent with the results in the simulation studies.

7. Concluding

The log-normal distribution was positively skewed and, for the skewed distributions and the median, was a better estimator than the mean. In medical studies, the median survival time was often reported to be more than the mean survival time. The salient feature of the log location–scale family is that the coefficient of variation of the distribution depends only on the scale parameter and not on the location parameter. Therefore, the median has more meaning than the mean, and consideration of the log-normal median has many more advantages. Although many scholars have studied the inference of the median with different methods, there are few research works on the inference of the difference between two medians of two or more independent log-normal distributions. In this paper, we constructed the confidence intervals for the difference of two independent log-normal distributions based on the PB approach, NA approach, FGCI approach and MOVER approach. Furthermore, we studied the SCIs for the difference of more than two independent log-normal distributions with the proposed approaches. From the simulation studies, we found that the PB and MOVER approaches had better CPs when the sample sizes were small, and the NA approach performed better for the large sample sizes. The values of the AL and SD of the confidence interval decreased as the sample size increased for all methods, and the NA method had the smallest AL values, followed by the MOVER and PB methods. We also found that the PB approach would be a suitable choice for smaller sample sizes for the two independent distributions and multiple independent distributions, and the MOVER and NA approaches were alternative competitors for constructing SCIs, especially when the populations had more variance. Although these approaches provide some good ideas to construct the confidence interval for the difference of two or more independent log-normal distributions, there are still some limitations that need to be discussed further. First, the confidence intervals for some approaches may lead to over-coverage when the sample size of $n_i$ gradually increases, such as with the NA and MOVER approaches. Second, the higher CPs for some confidence intervals may be caused by sacrificing the length of the confidence intervals. It is better to make the CPs close to the nominal level without increasing the confidence interval length too much. In the future, we will modify these approaches or propose some new approaches to make the CPs close to the nominal level without scarifying the ALs of the confidence intervals. Furthermore, we will extend these approaches to the skewed normal distribution and skewed slash distribution and study some financial time series data.