Confidence Intervals for Function of Percentiles of Birnbaum-Saunders Distributions Containing Zero Values with Application to Wind Speed Modelling

Abstract

The Birnbaum–Saunders (BS) distribution, defined only for non-negative values, is asymmetrical. However, it can be transformed into a normal distribution, which is symmetric. The BS distribution is particularly useful for analyzing data consisting of values greater than zero. This study aims to introduce six approaches for constructing confidence intervals for the difference and ratio of percentiles in Birnbaum–Saunders distributions containing zero values. The proposed approaches include the generalized confidence interval (GCI) approach, the bootstrap approach, the highest posterior density (HPD) approach based on the bootstrap method, the Bayesian approach, the HPD approach based on the Bayesian method, and the method of variance estimates recovery (MOVER) approach. To assess their performance, a Monte Carlo simulation study is conducted, focusing on coverage probability and average length. The results indicate that the MOVER approach and the HPD approach based on the Bayesian method perform better than other approaches for constructing confidence intervals for the difference between percentiles. Moreover, the GCI and Bayesian approaches outperform others when constructing confidence intervals for the ratio of percentiles. Finally, daily wind speed data from the Rayong and Prachin Buri provinces are used to demonstrate the efficacy of the proposed approaches.

1. Introduction

In meteorology, wind speed, or wind flow velocity, is a key atmospheric parameter resulting from air movement between regions of high and low pressure, typically driven by temperature variations. It is commonly measured using an anemometer. Wind speed plays a critical role in various fields, including weather forecasting, aviation, maritime operations, construction, plant growth and metabolism, and numerous other applications. It also significantly affects the dispersion and dilution of atmospheric pollutants, such as particulate matter 2.5 (PM_2.5). Higher wind speeds are often associated with improved pollutant dispersion, leading to lower concentrations of PM_2.5 in the air. Therefore, accurate wind speed prediction is essential for mitigating PM_2.5 levels and managing air quality. In the Rayong and Prachin Buri provinces, Thailand, daily wind speed data, comprising both zero and positive values, align with the Birnbaum–Saunders distribution containing zero values.

The Birnbaum–Saunders distribution was originally developed to model failure times in fatigue life scenarios. Its variant, the Birnbaum–Saunders distribution containing zero values, extends this model to accommodate datasets containing both zero and positive values. The Birnbaum–Saunders distribution containing zero values is particularly useful in fields, such as meteorology, environmental sciences, and biostatistics, where variables like wind speed or pollutant concentrations often consist of a mixture of zero and positive values. Its versatility makes it ideal for modeling real-world scenarios, such as wind speed measurements, where some days have no wind (zero values) while others record positive speeds. Researchers have shown considerable interest in estimating the parameters of the Birnbaum–Saunders distribution containing zero values. For instance, Ratasukharom et al. [1] proposed methods for estimating the variance of the Birnbaum–Saunders distribution with zero values. No researchers have constructed confidence intervals for the percentiles of the Birnbaum–Saunders distribution containing zero values, highlighting a significant gap in the literature.

Percentiles can be more suitable than averages in certain contexts, such as analyzing wind speed or PM_2.5 data. Specifically, the 2.5th and 97.5th percentiles are valuable tools for understanding data distributions. For example, percentiles provide insights into wind speed or PM_2.5 levels, often offering more meaningful information than average values. They can also be used to compare wind speed or PM_2.5 data between regions. Additionally, the difference or ratio between two percentiles can help study the variability in wind speeds or PM_2.5 levels across different locations or time periods. Several statisticians have developed methods for estimating confidence intervals for population percentiles or quantiles. For instance, Padgett and Tomlinson [2] introduced parametric bootstrap methods for estimating lower confidence bounds for percentiles of Weibull and Birnbaum–Saunders distributions. Moreover, Hasan and Krishnamoorthy [3] proposed confidence interval estimates for the mean and percentiles of zero-inflated lognormal data. No studies have developed confidence intervals for the percentiles of the Birnbaum–Saunders distribution that include zero values, particularly in the context of wind speed modeling.

The construction of confidence intervals for the difference and ratio of percentiles is a key area of interest. Various approaches are used to create these confidence intervals for the Birnbaum–Saunders distribution with zero values, including the generalized confidence interval (GCI) approach, bootstrap approach, highest posterior density (HPD) approach based on the bootstrap method, Bayesian approach, HPD approach based on the Bayesian method, and the method of variance estimates recovery (MOVER) approach. The GCI approach is particularly useful when the underlying data distribution is unknown or complex. Researchers have applied it in various contexts. For instance, Ye et al. [4] presented procedures for hypothesis testing and interval estimation for the common mean of inverse Gaussian distributions. The bootstrap approach is a resampling method where multiple samples are drawn with replacements from the original dataset. This method is particularly useful for complex or unknown data distributions, as it makes minimal assumptions about the population. Researchers have widely applied the bootstrap approach. For example, Singhasomboon et al. [5] proposed bootstrap confidence intervals for the ratio of modes of log-normal distributions. A common Bayesian technique involves using percentiles of the posterior distribution to define the credible interval, which serves as the Bayesian counterpart to the confidence interval. This approach is particularly beneficial when prior knowledge or expert opinion is available, as it directly incorporates uncertainty into the analysis. The HPD interval is used to determine the most credible values of a parameter based on the posterior distribution and offers more flexibility than traditional methods in capturing the most likely parameter values. Many researchers have used the Bayesian and HPD approaches to construct confidence intervals. For instance, Wu [6] introduced Bayesian interval estimation of the scale parameter for a two-parameter exponential distribution based on a right Type II censored sample. The MOVER approach aims to recover accurate variance estimates by combining model-based assumptions with empirical data. It is particularly useful for complex models, such as those with heteroscedasticity (non-constant variance), or in cases where variance estimation is difficult due to small sample sizes. Many researchers have used the MOVER approach to construct confidence intervals. For example, Tang [7] proposed MOVER confidence intervals for a difference or ratio effect parameter under stratified sampling. Additionally, Thangjai et al. [8] proposed confidence intervals for the percentile of delta-Birnbaum–Saunders distribution. This study extends the research of Thangjai et al. [8] by constructing confidence intervals for the difference between percentiles and the ratio of percentiles of Birnbaum–Saunders distributions containing zero values. The approach encompasses six methods: the GCI approach, the bootstrap approach, the HPD approach based on the bootstrap method, the Bayesian approach, the HPD approach based on the Bayesian method, and the MOVER approach.

2. Confidence Intervals for Difference Between Percentiles of Birnbaum- Saunders Distributions Containing Zero Values

For one population, let $n=n_{\left(0\right)}+n_{\left(1\right)}$ be the sample size, where $n_{\left(0\right)}$ is the number of zero observations and $n_{\left(1\right)}$ is the number of positive observations. The number of zero observations has a binomial distribution with the probability of observing zero values $\delta$, where 0$<\delta <$ 1. The count of positive observations should follow a binomial distribution since it represents discrete events. However, the individual positive observations might be modeled by a Birnbaum–Saunders distribution if they exhibit positive skewness, are non-negative, and correspond to reliability-related data, which are frequently encountered in survival analysis, reliability engineering, and fatigue modeling. In this paper, the number of positive observations has a Birnbaum–Saunders distribution with the probability of observing positive values ${\delta }^{\prime }=$ 1 $-\delta$. Data containing zero and positive observations have Birnbaum–Saunders distribution containing zero values with the probability of observing zero values $\delta$ and the probability of observing positive values ${\delta }^{\prime }=1-\delta$. Let $X=\left(X_1,X_2,\dots ,X_n\right)$ be a non-negative random variable of size n drawn from a Birnbaum–Saunders distribution containing zero values with shape parameter $\alpha$, scale parameter $\beta$, and probability of observing positive value $\delta$. Let $x=\left(x_1,x_2,\dots ,x_n\right)$ be the observed value of $X=\left(X_1,X_2,\dots ,X_n\right)$. According to Aitchison [9] and Birnbaum and Saunders [10], the distribution of X is

$$G\left(x_j;\delta ,\alpha ,\beta \right)=\left\{\begin{array}{cc}\delta ;\hfill & x_j=0\hfill \\ \delta +\left(1-\delta \right)F\left(x_j;\alpha ,\beta \right);\hfill & x_j>0,\hfill \end{array}\right.$$

(1)

where $F\left(x_j;\alpha ,\beta \right)$ is the Birnbaum–Saunders cumulative distribution function.

Let $\theta$ be the p-th percentile of Birnbaum–Saunders distribution containing zero values. According to Thangjai et al. [8], the percentile of Birnbaum–Saunders distribution containing zero values can be written as

$$\theta =\left\{\begin{array}{cc}0;\hfill & p<\delta \hfill \\ {\displaystyle \frac{\beta }{4}}{\left(\alpha {\Phi }^{-1}\left({\displaystyle \frac{p-\delta }{1-\delta }}\right)+\sqrt{{\alpha }^2{\left({\Phi }^{-1}\left({\displaystyle \frac{p-\delta }{1-\delta }}\right)\right)}^2+4}\right)}^2;\hfill & p>\delta ,\hfill \end{array}\right.$$

(2)

where ${\Phi }^{-1}\left(p\right)$ is the standard normal p-th quantile.

For two populations, the difference between percentiles is of interest. For the first population, let $X_1=\left(X_{11},X_{12},\dots ,X_{1n_1}\right)$ be a non-negative random variable of size $n_1$ drawn from a Birnbaum–Saunders distribution containing zero values with shape parameter ${\alpha }_1$, scale parameter ${\beta }_1$, and probability of observing positive value ${\delta }_1^{\prime }=1-{\delta }_1$. Let $n_1=n_{1\left(0\right)}+n_{1\left(1\right)}$ be the sample size where $n_{1\left(0\right)}$ is the number of zero observations and $n_{1\left(1\right)}$ is the number of positive observations. The percentile is

$${\theta }_1=\left\{\begin{array}{cc}0;\hfill & p_1<{\delta }_1\hfill \\ {\displaystyle \frac{{\beta }_1}{4}}{\left({\alpha }_1{\Phi }^{-1}\left({\displaystyle \frac{p_1-{\delta }_1}{1-{\delta }_1}}\right)+\sqrt{{\alpha }_1^2{\left({\Phi }^{-1}\left({\displaystyle \frac{p_1-{\delta }_1}{1-{\delta }_1}}\right)\right)}^2+4}\right)}^2;\hfill & p_1>{\delta }_1,\hfill \end{array}\right.$$

(3)

where ${\Phi }^{-1}\left(p_1\right)$ is the standard normal $p_1$-th quantile.

For the second population, let $X_2=\left(X_{21},X_{22},\dots ,X_{2n_2}\right)$ be a non-negative random variable of size $n_2$ drawn from a Birnbaum–Saunders distribution containing zero values with shape parameter ${\alpha }_2$, scale parameter ${\beta }_2$, and probability of observing positive value ${\delta }_2^{\prime }=1-{\delta }_2$. Let $n_2=n_{2\left(0\right)}+n_{2\left(1\right)}$ be the sample size, where $n_{2\left(0\right)}$ is the number of zero observations and $n_{2\left(1\right)}$ is the number of positive observations. The percentile is

$${\theta }_2=\left\{\begin{array}{cc}0;\hfill & p_2<{\delta }_2\hfill \\ {\displaystyle \frac{{\beta }_2}{4}}{\left({\alpha }_2{\Phi }^{-1}\left({\displaystyle \frac{p_2-{\delta }_2}{1-{\delta }_2}}\right)+\sqrt{{\alpha }_2^2{\left({\Phi }^{-1}\left({\displaystyle \frac{p_2-{\delta }_2}{1-{\delta }_2}}\right)\right)}^2+4}\right)}^2;\hfill & p_2>{\delta }_2,\hfill \end{array}\right.$$

(4)

where ${\Phi }^{-1}\left(p_2\right)$ is the standard normal $p_2$-th quantile.

The difference between percentiles of Birnbaum–Saunders distributions containing zero values is

$$\omega ={\theta }_1-{\theta }_2,$$

(5)

where ${\theta }_1$ and ${\theta }_2$ are defined in Equation (3) and Equation (4), respectively.

The confidence intervals for the difference between percentiles of Birnbaum–Saunders distributions containing zero values are constructed using the GCI approach, the bootstrap approach, the HPD approach based on the bootstrap method, the Bayesian approach, the HPD approach based on the Bayesian method, and the MOVER approach.

2.1. GCI for Difference Between Percentiles

The GCI approach is a statistical method for estimating confidence intervals for parameters. It is particularly useful for complex models, non-standard distributions, or parameters that depend on multiple variables. Introduced by Weerahandi [11], this method relies on the concept of the generalized pivotal quantity (GPQ), a random variable derived from sample data and model parameters. The GPQ is constructed to remain invariant under specific transformations and is used to simulate its distribution based on the statistical model. Confidence intervals are then determined by identifying the percentiles of the simulated GPQ distribution that correspond to the desired confidence level. The GCI approach is especially effective for small sample sizes or skewed data but can be computationally demanding due to its reliance on simulations. Additionally, careful construction of the GPQ is essential to ensure the validity of the method.

Definition 1.

Let $f(x;\delta ,\alpha ,\beta )$ be the probability density function with unknown parameters δ, α, and β. Let $X=\left(X_1,X_2,\dots ,X_n\right)$ be the random variable and let $x=\left(x_1,x_2,\dots ,x_n\right)$ be the observed value of $X=\left(X_1,X_2,\dots ,X_n\right)$. Suppose that $R(X;x,\delta ,\alpha ,\beta )$ is the function of X, x, δ, α, and β. It satisfies the following two conditions:

1. 

For $X=x$, $R(X;x,\delta ,\alpha ,\beta )$ it has a probability distribution free of unknown parameters.

2. 

For $X=x$, the observed value of $R(X;x,\delta ,\alpha ,\beta )$ does not depend on the nuisance parameter.

For the first population, suppose that $S_{11}={\displaystyle \sum _{i=1}^{n_1}}{X}_{1i}$ and $S_{12}={\displaystyle \sum _{i=1}^{n_1}}1/X_{1i}$. Let $s_{11}$ be the observed value of $S_{11}$ and let $s_{12}$ be the observed value of $S_{12}$. Suppose that $V_1\sim {\chi }^2\left(n_1\right)$, $A_1=(n_1-1)J_1^2-\left(L_1{T}_1^2\right)/n_1$, $B_1=(n_1-1)I_1{J}_1-(1-I_1{J}_1)T_1^2$, $C_1=(n_1-1)I_1^2-\left(K_1{T}_1^2\right)/n_1$, $I_1={\displaystyle \sum _{i=1}^{n_1}}\sqrt{X_{1i}}/n_1$, $J_1=\left({\displaystyle \sum _{i=1}^{n_1}}{\displaystyle \frac{1}{\sqrt{X_{1i}}}}\right)/n_1$, $K_1={\displaystyle \sum _{i=1}^{n_1}}{(\sqrt{X_{1i}}-I_1)}^2$, $L_1={\displaystyle \sum _{i=1}^{n_1}}{((1/\sqrt{X_{1i}})-J_1)}^2$, and $T_1\sim t(n_1-1)$. Let ${\beta }_{11}$ and ${\beta }_{12}$ be solutions for ${\beta }_1$ can be derived by solving $A_1{\beta }_1^2-2B_1{\beta }_1+C_1=0$. According to Thangjai et al. [8], the GPQ for ${\theta }_1$ is

$$R_{{\theta }_1}={\displaystyle \frac{R_{{\beta }_1}}{4}}{\left(R_{{\alpha }_1}{\Phi }^{-1}\left({\displaystyle \frac{p_1-{\delta }_1}{1-{\delta }_1}}\right)+\sqrt{{\left(R_{{\alpha }_1}\right)}^2{\left({\Phi }^{-1}\left({\displaystyle \frac{p_1-{\delta }_1}{1-{\delta }_1}}\right)\right)}^2+4}\right)}^2,$$

(6)

where

$$R_{{\beta }_1}=\left\{\begin{array}{cc}max\left({\beta }_{11},{\beta }_{12}\right);\hfill & T_1⩽0\hfill \\ min\left({\beta }_{11},{\beta }_{12}\right);\hfill & T_1>0\hfill \end{array}\right.$$

(7)

and

$$R_{{\alpha }_1}=\sqrt{{\displaystyle \frac{s_{12}{\left(R_{{\beta }_1}\right)}^2-2n_{1\left(1\right)}{R}_{{\beta }_1}+s_{11}}{R_{{\beta }_1}{V}_1}}}.$$

(8)

For the second population, suppose that $S_{21}={\displaystyle \sum _{i=1}^{n_2}}{X}_{2i}$ and $S_{22}={\displaystyle \sum _{i=1}^{n_2}}1/X_{2i}$. Let $s_{21}$ be the observed value of $S_{21}$ and let $s_{22}$ be the observed value of $S_{22}$. Suppose that $V_2\sim {\chi }^2\left(n_2\right)$, $A_2=(n_2-1)J_2^2-L_2{T}_2^2/n_2$, $B_2=(n_2-1)I_2{J}_2-(1-I_2{J}_2)T_2^2$, $C_2=(n_2-1)I_2^2-K_2{T}_2^2/n_2$, $I_2={\displaystyle \sum _{i=1}^{n_2}}\sqrt{X_{2i}}/n_2$, $J_2=({\displaystyle \sum _{i=1}^{n_2}}{\displaystyle \frac{1}{\sqrt{X_{2i}}}})/n_2$, $K_2={\displaystyle \sum _{i=1}^{n_2}}{(\sqrt{X_{2i}}-I_2)}^2$, $L_2={\displaystyle \sum _{i=1}^{n_2}}{((1/\sqrt{X_{2i}})-J_2)}^2$, and $T_2\sim t(n_2-1)$. Let ${\beta }_{21}$ and ${\beta }_{22}$ be solutions for ${\beta }_2$ that can be derived by solving $A_2{\beta }_2^2-2B_2{\beta }_2+C_2=0$. The GPQ for ${\theta }_2$ is

$$R_{{\theta }_2}={\displaystyle \frac{R_{{\beta }_2}}{4}}{\left(R_{{\alpha }_2}{\Phi }^{-1}\left({\displaystyle \frac{p_2-{\delta }_2}{1-{\delta }_2}}\right)+\sqrt{{\left(R_{{\alpha }_2}\right)}^2{\left({\Phi }^{-1}\left({\displaystyle \frac{p_2-{\delta }_2}{1-{\delta }_2}}\right)\right)}^2+4}\right)}^2,$$

(9)

where

$$R_{{\beta }_2}=\left\{\begin{array}{cc}max\left({\beta }_{21},{\beta }_{22}\right);\hfill & T_2⩽0\hfill \\ min\left({\beta }_{21},{\beta }_{22}\right);\hfill & T_2>0\hfill \end{array}\right.$$

(10)

and

$$R_{{\alpha }_2}=\sqrt{{\displaystyle \frac{s_{22}{\left(R_{{\beta }_2}\right)}^2-2n_{2\left(1\right)}{R}_{{\beta }_2}+s_{21}}{R_{{\beta }_2}{V}_2}}}.$$

(11)

The GPQ of the difference between percentiles is

$$R_{\omega }=R_{{\theta }_1}-R_{{\theta }_2},$$

(12)

where $R_{{\theta }_1}$ and $R_{{\theta }_2}$ are defined in Equation (6) and Equation (9), respectively.

Therefore, the $100(1-\gamma )\%$ two-sided GCI for the difference between percentiles is

$$C{I}_{\omega .GCI}=[L_{\omega .GCI},U_{\omega .GCI}]=[R_{\omega }(\gamma /2),R_{\omega }(1-\gamma /2)],$$

(13)

where $R_{\omega }(\gamma /2)$ and $R_{\omega }(1-\gamma /2)$ denote the $100(\gamma /2)$-th and $100(1-\gamma /2)$-th percentiles of $R_{\omega }$, respectively.

Example 1.

For $m=$ 10, the GPQs of the difference between percentiles is $R_{\omega }=$ 0.0962, 0.4628, 0.2922, 0.0658, 0.1756, 0.2112, 0.1302, 0.1522, 0.1675, and 0.3235. The lower and upper limits of the 95% two-sided GCI for the difference between percentiles are $L_{\omega .GCI}=R_{\omega }\left(0.025\right)=$ 0.0726 and $U_{\omega .GCI}=R_{\omega }\left(0.975\right)=$ 0.4315, respectively.

2.2. Bootstrap Confidence Interval for Difference Between Percentiles

A bootstrap confidence interval is a statistical method for estimating the likely range of a population parameter at a specified confidence level. This method involves resampling the observed data with replacement and is particularly useful when conventional parametric methods are impractical. Its advantages include being non-parametric, as it does not rely on assumptions about the population’s distribution, and its suitability for small or complex datasets where analytical methods may fail. However, its limitations include the need for numerous resamples to ensure computational accuracy, sensitivity to the quality and size of the original sample, and reliance on the sample being representative of the population.

For the first population, let $x_1^{\ast }=\left(x_{11}^{\ast },x_{12}^{\ast },\dots ,x_{1n_1}^{\ast }\right)$ be a bootstrap sample drawn from a Birnbaum–Saunders distribution containing zero values with ${\widehat{\alpha }}_1$, ${\widehat{\beta }}_1$, and ${\delta }_1$. Therefore, ${\widehat{\alpha }}_1^{\ast }$ and ${\widehat{\beta }}_1^{\ast }$ are acquired by utilizing B bootstrap samples. Let $b({\widehat{\alpha }}_1,{\alpha }_1)$ be the bias estimator of ${\widehat{\alpha }}_1$ and let $b({\widehat{\beta }}_1,{\beta }_1)$ be the bias estimator of ${\widehat{\beta }}_1$. According to Thangjai et al. [8], the bootstrap estimator of ${\theta }_1$ is

$${\widehat{\theta }}_{1k}={\displaystyle \frac{{\tilde{\beta }}_{1k}}{4}}{\left({\tilde{\alpha }}_{1k}{\Phi }^{-1}\left({\displaystyle \frac{p_1-{\delta }_1}{1-{\delta }_1}}\right)+\sqrt{{\left({\tilde{\alpha }}_{1k}\right)}^2{\left({\Phi }^{-1}\left({\displaystyle \frac{p_1-{\delta }_1}{1-{\delta }_1}}\right)\right)}^2+4}\right)}^2,$$

(14)

where ${\tilde{\alpha }}_{1k}={\widehat{\alpha }}_{1k}^{\ast }-2\widehat{b}({\widehat{\alpha }}_1,{\alpha }_1)$, ${\tilde{\beta }}_{1k}={\widehat{\beta }}_{1k}^{\ast }-2\widehat{b}({\widehat{\beta }}_1,{\beta }_1)$, $\widehat{b}({\widehat{\alpha }}_1,{\alpha }_1)={\displaystyle \frac{1}{B}}{\displaystyle \sum _{k=1}^B}{\widehat{\alpha }}_{1k}^{\ast }-{\widehat{\alpha }}_1$, $\widehat{b}({\widehat{\beta }}_1,{\beta }_1)={\displaystyle \frac{1}{B}}{\displaystyle \sum _{k=1}^B}{\widehat{\beta }}_{1k}^{\ast }-{\widehat{\beta }}_1$, and $k=1,2,\dots ,B$.

For the second population, let $x_2^{\ast }=\left(x_{21}^{\ast },x_{22}^{\ast },\dots ,x_{2n_2}^{\ast }\right)$ be a bootstrap sample drawn from Birnbaum–Saunders distribution containing zero values with ${\widehat{\alpha }}_2$, ${\widehat{\beta }}_2$, and ${\delta }_2$. Therefore, ${\widehat{\alpha }}_2^{\ast }$ and ${\widehat{\beta }}_2^{\ast }$ are acquired by utilizing B bootstrap samples. Let $b({\widehat{\alpha }}_2,{\alpha }_2)$ be the bias estimator of ${\widehat{\alpha }}_2$ and let $b({\widehat{\beta }}_2,{\beta }_2)$ be the bias estimator of ${\widehat{\beta }}_2$. The bootstrap estimator of ${\theta }_2$ is

$${\widehat{\theta }}_{2k}={\displaystyle \frac{{\tilde{\beta }}_{2k}}{4}}{\left({\tilde{\alpha }}_{2k}{\Phi }^{-1}\left({\displaystyle \frac{p_2-{\delta }_2}{1-{\delta }_2}}\right)+\sqrt{{\left({\tilde{\alpha }}_{2k}\right)}^2{\left({\Phi }^{-1}\left({\displaystyle \frac{p_2-{\delta }_2}{1-{\delta }_2}}\right)\right)}^2+4}\right)}^2,$$

(15)

where ${\tilde{\alpha }}_{2k}={\widehat{\alpha }}_{2k}^{\ast }-2\widehat{b}({\widehat{\alpha }}_2,{\alpha }_2)$, ${\tilde{\beta }}_{2k}={\widehat{\beta }}_{2k}^{\ast }-2\widehat{b}({\widehat{\beta }}_2,{\beta }_2)$, $\widehat{b}({\widehat{\alpha }}_2,{\alpha }_2)={\displaystyle \frac{1}{B}}{\displaystyle \sum _{k=1}^B}{\widehat{\alpha }}_{2k}^{\ast }-{\widehat{\alpha }}_2$, $\widehat{b}({\widehat{\beta }}_2,{\beta }_2)={\displaystyle \frac{1}{B}}{\displaystyle \sum _{k=1}^B}{\widehat{\beta }}_{2k}^{\ast }-{\widehat{\beta }}_2$, and $k=1,2,\dots ,B$.

The bootstrap estimator of the difference between percentiles is

$${\widehat{\omega }}_k={\widehat{\theta }}_{1k}-{\widehat{\theta }}_{2k},$$

(16)

where ${\widehat{\theta }}_{1k}$ and ${\widehat{\theta }}_{2k}$ are defined in Equation (14) and Equation (15), respectively.

Therefore, the $100(1-\gamma )\%$ two-sided bootstrap confidence interval for the difference between percentiles is

$$C{I}_{\omega .B}=[L_{\omega .B},U_{\omega .B}]=[{\widehat{\omega }}_k(\gamma /2),{\widehat{\omega }}_k(1-\gamma /2)],$$

(17)

where ${\widehat{\omega }}_k(\gamma /2)$ and ${\widehat{\omega }}_k(1-\gamma /2)$ denote the $100(\gamma /2)$-th and $100(1-\gamma /2)$-th percentiles of ${\widehat{\omega }}_k$, respectively.

2.3. HPD Interval Based on the Bootstrap Method for Difference Between Percentiles

The HPD interval is a method in Bayesian statistics used to create a credible interval for a parameter. It represents the range of parameter values with the highest posterior probability density, based on the observed data and prior knowledge. Advantages include its property as the shortest interval satisfying the probability criterion, making it highly informative, and its direct representation of the region of greatest belief according to the posterior distribution. Limitations include the requirement for advanced numerical methods to compute the HPD interval for complex posterior distributions.

The bootstrap estimator of the difference between percentiles, as defined in Equation (16), is used to construct the HPD interval based on the bootstrap method. Therefore, the $100(1-\gamma )\%$ two-sided HPD interval based on the bootstrap method for the difference between percentiles is

$$C{I}_{\omega .BHPD}=[L_{\omega .BHPD},U_{\omega .BHPD}],$$

(18)

where $L_{\omega .BHPD}$ and $U_{\omega .BHPD}$ are computed using the hdi function within the HDInterval package of the R software suite (RStudio version 2024.12.0+467).

Example 2.

For $b=$ 10, the bootstrap estimators of the difference between percentiles are ${\widehat{\omega }}_k=$ 0.3046, −0.0746, 0.0505, 0.1915, 0.3794, 0.1413, 0.2395, 0.1465, 0.2846, and 0.2874. The lower and upper limits of the 95% two-sided bootstrap confidence interval for the difference between percentiles are $L_{\omega .B}={\widehat{\omega }}_k\left(0.025\right)=$−0.0464 and $U_{\omega .B}={\widehat{\omega }}_k\left(0.975\right)=$ 0.3626, respectively. Additionally, the lower and upper limits of the 95% two-sided HPD interval based on the bootstrap method for the difference between percentiles are $L_{\omega .BHPD}=$−0.0746 and $U_{\omega .BHPD}=$ 0.3794, respectively.

2.4. Bayesian Credible Interval for Difference Between Percentiles

A Bayesian credible interval represents the range of values within which a population parameter is believed to lie, based on the Bayesian framework and given the observed data and prior knowledge. It is derived from the posterior probability, which combines prior information and observed data using Bayes’ theorem. Advantages include its straightforward probabilistic interpretation and adaptability to complex models and small datasets. Limitations include the need to define a prior, which can introduce subjectivity, and the computational demands of complex models, often requiring techniques like Markov Chain Monte Carlo (MCMC).

Xu and Tang [12] showed that the reference prior, which is a type of independent Jeffreys prior, is inappropriate for Bayesian estimation as it leads to an improper posterior distribution. To ensure a proper posterior, Wang et al. [13] recommended the use of proper priors with known hyperparameters to estimate the confidence intervals of parameters from a Birnbaum–Saunders distribution. To ensure proper posterior distributions, inverse gamma priors with known hyperparameters are applied as priors for $\beta$ and ${\alpha }^2$, respectively.

For the first population, according to Thangjai et al. [8], the prior distribution of ${\beta }_1$ is inverse gamma distribution with parameters $a_{11}$ and $b_{11}$. The prior distribution of ${\alpha }_1^2$ is inverse gamma distribution with parameters $a_{12}$ and $b_{12}$. The posterior distribution of ${\theta }_1$ is

$${\theta }_{Baye1}={\displaystyle \frac{{\beta }_1}{4}}{\left({\alpha }_1{\Phi }^{-1}\left({\displaystyle \frac{p_1-{\delta }_1}{1-{\delta }_1}}\right)+\sqrt{{\alpha }_1^2{\left({\Phi }^{-1}\left({\displaystyle \frac{p_1-{\delta }_1}{1-{\delta }_1}}\right)\right)}^2+4}\right)}^2.$$

(19)

For the second population, the prior distribution of ${\beta }_2$ is inverse gamma distribution with parameters $a_{21}$ and $b_{21}$. The prior distribution of ${\alpha }_2^2$ is inverse gamma distribution with parameters $a_{22}$ and $b_{22}$. The posterior distribution of ${\theta }_2$ is

$${\theta }_{Baye2}={\displaystyle \frac{{\beta }_2}{4}}{\left({\alpha }_2{\Phi }^{-1}\left({\displaystyle \frac{p_2-{\delta }_2}{1-{\delta }_2}}\right)+\sqrt{{\alpha }_2^2{\left({\Phi }^{-1}\left({\displaystyle \frac{p_2-{\delta }_2}{1-{\delta }_2}}\right)\right)}^2+4}\right)}^2.$$

(20)

The posterior distribution of the difference between percentiles is

$${\omega }_{Baye}={\theta }_{Baye1}-{\theta }_{Baye2},$$

(21)

where ${\theta }_{Baye1}$ and ${\theta }_{Baye2}$ are defined in Equation (19) and Equation (20), respectively.

Therefore, the $100(1-\gamma )\%$ two-sided Bayesian credible interval for the difference between percentiles is

$$C{I}_{\omega .Baye}=[L_{\omega .Baye},U_{\omega .Baye}]=[{\omega }_{Baye}(\gamma /2),{\omega }_{Baye}(1-\gamma /2)],$$

(22)

where ${\omega }_{Baye}(\gamma /2)$ and ${\omega }_{Baye}(1-\gamma /2)$ denote the $100(\gamma /2)$-th and $100(1-\gamma /2)$-th percentiles of ${\omega }_{Baye}$, respectively.

2.5. HPD Interval Based on the Bayesian Method for Difference Between Percentiles

The posterior distribution of ${\omega }_{Baye}$ defined in Equation (21) is used to construct the HPD interval based on the Bayesian method. Therefore, the $100(1-\gamma )\%$ two-sided HPD interval based on the Bayesian method for the difference between percentiles is

$$C{I}_{\omega .HPD}=[L_{\omega .HPD},U_{\omega .HPD}],$$

(23)

where $L_{\omega .HPD}$ and $U_{\omega .HPD}$ are computed using the hdi function within the HDInterval package of the R software suite.

Example 3.

For $i=$ 10, the posterior distributions of the difference between percentiles are ${\omega }_{Baye}=$ 0.3424, 0.1383, 0.1966, 0.1870, 0.0621, 0.1636, 0.1619, 0.2096, 0.2384, and 0.4014. The lower and upper limits of the 95% two-sided Bayesian credible interval for the difference between percentiles are $L_{\omega .Baye}={\omega }_{Baye}\left(0.025\right)=$ 0.0792 and $U_{\omega .Baye}={\omega }_{Baye}\left(0.975\right)=$ 0.3882, respectively. Additionally, the lower and upper limits of the 95% two-sided HPD interval based on the Bayesian method for the difference between percentiles are $L_{\omega .HPD}=$ 0.0621 and $U_{\omega .HPD}=$ 0.4014, respectively.

2.6. MOVER Confidence Interval for Difference Between Percentiles

The MOVER interval is a statistical method used to construct confidence intervals for composite measures such as differences, ratios, or sums of two parameters. It is particularly useful in meta-analysis or situations where parameter estimates are combined to create a new interval. Rather than deriving a complex formula for the composite confidence interval, the MOVER method relies on the confidence intervals of the individual components. Advantages include its simplicity, as it only requires individual confidence intervals, avoiding complicated derivations, and its effectiveness even when the sample sizes of the parameters differ. Limitations include its reliance on the assumption that the sampling distributions of the estimates are approximately normal, which may not always hold, and its accuracy is dependent on the precision of the input confidence intervals.

The GCI method based on the variance-stabilizing transformation (VST) is used to compute the lower and upper limits of the percentile of the delta-Birnbaum–Saunders distribution. For the first population, according to Wu and Hsieh [14], the GPQ for ${\delta }_1$ is

$$F_{{\delta }_1}^{VST}={sin}^2\left[arcsin\sqrt{{\widehat{\delta }}_1-{\displaystyle \frac{L_1}{2\sqrt{n_1}}}}\right],$$

(24)

where $L_1=2\sqrt{n_1}\left(arcsin\sqrt{{\widehat{\delta }}_1}-arcsin\sqrt{{\delta }_1}\right)$ follows a standard normal distribution as $n_1$ approaches infinity. The estimate of the GPQ for ${\theta }_1$ is

$$F_{{\theta }_1}^{VST}={\displaystyle \frac{R_{{\beta }_1}}{4}}{\left(R_{{\alpha }_1}{\Phi }^{-1}\left({\displaystyle \frac{p_1-F_{{\delta }_1}^{VST}}{1-F_{{\delta }_1}^{VST}}}\right)+\sqrt{{\left(R_{{\alpha }_1}\right)}^2{\left({\Phi }^{-1}\left({\displaystyle \frac{p_1-F_{{\delta }_1}^{VST}}{1-F_{{\delta }_1}^{VST}}}\right)\right)}^2+4}\right)}^2,$$

(25)

where $R_{{\beta }_1}$, $R_{{\alpha }_1}$, $F_{{\delta }_1}^{VST}$ are defined in Equation (7), Equation (8), and Equation (24), respectively.

The lower and upper limits of the percentile using the GCI based on the VST method are

$$\left[l_1,u_1\right]=\left[F_{{\theta }_1}^{VST}(\gamma /2),F_{{\theta }_1}^{VST}(1-\gamma /2)\right],$$

(26)

where $F_{{\theta }_1}^{VST}(\gamma /2)$ and $F_{{\theta }_1}^{VST}(1-\gamma /2)$ denote the $100(\gamma /2)$-th and $100(1-\gamma /2)$-th percentiles of $F_{{\theta }_1}^{VST}$, respectively.

For the second population, the GPQ for ${\delta }_2$ is

$$F_{{\delta }_2}^{VST}={sin}^2\left[arcsin\sqrt{{\widehat{\delta }}_2-{\displaystyle \frac{L_2}{2\sqrt{n_2}}}}\right],$$

(27)

where $L_2=2\sqrt{n_2}\left(arcsin\sqrt{{\widehat{\delta }}_2}-arcsin\sqrt{{\delta }_2}\right)$ follows a standard normal distribution as $n_2$ approaches infinity. The estimate of the GPQ for ${\theta }_2$ is

$$F_{{\theta }_2}^{VST}={\displaystyle \frac{R_{{\beta }_2}}{4}}{\left(R_{{\alpha }_2}{\Phi }^{-1}\left({\displaystyle \frac{p_2-F_{{\delta }_2}^{VST}}{1-F_{{\delta }_2}^{VST}}}\right)+\sqrt{{\left(R_{{\alpha }_2}\right)}^2{\left({\Phi }^{-1}\left({\displaystyle \frac{p_2-F_{{\delta }_2}^{VST}}{1-F_{{\delta }_2}^{VST}}}\right)\right)}^2+4}\right)}^2,$$

(28)

where $R_{{\beta }_2}$, $R_{{\alpha }_2}$, $F_{{\delta }_2}^{VST}$ are defined in Equation (10), Equation (11), and Equation (27), respectively.

The lower and upper limits of the percentile using the GCI based on the VST method are

$$\left[l_2,u_2\right]=\left[F_{{\theta }_2}^{VST}(\gamma /2),F_{{\theta }_2}^{VST}(1-\gamma /2)\right],$$

(29)

where $F_{{\theta }_2}^{VST}(\gamma /2)$ and $F_{{\theta }_2}^{VST}(1-\gamma /2)$ denote the $100(\gamma /2)$-th and $100(1-\gamma /2)$-th percentiles of $F_{{\theta }_2}^{VST}$, respectively.

Donner and Zou [15] introduced the MOVER confidence interval for the difference between two parameters. Let $l_1$ and $u_1$ be lower and upper limits of ${\theta }_1$. Let $l_2$ and $u_2$ be lower and upper limits of ${\theta }_2$. The lower and upper limits of ${\theta }_1-{\theta }_2$ are

$$L={\widehat{\theta }}_1-{\widehat{\theta }}_2-\sqrt{{({\widehat{\theta }}_1-l_1)}^2+{(u_2-{\widehat{\theta }}_2)}^2}$$

(30)

and

$$U={\widehat{\theta }}_1-{\widehat{\theta }}_2+\sqrt{{(u_1-{\widehat{\theta }}_1)}^2+{({\widehat{\theta }}_2-l_2)}^2}.$$

(31)

The lower and upper limits of the difference between percentiles derived from the GCI based on the VST method are

$$L_{\omega .MOVER}=F_{{\theta }_1}^{VST}-F_{{\theta }_2}^{VST}-\sqrt{{(F_{{\theta }_1}^{VST}-l_1)}^2+{(u_2-F_{{\theta }_2}^{VST})}^2}$$

(32)

and

$$U_{\omega .MOVER}=F_{{\theta }_1}^{VST}-F_{{\theta }_2}^{VST}+\sqrt{{(u_1-F_{{\theta }_1}^{VST})}^2+{(F_{{\theta }_2}^{VST}-l_2)}^2},$$

(33)

where $F_{{\theta }_1}^{VST}$ is defined in Equation (25), $\left[l_1,u_1\right]$ is defined in Equation (26), $F_{{\theta }_2}^{VST}$ is defined in Equation (28), and $\left[l_2,u_2\right]$ is defined in Equation (29).

Therefore, the $100(1-\gamma )\%$ two-sided MOVER confidence interval for the difference between percentiles is

$$C{I}_{\omega .MOVER}=[L_{\omega .MOVER},U_{\omega .MOVER}],$$

(34)

where $L_{\omega .MOVER}$ and $U_{\omega .MOVER}$ are defined in Equation (32) and Equation (33), respectively.

Example 4.

The estimates of the GPQ for ${\theta }_1$ are $F_{{\theta }_1}^{VST}=$ 0.5057, 0.6208, 0.5826, 0.4808, 0.5372, 0.5540, 0.5088, 0.5285, 0.5501, and 0.5786. The estimates of the GPQ for ${\theta }_2$ are $F_{{\theta }_2}^{VST}=$ 0.4383, 0.2968, 0.3781, 0.4347, 0.4143, 0.4062, 0.4177, 0.4220, 0.4329, and 0.3521. The lower and upper limits of the percentile using the GCI based on the VST method are $l_1=F_{{\theta }_1}^{VST}\left(0.025\right)=$ 0.4864, $u_1=F_{{\theta }_1}^{VST}\left(0.975\right)=$ 0.6122, $l_2=F_{{\theta }_2}^{VST}\left(0.025\right)=$ 0.3092, and $u_2=F_{{\theta }_2}^{VST}\left(0.975\right)=$ 0.4375. The lower and upper limits of the 95% two-sided MOVER confidence interval for the difference between percentiles are $L_{\omega .MOVER}=$ 0.0480 and $U_{\omega .MOVER}=$ 0.2347, respectively.

Example 5.

For $M=$ 10, the difference between percentiles of delta-Birnbaum–Saunders distributions is $\omega =$ 0. The 95% two-sided confidence intervals for the difference between percentiles are $[L_{\omega },U_{\omega }]=$ [0.0726, 0.4315], [−0.0450, 0.2513], [−0.1673, 0.0658], [0.0470, 0.2583], [−0.1718, 0.0964], [−0.1334, 0.1745], [−0.1885, 0.1849], −0.0663, 0.2386], [−0.1308, 0.1450], and [−0.0305, 0.2184]. The values of $p_{\omega }$ are 0, 1, 1, 0, 1, 1, 1, 1, 1, and 1. The coverage probability is 0.8000. The values of $U_{\omega }-L_{\omega }$ are 0.3589, 0.2962, 0.2331, 0.2113, 0.2681, 0.3078, 0.3734, 0.3049, 0.2758, and 0.2489. The average length is 0.2878.

3. Confidence Intervals for Ratio of Percentiles of Birnbaum–Saunders Distributions Containing Zero Values

The ratio of percentiles of Birnbaum–Saunders distributions containing zero values is

$$\tau ={\displaystyle \frac{{\theta }_1}{{\theta }_2}},$$

(35)

where ${\theta }_1$ and ${\theta }_2$ are defined in Equation (3) and Equation (4), respectively.

The confidence intervals for the ratio of percentiles of Birnbaum–Saunders distributions containing zero values are constructed using the GCI approach, the bootstrap approach, the HPD approach based on the bootstrap method, the Bayesian approach, the HPD approach based on the Bayesian method, and the MOVER approach.

3.1. GCI for Ratio of Percentiles

The GPQ of the ratio of percentiles is

$$R_{\tau }={\displaystyle \frac{R_{{\theta }_1}}{R_{{\theta }_2}}},$$

(36)

where $R_{{\theta }_1}$ and $R_{{\theta }_2}$ are defined in Equation (6) and Equation (9), respectively.

Therefore, the $100(1-\gamma )\%$ two-sided GCI for the ratio of percentiles is

$$C{I}_{\tau .GCI}=[L_{\tau .GCI},U_{\tau .GCI}]=[R_{\tau }(\gamma /2),R_{\tau }(1-\gamma /2)],$$

(37)

where $R_{\tau }(\gamma /2)$ and $R_{\tau }(1-\gamma /2)$ denote the $100(\gamma /2)$-th and $100(1-\gamma /2)$-th percentiles of $R_{\tau }$, respectively. The GCI for the ratio of percentiles can be computed similarly to the GCI for the difference of percentiles in Algorithm 1.

Algorithm 1 GCI for difference between percentiles

Step 1: Generate $x_1=\left(x_{11},x_{12},\dots ,x_{1n_1}\right)$ from Birnbaum–Saunders distribution containing zero values and compute $A_1$, $B_1$, $C_1$, $S_{11}$, and $S_{12}$

Step 2: Generate $x_2=\left(x_{21},x_{22},\dots ,x_{2n_2}\right)$ from Birnbaum–Saunders distribution containing zero values and compute $A_2$, $B_2$, $C_2$, $S_{21}$, and $S_{22}$

Step 3: At the m step (a) Compute $R_{{\beta }_1}$, $R_{{\alpha }_1}$, and $R_{{\theta }_1}$ (b) Compute $R_{{\beta }_2}$, $R_{{\alpha }_2}$, and $R_{{\theta }_2}$ (c) Compute $R_{\omega }$

Step 4: Repeat step 3, a total M times and obtain an array of $R_{\omega }$’s

Step 5: Compute $L_{\omega .GCI}$ and $U_{\omega .GCI}$

Example 6.

For $m=$ 10, the GPQs of the ratio of percentiles are $R_{\tau }=$ 1.1537, 2.0916, 1.5410, 1.1059, 1.2966, 1.3639, 1.2183, 1.2525, 1.2708, and 1.6432. The lower and upper limits of the 95% two-sided GCI for the ratio of percentiles are $L_{\tau .GCI}=R_{\tau }\left(0.025\right)=$ 1.1167 and $U_{\tau .GCI}=R_{\tau }\left(0.975\right)=$ 1.9907, respectively.

3.2. Bootstrap Confidence Interval for Ratio of Percentiles

The bootstrap estimator of the ratio of percentiles is

$${\widehat{\tau }}_k={\displaystyle \frac{{\widehat{\theta }}_{1k}}{{\widehat{\theta }}_{2k}}},$$

(38)

where ${\widehat{\theta }}_{1k}$ and ${\widehat{\theta }}_{2k}$ are defined in Equation (14) and Equation (15), respectively.

Therefore, the $100(1-\gamma )\%$ two-sided bootstrap confidence interval for the ratio of percentiles is

$$C{I}_{\tau .B}=[L_{\tau .B},U_{\tau .B}]=[{\widehat{\tau }}_k(\gamma /2),{\widehat{\tau }}_k(1-\gamma /2)],$$

(39)

where ${\widehat{\tau }}_k(\gamma /2)$ and ${\widehat{\tau }}_k(1-\gamma /2)$ denote the $100(\gamma /2)$-th and $100(1-\gamma /2)$-th percentiles of ${\widehat{\tau }}_k$, respectively.

3.3. HPD Interval Based on the Bootstrap Method for Ratio of Percentiles

The bootstrap estimator of the ratio of percentiles defined in Equation (38) is used to construct the HPD interval based on the bootstrap method. Therefore, the $100(1-\gamma )\%$ two-sided HPD interval based on the bootstrap method for the ratio of percentiles is

$$C{I}_{\tau .BHPD}=[L_{\tau .BHPD},U_{\tau .BHPD}],$$

(40)

where $L_{\tau .BHPD}$ and $U_{\tau .BHPD}$ are computed using the hdi function within the HDInterval package of the R software suite. The bootstrap confidence interval and HPD interval based on the bootstrap method for the ratio of percentiles can be computed similarly to the bootstrap confidence interval and HPD interval based on the bootstrap method for the difference between percentiles in Algorithm 2.

Algorithm 2 Bootstrap confidence interval and HPD interval based on the bootstrap method for the difference between percentiles

Step 1: Generate $x_1=(x_{11},x_{12},\dots ,x_{1n_1})$ from Birnbaum–Saunders distribution containing zero values and generate $x_2=(x_{21},x_{22},\dots ,x_{2n_2})$ from Birnbaum–Saunders distribution containing zero values

Step 2: At the b step (a) Generate $x_1^{\ast }=(x_{11}^{\ast },x_{12}^{\ast },\dots ,x_{1n_1}^{\ast })$ (b) Compute $\widehat{b}({\widehat{\alpha }}_1,{\alpha }_1)$, $\widehat{b}({\widehat{\beta }}_1,{\beta }_1)$, ${\tilde{\alpha }}_{1k}$, ${\tilde{\beta }}_{1k}$, and ${\widehat{\theta }}_{1k}$ (c) Generate $x_2^{\ast }=(x_{21}^{\ast },x_{22}^{\ast },\dots ,x_{2n_2}^{\ast })$ (d) Compute $\widehat{b}({\widehat{\alpha }}_2,{\alpha }_2)$, $\widehat{b}({\widehat{\beta }}_2,{\beta }_2)$, ${\tilde{\alpha }}_{2k}$, ${\tilde{\beta }}_{2k}$, and ${\widehat{\theta }}_{2k}$ (e) Compute ${\widehat{\omega }}_k$

Step 3: Repeat step 2, a total B times and obtain an array of ${\widehat{\omega }}_k$’s

Step 4: Compute $L_{\omega .B}$ and $U_{\omega .B}$ and compute $L_{\omega .BHPD}$ and $U_{\omega .BHPD}$

Example 7.

For $b=$ 10, the bootstrap estimators of the ratio of percentiles are ${\widehat{\tau }}_k=$ 1.0910, 1.0312, 0.9365, 1.0130, 0.9491, 1.0183, 0.9653, 0.9903, 0.9325, and 0.9843. The lower and upper limits of the 95% two-sided bootstrap confidence interval for the ratio of percentiles are $L_{\tau .B}={\widehat{\tau }}_k\left(0.025\right)=$ 0.9334 and $U_{\tau .B}={\widehat{\tau }}_k\left(0.975\right)=$ 1.0776, respectively. Additionally, the lower and upper limits of the 95% two-sided HPD interval based on the bootstrap method for the ratio of percentiles are $L_{\tau .BHPD}=$ 0.9325 and $U_{\tau .BHPD}=$ 1.0910, respectively.

3.4. Bayesian Credible Interval for Ratio of Percentiles

The posterior distribution of the ratio of percentiles is

$${\tau }_{Baye}={\displaystyle \frac{{\theta }_{Baye1}}{{\theta }_{Baye2}}},$$

(41)

where ${\theta }_{Baye1}$ and ${\theta }_{Baye2}$ are defined in Equation (19) and Equation (20), respectively.

Therefore, the $100(1-\gamma )\%$ two-sided Bayesian credible interval for the ratio of percentiles is

$$C{I}_{\tau .Baye}=[L_{\tau .Baye},U_{\tau .Baye}]=[{\tau }_{Baye}(\gamma /2),{\tau }_{Baye}(1-\gamma /2)],$$

(42)

where ${\tau }_{Baye}(\gamma /2)$ and ${\tau }_{Baye}(1-\gamma /2)$ denote the $100(\gamma /2)$-th and $100(1-\gamma /2)$-th percentiles of ${\tau }_{Baye}$, respectively.

3.5. HPD Interval Based on the Bayesian Method for Ratio of Percentiles

The posterior distribution of ${\tau }_{Baye}$ defined in Equation (41) is used to construct the HPD interval based on the Bayesian method. Therefore, the $100(1-\gamma )\%$ two-sided HPD interval based on the Bayesian method for the ratio of percentiles is

$$C{I}_{\tau .HPD}=[L_{\tau .HPD},U_{\tau .HPD}],$$

(43)

where $L_{\tau .HPD}$ and $U_{\tau .HPD}$ are computed using the hdi function within the HDInterval package of the R software suite. The Bayesian credible interval and the HPD interval based on the Bayesian method for the ratio of percentiles can be computed similarly to the Bayesian credible interval and the HPD interval based on the Bayesian method for the difference between percentiles in Algorithm 3.

Algorithm 3 Bayesian credible interval and the HPD interval based on the Bayesian method for the difference between percentiles

Step 1: Compute $a\left(r_1\right)=\underset{{\beta }_1>0}{sup}\left\{{\left[p\left({\beta }_1\right|x_1)\right]}^{1/(r_1+1)}\right\}$, $b^{+}\left(r_1\right)={sup}_{{\beta }_1>0}\left\{{\beta }_1{\left[p\left({\beta }_1\right|x_1)\right]}^{r_1/(r_1+1)}\right\}$, $a\left(r_2\right)=\underset{{\beta }_2>0}{sup}\left\{{\left[p\left({\beta }_2\right|x_2)\right]}^{1/(r_2+1)}\right\}$, and $b^{+}\left(r_2\right)={sup}_{{\beta }_2>0}\left\{{\beta }_2{\left[p\left({\beta }_2\right|x_2)\right]}^{r_2/(r_2+1)}\right\}$

Step 2: At the i step (a) Generate $u_1$ and $v_1$ and compute ${\rho }_1=v_1/u_1^{r_1}$ (b) If the value of ${\rho }_1$ is accepted, the set ${\beta }_{1\left(i\right)}={\rho }_1$ if $u_1⩽{\left[p\left({\beta }_1\right|x_1)\right]}^{1/(r_1+1)}$; otherwise, repeat step (a) (c) Generate ${\lambda }_1$ and compute ${\alpha }_{1\left(i\right)}=\sqrt{{\lambda }_1}$ and ${\theta }_{Baye1}$ (d) Generate $u_2$ and $v_2$ and compute ${\rho }_2=v_2/u_2^{r_2}$ (e) If the value of ${\rho }_2$ is accepted, the set ${\beta }_{2\left(i\right)}={\rho }_2$ if $u_2⩽{\left[p\left({\beta }_2\right|x_2)\right]}^{1/(r_2+1)}$; otherwise, repeat step (d) (f) Generate ${\lambda }_2$ and compute ${\alpha }_{2\left(i\right)}=\sqrt{{\lambda }_2}$ and ${\theta }_{Baye2}$ (g) Compute ${\omega }_{Baye}$

Step 3: Repeat step 1–step 2, a total M times and obtain an array of ${\omega }_{Baye}$’s

Step 4: Compute $L_{\omega .Baye}$ and $U_{\omega .Baye}$ and compute $L_{\omega .HPD}$ and $U_{\omega .HPD}$

Example 8.

For $i=$ 10, The posterior distributions of the ratio of percentiles are ${\tau }_{Baye}=$ 1.3336, 1.4733, 1.3178, 1.4497, 1.2076, 1.2378, 1.4234, 1.3280, 1.7690, and 1.2502. The lower and upper limits of the 95% two-sided Bayesian credible interval for the ratio of percentiles are $L_{\tau .Baye}={\tau }_{Baye}\left(0.025\right)=$ 1.2144 and $U_{\tau .Baye}={\tau }_{Baye}\left(0.975\right)=$ 1.7025, respectively. Additionally, the lower and upper limits of the 95% two-sided HPD interval based on the Bayesian method for the ratio of percentiles are $L_{\tau .HPD}=$ 1.2076 and $U_{\tau .HPD}=$ 1.7690, respectively.

3.6. MOVER Confidence Interval for Ratio of Percentiles

The MOVER confidence interval serves as an extension of Fieller’s theorem [16], reducing to Fieller’s method under the assumption of symmetric confidence intervals for ${\theta }_1$ and ${\theta }_2$. Fieller’s theorem has been extensively used in health economics to estimate confidence intervals for incremental cost-effectiveness ratios (ICERs) [17]. However, recent research has emphasized that its reliability is highly contingent on the assumption that the numerator and denominator have symmetric sampling distributions [18]. This is particularly noteworthy since the numerator of an ICER, which represents the cost difference between two therapeutic interventions, is seldom symmetrically distributed.

Donner and Zou [15] proposed the MOVER confidence interval for the ratio of two parameters. Let $l_1$ and $u_1$ be lower and upper limits of ${\theta }_1$. Let $l_2$ and $u_2$ be lower and upper limits of ${\theta }_2$. The lower and upper limits of ${\theta }_1/{\theta }_2$ are

$$L={\displaystyle \frac{{\widehat{\theta }}_1{\widehat{\theta }}_2-\sqrt{{\left({\widehat{\theta }}_1{\widehat{\theta }}_2\right)}^2-l_1{u}_2(2{\widehat{\theta }}_1-l_1)(2{\widehat{\theta }}_2-u_2)}}{u_2(2{\widehat{\theta }}_2-u_2)}}$$

(44)

and

$$U={\displaystyle \frac{{\widehat{\theta }}_1{\widehat{\theta }}_2+\sqrt{{\left({\widehat{\theta }}_1{\widehat{\theta }}_2\right)}^2-u_1{l}_2(2{\widehat{\theta }}_1-u_1)(2{\widehat{\theta }}_2-l_2)}}{l_2(2{\widehat{\theta }}_2-l_2)}}.$$

(45)

The lower and upper limits of the ratio of percentiles derived from the GCI based on the VST method are

$$L_{\tau .MOVER}={\displaystyle \frac{F_{{\theta }_1}^{VST}{F}_{{\theta }_2}^{VST}-\sqrt{{\left(F_{{\theta }_1}^{VST}{F}_{{\theta }_2}^{VST}\right)}^2-l_1{u}_2(2F_{{\theta }_1}^{VST}-l_1)(2F_{{\theta }_2}^{VST}-u_2)}}{u_2(2F_{{\theta }_2}^{VST}-u_2)}}$$

(46)

and

$$U_{\tau .MOVER}={\displaystyle \frac{F_{{\theta }_1}^{VST}{F}_{{\theta }_2}^{VST}+\sqrt{{\left(F_{{\theta }_1}^{VST}{F}_{{\theta }_2}^{VST}\right)}^2-u_1{l}_2(2F_{{\theta }_1}^{VST}-u_1)(2F_{{\theta }_2}^{VST}-l_2)}}{l_2(2F_{{\theta }_2}^{VST}-l_2)}},$$

(47)

where $F_{{\theta }_1}^{VST}$ is defined in Equation (25), $\left[l_1,u_1\right]$ is defined in Equation (26), $F_{{\theta }_2}^{VST}$ is defined in Equation (28), and $\left[l_2,u_2\right]$ is defined in Equation (29).

Therefore, the $100(1-\gamma )\%$ two-sided MOVER confidence interval for the ratio of percentiles is

$$C{I}_{\tau .MOVER}=[L_{\tau .MOVER},U_{\tau .MOVER}],$$

(48)

where $L_{\tau .MOVER}$ and $U_{\tau .MOVER}$ are defined in Equation (46) and Equation (47), respectively.

Example 9.

The estimates of the GPQ for ${\theta }_1$ are $F_{{\theta }_1}^{VST}=$ 0.5057, 0.6208, 0.5826, 0.4808, 0.5372, 0.5540, 0.5088, 0.5285, 0.5501, and 0.5786. The estimates of the GPQ for ${\theta }_2$ are $F_{{\theta }_2}^{VST}=$ 0.4383, 0.2968, 0.3781, 0.4347, 0.4143, 0.4062, 0.4177, 0.4220, 0.4329, and 0.3521. The lower and upper limits of the percentile using the GCI based on the VST method are $l_1=F_{{\theta }_1}^{VST}\left(0.025\right)=$ 0.4864, $u_1=F_{{\theta }_1}^{VST}\left(0.975\right)=$ 0.6122, $l_2=F_{{\theta }_2}^{VST}\left(0.025\right)=$ 0.3092, and $u_2=F_{{\theta }_2}^{VST}\left(0.975\right)=$ 0.4375. The lower and upper limits of the 95% two-sided MOVER confidence interval for the ratio of percentiles are $L_{\tau .MOVER}=$ 1.1096 and $U_{\tau .MOVER}=$ 1.7138, respectively.

The coverage probability and average length of the confidence interval for the ratio of percentiles can be calculated in the same way as for the difference between percentiles in Algorithm 4.

Algorithm 4 Coverage probability and average length of confidence interval for difference between percentiles

Step 1: Construct the confidence interval for $\omega$

Step 2: If $L_{\omega }⩽\omega ⩽U_{\omega }$, set $p_{\omega }=$ 1; else set $p_{\omega }=$ 0

Step 3: Compute $U_{\omega }-L_{\omega }$

Step 4: Compute step 1–step 3, a total 5000 times

Step 5: Compute the coverage probability by evaluating the mean of $p_{\omega }$

Step 6: Compute the average length by evaluating the mean of $U_{\omega }-L_{\omega }$.

4. Results

A Monte Carlo simulation study was conducted using R software to assess the effectiveness of confidence intervals for the difference and ratio of percentiles in Birnbaum–Saunders distributions containing zero values. The best-performing confidence intervals were those with a coverage probability greater than or close to the nominal confidence level of 0.95 and the shortest average length.

Data were generated for two independent Birnbaum–Saunders distributions containing zero values. The scale parameters are fixed as $({\beta }_1,{\beta }_2)=$ (1.0,1.0), the values of the shape parameters are set as $({\alpha }_1,{\alpha }_2)=$ (0.5,0.5), (0.5,1.0), and (1.0,1.0), the probabilities of obtaining a positive observation are fixed as $({\delta }_1,{\delta }_2)=$ (0.1,0.1), (0.1,0.3), and (0.3,0.3), the probabilities are set as $(p_1,p_2)=$ (0.5,0.5), and the sample sizes are fixed as $(n_1,n_2)=$ (30,30), (30,50), (50,50), (50,100), (100,100), (100,200), and (200,200). In the simulation, values of $r_1=r_2=$ 2.00 and $a_{11}=a_{12}=a_{21}=a_{22}=b_{11}=b_{12}=b_{21}=b_{22}={10}^{-4}$ were assigned for both the Bayesian approach and the HPD approach derived from the Bayesian method. A total of 5000 replications were conducted, with 5000 iterations used for the GCI, 500 iterations for the bootstrap confidence interval and the HPD interval based on the bootstrap method, and 1000 iterations for the Bayesian credible interval and the HPD interval based on the Bayesian method.

For the difference between percentiles, the coverage probabilities and average lengths of the confidence intervals are presented in

Table 1. The coverage probabilities (CPs) and average lengths (ALs) of 95% two-sided confidence intervals for the difference between percentiles of Birnbaum–Saunders distributions containing zero values.

$({\mathit{n}}_1,{\mathit{n}}_2)$	$({\mathit{\alpha }}_1,{\mathit{\alpha }}_2)$	$({\mathit{\delta }}_1,{\mathit{\delta }}_2)$	CP (AL)
			${\mathit{CI}}_{\mathit{\omega }.\mathit{GCI}}$	${\mathit{CI}}_{\mathit{\omega }.\mathit{B}}$	${\mathit{CI}}_{\mathit{\omega }.\mathit{BHPD}}$	${\mathit{CI}}_{\mathit{\omega }.\mathit{Baye}}$	${\mathit{CI}}_{\mathit{\omega }.\mathit{HPD}}$	${\mathit{CI}}_{\mathit{\omega }.\mathit{MOVER}}$
(30,30)	(0.5,0.5)	(0.1,0.1)	0.9480	0.9360	0.9340	0.9420	0.9430	0.9490
			(0.5081)	(0.4735)	(0.4685)	(0.4951)	(0.4910)	(0.4893)
		(0.1,0.3)	0.9590	0.9270	0.9280	0.9450	0.9400	0.7660
			(0.5069)	(0.4720)	(0.4670)	(0.4935)	(0.4890)	(0.4380)
		(0.3,0.3)	0.9590	0.9350	0.9320	0.9530	0.9450	0.9560
			(0.5005)	(0.4654)	(0.4607)	(0.4861)	(0.4821)	(0.3739)
	(0.5,1.0)	(0.1,0.1)	0.9460	0.9340	0.9370	0.9400	0.9450	0.9510
			(0.7346)	(0.6744)	(0.6649)	(0.7003)	(0.6922)	(0.6982)
		(0.1,0.3)	0.9620	0.9390	0.9420	0.9500	0.9520	0.9110
			(0.6245)	(0.5868)	(0.5772)	(0.6009)	(0.5952)	(0.5146)
		(0.3,0.3)	0.9460	0.9310	0.9270	0.9410	0.9390	0.9230
			(0.6214)	(0.5852)	(0.5756)	(0.5976)	(0.5918)	(0.4627)
	(1.0,1.0)	(0.1,0.1)	0.9530	0.9450	0.9420	0.9430	0.9470	0.9600
			(0.9086)	(0.8358)	(0.8262)	(0.8630)	(0.8546)	(0.8720)
		(0.1,0.3)	0.9560	0.9400	0.9470	0.9520	0.9530	0.9190
			(0.8238)	(0.7682)	(0.7596)	(0.7856)	(0.7772)	(0.7269)
		(0.3,0.3)	0.9370	0.9210	0.9250	0.9360	0.9380	0.9520
			(0.7256)	(0.6919)	(0.6832)	(0.6945)	(0.6884)	(0.5401)
(30,50)	(0.5,0.5)	(0.1,0.1)	0.9530	0.9420	0.9390	0.9480	0.9480	0.9490
			(0.4536)	(0.4265)	(0.4217)	(0.4430)	(0.4390)	(0.4364)
		(0.1,0.3)	0.9510	0.9360	0.9320	0.9460	0.9400	0.7220
			(0.4521)	(0.4251)	(0.4206)	(0.4413)	(0.4374)	(0.3992)
		(0.3,0.3)	0.9540	0.9410	0.9440	0.9520	0.9500	0.9580
			(0.4470)	(0.4201)	(0.4153)	(0.4357)	(0.4321)	(0.3329)
	(0.5,1.0)	(0.1,0.1)	0.9510	0.9390	0.9280	0.9470	0.9480	0.9500
			(0.6005)	(0.5630)	(0.5570)	(0.5800)	(0.5745)	(0.5770)
		(0.1,0.3)	0.9620	0.9480	0.9470	0.9560	0.9570	0.9000
			(0.5304)	(0.5025)	(0.4966)	(0.5141)	(0.5099)	(0.4544)
		(0.3,0.3)	0.9460	0.9370	0.9390	0.9470	0.9430	0.9110
			(0.5253)	(0.4987)	(0.4931)	(0.5085)	(0.5042)	(0.3936)
	(1.0,1.0)	(0.1,0.1)	0.9550	0.9400	0.9390	0.9540	0.9550	0.9650
			(0.8103)	(0.7504)	(0.7413)	(0.7727)	(0.7645)	(0.7775)
		(0.1,0.3)	0.9610	0.9420	0.9460	0.9540	0.9600	0.9210
			(0.7560)	(0.7025)	(0.6935)	(0.7216)	(0.7130)	(0.6750)
		(0.3,0.3)	0.9560	0.9310	0.9390	0.9430	0.9400	0.9510
			(0.6431)	(0.6145)	(0.6054)	(0.6163)	(0.6107)	(0.4785)
(50,50)	(0.5,0.5)	(0.1,0.1)	0.9490	0.9450	0.9410	0.9500	0.9500	0.9520
			(0.3870)	(0.3702)	(0.3664)	(0.3801)	(0.3769)	(0.3719)
		(0.1,0.3)	0.9700	0.9580	0.9580	0.9640	0.9620 )	0.6320
			(0.3825)	(0.3655)	(0.3621)	(0.3757)	(0.3727	(0.3306)
		(0.3,0.3)	0.9620	0.9480	0.9430	0.9570	0.9580	0.9650
			(0.3793)	(0.3631)	(0.3597)	(0.3718)	(0.3688)	(0.2835)
	(0.5,1.0)	(0.1,0.1)	0.9490	0.9360	0.9330	0.9460	0.9440	0.9490
			(0.5553)	(0.5224)	(0.5159)	(0.5359)	(0.5305)	(0.5279)
		(0.1,0.3)	0.9630	0.9390	0.9430	0.9520	0.9510	0.8840
			(0.4750)	(0.4544)	(0.4484)	(0.4609)	(0.4569)	(0.3936)
		(0.3,0.3)	0.9440	0.9300	0.9250	0.9460	0.9430	0.9100
			(0.4675)	(0.4492)	(0.4433)	(0.4545)	(0.4504)	(0.3469)
	(1.0,1.0)	(0.1,0.1)	0.9640	0.9580	0.9570	0.9530	0.9600	0.9640
			(0.6800)	(0.6384)	(0.6320)	(0.6534)	(0.6478)	(0.6516)
		(0.1,0.3)	0.9740	0.9610	0.9620	0.9620	0.9640	0.9160
			(0.6176)	(0.5869)	(0.5807)	(0.5944)	(0.5888)	(0.5396)
		(0.3,0.3)	0.9410	0.9310	0.9290	0.9400	0.9390	0.9470
			(0.5446)	(0.5235)	(0.5176)	(0.5257)	(0.5212)	(0.4058)
(50,100)	(0.5,0.5)	(0.1,0.1)	0.9480	0.9420	0.9420	0.9480	0.9430	0.9500
			(0.3343)	(0.3209)	(0.3175)	(0.3293)	(0.3263)	(0.3193)
		(0.1,0.3)	0.9480	0.9370	0.9380	0.9490	0.9430	0.5570
			(0.3313)	(0.3190)	(0.3155)	(0.3263)	(0.3234)	(0.2949)
		(0.3,0.3)	0.9550	0.9440	0.9450	0.9480	0.9450	0.9590
			(0.3271)	(0.3136)	(0.3105)	(0.3219)	(0.3191)	(0.2434)
	(0.5,1.0)	(0.1,0.1)	0.9490	0.9440	0.9420	0.9460	0.9430	0.9440
			(0.4306)	(0.4121)	(0.4073)	(0.4198)	(0.4161)	(0.4125)
		(0.1,0.3)	0.9520	0.9470	0.9380	0.9440	0.9430	0.8440
			(0.3841)	(0.3701)	(0.3663)	(0.3758)	(0.3727)	(0.3307)
		(0.3,0.3)	0.9550	0.9430	0.9460	0.9520	0.9520	0.8920
			(0.3783)	(0.3638)	(0.3600)	(0.3693)	(0.3664)	(0.2828)
	(1.0,1.0)	(0.1,0.1)	0.9500	0.9340	0.9360	0.9340	0.9380	0.9440
			(0.5854)	(0.5511)	(0.5450)	(0.5633)	(0.5579)	(0.5589)
		(0.1,0.3)	0.9390	0.9320	0.9270	0.9390	0.9420	0.9030
			(0.5517)	(0.5224)	(0.5153)	(0.5316)	(0.5261)	(0.4940)
		(0.3,0.3)	0.9480	0.9450	0.9480	0.9520	0.9530	0.9580
			(0.4689)	(0.4508)	(0.4452)	(0.4535)	(0.4495)	(0.3472)
(100,100)	(0.5,0.5)	(0.1,0.1)	0.9520	0.9440	0.9420	0.9480	0.9470	0.9510
			(0.2703)	(0.2622)	(0.2596)	(0.2669)	(0.2646)	(0.2600)
		(0.1,0.3)	0.9430	0.9410	0.9370	0.9410	0.9390	0.3710
			(0.2672)	(0.2597)	(0.2570)	(0.2641)	(0.2619)	(0.2300)
		(0.3,0.3)	0.9500	0.9440	0.9400	0.9510	0.9510	0.9460
			(0.2648)	(0.2569)	(0.2544)	(0.2612)	(0.2591)	(0.1973)
	(0.5,1.0)	(0.1,0.1)	0.9590	0.9520	0.9510	0.9610	0.9580	0.9590
			(0.3847)	(0.3686)	(0.3645)	(0.3735)	(0.3701)	(0.3664)
		(0.1,0.3)	0.9460	0.9370	0.9420	0.9480	0.9480	0.7780
			(0.3284)	(0.3178)	(0.3141)	(0.3210)	(0.3184)	(0.2708)
		(0.3,0.3)	0.9560	0.9450	0.9400	0.9550	0.9510	0.8660
			(0.3266)	(0.3164)	(0.3130)	(0.3189)	(0.3162)	(0.2422)
	(1.0,1.0)	(0.1,0.1)	0.9480	0.9430	0.9380	0.9480	0.9440	0.9550
			(0.4702)	(0.4492)	(0.4450)	(0.4555)	(0.4516)	(0.4544)
		(0.1,0.3)	0.9590	0.9480	0.9510	0.9520	0.9500	0.8390
			(0.4272)	(0.4107)	(0.4063)	(0.4132)	(0.4095)	(0.3747)
		(0.3,0.3)	0.9500	0.9400	0.9430	0.9460	0.9400	0.9420
			(0.3787)	(0.3667)	(0.3629)	(0.3680)	(0.3650)	(0.2836)
(100,200)	(0.5,0.5)	(0.1,0.1)	0.9480	0.9470	0.9480	0.9470	0.9480	0.9500
			(0.2330)	(0.2278)	(0.2255)	(0.2304)	(0.2285)	(0.2240)
		(0.1,0.3)	0.9590	0.9510	0.9450	0.9590	0.9510	0.2520
			(0.2317)	(0.2261)	(0.2238)	(0.2293)	(0.2272)	(0.2048)
		(0.3,0.3)	0.9570	0.9480	0.9440	0.9520	0.9450	0.9450
			(0.2293)	(0.2235)	(0.2212)	(0.2269)	(0.2250)	(0.1705)
	(0.5,1.0)	(0.1,0.1)	0.9420	0.9410	0.9340	0.9410	0.9440	0.9450
			(0.3006)	(0.2902)	(0.2874)	(0.2941)	(0.2916)	(0.2893)
		(0.1,0.3)	0.9470	0.9470	0.9420	0.9510	0.9460	0.7030
			(0.2667)	(0.2592)	(0.2564)	(0.2616)	(0.2595)	(0.2296)
		(0.3,0.3)	0.9490	0.9470	0.9450	0.9490	0.9480	0.8200
			(0.2649)	(0.2574)	(0.2548)	(0.2603)	(0.2581)	(0.1978)
	(1.0,1.0)	(0.1,0.1)	0.9490	0.9360	0.9400	0.9430	0.9440	0.9500
			(0.4064)	(0.3891)	(0.3848)	(0.3933)	(0.3900)	(0.3896)
		(0.1,0.3)	0.9490	0.9450	0.9400	0.9510	0.9460	0.8090
			(0.3820)	(0.3660)	(0.3619)	(0.3703)	(0.3668)	(0.3415)
		(0.3,0.3)	0.9600	0.9490	0.9500	0.9530	0.9520	0.9620
			(0.3255)	(0.3163)	(0.3126)	(0.3167)	(0.3141)	(0.2416)
(200,200)	(0.5,0.5)	(0.1,0.1)	0.9630	0.9620	0.9570	0.9630	0.9620	0.9590
			(0.1895)	(0.1858)	(0.1839)	(0.1878)	(0.1862)	(0.1819)
		(0.1,0.3)	0.9510	0.9520	0.9470	0.9470	0.9440	0.0940
			(0.1879)	(0.1847)	(0.1829)	(0.1859)	(0.1844)	(0.1621)
		(0.3,0.3)	0.9640	0.9650	0.9570	0.9640	0.9550	0.9590
			(0.1862)	(0.1826)	(0.1810)	(0.1848)	(0.1833)	(0.1389)
	(0.5,1.0)	(0.1,0.1)	0.9500	0.9350	0.9340	0.9410	0.9350	0.9420
			(0.2683)	(0.2588)	(0.2559)	(0.2619)	(0.2595)	(0.2542)
		(0.1,0.3)	0.9480	0.9410	0.9370	0.9430	0.9440	0.6460
			(0.2302)	(0.2237)	(0.2214)	(0.2254)	(0.2236)	(0.1894)
		(0.3,0.3)	0.9560	0.9470	0.9420	0.9540	0.9480	0.7630
			(0.2287)	(0.2223)	(0.2201)	(0.2242)	(0.2223)	(0.1699)
	(1.0,1.0)	(0.1,0.1)	0.9530	0.9490	0.9470	0.9510	0.9510	0.9610
			(0.3290)	(0.3157)	(0.3128)	(0.3196)	(0.3170)	(0.3159)
		(0.1,0.3)	0.9670	0.9590	0.9610	0.9630	0.9640	0.7800
			(0.2983)	(0.2885)	(0.2856)	(0.2897)	(0.2873)	(0.2628)
		(0.3,0.3)	0.9500	0.9460	0.9460	0.9500	0.9470	0.9570
			(0.2645)	(0.2571)	(0.2546)	(0.2577)	(0.2554)	(0.1973)

. The results indicate that the coverage probabilities of the GCI approach, the Bayesian approach, the HPD approach based on the Bayesian method, and the MOVER approach are close to the nominal confidence level of 0.95 for almost all cases, except that the coverage probabilities of the MOVER approach are below the nominal confidence level of 0.95 for ${\delta }_1\ne {\delta }_2$. Moreover, the coverage probabilities of the bootstrap approach and the HPD approach based on the bootstrap method are below the nominal confidence level of 0.95 for almost all cases. For ${\delta }_1={\delta }_2$, the average lengths of the MOVER approach are the shortest. Therefore, the MOVER approach is recommended for constructing the confidence interval for the difference between percentiles for ${\delta }_1={\delta }_2$. For ${\delta }_1\ne {\delta }_2$, the average lengths of the HPD approach based on the Bayesian method are the shortest. Therefore, the HPD approach based on the Bayesian method is recommended for constructing the confidence interval for the difference between percentiles for ${\delta }_1\ne {\delta }_2$. Figure 1, Figure 2 and Figure 3 illustrate the coverage probabilities and average lengths of the confidence intervals for the differences between percentiles, considering various sample sizes, shape parameters, and probabilities of obtaining a positive observation, respectively. In Figure 1, the coverage probabilities approached the nominal confidence level of 0.95 as the sample size increased, while the average lengths of all the approaches decreased with larger sample sizes. Figure 2 shows that as the shape parameter increased, the coverage probabilities remained close to the nominal confidence level of 0.95, but the average lengths of all approaches increased. In Figure 3, the coverage probabilities were close to the nominal confidence level of 0.95 for $({\delta }_1,{\delta }_2)=$ (0.1,0.1) and $({\delta }_1,{\delta }_2)=$ (0.1,0.3), except for the MOVER approach, which deviated from this pattern. Additionally, the average lengths of all the approaches decreased as the probabilities of obtaining a positive observation increased.

For the ratio of percentiles, the coverage probabilities and average lengths of the confidence intervals are provided in

Table 2. The CPs and ALs of 95% two-sided confidence intervals for the ratio of percentiles of Birnbaum–Saunders distributions containing zero values.

$({\mathit{n}}_1,{\mathit{n}}_2)$	$({\mathit{\alpha }}_1,{\mathit{\alpha }}_2)$	$({\mathit{\delta }}_1,{\mathit{\delta }}_2)$	CP (AL)
			${\mathit{CI}}_{\mathit{\tau }.\mathit{GCI}}$	${\mathit{CI}}_{\mathit{\tau }.\mathit{B}}$	${\mathit{CI}}_{\mathit{\tau }.\mathit{BHPD}}$	${\mathit{CI}}_{\mathit{\tau }.\mathit{Baye}}$	${\mathit{CI}}_{\mathit{\tau }.\mathit{HPD}}$	${\mathit{CI}}_{\mathit{\tau }.\mathit{MOVER}}$
(30,30)	(0.5,0.5)	(0.1,0.1)	0.9480	0.9360	0.9290	0.9420	0.9430	0.9490
			(0.5494)	(0.5123)	(0.5029)	(0.5346)	(0.5253)	(0.5887)
		(0.1,0.3)	0.9590	0.9240	0.9220	0.9400	0.9420	0.6330
			(0.8082)	(0.7253)	(0.7115)	(0.7841)	(0.7600)	(1.1193)
		(0.3,0.3)	0.9590	0.9350	0.9250	0.9530	0.9490	0.9560
			(0.7044)	(0.6338)	(0.6200)	(0.6821)	(0.6662)	(0.7558)
	(0.5,1.0)	(0.1,0.1)	0.9440	0.9340	0.9320	0.9430	0.9400	0.9510
			(0.9179)	(0.8620)	(0.8381)	(0.8755)	(0.8474)	(0.9852)
		(0.1,0.3)	0.9600	0.9370	0.9380	0.9480	0.9580	0.7890
			(1.8608)	(1.6175)	(1.5680)	(1.7599)	(1.6422)	(1.5668)
		(0.3,0.3)	0.9480	0.9300	0.9320	0.9430	0.9410	0.9440
			(1.5767)	(1.3610)	(1.3173)	(1.4731)	(1.3794)	(1.6728)
	(1.0,1.0)	(0.1,0.1)	0.9530	0.9450	0.9380	0.9430	0.9440	0.9600
			(1.1008)	(1.0203)	(0.9828)	(1.0403)	(1.0015)	(1.1884)
		(0.1,0.3)	0.9570	0.9400	0.9460	0.9520	0.9590	0.8570
			(2.0449)	(1.7863)	(1.7154)	(1.9301)	(1.8048)	(2.6074)
		(0.3,0.3)	0.9370	0.9210	0.9200	0.9360	0.9460	0.9520
			(1.4935)	(1.3238)	(1.2594)	(1.4058)	(1.3247)	(1.6159)
(30,50)	(0.5,0.5)	(0.1,0.1)	0.9530	0.9400	0.9370	0.9480	0.9440	0.9490
			(0.4904)	(0.4608)	(0.4532)	(0.4793)	(0.4725)	(0.5270)
		(0.1,0.3)	0.9520	0.9350	0.9320	0.9420	0.9430	0.5640
			(0.6719)	(0.6226)	(0.6116)	(0.6560)	(0.6434)	(0.9312)
		(0.3,0.3)	0.9540	0.9410	0.9400	0.9520	0.9470	0.9580
			(0.6116)	(0.5652)	(0.5540)	(0.5958)	(0.5873)	(0.6527)
	(0.5,1.0)	(0.1,0.1)	0.9500	0.9360	0.9310	0.9520	0.9400	0.9500
			(0.7432)	(0.7032)	(0.6880)	(0.7149)	(0.6987)	(0.7985)
		(0.1,0.3)	0.9530	0.9450	0.9440	0.9550	0.9530	0.7460
			(1.3899)	(1.2629)	(1.2334)	(1.3323)	(1.2778)	(1.9234)
		(0.3,0.3)	0.9430	0.9350	0.9290	0.9470	0.9360	0.9470
			(1.1674)	(1.0651)	(1.0382)	(1.1198)	(1.0783)	(1.2236)
	(1.0,1.0)	(0.1,0.1)	0.9550	0.9400	0.9370	0.9540	0.9470	0.9650
			(0.9455)	(0.8773)	(0.8487)	(0.8988)	(0.8729)	(1.0223)
		(0.1,0.3)	0.9550	0.9470	0.9450	0.9510	0.9510	0.8390
			(1.6629)	(1.5013)	(1.4501)	(1.5813)	(1.5166)	(2.3128)
		(0.3,0.3)	0.9560	0.9310	0.9330	0.9430	0.9430	0.9510
			(1.2218)	(1.1291)	(1.0778)	(1.1671)	(1.1278)	(1.3150)
(50,50)	(0.5,0.5)	(0.1,0.1)	0.9490	0.9450	0.9420	0.9500	0.9430	0.9520
			(0.4182)	(0.4005)	(0.3943)	(0.4109)	(0.4052)	(0.4468)
		(0.1,0.3)	0.9660	0.9560	0.9460	0.9620	0.9630	0.4370
			(0.5910)	(0.5536)	(0.5449)	(0.5799)	(0.5682)	(0.8147)
		(0.3,0.3)	0.9620	0.9480	0.9470	0.9570	0.9570	0.9650
			(0.5205)	(0.4889)	(0.4805)	(0.5094)	(0.5007)	(0.5578)
	(0.5,1.0)	(0.1,0.1)	0.9530	0.9380	0.9420	0.9460	0.9450	0.9490
			(0.6875)	(0.6544)	(0.6412)	(0.6621)	(0.6475)	(0.7318)
		(0.1,0.3)	0.9530	0.9410	0.9440	0.9510	0.9450	0.7390
			(1.2924)	(1.1772)	(1.1526)	(1.2369)	(1.1870)	(1.7610)
		(0.3,0.3)	0.9420	0.9300	0.9270	0.9450	0.9340	0.9530
			(1.0769)	(0.9900)	(0.9666)	(1.0351)	(0.9947)	(1.1395)
	(1.0,1.0)	(0.1,0.1)	0.9640	0.9580	0.9550	0.9530	0.9600	0.9640
			(0.8014)	(0.7575)	(0.7369)	(0.7687)	(0.7496)	(0.8653)
		(0.1,0.3)	0.9690	0.9570	0.9470	0.9580	0.9570	0.8040
			(1.4323)	(1.3086)	(1.2727)	(1.3650)	(1.3080)	(1.9684)
		(0.3,0.3)	0.9410	0.9310	0.9260	0.9400	0.9430	0.9470
			(1.0229)	(0.9433)	(0.9116)	(0.9786)	(0.9430)	(1.1033)
(50,100)	(0.5,0.5)	(0.1,0.1)	0.9480	0.9420	0.9430	0.9480	0.9470	0.9500
			(0.3614)	(0.3467)	(0.3420)	(0.3558)	(0.3516)	(0.3845)
		(0.1,0.3)	0.9510	0.9370	0.9350	0.9470	0.9430	0.3290
			(0.4811)	(0.4607)	(0.4539)	(0.4739)	(0.4677)	(0.6634)
		(0.3,0.3)	0.9550	0.9440	0.9420	0.9480	0.9450	0.9590
			(0.4380)	(0.4167)	(0.4107)	(0.4309)	(0.4261)	(0.4662)
	(0.5,1.0)	(0.1,0.1)	0.9500	0.9440	0.9380	0.9420	0.9390	0.9460
			(0.5212)	(0.5000)	(0.4920)	(0.5076)	(0.4998)	(0.5554)
		(0.1,0.3)	0.9500	0.9460	0.9470	0.9470	0.9470	0.5570
			(0.9358)	(0.8838)	(0.8685)	(0.9092)	(0.8874)	(1.2847)
		(0.3,0.3)	0.9510	0.9440	0.9360	0.9440	0.9430	0.9560
			(0.8088)	(0.7652)	(0.7516)	(0.7873)	(0.7710)	(0.8666)
	(1.0,1.0)	(0.1,0.1)	0.9500	0.9340	0.9270	0.9340	0.9390	0.9440
			(0.6832)	(0.6423)	(0.6280)	(0.6553)	(0.6432)	(0.7304)
		(0.1,0.3)	0.9410	0.9370	0.9390	0.9410	0.9440	0.7100
			(1.1484)	(1.0752)	(1.0507)	(1.1076)	(1.0796)	(1.5954)
		(0.3,0.3)	0.9480	0.9450	0.9440	0.9520	0.9480	0.9580
			(0.8530)	(0.8052)	(0.7821)	(0.8235)	(0.8067)	(0.9092)
(100,100)	(0.5,0.5)	(0.1,0.1)	0.9520	0.9440	0.9450	0.9480	0.9430	0.9510
			(0.2926)	(0.2841)	(0.2806)	(0.2889)	(0.2858)	(0.3135)
		(0.1,0.3)	0.9420	0.9350	0.9320	0.9410	0.9400	0.1650
			(0.4040)	(0.3886)	(0.3838)	(0.3992)	(0.3935)	(0.5539)
		(0.3,0.3)	0.9500	0.9440	0.9420	0.9510	0.9500	0.9460
			(0.3584)	(0.3446)	(0.3402)	(0.3537)	(0.3491)	(0.3821)
	(0.5,1.0)	(0.1,0.1)	0.9580	0.9510	0.9480	0.9590	0.9540	0.9570
			(0.4716)	(0.4549)	(0.4481)	(0.4578)	(0.4510)	(0.4985)
		(0.1,0.3)	0.9410	0.9310	0.9290	0.9380	0.9340	0.4850
			(0.8651)	(0.8165)	(0.8034)	(0.8382)	(0.8172)	(1.1649)
		(0.3,0.3)	0.9550	0.9470	0.9420	0.9550	0.9490	0.9510
			(0.7280)	(0.6894)	(0.6789)	(0.7073)	(0.6910)	(0.7668)
	(1.0,1.0)	(0.1,0.1)	0.9480	0.9430	0.9320	0.9480	0.9420	0.9550
			(0.5482)	(0.5258)	(0.5169)	(0.5318)	(0.5231)	(0.5911)
		(0.1,0.3)	0.9560	0.9460	0.9500	0.9480	0.9520	0.6410
			(0.9514)	(0.9002)	(0.8835)	(0.9209)	(0.8983)	(1.3156)
		(0.3,0.3)	0.9500	0.9420	0.9410	0.9460	0.9460	0.9420
			(0.6899)	(0.6564)	(0.6412)	(0.6679)	(0.6541)	(0.7433)
(100,200)	(0.5,0.5)	(0.1,0.1)	0.9480	0.9470	0.9440	0.9470	0.9470	0.9500
			(0.2507)	(0.2453)	(0.2422)	(0.2479)	(0.2456)	(0.2680)
		(0.1,0.3)	0.9570	0.9510	0.9490	0.9520	0.9500	0.0730
			(0.3339)	(0.3249)	(0.3212)	(0.3303)	(0.3266)	(0.4568)
		(0.3,0.3)	0.9570	0.9480	0.9450	0.9520	0.9500	0.9450
			(0.3064)	(0.2975)	(0.2938)	(0.3032)	(0.3004)	(0.3258)
	(0.5,1.0)	(0.1,0.1)	0.9440	0.9430	0.9370	0.9450	0.9420	0.9440
			(0.3627)	(0.3512)	(0.3467)	(0.3547)	(0.3503)	(0.3887)
		(0.1,0.3)	0.9540	0.9500	0.9450	0.9560	0.9550	0.2780
			(0.6464)	(0.6210)	(0.6128)	(0.6316)	(0.6214)	(0.8818)
		(0.3,0.3)	0.9560	0.9500	0.9440	0.9510	0.9490	0.9530
			(0.5552)	(0.5340)	(0.5267)	(0.5436)	(0.5352)	(0.5916)
	(1.0,1.0)	(0.1,0.1)	0.9490	0.9360	0.9320	0.9430	0.9460	0.9500
			(0.4704)	(0.4506)	(0.4432)	(0.4548)	(0.4487)	(0.5035)
		(0.1,0.3)	0.9570	0.9530	0.9470	0.9540	0.9520	0.5300
			(0.7804)	(0.7436)	(0.7313)	(0.7570)	(0.7446)	(1.0727)
		(0.3,0.3)	0.9600	0.9490	0.9480	0.9530	0.9490	0.9620
			(0.5784)	(0.5568)	(0.5454)	(0.5626)	(0.5549)	(0.6152)
(200,200)	(0.5,0.5)	(0.1,0.1)	0.9630	0.9620	0.9580	0.9630	0.9610	0.9590
			(0.2042)	(0.2002)	(0.1980)	(0.2024)	(0.2004)	(0.2182)
		(0.1,0.3)	0.9540	0.9550	0.9530	0.9530	0.9540	0.0110
			(0.2831)	(0.2770)	(0.2738)	(0.2801)	(0.2770)	(0.3895)
		(0.3,0.3)	0.9640	0.9650	0.9550	0.9640	0.9580	0.9590
			(0.2493)	(0.2435)	(0.2409)	(0.2473)	(0.2447)	(0.2655)
	(0.5,1.0)	(0.1,0.1)	0.9500	0.9340	0.9310	0.9410	0.9400	0.9480
			(0.3277)	(0.3164)	(0.3128)	(0.3195)	(0.3156)	(0.3440)
		(0.1,0.3)	0.9560	0.9460	0.9450	0.9510	0.9460	0.2180
			(0.5902)	(0.5665)	(0.5592)	(0.5754)	(0.5663)	(0.7924)
		(0.3,0.3)	0.9570	0.9490	0.9430	0.9520	0.9480	0.9530
			(0.4996)	(0.4806)	(0.4747)	(0.4880)	(0.4807)	(0.5249)
	(1.0,1.0)	(0.1,0.1)	0.9530	0.9480	0.9470	0.9510	0.9480	0.9610
			(0.3817)	(0.3661)	(0.3612)	(0.3700)	(0.3654)	(0.4088)
		(0.1,0.3)	0.9640	0.9630	0.9580	0.9650	0.9590	0.4160
			(0.6529)	(0.6260)	(0.6165)	(0.6339)	(0.6239)	(0.9040)
		(0.3,0.3)	0.9500	0.9460	0.9420	0.9500	0.9470	0.9570
			(0.4667)	(0.4488)	(0.4422)	(0.4535)	(0.4466)	(0.4980)

. The results indicate that the coverage probabilities of the GCI approach, Bayesian approach, HPD approach based on the Bayesian method, and the MOVER approach are close to the nominal confidence level of 0.95, except for the MOVER approach, where the coverage probabilities fall below the nominal confidence level of 0.95 for ${\delta }_1\ne {\delta }_2$. Additionally, the coverage probabilities for the bootstrap approach and the HPD approach based on the bootstrap method are below the nominal confidence level of 0.95 when $(n_1,n_2)<$ (100,100), while they are close to the nominal confidence level of 0.95 when $(n_1,n_2)⩾$ (100,100). The GCI approach has the shortest average lengths when $(n_1,n_2)<$ (100,100), whereas the Bayesian approach has the shortest average lengths when $(n_1,n_2)⩾$ (100,100). Therefore, the GCI approach is recommended for constructing the confidence interval for the ratio of percentiles when $(n_1,n_2)<$ (100,100) and the Bayesian approach is recommended when $(n_1,n_2)⩾$ (100,100). Figure 4, Figure 5 and Figure 6 present the coverage probabilities and average lengths of confidence intervals for differences between percentiles, based on varying sample sizes, shape parameters, and probabilities of obtaining a positive observation. In Figure 4, the coverage probabilities converged toward the nominal confidence level of 0.95 as the sample size increased, while the average lengths of all methods decreased with larger sample sizes. Figure 5 indicates that as the shape parameter grew, the coverage probabilities remained near the nominal confidence level of 0.95, although the average lengths of all methods increased. In Figure 6, the coverage probabilities were close to the nominal confidence level of 0.95 for $({\delta }_1,{\delta }_2)=$ (0.1,0.1) and $({\delta }_1,{\delta }_2)=$ (0.1,0.3), except for the MOVER approach, which showed deviations. Furthermore, the average lengths of all methods decreased as the probabilities of obtaining a positive observation increased.

5. Empirical Application

Rayong and Prachin Buri are two provinces in Thailand known for their industrial expansion, which plays a major role in PM_2.5 air pollution. These provinces continue to face persistent air quality issues, as PM_2.5 pollution presents serious health threats and affects the well-being of their inhabitants. PM_2.5 concentrations in the air fluctuate due to various factors, with wind speed playing a significant role. Wind speed can influence PM_2.5 levels by dispersing particles, thereby reducing concentrations, or by transporting pollutants from nearby areas, which can increase local levels. Analyzing the relationship between PM_2.5 concentrations and wind speed in the Rayong and Prachin Buri provinces is crucial for effective air quality management and pollution control. By studying these data, valuable insights into pollution trends can be obtained, enabling the development of strategies to minimize harmful pollutant exposure in the region. This study estimates the difference between percentiles and the ratio of percentiles of daily wind speed data in the Rayong and Prachin Buri provinces.

The Thai Meteorological Department has provided daily wind speed data for the Rayong and Prachin Buri provinces, Thailand, covering the period from October to December 2023. The wind speed data consistently include both zero and positive values. The positive values can be modeled using either the Birnbaum–Saunders or Weibull distribution, as the Weibull distribution shares similar characteristics with the Birnbaum–Saunders distribution. Ratasukharom et al. [1] compared the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) values for these distributions when applied to daily wind speed data from the Rayong and Prachin Buri provinces. Their findings indicated that the Birnbaum–Saunders distribution had lower AIC and BIC values than the Weibull distribution. Consequently, the positive daily wind speed data for these provinces are best described by the Birnbaum–Saunders distribution. The sample statistics of the daily wind speed data for these provinces are presented in

Table 3. Sample statistics of the daily wind speed data for Rayong and Prachin Buri provinces.

Statistics	Rayong Province	Prachin Buri Province
$n_i$	92	92
$n_{i\left(0\right)}$	2	18
$n_{i\left(1\right)}$	90	74
${\delta }_i$	0.0217	0.1957
${\widehat{\alpha }}_i$	0.6680	0.5772
${\widehat{\beta }}_i$	1.2008	1.4072
$p_i$	0.5000	0.5000
${\widehat{\theta }}_i$	1.6122	0.9772

. The difference between the percentiles of daily wind speed data in the Rayong and Prachin Buri provinces is $\widehat{\omega }={\widehat{\theta }}_1-{\widehat{\theta }}_2=$ 0.6350, and the ratio of the percentiles is $\widehat{\tau }={\widehat{\theta }}_1/{\widehat{\theta }}_2=$ 1.6498.

Table 4. Lower limit and upper limit of the 95% confidence intervals for the difference between percentiles of the daily wind speed data in Rayong and Prachin Buri provinces.

Confidence Interval	Lower Limit	Upper Limit	Interval Length
GCI	0.4390	0.8812	0.4422
Bootstrap confidence interval	0.4240	0.8494	0.4254
HPD interval based on bootstrap method	0.4377	0.8614	0.4237
Bayesian credible interval	0.4161	0.8595	0.4434
HPD interval based on Bayesian method	0.4154	0.8532	0.4378
MOVER confidence interval	0.6127	1.0271	0.4144

presents the 95% two-sided confidence intervals for the difference between percentiles of daily wind speed data in the Rayong and Prachin Buri provinces. The findings show that the MOVER confidence interval yielded the shortest interval for these differences. Similarly,

Table 5. Lower limit and upper limit of the 95% confidence intervals for the ratio of percentiles of the daily wind speed data in Rayong and Prachin Buri provinces.

Confidence Interval	Lower Limit	Upper Limit	Interval Length
GCI	1.4212	1.9923	0.5711
Bootstrap confidence interval	1.3914	1.9392	0.5478
HPD interval based on bootstrap method	1.3737	1.9108	0.5371
Bayesian credible interval	1.3965	1.9568	0.5603
HPD interval based on Bayesian method	1.3670	1.9261	0.5591
MOVER confidence interval	1.7318	2.4267	0.6949

presents the 95% two-sided confidence intervals for the ratio of percentiles of daily wind speed data in the same provinces. The results reveal that the HPD interval based on the bootstrap method produced the shortest interval for the ratio of percentiles.

The results for the difference and ratio of percentiles align with the simulation results in terms of interval length discussed in the previous section. However, it is important to note that the coverage probability and average length in the simulations were calculated using 5000 random samples, whereas, in the real-world scenario, a single sample was used to determine the interval length. The coverage probability of the MOVER confidence interval falls below the nominal confidence level of 0.95 when ${\delta }_1\ne {\delta }_2$. Therefore, the MOVER approach is not recommended for constructing the confidence interval for the difference between percentiles when ${\delta }_1\ne {\delta }_2$. Additionally, the coverage probability of the HPD interval based on the bootstrap method also falls below the nominal confidence level of 0.95. As a result, the HPD interval based on the bootstrap method approach is not recommended for constructing the confidence interval for the ratio percentiles.

6. Discussion

In environmental science and air quality studies, percentiles are a valuable tool for describing PM_2.5 concentrations, with wind speed being a key influencing factor. These estimated wind speed percentiles can aid in strategic planning to reduce air pollution. Confidence intervals for the difference between percentiles and the ratio of percentiles can be used to compare wind speed percentiles in the Rayong and Prachin Buri provinces. When the ratio is one or the difference is zero, the percentiles of the Rayong and Prachin Buri provinces are not different.

For the difference between percentiles, the MOVER approach is recommended for constructing confidence intervals for the difference between percentiles of Birnbaum–Saunders distributions containing zero values when ${\delta }_1={\delta }_2$. The HPD approach based on the Bayesian method is recommended for constructing confidence intervals for the difference between percentiles of Birnbaum–Saunders distributions containing zero values when ${\delta }_1\ne {\delta }_2$. The findings show that both the MOVER approach and the HPD approach, which is based on the Bayesian method, are the most effective for this purpose, aligning with previous research by Wu [6].

For the ratio of percentiles, the GCI and Bayesian approaches are recommended for constructing confidence intervals for the ratio of percentiles of Birnbaum–Saunders distributions containing zero values. The results suggest that the GCI and Bayesian approaches are the most reliable methods for constructing confidence intervals, aligning with previous studies by Ye et al. [4] and Wu [6].

This study focuses on estimating the confidence interval between two different percentiles. Future research could extend this approach to include simultaneous confidence intervals for all pairwise differences between the percentiles of multiple delta-Birnbaum–Saunders distributions.

7. Conclusions

Confidence intervals for the difference and ratio of percentiles were constructed using the GCI approach, the bootstrap approach, the HPD approach based on the bootstrap method, the Bayesian approach, the HPD approach based on the Bayesian method, and the MOVER approach. The MOVER approach is recommended to construct the confidence interval for the difference between percentiles when ${\delta }_1={\delta }_2$. The HPD approach based on the Bayesian method is also recommended for constructing the confidence interval for the difference between percentiles when ${\delta }_1\ne {\delta }_2$. The GCI approach is recommended for constructing the confidence interval for the ratio of percentiles when $(n_1,n_2)<$ (100,100), whereas the Bayesian approach is recommended for constructing the confidence interval for the ratio of percentiles when $(n_1,n_2)⩾$ (100,100).