Confidence Intervals for the Difference and Ratio Means of Zero-Inflated Two-Parameter Rayleigh Distribution

Abstract

The analysis of road traffic accidents often reveals asymmetric patterns, providing insights that support the development of preventive measures, reduce fatalities, and improve road safety interventions. The Rayleigh distribution, a continuous distribution with inherent asymmetry, is well suited for modeling right-skewed data and is widely used in scientific and engineering fields. It also shares structural characteristics with other skewed distributions, such as the Weibull and exponential distributions, and is particularly effective for analyzing right-skewed accident data. This study considers several approaches for constructing confidence intervals, including the percentile bootstrap, bootstrap with standard error, generalized confidence interval, method of variance estimates recovery, normal approximation, Bayesian Markov Chain Monte Carlo, and Bayesian highest posterior density methods. Their performance was evaluated through Monte Carlo simulation based on coverage probabilities and expected lengths. The results show that the HPD method achieved coverage probabilities at or above the nominal confidence level while providing the shortest expected lengths. Finally, all proposed confidence intervals were applied to fatalities recorded during the seven hazardous days of Thailand’s Songkran festival in 2024 and 2025.

1. Introduction

In practical studies involving mortality rates and road traffic accidents, the observed datasets typically consist of non-negative values and often contain a substantial proportion of zeros. Such data characteristics commonly lead to right-skewed and asymmetric distributions, driven by an excess of zero counts and the low probability of severe events. This pattern is consistent with empirical evidence reported in accident data analyses, where a high frequency of zero observations and occasional extreme values are frequently observed due to the rarity of severe crashes [1]. For instance, road-traffic fatality records may include numerous zero observations, representing days or locations with no recorded deaths, alongside occasional extreme values arising from severe accidents. This pronounced asymmetry presents challenges for statistical inference, as zero-inflated and highly skewed data structures are frequently encountered in real-world applications and can substantially influence analytical results and policy interpretations. These challenges have motivated the development and use of specialized statistical models, such as zero-inflated distributions, which accommodate excess zeros while adequately characterizing the variability in positive observations. Previous studies have also highlighted the importance of zero-adjusted or zero-inflated modeling in statistical inference. For example, Jana and Gautam [2] proposed confidence intervals for the difference and ratio of means under the zero-adjusted inverse Gaussian distribution, demonstrating that models designed to handle excess zeros can effectively manage skewness and zero-dominated structures in real-world data. Among these models, the Rayleigh distribution and its zero-inflated extensions have gained considerable attention owing to their flexibility in representing right-skewed phenomena and their suitability for accident analysis and related scientific fields.

The mean is one of the most widely used measures of central tendency and serves as a fundamental summary statistic for describing a dataset. It also plays an important role in comparative studies involving multiple datasets. A substantial body of research in this area has examined methods for estimating means across different groups, utilizing both point estimators and confidence interval procedures under various distributions [3].

When comparing means, two common strategies involve examining either their difference or their ratio. For example, Gao and Lili [4] developed estimators for both the difference and the ratio of the means of two gamma distributions. Singhasomboon and Piladaeng [5] proposed point estimation procedures for the ratio of the means of two independent lognormal distributions based on a normal approximation. La-ongkaew et al. [6] introduced alternative confidence interval formulations for assessing both the difference and the ratio of means under Weibull distributions. Their approach was constructed using the generalized confidence interval (GCI) framework together with the method of variance estimates recovery (MOVER), which is an extension of the classical Wald-type confidence interval.

Fuxiang et al. [7] were the first to propose the zero-inflated Rayleigh distribution by integrating the zero-inflation mechanism with the traditional Rayleigh model originally introduced by Rayleigh [8]. In many real-world datasets, a considerable proportion of observations consist of zeros, which cannot be adequately modeled using the standard two-parameter Rayleigh distribution or its classical form alone. The classical Rayleigh distribution is a single-parameter model that has been widely applied in various scientific and engineering fields, including medical studies, quality control, and reliability analysis. A growing body of research has demonstrated its applicability to real-world datasets. For instance, Arikan et al. [9] investigated wind energy potential in Turkey using the Rayleigh distribution to identify suitable wind turbines and estimate cost efficiency. Similarly, Kilai et al. [10] employed the Rayleigh distribution to analyze COVID-19 mortality data from Italy and Canada.

The two-parameter Rayleigh distribution was first introduced by John and Balakrishnan [11]. Subsequent work by Khan, Provost, and Singh [12] examined statistical inference for this distribution, which is characterized by a location parameter and a scale parameter. Later, Dey et al. [13] proposed estimation procedures for its unknown parameters, while Shang and Gui [14] developed point estimators and exact confidence intervals under progressive first-failure censoring using pivotal quantities. Krishnamoorthy et al. [15] further investigated interval estimation for both the parameters and the mean of the two-parameter Rayleigh distribution, contributing to the advancement of inference techniques for this model.

This paper focuses on constructing credible and confidence intervals for the difference and ratio of means under zero-inflated two-parameter Rayleigh distributions. Seven interval estimation techniques are examined: the percentile bootstrap, the bootstrap method based on standard error, the generalized confidence interval, the method of variance estimates recovery, the normal approximation, the Bayesian credible interval, and the Bayesian highest posterior density interval. Within the Bayesian framework, a gamma prior is assigned to the scale parameter, while a uniform prior is adopted for the location parameter. The performance of the proposed intervals is evaluated in terms of coverage probability and expected length, with results generated through Monte Carlo simulations implemented in R version 4.4.1 and OpenBUGS version 3.2.3. Finally, the practical utility of the methods is demonstrated using datasets of total road-traffic fatalities recorded during the seven hazardous days of Thailand’s Songkran festival in 2024 and 2025.

2. Materials and Methods

Let $x_i=(x_{i1},\dots ,x_{i{n}_i}),i=1,2,$ be two independent random samples drawn from the zero-inflated two-parameter Rayleigh distribution, denoted by $ZITR({\mu }_i,{\lambda }_i,{\delta }_i)$. Here, ${\mu }_i$, ${\lambda }_i$, and ${\delta }_i$ represent the location parameter, the scale parameter, and the probability of zero, respectively. The probability function of $x_i$ is given by

$$f\left(x_i;{\mu }_i,{\lambda }_i,{\delta }_i\right)=\left\{\begin{array}{c}{\delta }_i;x_i=0\\ \left(1-{\delta }_i\right)2{\lambda }_i({\mathrm{x}}_i-{\mu }_i){\mathrm{e}}^{-{\lambda }_i{\left(x_i-{\mu }_i\right)}^2};x_i>{\mu }_i.\end{array}\right.$$

(1)

For the zero-inflated two-parameter Rayleigh distribution, the probability of observing $x_i=0$ is ${\delta }_i$, while for $x_i$ > ${\mu }_i$, the values follow the two-parameter Rayleigh distribution. The parameters ${\mu }_i$ and ${\lambda }_i$ can be estimated using the maximum likelihood method. Since closed-form solutions for these estimators are not available, numerical procedures are required (see [13,15]). In particular, the estimator ${\widehat{\mu }}_i$ of ${\mu }_i$ can be obtained by solving the following equation:

$${\displaystyle \frac{2n_1^2\left(\overline{x}-\mu \right)}{{\displaystyle \sum _{i=1}^{n_1}}{\left(x_i-\mu \right)}^2}}-{\displaystyle \sum _{i=1}^{n_1}}{\left(x_i-\mu \right)}^{-1}=0,$$

(2)

similarly, the maximum likelihood estimators of ${\widehat{\lambda }}_i$ of ${\lambda }_i$ are defined by

$$\widehat{\lambda }={\displaystyle \frac{n_1}{{\displaystyle \sum _{i=1}^{n_1}}{(x_i-\widehat{\mu })}^2}}.$$

(3)

In this study, the maximum likelihood estimates are obtained by maximizing the profile log-likelihood function. The location parameter is estimated using the uniroot-based convergence procedure implemented in R. Specifically, the uniroot function is employed with an error tolerance set to ${10}^{-5}$, following the recommendation of Krishnamoorthy. This approach ensures stable numerical convergence and provides reliable parameter estimates for the two-parameter Rayleigh distribution. Finally, the maximum likelihood estimate of the probability of zero is computed as ${\widehat{\delta }}_i={\displaystyle \frac{n_{0i}}{n_i}}$, where $n_{0i}$ denotes the number of zero observations and $n_i$ represents the total sample size. Furthermore, the mean obtained as

$${\psi }_i=\left(1-{\delta }_i\right)\left({\mu }_i+{\displaystyle \frac{1}{\sqrt{{\lambda }_i}}}\Gamma \left({\displaystyle \frac{3}{2}}\right)\right).$$

(4)

Two independent random samples of sizes $n_1$ and $n_2$ are denoted by $x_1=(x_{11},\dots ,x_{1n_1})$ and $x_2=(x_{21},\dots ,x_{2n_2})$, respectively, each drawn from a zero-inflated two-parameter Rayleigh distribution with location parameters ${\mu }_1$ and ${\mu }_2$ and scale parameters ${\lambda }_1$ and ${\lambda }_2$. The probabilities of zero are denoted by ${\delta }_1$ and ${\delta }_2$. Accordingly, the expressions for the difference and ratio of the means are given by

$$\theta ={\psi }_1-{\psi }_2=\left(1-{\delta }_1\right)\left({\mu }_1+{\displaystyle \frac{1}{\sqrt{{\lambda }_1}}}\Gamma \left({\displaystyle \frac{3}{2}}\right)\right)-\left(1-{\delta }_2\right)\left({\mu }_2+{\displaystyle \frac{1}{\sqrt{{\lambda }_2}}}\Gamma \left({\displaystyle \frac{3}{2}}\right)\right),$$

(5)

and

$$\tau ={\displaystyle \frac{{\psi }_1}{{\psi }_2}}={\displaystyle \frac{\left(1-{\delta }_1\right)\left({\mu }_1+\frac{1}{\sqrt{{\lambda }_1}}\Gamma \left(\frac{3}{2}\right)\right)}{\left(1-{\delta }_2\right)\left({\mu }_2+\frac{1}{\sqrt{{\lambda }_2}}\Gamma \left(\frac{3}{2}\right)\right)}}.$$

(6)

Subsequently, seven different approaches for constructing the confidence interval for $\theta$ and $\tau$ are developed.

2.1. The Bootstrap Method

The bootstrap technique, originally proposed by [16], is a resampling procedure designed to assess the accuracy of statistical estimators. Let $x_i=(x_{i1},\dots ,x_{i{n}_i}),i=1,2,$ be two independent random samples drawn from the zero-inflated two-parameter Rayleigh distribution. Let $x_i^{*b}=(x_{i1}^{*b},\dots ,x_{i{n}_i}^{*b}),i=1,2,$ denote the bootstrap samples drawn with replacement from the original datasets, each having the same sample size as the corresponding original sample. After generating B bootstrap replicates, the difference and ratio of the means are computed for each replicate, denoted by ${\widehat{\theta }}^{*b}={\widehat{\psi }}_1^{*b}-{\widehat{\psi }}_2^{*b}$ and ${\widehat{\tau }}^{*b}={\displaystyle \frac{{{\psi }_1}^{*b}}{{{\psi }_2}^{*b}}}$, $b=1,\dots ,B$.

2.1.1. The Percentile Bootstrap Confidence Interval

The percentile bootstrap approach utilizes resampling from the empirical distribution to construct confidence intervals. In this method, repeated samples are drawn with replacement from the original data to estimate the sampling distribution of the statistic of interest. The percentile bootstrap method is distribution-free and does not rely on assumptions regarding the underlying distribution. Moreover, it can naturally capture asymmetry in the sampling distribution, making it particularly suitable for skewed data or distributions [16]. In addition, the method is simple to implement and has been widely used in the literature. The mean of the zero-inflated two-parameter Rayleigh distribution can be derived using the corresponding analytical expression. The two-sided percentile bootstrap confidence intervals for both the difference and the ratio of the means are given by

$$C{I}_{\theta \left(PB\right)}=\left(L_{\theta \left(PB\right)},U_{\theta \left(PB\right)}\right)=\left({\widehat{\theta }}^{*b}\left(\alpha /2\right),{\widehat{\theta }}^{*b}\left(1-\alpha /2\right)\right),$$

(7)

and

$$C{I}_{\tau \left(PB\right)}=\left(L_{\tau \left(PB\right)},U_{\tau \left(PB\right)}\right)=\left({\widehat{\tau }}^{*b}\left(\alpha /2\right),{\widehat{\tau }}^{*b}\left(1-\alpha /2\right)\right),$$

(8)

where ${\widehat{\theta }}^{*b}\left({\displaystyle \frac{\alpha }{2}}\right)$ and ${\widehat{\tau }}^{*b}\left({\displaystyle \frac{\alpha }{2}}\right)$ denote the $100(1-\alpha )$th percentiles of $\widehat{\theta }$ and $\widehat{\tau }$, respectively. Algorithm 1 outlines the steps for constructing percentile bootstrap confidence intervals for both the difference and the ratio of means in the zero-inflated two-parameter Rayleigh distribution.

Algorithm 1 The percentile bootstrap method.

1. Draw a random sample $x_i\sim ZITR({\mu }_i,{\lambda }_i,{\delta }_i)$ for $i=1,2$. 2. Generate a bootstrap resample $x_i^{*b}=\left(x_{i1}^{*b},x_{i2}^{*b},\dots ,x_{i{n}_i}^{*b}\right)$ from the original data set $x_i=(x_{i1},x_{i2},\dots ,x_{i{n}_i})$. 3. Estimate the parameters ${\widehat{\mu }}_i^{*b},{\widehat{\lambda }}_i^{*b}$, and ${\widehat{\delta }}_i^{*b}$ using MLE. 4. Calculate the bootstrap estimates of the difference ${\widehat{\theta }}^{*b}$ and the ratio ${\widehat{\tau }}^{*b}$ of the means. 5. Perform Steps 2–4 repeatedly for B iterations, where B is the total count of bootstrap samples generated. 6. Order the bootstrap replications ${\widehat{\theta }}^{*b}$ and ${\widehat{\tau }}^{*b}$ in ascending sequence. 7. Construct the 95% percentile bootstrap confidence intervals for $\theta$ and $\tau$ based on Equations (7) and (8).

2.1.2. The Bootstrap Method with Standard Error

The bootstrap method with standard error is conceptually simple and differs from classical confidence interval approaches in that it does not require explicit analytical derivation of the variance for the parameter of interest. By repeatedly resampling from the observed data, the method provides empirical estimates of the estimator’s variability, resulting in reliable standard error measures. This approach is especially suitable for situations with small sample sizes or other data limitations [16].

Given B bootstrap replications, let ${\widehat{\theta }}^{*b}$ and ${\widehat{\tau }}^{*b}$, for $b=1,2,\dots ,B$, denote the corresponding bootstrap estimates. The standard error of each statistic is then approximated by the sample standard deviation of its bootstrap distribution. Based on this, the two-sided $100(1-\alpha )\%$ confidence intervals for the difference and ratio of the means of the zero-inflated two-parameter Rayleigh distribution, derived using the bootstrap method with standard error estimation, can be expressed as

$$C{I}_{\theta \left(BS\right)}=\left(L_{\theta \left(BS\right)},U_{\theta \left(BS\right)}\right)=\left({\overline{\theta }}^{*}-Z_{\alpha /2}SE\left(\widehat{\theta }\right),{\overline{\theta }}^{*}+Z_{\alpha /2}SE\left(\widehat{\theta }\right)\right),$$

(9)

and

$$C{I}_{\tau \left(BS\right)}=\left(L_{\tau \left(BS\right)},U_{\tau \left(BS\right)}\right)=\left({\overline{\tau }}^{*}-Z_{\alpha /2}SE\left(\widehat{\tau }\right),{\overline{\tau }}^{*}+Z_{\alpha /2}SE\left(\widehat{\tau }\right)\right),$$

(10)

where $SE\left(\widehat{\theta }\right)={\displaystyle \sum _{i=1}^b}{\displaystyle \frac{\left({\widehat{\theta }}^{*b}-{\overline{\theta }}^{*}\right)}{b-1}}$ and $SE\left(\widehat{\tau }\right)={\displaystyle \sum _{i=1}^b}{\displaystyle \frac{\left({\widehat{\tau }}^{*b}-{\overline{\tau }}^{*}\right)}{b-1}}$ be the standard error of $\theta$ and $\tau$. The following Algorithm 2 is employed to construct the bootstrap confidence interval based on standard error for the difference between and the ratio of means of the zero-inflated two-parameter Rayleigh distribution.

Algorithm 2 The bootstrap method with standard error method.

1. Draw a random sample $x_i\sim ZITR({\mu }_i,{\lambda }_i,{\delta }_i)$ for $i=1,2$. 2. Form a bootstrap sample $x_i^{*b}=\left(x_{i1}^{*b},x_{i2}^{*b},\dots ,x_{i{n}_i}^{*b}\right)$ by resampling with replacement from the original observations. 3. For each bootstrap sample, obtain the corresponding estimates of the difference ${\widehat{\theta }}^{*b}$ and the ratio ${\widehat{\tau }}^{*b}$ of the means. 4. Repeat steps 2 and 3 for B replications, where B represents the total number of bootstrap iterations. 5. Compute the standard errors of the bootstrap estimates as $SE\left(\widehat{\theta }\right)={\displaystyle \sum _{i=1}^b}{\displaystyle \frac{\left({\widehat{\theta }}^{*b}-{\overline{\theta }}^{*}\right)}{b-1}}$ and $SE\left(\widehat{\tau }\right)={\displaystyle \sum _{i=1}^b}{\displaystyle \frac{\left({\widehat{\tau }}^{*b}-{\overline{\tau }}^{*}\right)}{b-1}}$, where $\overline{\theta }$ and $\overline{\tau }$ denote the means of the bootstrap estimates for $\theta$ and $\tau$, respectively. 6. Construct the 95% confidence intervals for $\theta$ and $\tau$ based on the standard error method as specified in Equations (9) and (10).

2.2. The Generalized Confidence Interval (GCI)

Weerahandi [17] initially introduced the concept of the generalized confidence interval (GCI), which is based on the idea of the generalized pivotal quantity (GPQ). This approach provides an alternative framework for interval estimation, particularly when the exact distribution of the estimator is difficult to derive or when classical methods fail to maintain the desired coverage probability. By utilizing simulated pivotal quantities that are free from unknown parameters, the GCI method allows the construction of confidence intervals applicable to a wide range of complex statistical models. In the context of this study, the GCI technique is employed to estimate confidence intervals for the difference and ratio of means under the zero-inflated two-parameter Rayleigh distribution.

The key concept underlying generalized confidence intervals (GCIs) is the generalized pivotal quantity (GPQ), defined as follows. Let $\left(X_1,\dots ,X_n\right)$ be a random variables with the probability density function $f\left(x;\vartheta ,\varsigma \right)$. The $\vartheta$ and $\varsigma$ are the parameter of interest and the vector of nuisance parameter, respectively. The GPQ $R\left(X;x,\vartheta ,\varsigma \right)$ for confidence interval estimation, must satisfy the following two conditions.

(i)

The probability distribution of $R\left(X;x,\vartheta ,\varsigma \right)$ is free unknown parameters.

(ii)

The observed value of $R\left(X;x,\vartheta ,\varsigma \right)$, depend on the parameter of interest $\vartheta$.

This method is employed for constructing confidence intervals when classical approaches may fail due to the complexity of variance derivation. By using GPQs, which are free from unknown parameters, it can handle skewed distributions and small sample sizes, offering a flexible framework for parameter estimation.

In the framework of the two-parameter Rayleigh distribution, Krishnamoorthy et al. [15] developed pivotal quantities corresponding to its unknown parameters. These parameters can be defined as $a=\mu \mathrm{and}\lambda ={\displaystyle \frac{1}{2b^2}}.$ Let ${\widehat{a}}_0$ and ${\widehat{b}}_0$ denote the observed estimates of $\widehat{a}$ and $\widehat{b}$, respectively. The pivotal quantities associated with these parameters are then formulated as follows:

$${\displaystyle \frac{\widehat{a}-a}{b}}\sim {\widehat{a}}^{*},$$

(11)

and

$${\displaystyle \frac{\widehat{b}}{b}}\sim {\widehat{b}}^{*},$$

(12)

where $\widehat{a}$ and $\widehat{b}$ denote the maximum likelihood estimators of the two-parameter Rayleigh distribution under the setting $a=0$ and $b=1$, obtained from a random sample of size n. The associated pivotal quantities corresponding to a and b are characterized by the distribution of

$$Q_a=\widehat{a}-\widehat{b}{\displaystyle \frac{{\widehat{a}}^{*}}{{\widehat{b}}^{*}}},$$

(13)

and

$$Q_b={\displaystyle \frac{\widehat{b}}{{\widehat{b}}^{*}}}.$$

(14)

Accordingly, the pivotal quantity for $\lambda$ is expressed as $Q_a=Q_{\mu }$ and $Q_{\lambda }=\frac{1}{2Q_b^2}$. In addition, Wu and Hsieh [18] proposed the use of pivotal quantities based on a variance-stabilizing transformation (VST) as an alternative approach for constructing confidence intervals. The VST-based pivotal quantity is defined as

$$Q_{\delta }={sin}^2\left(arcsin\sqrt{\widehat{\delta }}-{\displaystyle \frac{T}{2\sqrt{n}}}\right),$$

(15)

where $T=2\sqrt{n}\left(arcsin\sqrt{\widehat{\delta }}-arcsin\sqrt{\delta }\right)\sim N(0,1)$. Thus, by incorporating the three pivotal quantities $Q_a$, $Q_{\lambda }$, and $Q_{\delta }$, all of which are free from dependence on the underlying parameter values, the generalized pivotal quantity (GPQ) for the difference between means and ratio means of the zero-inflated two-parameter Rayleigh distribution can be formulated as

$$Q_{\theta }=Q_{{\psi }_1}-Q_{{\psi }_1}=\left(1-Q_{{\delta }_1}\right)\left(Q_{{\mu }_1}+{\displaystyle \frac{1}{\sqrt{Q_{{\lambda }_1}}}}\Gamma \left({\displaystyle \frac{3}{2}}\right)\right)-\left(1-Q_{{\delta }_2}\right)\left(Q_{{\mu }_2}+{\displaystyle \frac{1}{\sqrt{Q_{{\lambda }_2}}}}\Gamma \left({\displaystyle \frac{3}{2}}\right)\right),$$

(16)

and

$$Q_{\tau }={\displaystyle \frac{Q_{{\psi }_1}}{Q_{{\psi }_2}}}={\displaystyle \frac{\left(1-Q_{{\delta }_1}\right)\left(Q_{{\mu }_1}+\frac{1}{\sqrt{Q_{{\lambda }_1}}}\Gamma \left(\frac{3}{2}\right)\right)}{\left(1-Q_{{\delta }_2}\right)\left(Q_{{\mu }_2}+\frac{1}{\sqrt{Q_{{\lambda }_2}}}\Gamma \left(\frac{3}{2}\right)\right)}}.$$

(17)

Therefore, GCI for the difference between means and ratio means of the zero-inflated two-parameter Rayleigh distribution is given by Equations (18) and (19):

$$C{I}_{\theta \left(GCI\right)}=\left(L_{\theta \left(GCI\right)},U_{\theta \left(GCI\right)}\right)=\left(Q_{\theta }\left(\alpha /2\right),Q_{\theta }\left(1-\alpha /2\right)\right)$$

(18)

and

$$C{I}_{\tau \left(GCI\right)}=\left(L_{\tau \left(GCI\right)},U_{\tau \left(GCI\right)}\right)=\left(Q_{\tau }\left(\alpha /2\right),Q_{\tau }\left(1-\alpha /2\right)\right)$$

(19)

The confidence intervals for the difference means and ratio means of the zero-inflated two-parameter Rayleigh distribution, based on the GCI method, are constructed according to Algorithm 3.

Algorithm 3 The generalized confidence interval.

1. Begin by drawing two random samples from the zero-inflated two-parameter Rayleigh distribution, denoted as $ZITR({\mu }_i,{\lambda }_i,{\delta }_i)$. 2. Estimate the parameters ${\widehat{a}}_i$ and ${\widehat{b}}_i$ using the maximum likelihood approach. 3. Generate the samples ${x_i}^{*}=\left({x_{i1}}^{*},{x_{i2}}^{*},\dots ,{x_{in}}^{*}\right)$ from the two-parameter Rayleigh model, for example with parameters $\left(a=0,b=1\right)$ or equivalently $\left(\mu =0,\lambda =0.5\right)$. 4. Obtain the resampled maximum likelihood estimates ${\widehat{a}}_i^{*}$ and ${\widehat{b}}_i^{*}$ from MLE. 5. Using these estimates, calculate the generalized pivotal quantities $Q_{{\delta }_i}$, $Q_{{\mu }_i}$, $Q_{{\lambda }_i}$ and $Q_{{\psi }_i}$. 6. Repeat the resampling and computation process a sufficiently large number of times, say q, to approximate the distribution of the pivotal quantities. 7. Order the simulated values of $Q_{\theta }$ and $Q_{\tau }$ and extract the appropriate quantiles to form the 95% generalized confidence interval, as expressed in Equations (18) and (19).

2.3. The Method of Variance Estimates Recovery (MOVER)

Donner and Zou [19] constructed confidence intervals for the difference and ratio of two distribution parameters. This method is particularly well suited for constructing confidence intervals for the difference and ratio of means when the variance of these parameters is difficult to derive analytically. For the difference and ratio of means, the confidence intervals are defined as

$$C{I}_{\theta \left(MOVER\right)}=\left(L_{\theta \left(MOVER\right)},U_{\theta \left(MOVER\right)}\right),$$

(20)

and

$$C{I}_{\tau \left(MOVER\right)}=\left(L_{\tau \left(MOVER\right)},U_{\tau \left(MOVER\right)}\right),$$

(21)

respectively, where the lower limit and the upper limit for $\theta$ are given by

$$L_{\theta \left(MOVER\right)}=\left({\widehat{\psi }}_1-{\widehat{\psi }}_2\right)-\sqrt{{\left({\widehat{\psi }}_1-l_1\right)}^2-{\left(u_2-{\widehat{\psi }}_2\right)}^2},$$

(22)

and

$$U_{\theta \left(MOVER\right)}=\left({\widehat{\psi }}_1-{\widehat{\psi }}_2\right)+\sqrt{{\left(u_1-{\widehat{\psi }}_1\right)}^2-{\left({\widehat{\psi }}_2-l_2\right)}^2},$$

(23)

where the lower limit and upper limit for ${\psi }_x/{\psi }_y$ are given by

$$L_{\tau \left(MOVER\right)}={\displaystyle \frac{\left({\widehat{\psi }}_1{\widehat{\psi }}_2\right)-\sqrt{{\left({\widehat{\psi }}_1{\widehat{\psi }}_2\right)}^2-l_1{u}_2\left(2{\widehat{\psi }}_1-l_1\right)\left(2{\widehat{\psi }}_2-u_2\right)}}{u_2\left(2{\widehat{\psi }}_2-u_2\right)}}$$

(24)

and

$$U_{\tau \left(MOVER\right)}={\displaystyle \frac{\left({\widehat{\psi }}_1{\widehat{\psi }}_2\right)+\sqrt{{\left({\widehat{\psi }}_1{\widehat{\psi }}_2\right)}^2-u_1{l}_2\left(2{\widehat{\psi }}_1-u_1\right)\left(2{\widehat{\psi }}_2-l_2\right)}}{l_2\left(2{\widehat{\psi }}_2-l_2\right)}},$$

(25)

where for $\left(l_1,u_1\right)$ and $\left(l_2,u_2\right)$, we construct the confidence interval using the GCI method. The confidence intervals for the difference means and ratio means of the zero-inflated two-parameter Rayleigh distribution, based on MOVER method, are constructed according to Algorithm 4.

Algorithm 4 The method of variance estimates recovery.

1. Generate random sample $x_1\sim ZITR({\mu }_1,{\lambda }_1,{\delta }_1)$ and $x_2\sim ZITR({\mu }_2,{\lambda }_2,{\delta }_2)$. 2. Compute ${\widehat{\psi }}_1$ and ${\widehat{\psi }}_2$ from the GCI method 3. Compute the intervals for $\theta$ and $\tau$ from the GCI method. 4. Compute the intervals for $\theta$ and $\tau$ from Equations (20) and (21).

2.4. Normal Approximation

The Delta method is a well-established procedure within the normal approximation framework, commonly used to approximate the variance of a function of parameters. This approach is grounded in classical asymptotic theory based on Taylor series expansion (see, for example, Cramér [20]). In Appendix A, we derive the variances for both the difference of means and the ratio of means. The Delta method approximates the variance of parameters using a Taylor series expansion. It is effective for moderate to large sample sizes and enables analysis without the need for resampling. Using this approach, the variances of $\widehat{\theta }$ and $\widehat{\tau }$ can be approximated as follows:

$$\begin{array}{c}V\left(\widehat{\theta }\right)={\left({\mu }_1+{\displaystyle \frac{1}{{{\lambda }_1}^{\frac{1}{2}}}}\Gamma \left({\displaystyle \frac{3}{2}}\right)\right)}^2{\displaystyle \frac{{\delta }_1\left(1-{\delta }_1\right)}{n_1}}+{\left(1-{\delta }_1\right)}^2{\displaystyle \frac{1}{2{\mu }_1+\frac{1}{n_{11}}{\displaystyle \sum _{i=1}^{n_{11}}}{\left(x_{11i}-{\mu }_1\right)}^{-2}}}\hfill \\ +{\left(\left(1-{\delta }_1\right)\left({\displaystyle \frac{1}{2}}\right){{\lambda }_1}^{-{\displaystyle \frac{3}{2}}}\Gamma \left({\displaystyle \frac{3}{2}}\right)\right)}^2{\displaystyle \frac{{{\lambda }_1}^2}{n_{11}}}+{\left({\mu }_2+{\displaystyle \frac{1}{{{\lambda }_2}^{\frac{1}{2}}}}\Gamma \left({\displaystyle \frac{3}{2}}\right)\right)}^2{\displaystyle \frac{{\delta }_2\left(1-{\delta }_2\right)}{n_2}}\hfill \\ +{\left(1-{\delta }_2\right)}^2{\displaystyle \frac{1}{2{\mu }_2+\frac{1}{n_{12}}{\displaystyle \sum _{i=1}^{n_{12}}}{\left(x_{12i}-{\mu }_2\right)}^{-2}}}+{\left(\left(1-{\delta }_2\right)\left({\displaystyle \frac{1}{2}}\right){{\lambda }_2}^{-{\displaystyle \frac{3}{2}}}\Gamma \left({\displaystyle \frac{3}{2}}\right)\right)}^2{\displaystyle \frac{{{\lambda }_2}^2}{n_{12}}}\hfill \end{array}$$

(26)

and

$$\begin{array}{c}V\left(\widehat{\tau }\right)={\left(\left({\displaystyle \frac{\left(-1\right)\left({\widehat{\mu }}_1+\frac{1}{{{\widehat{\lambda }}_1}^{\frac{1}{2}}}\Gamma \left(\frac{3}{2}\right)\right)}{\left(1-{\widehat{\delta }}_2\right)\left({\widehat{\mu }}_2+\frac{1}{{{\widehat{\lambda }}_2}^{\frac{1}{2}}}\Gamma \left(\frac{3}{2}\right)\right)}}\right)\right)}^2{\displaystyle \frac{{\delta }_1\left(1-{\delta }_1\right)}{n_1}}+{\left({\displaystyle \frac{\left(1-{\widehat{\delta }}_1\right)}{\left(1-{\widehat{\delta }}_2\right)\left({\widehat{\mu }}_2+\frac{1}{{{\widehat{\lambda }}_2}^{\frac{1}{2}}}\Gamma \left(\frac{3}{2}\right)\right)}}\right)}^2{\displaystyle \frac{1}{2{\mu }_1+\frac{1}{n_{11}}{\displaystyle \sum _{i=1}^{n_{11}}}{\left(x_{11i}-{\mu }_1\right)}^{-2}}}\hfill \\ +{\left({\displaystyle \frac{\left(1-{\widehat{\delta }}_1\right)\left(\left(-\frac{1}{2}\right){{\widehat{\lambda }}_1}^{-\frac{3}{2}}\Gamma \left(\frac{3}{2}\right)\right)}{\left(1-{\widehat{\delta }}_2\right)\left({\widehat{\mu }}_2+\frac{1}{{{\widehat{\lambda }}_2}^{\frac{1}{2}}}\Gamma \left(\frac{3}{2}\right)\right)}}\right)}^2{\displaystyle \frac{{{\lambda }_1}^2}{n_{11}}}+{\left({\displaystyle \frac{\left(1-{\widehat{\delta }}_1\right)\left({\widehat{\mu }}_1+\frac{1}{{{\widehat{\lambda }}_1}^{\frac{1}{2}}}\Gamma \left(\frac{3}{2}\right)\right)\left({\widehat{\mu }}_2+\frac{1}{{{\widehat{\lambda }}_2}^{\frac{1}{2}}}\Gamma \left(\frac{3}{2}\right)\right)}{{\left(\left(1-{\widehat{\delta }}_2\right)\left({\widehat{\mu }}_2+\frac{1}{{{\widehat{\lambda }}_2}^{\frac{1}{2}}}\Gamma \left(\frac{3}{2}\right)\right)\right)}^2}}\right)}^2{\displaystyle \frac{{\delta }_2\left(1-{\delta }_2\right)}{n_2}}\hfill \\ +{\left({\displaystyle \frac{\left(1-{\widehat{\delta }}_1\right)\left({\widehat{\mu }}_1+\frac{1}{{{\widehat{\lambda }}_1}^{\frac{1}{2}}}\Gamma \left(\frac{3}{2}\right)\right)\left(-1\right)\left(1-{\widehat{\delta }}_2\right)}{{\left(\left(1-{\widehat{\delta }}_2\right)\left({\widehat{\mu }}_2+\frac{1}{{{\widehat{\lambda }}_2}^{\frac{1}{2}}}\Gamma \left(\frac{3}{2}\right)\right)\right)}^2}}\right)}^2{\displaystyle \frac{1}{2{\mu }_2+\frac{1}{n_{12}}{\displaystyle \sum _{i=1}^{n_{12}}}{\left(x_{12i}-{\mu }_2\right)}^{-2}}}\hfill \\ +{\left({\displaystyle \frac{\left(1-{\widehat{\delta }}_1\right)\left({\widehat{\mu }}_1+\frac{1}{{{\widehat{\lambda }}_1}^{\frac{1}{2}}}\Gamma \left(\frac{3}{2}\right)\right)\left(-1\right)\left(1-{\widehat{\delta }}_2\right)\left(-\frac{1}{2}\right){\lambda }^{-\frac{3}{2}}\Gamma \left(\frac{3}{2}\right)}{{\left(\left(1-{\widehat{\delta }}_2\right)\left({\widehat{\mu }}_2+\frac{1}{{{\widehat{\lambda }}_2}^{\frac{1}{2}}}\Gamma \left(\frac{3}{2}\right)\right)\right)}^2}}\right)}^2{\displaystyle \frac{{{\lambda }_2}^2}{n_{12}}},\hfill \end{array}$$

(27)

respectively. The asymptotic standard normal distribution is $Z={\displaystyle \frac{\widehat{\theta }-\theta }{\sqrt{\widehat{V}\left(\widehat{\theta }\right)}}}$ and $Z={\displaystyle \frac{\widehat{\tau }-\tau }{\sqrt{\widehat{V}\left(\widehat{\tau }\right)}}}$.

Therefore, the confidence interval for approximately normal based on the delta method for the difference and ratio means constructed by Equations (31) and (32):

$$C{I}_{\theta \left(AN\right)}=\left(L_{\theta \left(AN\right)},U_{\theta \left(AN\right)}\right)=\left(\widehat{\theta }-z_{1-\alpha /2}\sqrt{\widehat{V}\left(\widehat{\theta }\right)},\widehat{\theta }+z_{1-\alpha /2}\sqrt{\widehat{V}\left(\widehat{\theta }\right)}\right)$$

(28)

and

$$C{I}_{\tau \left(AN\right)}=\left(L_{\tau \left(AN\right)},U_{\tau \left(AN\right)}\right)=\left(\widehat{\tau }-z_{1-\alpha /2}\sqrt{\widehat{V}\left(\widehat{\tau }\right)},\widehat{\tau }+z_{1-\alpha /2}\sqrt{\widehat{V}\left(\widehat{\tau }\right)}\right),$$

(29)

respectively, where $\widehat{\theta }=\left(1-{\widehat{\delta }}_1\right)\left({\widehat{\mu }}_1+{\displaystyle \frac{1}{\sqrt{{\widehat{\lambda }}_1}}}\Gamma \left({\displaystyle \frac{3}{2}}\right)\right)-\left(1-{\widehat{\delta }}_2\right)\left({\widehat{\mu }}_2+{\displaystyle \frac{1}{\sqrt{{\widehat{\lambda }}_2}}}\Gamma \left({\displaystyle \frac{3}{2}}\right)\right)$ and $\widehat{\tau }={\displaystyle \frac{\left(1-{\widehat{\delta }}_1\right)\left({\widehat{\mu }}_1+\frac{1}{\sqrt{{\widehat{\lambda }}_1}}\Gamma \left(\frac{3}{2}\right)\right)}{\left(1-{\widehat{\delta }}_2\right)\left({\widehat{\mu }}_2+\frac{1}{\sqrt{{\widehat{\lambda }}_2}}\Gamma \left(\frac{3}{2}\right)\right)}}$ with ${\widehat{\delta }}_1,{\widehat{\delta }}_2,{\widehat{\lambda }}_1,{\widehat{\lambda }}_2,{\widehat{\mu }}_1$ and ${\widehat{\mu }}_2$ being the maximum likelihood estimators. The variances of $\widehat{\theta }$ and $\widehat{\tau }$, denoted by $V\left(\widehat{\theta }\right)$ and $V\left(\widehat{\tau }\right)$ are approximated using the Delta method and calculated as shown in Equations (26) and (27). Based on the approximate normality derived from this method, the confidence interval for the difference and ratio means of the zero-inflated two-parameter Rayleigh distribution is then constructed following the procedure outlined in Algorithm 5.

Algorithm 5 The normal approximation method.

1. Generate random sample $x_i\sim ZITR({\mu }_i,{\lambda }_i,{\delta }_i);i=1,2$. 2. Estimation parameters ${\widehat{\delta }}_1,{\widehat{\lambda }}_1,{\widehat{\mu }}_1,{\widehat{\delta }}_2,{\widehat{\lambda }}_2$ and ${\widehat{\mu }}_2$ by using maximum likelihood. 3. Compute the variance for $\widehat{\theta }$ and $\widehat{\tau }$ based on the delta method from Equation denotes by $V\left(\widehat{\theta }\right)$ and $V\left(\widehat{\tau }\right)$, respectively. 4. Evaluate the confidence interval for $\widehat{\theta }$ and $\widehat{\tau }$ with the approximate normal distribution, as shown in Equations (28) and (29).

2.5. Bayesian Credible Intervals

The Bayesian framework provides a structured approach for updating prior beliefs and making inferences when new data are introduced. It is based on Bayes’ theorem, which combines prior probabilities with the likelihood function to yield the posterior distribution. The prior distribution reflects the uncertainty about parameters before any observations are made. The posterior density function is given by

$$\pi (\lambda ,\mu \mid x)\propto \ell (\lambda ,\mu \mid x),\pi \left(\lambda \right),\pi \left(\mu \right),$$

(30)

where $\ell (\lambda ,\mu \mid x)$ denotes the likelihood function constructed from the observed data. The prior distribution for the scale parameter is expressed as $\pi \left(\lambda \right)$, while $\pi \left(\mu \right)$ represents the prior for the location parameter. Consequently, the posterior distribution is given by $\pi (\lambda ,\mu \mid x)$. Because the posterior distributions of the two-parameter Rayleigh model are not available in closed form, Gibbs sampling is employed to generate posterior samples and approximate the unknown parameters [21].

This study introduces two approaches for constructing Bayesian credible intervals: the Bayesian Markov chain Monte Carlo (MCMC) interval and the Bayesian highest posterior density (HPD) interval.

2.5.1. Bayesian Markov Chain Monte Carlo Interval

Let $x=(x_1,x_2,\dots ,x_n)$ denote a random sample independently drawn from a zero-inflated two-parameter Rayleigh distribution. To obtain the posterior distribution and estimate the parameters of interest, Markov chain Monte Carlo (MCMC) techniques were employed. For the prior specifications, a gamma prior was assigned to the scale parameter $\lambda$, a uniform prior was chosen for the location parameter $\mu$ [13], and a beta prior was specified for the parameter $\delta$. Informative priors were incorporated to reflect prior knowledge, reduce posterior variability, and improve the precision of the resulting parameter estimates. This strategy also enhances the reliability of inference, particularly when the sample size is limited. The Bayesian computations were carried out using the R2OpenBUGS package in R program.

This analysis focuses on the difference and ratio of the means of the zero-inflated two-parameter Rayleigh distribution. The Bayesian MCMC approach is then used to construct the corresponding two-sided credible intervals. By estimating the posterior distributions of the parameters of interest, inference is not driven solely by the likelihood or the observed data, resulting in more appropriate parameter estimates ([22,23]). Consequently, the two-sided credible intervals for the difference and ratio of means are obtained from the appropriate quantiles of the posterior samples, as follows:

$$C{I}_{\theta \left(MCMC\right)}=\left(L_{\theta \left(MCMC\right)},U_{\theta \left(MCMC\right)}\right)=\left({\theta }_{MCMC}\left(\alpha /2\right),{\theta }_{MCMC}\left(1-\alpha /2\right)\right),$$

(31)

and

$$C{I}_{\tau \left(MCMC\right)}=\left(L_{\tau \left(MCMC\right)},U_{\tau \left(MCMC\right)}\right)=\left({\tau }_{MCMC}\left(\alpha /2\right),{\tau }_{MCMC}\left(1-\alpha /2\right)\right).$$

(32)

The credible intervals for the difference of means and the ratio of means of the zero-inflated two-parameter Rayleigh distribution are constructed using Bayesian MCMC, as outlined in Algorithm 6.

Algorithm 6 Bayesian Markov chain Monte Carlo method.

1. Generate a random sample $x_i\sim ZITR({\mu }_i,{\lambda }_i,{\delta }_i);i=1,2$. 2. Specify prior distributions for the parameters using trial hyperparameters: ${\delta }_i\sim \mathrm{Beta}({\alpha }_{1i},{\beta }_{1i})$, ${\lambda }_i\sim \mathrm{Gamma}({\alpha }_{2i},{\beta }_{2i})$, and ${\mu }_i\sim \mathrm{Uniform}(a_{1i},b_{1i})$. 3. Compute the difference of means $\theta$ and the ratio of means $\tau$. 4. Repeat steps 1–3 for T iterations, where T represents the total number of Gibbs sampling replications. 5. Apply a burn-in by discarding the first C samples to ensure convergence. 6. Sort the sampled values of ${\widehat{\theta }}_{MCMC}$ and ${\widehat{\tau }}_{MCMC}$ in ascending order. 7. Construct the 95% Bayesian MCMC credible intervals for $\theta$ and $\tau$.

2.5.2. The Bayesian Highest Posterior Density (HPD) Interval

The highest posterior density (HPD) interval is defined such that the posterior density at any point within the interval is greater than that at any point outside the interval, and it represents the narrowest interval satisfying this property [24]. In this study, the HPD interval was computed using the HDInterval package in R program. Accordingly, the $100\left(1-\alpha \right)\%$ two-sided credible intervals for the difference of means and the ratio of means can be obtained as follows:

$$C{I}_{\theta \left(HPD\right)}=\left(L_{\theta \left(HPD\right)},U_{\theta \left(HPD\right)}\right)=\left({\theta }_{HPD}\left(\alpha /2\right),{\theta }_{HPD}\left(1-\alpha /2\right)\right),$$

(33)

and

$$C{I}_{\tau \left(HPD\right)}=\left(L_{\tau \left(HPD\right)},U_{\tau \left(HPD\right)}\right)=\left({\tau }_{HPD}\left(\alpha /2\right),{\tau }_{HPD}\left(1-\alpha /2\right)\right)$$

(34)

The Bayesian HPD approach is utilized to develop credible intervals for the difference and ratio means of the zero-inflated two-parameter Rayleigh distribution, following the procedure described in Algorithm 7.

Algorithm 7 Bayesian highest posterior density method.

1. Generate a random sample $x_i\sim ZITR({\mu }_i,{\lambda }_i,{\delta }_i);i=1,2$. 2. Specify prior distributions for the parameters with trial hyperparameters: ${\delta }_i\sim \mathrm{Beta}({\alpha }_{1i},{\beta }_{1i})$, ${\lambda }_i\sim \mathrm{Gamma}({\alpha }_{2i},{\beta }_{2i})$, and ${\mu }_i\sim \mathrm{Uniform}(a_{1i},b_{1i})$. 3. Use Gibbs sampling, implemented in R with OpenBUGS, to obtain posterior draws of $\delta$, $\lambda$, and $\mu$. 4. From each posterior sample, compute the difference of means $\theta$ and the ratio of means $\tau$. 5. Repeat steps 1–4 for T replications, where T denotes the total number of Gibbs sampling iterations. 6. Apply a burn-in by discarding the first C samples, with C representing the number of iterations removed to ensure convergence. 7. Estimate the highest posterior density (HPD) intervals using the HDInterval package in R. 8. Arrange the sampled values of ${\widehat{\theta }}_{HPD}$ and ${\widehat{\tau }}_{HPD}$ in ascending order. 9. Construct the 95% Bayesian HPD credible intervals for $\theta$ and $\tau$.

3. Results

This study evaluates the efficiency of various confidence interval methods by examining the coverage probability (CP) and expected length (EL) through Monte Carlo simulations performed in the R program. The primary objective is to propose procedures for constructing confidence intervals for the difference of means and the ratio of means of the zero-inflated two-parameter Rayleigh distribution. In addition, the study investigates the performance of several interval estimation techniques, including the percentile bootstrap (PB), bootstrap with standard error (BS), generalized confidence interval (GCI), method of variance estimates recovery (MOVER), normal approximation (AN), Bayesian Markov chain Monte Carlo (MCMC), and Bayesian highest posterior density (HPD). To evaluate the performance of the proposed methods, simulations were conducted under a variety of parameter settings. In this paper, only a selected subset of parameters is presented to provide an overview and facilitate comparison of the confidence interval methods across different scenarios. The presented parameters are as follows:

Scale parameter $\left({\lambda }_1,{\lambda }_2\right)=\left(0.1,0.1\right),(0.1,0.5),\left(0.5,0.5\right),(0.5,1)$ and $\left(1,1\right)$. Probability of zero values $\left({\delta }_1,{\delta }_2\right)=\left(0.1,0.1\right),\left(0.1,0.3\right)$ and $\left(0.3,0.3\right)$. Sample size $\left(n_1,n_2\right)=\left(10,10\right),\left(10,40\right),\left(10,70\right),\left(10,100\right),\left(40,40\right),\left(40,70\right),\left(40,100\right),\left(70,70\right),\left(70,100\right),\left(100,100\right),\left(100,200\right),\left(200,200\right)$. In the Monte Carlo simulations, a total of M = 1000 replications was performed. For the bootstrap procedures, each scenario was evaluated using B = 1000 bootstrap samples along with q = 2500 pivotal quantities. In the Bayesian framework, T = 20,000 iterations were generated via Gibbs sampling, with the first C = 5000 iterations discarded as burn-in. The criterion for identifying the most effective method was based on both the CP and EL. The method was considered optimal if its CP was not lower than the nominal confidence level of 0.95 while simultaneously achieving the shortest EL. The CP and EL of the confidence intervals for the difference of means and ratio of means of the zero-inflated two-parameter Rayleigh distribution were evaluated based on Algorithm 8.

Algorithm 8 CP and EL for difference and ratio means of $ZITR$.

1. Specify the number of simulation replications and assign the distributional parameters, namely M, B, T, C, n, $\mu$, $\lambda$, and $\delta$. 2. Generate a random sample $x_i=(x_{i1},\dots ,x_{i{n}_i})$ from the zero-inflated two-parameter Rayleigh distribution $ZITR({\mu }_i,{\lambda }_i,{\delta }_i)$ for $i=1,2$. 3. Apply Algorithm 1 to construct the 95% percentile bootstrap confidence interval for the difference means and ratio means. 4. Apply Algorithm 2 to obtain the 95% bootstrap with standard error for the difference means and ratio means. 5. Apply Algorithm 3 to obtain the 95% generalized confidence interval for the difference means and ratio means. 6. Apply Algorithm 4 to obtain the 95% of the MOVER method to construct confidence interval for the difference means and ratio means. 7. Employ Algorithm 5 to derive the 95% approximate normal confidence interval for the difference means and ratio means. 8. Use Algorithm 6 to generate the 95% Bayesian MCMC confidence interval for the difference means and ratio means. 9. Use Algorithm 7 to determine the 95% Bayesian HPD interval for the difference means and ratio means. 10.  For each constructed interval, set $P=1$ if $L\le \mu \le U$, and set $P=0$ otherwise. 11.  Record the length of the interval as $U-L$. 12.  Repeat steps 2–11 across M replications. 13.  Estimate the coverage probability by averaging the values of P. 14.  Estimate the expected length by averaging the interval lengths $U-L$.

The CP and EL of the 95% confidence intervals for the difference and ratio of the two means are presented in

Table 1. CP and EL for 95% confidence intervals for the difference between two means of zero-inflated two-parameter Rayleigh distribution when location parameter $\left({\mu }_1,{\mu }_2\right)=\left(0.1,0.1\right)$ and scale parameter $\left({\lambda }_1,{\lambda }_2\right)=\left(0.1,0.1\right)$.

Scale	Prob	Coverage Probability
		Samplesize	PB	BS	GCI	MOVER	AN	MCMC	HPD
0.1:0.1	0.1:0.1	10:10	0.898	0.784	0.956	0.957	0.986	0.948	0.953
			(2.6068)	(2.5653)	(3.2062)	(3.2172)	(7.3751)	(2.3475)	(2.3373)
		10:40	0.887	0.788	0.958	0.956	0.987	0.939	0.943
			(2.5166)	(2.5331)	(3.0969)	(3.1071)	(6.9820)	(2.3648)	(2.3550)
		10:70	0.907	0.774	0.95	0.953	0.982	0.945	0.952
			(2.5077)	(2.5211)	(3.0754)	(3.0882)	(7.7223)	(2.3562)	(2.3466)
		10:100	0.884	0.788	0.953	0.953	0.988	0.942	0.946
			(2.4863)	(2.5372)	(3.0426)	(3.0535)	(7.5441)	(2.3474)	(2.3375)
		40:40	0.94	0.816	0.953	0.953	0.999	0.948	0.948
			(1.3869)	(1.3854)	(1.4485)	(1.4517)	(10.9208)	(1.3034)	(1.2998)
		40:70	0.924	0.817	0.94	0.941	1	0.955	0.952
			(1.3451)	(1.3668)	(1.4096)	(1.4111)	(10.7472)	(1.3023)	(1.2986)
		40:100	0.905	0.795	0.934	0.937	1	0.949	0.952
			(1.3252)	(1.3203)	(1.3900)	(1.3938)	(10.8491)	(1.3019)	(1.2983)
		70:70	0.949	0.83	0.954	0.954	0.999	0.956	0.955
			(1.0527)	(1.0554)	(1.0864)	(1.0872)	(11.3662)	(1.0060)	(1.0032)
		70:100	0.934	0.83	0.941	0.944	0.997	0.946	0.946
			(1.0303)	(1.0363)	(1.0639)	(1.0643)	(12.0405)	(1.0096)	(1.0068)
		100:100	0.94	0.825	0.947	0.945	0.999	0.94	0.939
			(0.8865)	(0.8689)	(0.9016)	(0.9021)	(13.0117)	(0.8527)	(0.8504)
		100::200	0.884	0.79	0.926	0.927	1	0.947	0.947
			(0.8559)	(0.8495)	(0.8682)	(0.8684)	(11.5592)	(0.8525)	(0.8503)
		200:200	0.937	0.816	0.949	0.951	1	0.943	0.943
			(0.6296)	(0.6281)	(0.6321)	(0.6318)	(12.9709)	(0.6099)	(0.6084)
	0.1:0.3	10:10	0.922	0.804	0.956	0.959	0.976	0.916	0.921
			(2.7630)	(2.7533)	(3.3452)	(3.3537)	(6.4866)	(2.5042)	(2.4929)
		10:40	0.803	0.707	0.936	0.937	0.974	0.917	0.925
			(2.5902)	(2.5953)	(3.0856)	(3.0917)	(6.3127)	(2.5018)	(2.4902)
		10:70	0.768	0.675	0.935	0.938	0.975	0.923	0.93
			(2.6198)	(2.6683)	(3.0278)	(3.0377)	(6.6101)	(2.4991)	(2.4880)
		10:100	0.735	0.666	0.919	0.917	0.962	0.911	0.917
			(2.6421)	(2.5941)	(3.0552)	(3.0660)	(6.2433)	(2.4996)	(2.4885)
		40:40	0.933	0.816	0.951	0.956	0.999	0.933	0.932
			(1.4533)	(1.4506)	(1.5255)	(1.5274)	(9.1961)	(1.3819)	(1.3782)
		40:70	0.788	0.689	0.939	0.94	1	0.931	0.932
			(1.3784)	(1.3731)	(1.4319)	(1.4340)	(9.2486)	(1.3876)	(1.3837)
		40:100	0.693	0.621	0.939	0.939	0.996	0.936	0.937
			(1.3775)	(1.3595)	(1.3913)	(1.3930)	(8.5717)	(1.3896)	(1.3858)
		70:70	0.942	0.829	0.95	0.948	0.997	0.944	0.944
			(1.1125)	(1.1158)	(1.1420)	(1.1435)	(10.2608)	(1.0705)	(1.0677)
		70:100	0.846	0.738	0.949	0.947	0.997	0.939	0.941
			(1.0651)	(1.0527)	(1.0889)	(1.0893)	(10.0585)	(1.0709)	(1.0681)
		100::100	0.942	0.829	0.951	0.951	1	0.942	0.939
			(0.9384)	(0.9449)	(0.9519)	(0.9525)	(10.6172)	(0.9077)	(0.9054)
		100::200	0.519	0.5	0.934	0.933	1	0.937	0.935
			(0.8804)	(0.8817)	(0.8780)	(0.8780)	(11.2478)	(0.9067)	(0.9043)
		200:200	0.953	0.847	0.961	0.959	1	0.951	0.951
			(0.6639)	(0.6584)	(0.6688)	(0.6690)	(10.9930)	(0.6507)	(0.6490)
	0.3:0.3	10:10	0.898	0.788	0.968	0.966	0.971	0.941	0.947
			(2.8797)	(2.8687)	(3.5343)	(3.5290)	(5.5203)	(2.5406)	(2.5288)
		10:40	0.828	0.719	0.95	0.949	0.976	0.96	0.967
			(2.7290)	(2.7502)	(3.2761)	(3.2684)	(5.7079)	(2.5323)	(2.5202)
		10:70	0.786	0.715	0.944	0.946	0.974	0.951	0.956
			(2.7622)	(2.8198)	(3.2682)	(3.2686)	(6.0703)	(2.5303)	(2.5184)
		10:100	0.789	0.708	0.943	0.944	0.971	0.973	0.977
			(2.7630)	(2.8275)	(3.1730)	(3.1766)	(5.4823)	(2.5311)	(2.5194)
		40:40	0.946	0.842	0.949	0.947	0.993	0.949	0.949
			(1.5376)	(1.5731)	(1.6033)	(1.6045)	(7.6538)	(1.4139)	(1.4099)
		40:70	0.809	0.707	0.939	0.943	0.996	0.945	0.943
			(1.4605)	(1.4402)	(1.5044)	(1.5075)	(8.6319)	(1.4070)	(1.4030)
		40:100	0.695	0.622	0.937	0.934	0.997	0.951	0.952
			(1.4556)	(1.4415)	(1.4741)	(1.4743)	(7.9030)	(1.4099)	(1.4059)
		70:70	0.948	0.844	0.953	0.952	0.999	0.942	0.944
			(1.1726)	(1.1819)	(1.1962)	(1.1966)	(8.9083)	(1.0928)	(1.0898)
70:100		0.836	0.74	0.938	0.937	0.999	0.941	0.939
		(1.1264)	(1.1130)	(1.1491)	(1.1500)	(8.6708)	(1.0940)	(1.0911)
100::100		0.936	0.831	0.943	0.946	0.997	0.938	0.934
		(0.9823)	(1.0009)	(0.9994)	(1.0002)	(9.4559)	(0.9303)	(0.9279)
100::200		0.549	0.506	0.935	0.935	1	0.933	0.933
		(0.9357)	(0.9281)	(0.9311)	(0.9301)	(9.5176)	(0.9265)	(0.9241)
200:200		0.948	0.861	0.951	0.947	1	0.944	0.946
		(0.6965)	(0.6977)	(0.7019)	(0.7022)	(9.5204)	(0.6672)	(0.6655)

Values in bold correspond to the shortest expected lengths among those whose coverage probabilities are equal to or greater than the nominal confidence level.

,

Table 2. CP and EL for 95% confidence intervals for the difference between two means of zero-inflated two-parameter Rayleigh distribution when location parameter $\left({\mu }_1,{\mu }_2\right)=\left(0.1,0.1\right)$ and scale parameter $\left({\lambda }_1,{\lambda }_2\right)=\left(0.1,0.5\right)$.

Scale	Prob	Coverage Probability
		Samplesize	PB	BS	GCI	MOVER	AN	MCMC	HPD
0.1:0.5	0.1:0.1	10:10	0.883	0.801	0.955	0.956	0.996	0.941	0.932
			(2.0097)	(2.0398)	(2.4380)	(2.4452)	(8.2624)	(1.9042)	(1.8868)
		10:40	0.893	0.79	0.966	0.966	0.997	0.938	0.93
			(1.9769)	(1.9819)	(2.4416)	(2.4473)	(8.0877)	(1.9016)	(1.8841)
		10:70	0.9	0.805	0.952	0.951	0.999	0.942	0.939
			(2.0093)	(2.0082)	(2.4309)	(2.4368)	(9.5710)	(1.9176)	(1.9002)
		10:100	0.891	0.787	0.951	0.949	0.996	0.937	0.932
			(1.9898)	(2.0102)	(2.4161)	(2.4235)	(8.8907)	(1.9121)	(1.8949)
		40:40	0.937	0.82	0.954	0.95	1	0.948	0.947
			(1.0704)	(1.0762)	(1.1246)	(1.1266)	(11.7596)	(1.0211)	(1.0167)
		40:70	0.933	0.834	0.951	0.953	1	0.95	0.944
			(1.0630)	(1.0894)	(1.1111)	(1.1120)	(10.9058)	(1.0283)	(1.0239)
		40:100	0.931	0.811	0.94	0.941	0.999	0.955	0.951
			(1.0629)	(1.0540)	(1.1054)	(1.1076)	(10.8462)	(1.0290)	(1.0246)
		70:70	0.954	0.825	0.948	0.947	0.999	0.962	0.96
			(0.8184)	(0.8038)	(0.8388)	(0.8396)	(11.6646)	(0.7895)	(0.7867)
		70:100	0.932	0.848	0.948	0.948	1	0.942	0.942
			(0.8106)	(0.8107)	(0.8348)	(0.8359)	(11.7842)	(0.7904)	(0.7876)
		100::100	0.948	0.843	0.939	0.939	1	0.945	0.942
			(0.6876)	(0.6853)	(0.7002)	(0.7006)	(11.9455)	(0.6647)	(0.6626)
		100::200	0.923	0.804	0.944	0.947	1	0.943	0.948
			(0.6750)	(0.6750)	(0.6913)	(0.6911)	(12.7108)	(0.6658)	(0.6636)
		200:200	0.94	0.83	0.952	0.952	1	0.941	0.938
			(0.4855)	(0.4882)	(0.4914)	(0.4913)	(11.9212)	(0.4766)	(0.4752)
	0.1:0.3	10:10	0.913	0.801	0.968	0.969	0.99	0.905	0.896
			(2.0560)	(2.0867)	(2.5327)	(2.5299)	(7.1277)	(1.9866)	(1.9728)
		10:40	0.866	0.754	0.95	0.948	0.996	0.928	0.915
			(2.0008)	(2.0053)	(2.4330)	(2.4392)	(7.2689)	(1.9992)	(1.9860)
		10:70	0.826	0.756	0.949	0.947	0.993	0.901	0.887
			(2.0026)	(2.0557)	(2.4388)	(2.4428)	(7.1381)	(1.9959)	(1.9823)
		10:100	0.803	0.731	0.939	0.94	0.988	0.888	0.873
			(2.0144)	(2.0499)	(2.4307)	(2.4354)	(7.3348)	(1.9774)	(1.9636)
		40:40	0.93	0.82	0.952	0.95	0.999	0.927	0.924
			(1.0992)	(1.1002)	(1.1495)	(1.1494)	(9.3931)	(1.0678)	(1.0638)
		40:70	0.892	0.775	0.948	0.948	1	0.933	0.931
			(1.0694)	(1.0735)	(1.1156)	(1.1169)	(11.9177)	(1.0651)	(1.0612)
		40:100	0.83	0.732	0.938	0.934	0.999	0.931	0.923
			(1.0623)	(1.0658)	(1.1048)	(1.1064)	(9.9227)	(1.0600)	(1.0559)
		70:70	0.948	0.858	0.956	0.954	1	0.952	0.952
			(0.8361)	(0.8467)	(0.8578)	(0.8584)	(10.5498)	(0.8180)	(0.8154)
		70:100	0.894	0.79	0.955	0.954	1	0.949	0.945
			(0.8212)	(0.8167)	(0.8452)	(0.8454)	(10.3027)	(0.8160)	(0.8135)
		100::100	0.947	0.822	0.951	0.952	1	0.94	0.938
			(0.7022)	(0.6915)	(0.7124)	(0.7124)	(10.6920)	(0.6879)	(0.6858)
		100::200	0.794	0.703	0.94	0.939	1	0.952	0.953
			(0.6844)	(0.6831)	(0.6913)	(0.6920)	(10.6641)	(0.6878)	(0.6857)
		200:200	0.958	0.822	0.953	0.957	1	0.957	0.952
			(0.4984)	(0.4863)	(0.5032)	(0.5030)	(11.4851)	(0.4916)	(0.4902)
	0.3:0.3	10:10	0.907	0.809	0.959	0.96	0.99	0.966	0.968
			(2.2497)	(2.2369)	(2.7192)	(2.7144)	(6.7539)	(2.0710)	(2.0513)
		10:40	0.879	0.766	0.959	0.961	0.995	0.964	0.961
			(2.2067)	(2.2331)	(2.6635)	(2.6621)	(6.4714)	(2.0712)	(2.0509)
		10:70	0.863	0.77	0.953	0.956	0.99	0.966	0.968
			(2.2102)	(2.2113)	(2.6407)	(2.6408)	(6.3714)	(2.0720)	(2.0522)
		10:100	0.86	0.757	0.944	0.946	0.989	0.964	0.961
			(2.2130)	(2.2227)	(2.5776)	(2.5776)	(6.8369)	(2.0692)	(2.0490)
		40:40	0.942	0.832	0.939	0.938	0.999	0.961	0.961
			(1.2012)	(1.2327)	(1.2421)	(1.2424)	(8.1930)	(1.1146)	(1.1096)
		40:70	0.899	0.784	0.944	0.949	1	0.953	0.952
			(1.1722)	(1.1760)	(1.2178)	(1.2183)	(8.6506)	(1.1071)	(1.1022)
		40:100	0.86	0.744	0.941	0.94	1	0.959	0.956
			(1.1668)	(1.1910)	(1.2115)	(1.2126)	(8.2399)	(1.1081)	(1.1033)
		70:70	0.949	0.839	0.943	0.941	0.999	0.948	0.947
			(0.9167)	(0.9102)	(0.9252)	(0.9261)	(8.8207)	(0.8659)	(0.8628)
70:100		0.924	0.832	0.961	0.962	0.999	0.956	0.956
		(0.8968)	(0.9092)	(0.9174)	(0.9179)	(9.3795)	(0.8590)	(0.8559)
100::100		0.949	0.83	0.941	0.938	1	0.952	0.953
		(0.7663)	(0.7680)	(0.7761)	(0.7764)	(9.1519)	(0.7284)	(0.7260)
100::200		0.792	0.726	0.947	0.948	1	0.941	0.94
		(0.7493)	(0.7542)	(0.7594)	(0.7598)	(9.3836)	(0.7293)	(0.7269)
200:200		0.952	0.824	0.952	0.951	1	0.955	0.953
		(0.3198)	(0.3165)	(0.5471)	(0.5472)	(9.0699)	(0.3099)	(0.3090)

Values in bold correspond to the shortest expected lengths among those whose coverage probabilities are equal to or greater than the nominal confidence level.

,

Table 3. CP and EL for 95% confidence intervals for the difference between two means of zero-inflated two-parameter Rayleigh distribution when location parameter $\left({\mu }_1,{\mu }_2\right)=\left(0.1,0.1\right)$ and scale parameter $\left({\lambda }_1,{\lambda }_2\right)=\left(0.5,0.5\right)$.

Scale	Prob	Coverage Probability
		Samplesize	PB	BS	GCI	MOVER	AN	MCMC	HPD
0.5:0.5	0.1:0.1	10:10	0.898	0.79	0.966	0.967	1	0.986	0.991
			(1.1877)	(1.1788)	(1.4434)	(1.4512)	(10.7351)	(1.2823)	(1.2778)
		10:40	0.897	0.787	0.957	0.958	1	0.991	0.99
			(1.1197)	(1.1133)	(1.3790)	(1.3857)	(9.3153)	(1.2711)	(1.2665)
		10:70	0.904	0.815	0.951	0.953	0.999	0.995	0.997
			(1.1277)	(1.1334)	(1.3898)	(1.3960)	(9.5199)	(1.2759)	(1.2716)
		10:100	0.885	0.794	0.952	0.952	1	0.99	0.992
			(1.1292)	(1.1183)	(1.3645)	(1.3694)	(10.3862)	(1.2699)	(1.2654)
		40:40	0.946	0.831	0.95	0.951	1	0.968	0.971
			(0.6288)	(0.6338)	(0.6539)	(0.6546)	(12.5557)	(0.6238)	(0.6220)
		40:70	0.926	0.795	0.933	0.935	1	0.973	0.972
			(0.6056)	(0.6057)	(0.6367)	(0.6370)	(12.3534)	(0.6237)	(0.6220)
		40:100	0.897	0.798	0.95	0.949	1	0.964	0.962
			(0.6017)	(0.5999)	(0.6286)	(0.6296)	(12.0819)	(0.6231)	(0.6214)
		70:70	0.943	0.828	0.963	0.964	1	0.958	0.959
			(0.4779)	(0.4768)	(0.4898)	(0.4908)	(11.9157)	(0.4706)	(0.4694)
		70:100	0.931	0.807	0.933	0.932	1	0.96	0.957
			(0.4643)	(0.4624)	(0.4795)	(0.4801)	(12.8689)	(0.4715)	(0.4702)
		100::100	0.952	0.827	0.954	0.953	1	0.957	0.959
			(0.4010)	(0.3988)	(0.4073)	(0.4078)	(11.7305)	(0.3954)	(0.3944)
		100::200	0.889	0.772	0.931	0.937	1	0.949	0.947
			(0.3853)	(0.3828)	(0.3931)	(0.3934)	(11.6899)	(0.3968)	(0.3958)
		200:200	0.946	0.847	0.957	0.961	1	0.952	0.952
			(0.2842)	(0.2854)	(0.2870)	(0.2871)	(11.1239)	(0.2816)	(0.2809)
	0.1:0.3	10:10	0.912	0.815	0.958	0.96	0.999	0.978	0.98
			(1.2637)	(1.2582)	(1.5229)	(1.5250)	(8.7031)	(1.3864)	(1.3804)
		10:40	0.778	0.692	0.948	0.948	1	0.974	0.978
			(1.1657)	(1.1693)	(1.4117)	(1.4146)	(8.7225)	(1.3729)	(1.3672)
		10:70	0.761	0.679	0.934	0.937	0.999	0.972	0.973
			(1.1835)	(1.2051)	(1.3643)	(1.3691)	(8.5678)	(1.3810)	(1.3750)
		10:100	0.753	0.657	0.913	0.914	0.999	0.974	0.976
			(1.1973)	(1.1900)	(1.3530)	(1.3565)	(8.5176)	(1.3826)	(1.3768)
		40:40	0.924	0.827	0.948	0.949	1	0.945	0.943
			(0.6661)	(0.6608)	(0.6936)	(0.6945)	(11.1649)	(0.6667)	(0.6649)
		40:70	0.812	0.728	0.928	0.927	1	0.961	0.961
			(0.6237)	(0.6287)	(0.6481)	(0.6489)	(10.1251)	(0.6686)	(0.6668)
		40:100	0.659	0.61	0.932	0.933	1	0.961	0.964
			(0.6178)	(0.6210)	(0.6243)	(0.6257)	(10.5625)	(0.6681)	(0.6662)
		70:70	0.944	0.836	0.942	0.944	1	0.952	0.95
			(0.5079)	(0.5058)	(0.5212)	(0.5209)	(10.6052)	(0.5052)	(0.5039)
		70:100	0.808	0.725	0.927	0.928	1	0.946	0.948
			(0.4838)	(0.4869)	(0.4959)	(0.4962)	(17.8627)	(0.5070)	(0.5057)
		100::100	0.939	0.84	0.939	0.941	1	0.949	0.948
			(0.4270)	(0.4272)	(0.4341)	(0.4342)	(10.8555)	(0.4243)	(0.4233)
		100::200	0.484	0.469	0.922	0.926	1	0.945	0.948
			(0.3974)	(0.3927)	(0.3978)	(0.3976)	(10.6879)	(0.4240)	(0.4229)
		200:200	0.95	0.823	0.952	0.953	1	0.955	0.949
			(0.3030)	(0.2983)	(0.3052)	(0.3049)	(10.0221)	(0.3014)	(0.3006)
	0.3:0.3	10:10	0.917	0.792	0.959	0.96	0.999	0.999	1
			(1.3132)	(1.3114)	(2.7192)	(2.7144)	(7.3613)	(1.4245)	(1.4182)
		10:40	0.8	0.708	0.959	0.961	0.996	0.992	0.994
			(1.2514)	(1.2607)	(2.6635)	(2.6621)	(6.8516)	(1.4246)	(1.4181)
		10:70	0.776	0.693	0.934	0.956	0.997	0.996	0.996
			(1.2600)	(1.2390)	(2.6407)	(2.6408)	(7.3056)	(1.4314)	(1.4252)
		10:100	0.781	0.693	0.944	0.946	1	0.998	0.998
			(1.2684)	(1.2851)	(2.5776)	(2.5776)	(7.5875)	(1.4219)	(1.4159)
		40:40	0.954	0.841	0.939	0.938	1	0.966	0.969
			(0.7040)	(0.6956)	(1.2421)	(1.2424)	(9.0816)	(0.6796)	(0.6777)
		40:70	0.819	0.742	0.944	0.949	1	0.968	0.964
			(0.6663)	(0.6768)	(1.2178)	(1.2183)	(9.7567)	(0.6804)	(0.6785)
		40:100	0.691	0.63	0.941	0.94	1	0.964	0.962
			(0.6567)	(0.6557)	(1.2115)	(1.2126)	(9.2203)	(0.6786)	(0.6767)
		70:70	0.934	0.821	0.943	0.941	0.999	0.951	0.951
			(0.5374)	(0.5367)	(0.9252)	(0.9261)	(10.4404)	(0.5172)	(0.5159)
70:100		0.83	0.729	0.961	0.962	1	0.96	0.963
		(0.5128)	(0.5123)	(0.9174)	(0.9179)	(9.6943)	(0.5175)	(0.5161)
100::100		0.939	0.834	0.941	0.938	1	0.951	0.948
		(0.4497)	(0.4506)	(0.7761)	(0.7764)	(9.2568)	(0.4336)	(0.4324)
100::200		0.519	0.504	0.947	0.948	1	0.958	0.956
		(0.4251)	(0.4274)	(0.7594)	(0.7598)	(9.3048)	(0.4344)	(0.4332)
200:200		0.952	0.824	0.952	0.951	1	0.955	0.953
		(0.3198)	(0.3165)	(0.5471)	(0.5472)	(9.0699)	(0.3099)	(0.3090)

Values in bold correspond to the shortest expected lengths among those whose coverage probabilities are equal to or greater than the nominal confidence level.

,

Table 4. CP and EL for 95% confidence intervals for the difference between two means of zero-inflated two-parameter Rayleigh distribution when location parameter $\left({\mu }_1,{\mu }_2\right)=\left(0.1,0.1\right)$ and scale parameter $\left({\lambda }_1,{\lambda }_2\right)=\left(0.5,1\right)$.

Scale	Prob	Coverage Probability
		Samplesize	PB	BS	GCI	MOVER	AN	MCMC	HPD
0.5:1	0.1:0.1	10:10	0.896	0.817	0.949	0.949	1	0.991	0.992
			(1.0285)	(1.0522)	(1.2341)	(1.2397)	(10.0113)	(1.1726)	(1.1676)
		10:40	0.881	0.786	0.959	0.955	1	0.994	0.996
			(0.9880)	(0.9988)	(1.2239)	(1.2267)	(9.5570)	(1.1695)	(1.1647)
		10:70	0.901	0.821	0.931	0.933	1	0.992	0.992
			(1.0013)	(1.0347)	(1.2154)	(1.2205)	(9.9700)	(1.1706)	(1.1659)
		10:100	0.881	0.795	0.949	0.948	1	0.992	0.993
			(0.9998)	(1.0166)	(1.2057)	(1.2112)	(9.5573)	(1.1766)	(1.1717)
		40:40	0.954	0.835	0.942	0.942	1	0.974	0.974
			(0.5491)	(0.5573)	(0.5674)	(0.5686)	(11.5186)	(0.5556)	(0.5539)
		40:70	0.926	0.817	0.943	0.946	1	0.958	0.959
			(0.5271)	(0.5236)	(0.5554)	(0.5562)	(11.7558)	(0.5507)	(0.5490)
		40:100	0.92	0.796	0.946	0.945	1	0.963	0.964
			(0.5275)	(0.5181)	(0.5508)	(0.5525)	(12.3368)	(0.5528)	(0.5511)
		70:70	0.946	0.844	0.957	0.957	1	0.964	0.962
			(0.4139)	(0.4137)	(0.4267)	(0.4272)	(11.5407)	(0.4130)	(0.4119)
		70:100	0.918	0.791	0.948	0.948	1	0.958	0.959
			(0.4047)	(0.3964)	(0.4179)	(0.4184)	(11.5026)	(0.4141)	(0.4130)
		100::100	0.945	0.827	0.952	0.951	1	0.963	0.959
			(0.3489)	(0.3521)	(0.3538)	(0.3542)	(11.5221)	(0.3475)	(0.3465)
		100::200	0.905	0.812	0.94	0.938	1	0.957	0.956
			(0.3379)	(0.3475)	(0.3451)	(0.3454)	(11.3463)	(0.3463)	(0.3453)
		200:200	0.945	0.834	0.967	0.968	1	0.953	0.954
			(0.2471)	(0.2490)	(0.2492)	(0.2494)	(11.1933)	(0.2466)	(0.2460)
	0.1:0.3	10:10	0.931	0.82	0.965	0.966	0.999	0.986	0.989
			(1.0833)	(1.0766)	(1.3003)	(1.3013)	(8.8081)	(1.2705)	(1.2657)
		10:40	0.83	0.722	0.946	0.941	0.999	0.977	0.975
			(1.0224)	(1.0260)	(1.2239)	(1.2268)	(9.4244)	(1.2608)	(1.2560)
		10:70	0.778	0.698	0.945	0.941	0.998	0.966	0.965
			(1.0146)	(1.0205)	(1.2095)	(1.2191)	(8.6714)	(1.2564)	(1.2515)
		10:100	0.769	0.691	0.941	0.941	1	0.969	0.966
			(1.0215)	(0.9983)	(1.2026)	(1.2065)	(8.5638)	(1.2561)	(1.2510)
		40:40	0.938	0.838	0.943	0.942	1	0.951	0.951
			(0.5720)	(0.5714)	(0.5940)	(0.5949)	(10.8970)	(0.5891)	(0.5875)
		40:70	0.854	0.744	0.932	0.93	1	0.964	0.963
			(0.5452)	(0.5437)	(0.5657)	(0.5666)	(10.8200)	(0.5890)	(0.5874)
		40:100	0.732	0.663	0.939	0.94	1	0.954	0.952
			(0.5396)	(0.5448)	(0.5534)	(0.5542)	(10.4897)	(0.5870)	(0.5854)
		70:70	0.937	0.822	0.95	0.949	1	0.952	0.951
			(0.4345)	(0.4365)	(0.4455)	(0.4459)	(10.5684)	(0.4400)	(0.4388)
		70:100	0.839	0.748	0.934	0.936	1	0.945	0.949
			(0.4173)	(0.4100)	(0.4307)	(0.4307)	(10.4528)	(0.4404)	(0.4392)
		100::100	0.944	0.83	0.944	0.948	1	0.949	0.949
			(0.3665)	(0.3652)	(0.3707)	(0.3709)	(10.5046)	(0.3683)	(0.3674)
		100::200	0.602	0.554	0.936	0.933	1	0.951	0.951
			(0.3465)	(0.3393)	(0.3487)	(0.3488)	(10.3036)	(0.3671)	(0.3662)
		200:200	0.955	0.827	0.953	0.954	1	0.958	0.958
			(0.2581)	(0.2566)	(0.2611)	(0.2611)	(9.9241)	(0.2598)	(0.2591)
	0.3:0.3	10:10	0.915	0.815	0.97	0.972	1	1	1
			(1.1538)	(1.1389)	(1.4110)	(1.4078)	(7.3772)	(1.3262)	(1.3196)
		10:40	0.837	0.744	0.941	0.942	1	0.996	0.997
			(1.0985)	(1.1306)	(1.3130)	(1.3143)	(7.3403)	(1.3196)	(1.3130)
		10:70	0.809	0.717	0.936	0.941	0.999	0.996	0.997
			(1.0942)	(1.0907)	(1.3069)	(1.3059)	(8.0045)	(1.3117)	(1.3052)
		10:100	0.796	0.712	0.937	0.938	0.999	1	1
			(1.1194)	(1.0977)	(1.2825)	(1.2850)	(7.7049)	(1.3226)	(1.3162)
		40:40	0.925	0.819	0.946	0.949	1	0.96	0.961
			(0.6136)	(0.6199)	(0.6344)	(0.6355)	(9.2715)	(0.6069)	(0.6050)
		40:70	0.834	0.738	0.939	0.942	1	0.97	0.969
			(0.5879)	(0.5867)	(0.6096)	(0.6099)	(9.1310)	(0.6066)	(0.6047)
		40:100	0.769	0.682	0.942	0.94	1	0.975	0.974
			(0.5817)	(0.5840)	(0.5968)	(0.5978)	(9.3129)	(0.6054)	(0.6036)
		70:70	0.944	0.829	0.955	0.953	1	0.967	0.965
			(0.4683)	(0.4640)	(0.4758)	(0.4763)	(9.0969)	(0.4562)	(0.4549)
70:100		0.882	0.759	0.944	0.941	1	0.952	0.954
		(0.4519)	(0.4503)	(0.4651)	(0.4654)	(9.0945)	(0.4589)	(0.4576)
100::100		0.942	0.849	0.951	0.953	1	0.959	0.956
		(0.3922)	(0.3923)	(0.3976)	(0.3978)	(9.8030)	(0.3818)	(0.3808)
100::200		0.671	0.617	0.952	0.953	1	0.967	0.966
		(0.3760)	(0.3736)	(0.3787)	(0.3785)	(8.9837)	(0.3844)	(0.3833)
200:200		0.947	0.834	0.941	0.944	1	0.952	0.952
		(0.2787)	(0.2787)	(0.2798)	(0.2801)	(8.7010)	(0.2716)	(0.2708)

Values in bold correspond to the shortest expected lengths among those whose coverage probabilities are equal to or greater than the nominal confidence level.

,

Table 5. CP and EL for 95% confidence intervals for the difference between two means of zero-inflated two-parameter Rayleigh distribution when location parameter $\left({\mu }_1,{\mu }_2\right)=\left(0.1,0.1\right)$ and scale parameter $\left({\lambda }_1,{\lambda }_2\right)=\left(1,1\right)$.

Scale	Prob	Coverage Probability
		Samplesize	PB	BS	GCI	MOVER	AN	MCMC	HPD
1:1	0.1:0.1	10:10	0.895	0.803	0.958	0.958	0.999	0.999	1
			(0.8471)	(0.8497)	(1.0271)	(1.0317)	(10.3602)	(1.0545)	(1.0508)
		10:40	0.906	0.795	0.961	0.955	1	1	1
			(0.8037)	(0.7976)	(0.9798)	(0.9840)	(10.4420)	(1.0580)	(1.0546)
		10:70	0.887	0.753	0.964	0.965	0.999	1	1
			(0.8055)	(0.7921)	(0.9796)	(0.9832)	(10.9027)	(1.0509)	(1.0472)
		10:100	0.88	0.789	0.948	0.947	1	1	1
			(0.8098)	(0.8044)	(0.9730)	(0.9772)	(11.1338)	(1.0570)	(1.0535)
		40:40	0.939	0.828	0.959	0.959	1	0.984	0.985
			(0.4473)	(0.4468)	(0.4674)	(0.4684)	(11.7476)	(0.4695)	(0.4682)
		40:70	0.926	0.809	0.949	0.948	1	0.975	0.975
			(0.4323)	(0.4259)	(0.4554)	(0.4562)	(11.5642)	(0.4712)	(0.4699)
		40:100	0.917	0.805	0.942	0.943	1	0.984	0.982
			(0.4274)	(0.4290)	(0.4442)	(0.4451)	(11.6768)	(0.4701)	(0.4688)
		70:70	0.934	0.801	0.956	0.954	1	0.961	0.964
			(0.3413)	(0.3272)	(0.3493)	(0.3499)	(11.1939)	(0.3483)	(0.3473)
		70:100	0.931	0.819	0.956	0.956	1	0.975	0.974
			(0.3305)	(0.3299)	(0.3418)	(0.3423)	(11.2472)	(0.3485)	(0.3476)
		100::100	0.942	0.832	0.947	0.948	1	0.957	0.958
			(0.2865)	(0.2818)	(0.2915)	(0.2919)	(11.1646)	(0.2907)	(0.2899)
		100::200	0.861	0.773	0.933	0.935	1	0.965	0.969
			(0.2742)	(0.2756)	(0.2796)	(0.2800)	(11.2008)	(0.2909)	(0.2901)
		200:200	0.948	0.826	0.952	0.95	1	0.958	0.957
			(0.2035)	(0.2035)	(0.2048)	(0.2049)	(11.0479)	(0.2043)	(0.2038)
	0.1:0.3	10:10	0.919	0.789	0.963	0.964	1	1	1
			(0.9594)	(0.9407)	(1.0855)	(1.0869)	(8.0349)	(1.1936)	(1.1886)
		10:40	0.795	0.682	0.937	0.939	1	0.994	0.994
			(0.8368)	(0.8336)	(0.9902)	(0.9918)	(9.6171)	(1.1555)	(1.1505)
		10:70	0.769	0.711	0.927	0.926	1	0.995	0.997
			(0.8442)	(0.8430)	(0.9748)	(0.9748)	(9.1378)	(1.1458)	(1.1409)
		10:100	0.713	0.631	0.924	0.927	1	0.992	0.996
			(0.8487)	(0.8434)	(0.9666)	(0.9698)	(9.2505)	(1.1515)	(1.1462)
		40:40	0.937	0.812	0.945	0.943	1	0.968	0.969
			(0.4782)	(0.4816)	(0.4981)	(0.4984)	(10.6903)	(0.5070)	(0.5055)
		40:70	0.788	0.708	0.93	0.928	1	0.975	0.974
			(0.4446)	(0.4378)	(0.4649)	(0.4653)	(10.5807)	(0.5067)	(0.5053)
		40:100	0.637	0.597	0.929	0.927	1	0.977	0.975
			(0.4412)	(0.4384)	(0.4493)	(0.4497)	(10.5507)	(0.5093)	(0.5079)
		70:70	0.949	0.831	0.953	0.954	1	0.971	0.971
			(0.3649)	(0.3624)	(0.3726)	(0.3729)	(10.1877)	(0.3761)	(0.3751)
		70:100	0.813	0.735	0.944	0.944	1	0.958	0.957
			(0.3443)	(0.3474)	(0.3545)	(0.3549)	(10.3120)	(0.3766)	(0.3756)
		100::100	0.949	0.831	0.945	0.943	1	0.958	0.959
			(0.3064)	(0.3080)	(0.3104)	(0.3106)	(10.1941)	(0.3131)	(0.3123)
		100::200	0.494	0.481	0.927	0.924	1	0.957	0.962
			(0.2842)	(0.2864)	(0.2846)	(0.2848)	(10.2865)	(0.3128)	(0.3120)
		200:200	0.947	0.809	0.945	0.945	1	0.954	0.952
			(0.2174)	(0.2129)	(0.2189)	(0.2191)	(9.8696)	(0.2199)	(0.2194)
	0.3:0.3	10:10	0.923	0.814	0.977	0.98	0.999	1	1
			(0.9600)	(0.9701)	(1.1471)	(1.1444)	(8.2591)	(1.2028)	(1.1977)
		10:40	0.782	0.705	0.943	0.941	1	1	1
			(0.8831)	(0.8827)	(1.0550)	(1.0534)	(7.4316)	(1.1996)	(1.1942)
		10:70	0.776	0.681	0.936	0.938	1	0.999	1
			(0.8868)	(0.8935)	(1.0383)	(1.0386)	(8.4906)	(1.1983)	(1.1929)
		10:100	0.767	0.675	0.921	0.924	1	1	1
			(0.8946)	(0.8991)	(1.0260)	(1.0262)	(7.4111)	(1.1979)	(1.1927)
		40:40	0.943	0.802	0.947	0.948	1	0.978	0.978
			(0.5074)	(0.5077)	(0.5244)	(0.5250)	(9.1198)	(0.5180)	(0.5165)
		40:70	0.821	0.733	0.932	0.931	1	0.987	0.989
			(0.4777)	(0.4810)	(0.4919)	(0.4923)	(9.1155)	(0.5246)	(0.5231)
		40:100	0.663	0.625	0.927	0.931	1	0.982	0.984
			(0.4742)	(0.4697)	(0.4801)	(0.4800)	(9.2141)	(0.5192)	(0.5178)
		70:70	0.935	0.84	0.956	0.954	1	0.969	0.968
			(0.3882)	(0.3943)	(0.3937)	(0.3940)	(9.1637)	(0.3879)	(0.3869)
70:100		0.836	0.767	0.943	0.947	1	0.97	0.972
		(0.3693)	(0.3721)	(0.3769)	(0.3774)	(8.8447)	(0.3879)	(0.3869)
100::100		0.938	0.822	0.959	0.956	1	0.956	0.954
		(0.3255)	(0.3221)	(0.3281)	(0.3285)	(8.8167)	(0.3220)	(0.3212)
100::200		0.509	0.502	0.923	0.923	1	0.958	0.956
		(0.3033)	(0.2971)	(0.3043)	(0.3050)	(8.8153)	(0.3214)	(0.3205)
200:200		0.952	0.824	0.949	0.95	1	0.955	0.953
		(0.3198)	(0.3165)	(0.2313)	(0.2314)	(9.0699)	(0.3099)	(0.3090)

Values in bold correspond to the shortest expected lengths among those whose coverage probabilities are equal to or greater than the nominal confidence level.

,

Table 6. CP and EL for 95% confidence intervals for the ratio of two means for zero-inflated two-parameter Rayleigh distribution when location parameter $\left({\mu }_1,{\mu }_2\right)=\left(0.1,0.1\right)$ and scale parameter $\left({\lambda }_1,{\lambda }_2\right)=\left(0.1,0.1\right)$.

Scale	Prob	Coverage Probability
		Samplesize	PB	BS	GCI	MOVER	AN	MCMC	HPD
0.1:0.1	0.1:0.1	10:10	0.898	0.801	0.956	0.958	0.948	0.935	1
			(1.2052)	(1.2072)	(1.4395)	(1.4492)	(2.9824)	(0.9513)	(0.9229)
		10:40	0.887	0.790	0.958	0.956	0.986	0.939	0.932
			(0.9467)	(0.9460)	(1.3086)	(1.3147)	(2.8736)	(0.9556)	(0.9272)
		10:70	0.907	0.779	0.950	0.952	0.986	0.945	0.943
			(0.9434)	(0.9460)	(1.2594)	(1.2657)	(3.2472)	(0.9631)	(0.9348)
		10:100	0.884	0.791	0.953	0.953	0.996	0.942	0.937
			(0.9218)	(0.9405)	(1.2868)	(1.2943)	(7.5441)	(0.9626)	(0.9344)
		40:40	0.940	0.825	0.953	0.954	0.999	0.948	0.949
			(0.5537)	(0.5542)	(0.5677)	(0.5695)	(4.2702)	(0.5150)	(0.5093)
		40:70	0.924	0.810	0.940	0.941	1	0.955	0.952
			(0.4940)	(0.5029)	(0.5510)	(0.5520)	(10.7472)	(0.5134)	(0.5077)
		40:100	0.905	0.787	0.934	0.937	1	0.949	0.948
			(0.4717)	(0.4695)	(0.5428)	(0.5446)	(10.8491)	(0.5116)	(0.5059)
		70:70	0.949	0.835	0.954	0.954	1	0.956	0.952
			(0.4129)	(0.4136)	(0.4214)	(0.4220)	(4.3749)	(0.3933)	(0.3902)
		70:100	0.934	0.824	0.941	0.945	0.999	0.946	0.94
			(0.3795)	(0.3825)	(0.4122)	(0.4123)	(12.0405)	(0.3917)	(0.3886)
		100::100	0.940	0.828	0.947	0.945	1	0.94	0.939
			(0.3454)	(0.3387)	(0.3486)	(0.3492)	(5.0112)	(0.3321)	(0.3282)
		100::200	0.884	0.772	0.926	0.928	1	0.947	0.947
			(0.3049)	(0.3026)	(0.3348)	(0.3348)	(11.5592)	(0.3302)	(0.3282)
		200:200	0.937	0.817	0.949	0.951	1	0.943	0.943
			(0.2421)	(0.2415)	(0.2450)	(0.2449)	(12.9709)	(0.2348)	(0.2338)
	0.1:0.3	10:10	0.920	0.84	0.959	0.955	0.968	0.914	0.895
			(2.4029)	(2.4899)	(2.8877)	(3.0190)	(4.0483)	(1.2723)	(1.2207)
		10:40	0.782	0.662	0.93	0.926	0.974	0.915	0.881
			(1.1149)	(1.1277)	(2.2141)	(2.3829)	(4.1506)	(1.2838)	(1.2321)
		10:70	0.74	0.621	0.931	0.921	0.979	0.925	0.901
			(1.0946)	(1.1113)	(2.3544)	(2.0362)	(4.1986)	(1.2988)	(1.2452)
		10:100	0.704	0.588	0.918	0.906	0.983	0.905	0.874
			(1.0348)	(1.0182)	(2.2334)	(2.2629)	(6.2433)	(1.2512)	(1.2006)
		40:40	0.938	0.827	0.952	0.951	1	0.938	0.926
			(0.9317)	(0.9348)	(0.9278)	(0.92889)	(5.2578)	(0.7772)	(0.7628)
		40:70	0.765	0.638	0.94	0.94	1	0.931	0.915
			(0.6268)	(0.6189)	(0.8466)	(0.8481)	(9.2486)	(0.7731)	(0.7590)
		40:100	0.623	0.535	0.933	0.933	0.999	0.935	0.924
			(0.5655)	(0.5578)	(0.8159)	(0.8169)	(8.5717)	(0.7700)	(0.7561)
		70:70	0.942	0.849	0.949	0.95	0.999	0.943	0.938
			(0.6790)	(0.6794)	(0.6865)	(0.6879)	(10.1438)	(0.6104)	(0.6031)
		70:100	0.811	0.693	0.946	0.946	0.997	0.94	0.93
			(0.5139)	(0.5080)	(0.6389)	(0.6397)	(10.0585)	(0.6047)	(0.5973)
		100::100	0.948	0.834	0.948	0.946	1	0.942	0.935
			(0.5543)	(0.5583)	(0.5603)	(0.5611)	(10.6036)	(0.5108)	(0.5060)
		100::200	0.404	0.385	0.927	0.924	1	0.936	0.926
			(0.3712)	(0.3731)	(0.5015)	(0.5016)	(11.2478)	(0.5119)	(0.5069)
		200:200	0.953	0.844	0.954	0.954	1	0.949	0.947
			(0.3862)	(0.3834)	(0.3889)	(0.3892)	(10.9930)	(0.3700)	(0.3676)
	0.3:0.3	10:10	0.898	0.847	0.97	0.963	0.957	0.941	0.943
			(2.2042)	(2.2803)	(2.6098)	(2.4737)	(3.0905)	(1.2364)	(1.1822)
		10:40	0.828	0.687	0.953	0.946	0.961	0.96	0.96
			(1.0540)	(1.0609)	(2.1257)	(1.3000)	(3.1736)	(1.1936)	(1.1423)
		10:70	0.786	0.692	0.946	0.944	0.96	0.951	0.95
			(1.0343)	(1.0554)	(2.0280)	(2.1346)	(3.5037)	(1.2122)	(1.1608)
		10:100	0.789	0.685	0.945	0.937	0.985	0.973	0.963
			(0.9942)	(1.0215)	(2.0280)	(2.1346)	(3.5037)	(1.2122)	(1.1608)
		40:40	0.946	0.86	0.949	0.948	0.997	0.949	0.953
			(0.8312)	(0.8492)	(0.8386)	(0.8405)	(3.8788)	(0.7021)	(0.6895)
		40:70	0.809	0.675	0.939	0.943	1	0.945	0.949
			(0.5872)	(0.5782)	(0.7773)	(0.7796)	(8.6319)	(0.6908)	(0.6786)
		40:100	0.695	0.586	0.937	0.934	1	0.951	0.952
			(0.5388)	(0.5336)	(0.7607)	(0.7609)	(7.9030)	(0.6948)	(0.6824)
		70:70	0.948	0.847	0.953	0.951	0.999	0.942	0.942
			(0.6023)	(0.6087)	(0.6085)	(0.6089)	(4.4774)	(0.5382)	(0.5320)
70:100		0.836	0.719	0.938	0.936	0.999	0.941	0.942
		(0.4742)	(0.4696)	(0.5877)	(0.5885)	(8.6708)	(0.5333)	(0.5269)
100::100		0.936	0.848	0.943	0.947	0.999	0.938	0.943
		(0.5010)	(0.5095)	(0.5058)	(0.5065)	(4.7516)	(0.4601)	(0.4559)
100::200		0.549	0.468	0.935	0.935	1	0.933	0.928
		(0.3558)	(0.3533)	(0.4638)	(0.4635)	(9.5176)	(0.4583)	(0.4540)
200:200		0.948	0.962	0.951	0.947	1	0.944	0.953
		(0.3488)	(0.3497)	(0.3508)	(0.3510)	(9.5204)	(0.3297)	(0.3277)

Values in bold correspond to the shortest expected lengths among those whose coverage probabilities are equal to or greater than the nominal confidence level.

,

Table 7. CP and EL for 95% confidence intervals for the ratio of two means for zero-inflated two-parameter Rayleigh distribution when location parameter $\left({\mu }_1,{\mu }_2\right)=\left(0.1,0.1\right)$ and scale parameter $\left({\lambda }_1,{\lambda }_2\right)=\left(0.1,0.5\right)$.

Scale	Prob	Coverage Probability
		Samplesize	PB	BS	GCI	MOVER	AN	MCMC	HPD
0.1:0.5	0.1:0.1	10:10	0.899	0.822	0.958	0.958	1	0.942	0.917
			(2.5124)	(2.5925)	(3.0610)	(3.0894)	(13.6701)	(1.8539)	(1.7989)
		10:40	0.893	0.791	0.965	0.967	0.993	0.944	0.92
			(1.9601)	(1.9751)	(2.6672)	(2.6805)	(12.8757)	(1.8496)	(1.7941)
		10:70	0.899	0.802	0.946	0.946	0.998	0.954	0.931
			(1.9460)	(1.9429)	(2.6814)	(2.6875)	(14.2668)	(1.8460)	(1.7911)
		10:100	0.89	0.78	0.952	0.952	0.987	0.942	0.915
			(1.9256)	(1.9281)	(2.6880)	(2.6980)	(8.8907)	(1.8628)	(1.8066)
		40:40	0.946	0.84	0.961	0.964	1	0.949	0.94
			(1.1529)	(1.1610)	(1.2106)	(1.2149)	(16.9036)	(1.0563)	(1.0446)
		40:70	0.928	0.823	0.948	0.949	1	0.956	0.946
			(1.0370)	(1.0553)	(1.1715)	(1.1740)	(10.9058)	(1.0725)	(1.0605)
		40:100	0.906	0.804	0.936	0.94	0.999	0.953	0.949
			(1.0115)	(1.0055)	(1.1349)	(1.1372)	(10.8462)	(1.0779)	(1.0659)
		70:70	0.964	0.832	0.946	0.944	1	0.963	0.958
			(0.8752)	(0.8624)	(0.8958)	(0.8980)	(16.1578)	(0.8278)	(0.8215)
		70:100	0.92	0.834	0.945	0.945	1	0.939	0.935
			(0.8047)	(0.8030)	(0.8628)	(0.8636)	(11.7842)	(0.8269)	(0.8206)
		100::100	0.951	0.853	0.946	0.946	1	0.949	0.944
			(0.7260)	(0.7264)	(0.7372)	(0.7385)	(16.2402)	(0.6931)	(0.6889)
		100::200	0.884	0.763	0.943	0.939	1	0.943	0.94
			(0.6382)	(0.6360)	(0.7088)	(0.7087)	(17.1126)	(0.6931)	(0.6888)
		200:200	0.935	0.813	0.952	0.957	1	0.936	0.934
			(0.5116)	(0.5131)	(0.5149)	(0.5146)	(15.8400)	(0.5004)	(0.4982)
	0.1:0.3	10:10	0.923	0.866	0.966	0.962	0.994	0.857	0.796
			(5.0020)	(5.1631)	(6.5869)	(8.5412)	(18.3253)	(2.3842)	(2.2873)
		10:40	0.775	0.66	0.937	0.93	0.994	0.869	0.812
			(2.3925)	(2.3869)	(4.6888)	(4.5245)	(18.7880)	(2.4419)	(2.3423)
		10:70	0.717	0.606	0.943	0.936	0.994	0.854	0.782
			(2.1867)	(2.2729)	(4.5653)	(3.9663)	(16.7959)	(2.3824)	(2.2854)
		10:100	0.685	0.564	0.924	0.914	0.938	0.828	0.775
			(2.1642)	(2.1659)	(4.7188)	(6.0957)	(7.3348)	(2.3689)	(2.2706)
		40:40	0.944	0.844	0.95	0.948	1	0.926	0.914
			(1.9878)	(2.0071)	(1.9817)	(1.9857)	(19.4643)	(1.6146)	(1.5850)
		40:70	0.776	0.631	0.932	0.931	0.999	0.917	0.904
			(1.3175)	(1.3164)	(1.7943)	(1.7957)	(11.9177)	(1.6096)	(1.5796)
		40:100	0.585	0.515	0.909	0.914	0.999	0.925	0.908
			(1.1799)	(1.1823)	(1.7121)	(1.7151)	(9.9227)	(1.5910)	(1.5618)
		70:70	0.955	0.85	0.956	0.957	1	0.939	0.925
			(1.4222)	(1.4454)	(1.4453)	(1.4458)	(21.1815)	(1.2678)	(1.2522)
		70:100	0.797	0.672	0.932	0.932	1	0.934	0.92
			(1.0749)	(1.0740)	(1.3618)	(1.3635)	(10.3027)	(1.2606)	(1.2452)
		100::100	0.948	0.836	0.953	0.955	1	0.945	0.937
			(1.1771)	(1.1572)	(1.1782)	(1.1791)	(21.3230)	(1.0840)	(1.0737)
		100::200	0.39	0.401	0.927	0.926	1	0.94	0.934
			(0.7839)	(0.7851)	(1.0566)	(1.0566)	(10.6641)	(1.0860)	(1.0757)
		200:200	0.956	0.839	0.955	0.951	0.999	0.952	0.947
			(0.8179)	(0.8043)	(0.8238)	(0.8240)	(21.1228)	(0.7834)	(0.7785)
	0.3:0.3	10:10	0.922	0.856	0.968	0.965	0.957	0.953	0.923
			(4.4622)	(4.5134)	(5.3097)	(6.3643)	(6.7539)	(2.2683)	(2.1711)
		10:40	0.801	0.712	0.947	0.941	0.993	0.948	0.924
			(2.2505)	(2.2485)	(4.2830)	(1.0791)	(13.5859)	(2.2813)	(2.1841)
		10:70	0.79	0.676	0.947	0.941	0.988	0.947	0.927
			(2.1603)	(2.1684)	(4.5351)	(1.4561)	(13.4526)	(2.2882)	(2.1889)
		10:100	0.758	0.669	0.944	0.938	0.95	0.94	0.915
			(2.1041)	(2.1183)	(4.0039)	(2.3349)	(6.8369)	(2.2585)	(2.1613)
		40:40	0.949	0.842	0.948	0.949	1	0.957	0.948
			(1.7643)	(1.8200)	(1.7857)	(1.7893)	(15.5408)	(1.4387)	(1.4127)
		40:70	0.834	0.706	0.945	0.954	0.997	0.947	0.939
			(1.2537)	(1.2581)	(1.6737)	(1.6747)	(8.6506)	(1.4325)	(1.4071)
		40:100	0.697	0.589	0.92	0.922	0.998	0.957	0.947
			(1.1370)	(1.1571)	(1.5997)	(1.5996)	(8.2399)	(1.4326)	(1.4072)
		70:70	0.944	0.852	0.951	0.951	1	0.945	0.944
			(1.2906)	(1.2804)	(1.2894)	(1.2910)	(15.9439)	(1.1402)	(1.1264)
70:100		0.878	0.759	0.949	0.947	0.999	0.951	0.949
		(1.0159)	(1.0332)	(1.2324)	(1.2335)	(9.3795)	(1.1337)	(1.1203)
100::100		0.949	0.825	0.948	0.948	0.999	0.953	0.945
		(1.0601)	(1.0568)	(1.0751)	(1.0758)	(9.1519)	(0.9587)	(0.9496)
100::200		0.544	0.487	0.936	0.933	1	0.938	0.937
		(0.7508)	(0.7511)	(0.9926)	(0.9931)	(16.8525)	(0.9643)	(0.9555)
200:200		0.952	0.828	0.943	0.94	1	0.955	0.957
		(0.3431)	(0.3393)	(0.7445)	(0.7453)	(9.6765)	(0.3258)	(0.3238)

Values in bold correspond to the shortest expected lengths among those whose coverage probabilities are equal to or greater than the nominal confidence level.

,

Table 8. CP and EL for 95% confidence intervals for the ratio of two means for zero-inflated two-parameter Rayleigh distribution when location parameter $\left({\mu }_1,{\mu }_2\right)=\left(0.1,0.1\right)$ and scale parameter $\left({\lambda }_1,{\lambda }_2\right)=\left(0.5,0.5\right)$.

Scale	Prob	Coverage Probability
		Samplesize	PB	BS	GCI	MOVER	AN	MCMC	HPD
0.5:0.5	0.1:0.1	10:10	0.898	0.828	0.966	0.967	0.997	0.986	0.986
			(1.1891)	(1.2009)	(1.3940)	(1.4063)	(9.7894)	(0.9879)	(0.9570)
		10:40	0.897	0.798	0.957	0.958	0.999	0.991	0.99
			(0.9039)	(0.9003)	(1.2602)	(1.2655)	(8.2771)	(0.9807)	(0.9503)
		10:70	0.904	0.801	0.951	0.952	0.999	0.995	0.993
			(0.8779)	(0.8854)	(1.2226)	(1.2304)	(9.5199)	(0.9757)	(0.9449)
		10:100	0.885	0.799	0.952	0.952	0.997	0.99	0.989
			(0.8898)	(0.8749)	(1.2308)	(1.2346)	(9.2473)	(0.9807)	(0.9505)
		40:40	0.946	0.827	0.95	0.953	1	0.968	0.97
			(0.5316)	(0.5353)	(0.5610)	(0.5620)	(10.4940)	(0.5082)	(0.5027)
		40:70	0.926	0.788	0.933	0.935	1	0.973	0.971
			(0.4746)	(0.4744)	(0.5334)	(0.5338)	(10.3271)	(0.5104)	(0.5048)
		40:100	0.897	0.78	0.95	0.948	1	0.964	0.964
			(0.4562)	(0.4541)	(0.5288)	(0.5298)	(10.0574)	(0.5082)	(0.5026)
		70:70	0.943	0.825	0.963	0.964	1	0.958	0.953
			(0.4014)	(0.4007)	(0.4094)	(0.4104)	(9.9949)	(0.3885)	(0.3856)
		70:100	0.931	0.812	0.933	0.933	1	0.96	0.962
			(0.3687)	(0.3669)	(0.3980)	(0.3986)	(10.6754)	(0.3885)	(0.3855)
		100::100	0.952	0.829	0.954	0.952	1	0.957	0.96
			(0.3336)	(0.3320)	(0.3390)	(0.3396)	(9.7897)	(0.3259)	(0.3239)
		100:200	0.889	0.755	0.931	0.937	1	0.949	0.951
			(0.2947)	(0.2925)	(0.3248)	(0.3253)	(9.7177)	(0.3270)	(0.3250)
		200:200	0.946	0.847	0.957	0.96	1	0.952	0.951
			(0.2338)	(0.2351)	(0.2361)	(0.2362)	(9.1347)	(0.2309)	(0.2299)
	0.1:0.3	10:10	0.906	0.869	0.959	0.96	0.999	0.961	0.938
			(2.4326)	(2.5332)	(2.9349)	(2.2742)	(11.7912)	(1.2824)	(1.2278)
		10:40	0.75	0.637	0.935	0.928	0.999	0.958	0.932
			(1.0823)	(1.0860)	(2.2574)	(1.4859)	(12.5057)	(1.2739)	(1.2210)
		10:70	0.737	0.617	0.927	0.924	0.991	0.966	0.944
			(1.0023)	(1.0277)	(2.0667)	(2.1796)	(8.5678)	(1.2496)	(1.1975)
		10:100	0.72	0.589	0.906	0.902	0.999	0.965	0.942
			(1.0153)	(1.0145)	(2.0539)	(1.5965)	(11.3867)	(1.2692)	(1.2165)
		40:40	0.933	0.839	0.945	0.947	1	0.94	0.922
			(0.8984)	(0.8983)	(0.9308)	(0.9330)	(13.6018)	(0.7610)	(0.7470)
		40:70	0.781	0.667	0.928	0.929	1	0.958	0.945
			(0.6094)	(0.6146)	(0.8287)	(0.8298)	(12.4913)	(0.7727)	(0.7585)
		40:100	0.587	0.517	0.927	0.929	1	0.954	0.937
			(0.5437)	(0.5459)	(0.7847)	(0.7861)	(12.7442)	(0.7642)	(0.7504)
		70:70	0.951	0.846	0.945	0.946	1	0.948	0.936
			(0.6572)	(0.6540)	(0.6645)	(0.6645)	(12.8456)	(0.5966)	(0.5893)
		70:100	0.772	0.672	0.923	0.926	1	0.94	0.936
			(0.4957)	(0.4999)	(0.6244)	(0.6250)	(20.5803)	(0.5956)	(0.5882)
		100::100	0.933	0.849	0.943	0.941	1	0.946	0.936
			(0.5494)	(0.5492)	(0.5522)	(0.5523)	(13.0645)	(0.5112)	(0.5062)
		100::200	0.405	0.351	0.92	0.921	1	0.948	0.942
			(0.3596)	(0.3553)	(0.4924)	(0.4921)	(12.8408)	(0.5073)	(0.5025)
		200:200	0.948	0.826	0.953	0.954	1	0.949	0.942
			(0.3768)	(0.3709)	(0.3823)	(0.3818)	(11.7710)	(0.3638)	(0.3616)
	0.3:0.3	10:10	0.917	0.841	0.968	0.965	0.995	0.999	0.995
			(2.0623)	(2.1062)	(5.3097)	(6.3643)	(8.9779)	(1.2329)	(1.1771)
		10:40	0.8	0.683	0.947	0.941	0.999	0.992	0.994
			(1.0287)	(1.0431)	(4.2830)	(1.0791)	(8.2850)	(1.2260)	(1.1711)
		10:70	0.776	0.664	0.947	0.941	0.994	0.996	0.991
			(0.9798)	(0.9664)	(4.5351)	(1.4561)	(7.3056)	(1.2399)	(1.1833)
		10:100	0.781	0.663	0.944	0.938	0.997	0.998	0.995
			(0.9773)	(0.9897)	(4.0039)	(2.3349)	(9.2067)	(1.2417)	(1.1853)
		40:40	0.954	0.854	0.948	0.949	1	0.966	0.971
			(0.8161)	(0.8097)	(1.7857)	(1.7893)	(9.9028)	(0.6965)	(0.6840)
		40:70	0.819	0.721	0.945	0.954	1	0.968	0.97
			(0.5767)	(0.5880)	(1.6737)	(1.6747)	(10.7333)	(0.6896)	(0.6775)
		40:100	0.692	0.593	0.92	0.922	1	0.964	0.964
			(0.5245)	(0.5245)	(1.5997)	(1.5996)	(10.3552)	(0.6953)	(0.6828)
		70:70	0.934	0.826	0.951	0.951	1	0.951	0.954
			(0.5957)	(0.5935)	(1.2894)	(1.2910)	(11.1853)	(0.5372)	(0.5308)
70:100		0.83	0.707	0.949	0.947	1	0.96	0.957
		(0.4663)	(0.4656)	(1.2324)	(1.2335)	(10.4539)	(0.5355)	(0.5293)
100::100		0.939	0.842	0.948	0.948	1	0.951	0.948
		(0.4904)	(0.4907)	(1.0751)	(1.0758)	(9.8490)	(0.4527)	(0.4486)
100::200		0.519	0.47	0.936	0.933	1	0.958	0.955
		(0.3446)	(0.3455)	(0.9926)	(0.9931)	(9.8866)	(0.4496)	(0.4454)
200:200		0.952	0.828	0.943	0.94	1	0.955	0.957
		(0.3431)	(0.3393)	(0.7445)	(0.7453)	(9.6765)	(0.3258)	(0.3238)

Values in bold correspond to the shortest expected lengths among those whose coverage probabilities are equal to or greater than the nominal confidence level.

,

Table 9. CP and EL for 95% confidence intervals for the ratio of two means for zero-inflated two-parameter Rayleigh distribution when location parameter $\left({\mu }_1,{\mu }_2\right)=\left(0.1,0.1\right)$ and scale parameter $\left({\lambda }_1,{\lambda }_2\right)=\left(0.5,1\right)$.

Scale	Prob	Coverage Probability
		Samplesize	PB	BS	GCI	MOVER	AN	MCMC	HPD
0.5:1	0.1:0.1	10:10	0.902	0.849	0.95	0.951	1	0.986	0.973
			(1.6085)	(1.6692)	(1.8641)	(1.8794)	(15.4192)	(1.2385)	(1.1994)
		10:40	0.882	0.774	0.957	0.954	0.999	0.991	0.978
			(1.2161)	(1.2259)	(1.6796)	(1.6862)	(14.5058)	(1.2234)	(1.1850)
		10:70	0.899	0.8	0.927	0.926	1	0.987	0.975
			(1.2063)	(1.2425)	(1.6681)	(1.6745)	(9.9700)	(1.2251)	(1.1864)
		10:100	0.869	0.784	0.945	0.943	0.999	0.992	0.978
			(1.1805)	(1.2033)	(1.6206)	(1.6274)	(13.9863)	(1.2224)	(1.1840)
		40:40	0.948	0.841	0.944	0.939	1	0.971	0.969
			(0.7250)	(0.7375)	(0.7492)	(0.7513)	(15.8592)	(0.6806)	(0.6731)
		40:70	0.917	0.814	0.939	0.943	1	0.96	0.952
			(0.6381)	(0.6325)	(0.7168)	(0.7179)	(16.0949)	(0.6784)	(0.6707)
		40:100	0.901	0.774	0.938	0.942	1	0.966	0.96
			(0.6176)	(0.6085)	(0.7111)	(0.7130)	(16.7339)	(0.6782)	(0.6706)
		70:70	0.94	0.847	0.956	0.957	1	0.958	0.952
			(0.5417)	(0.5424)	(0.5560)	(0.5570)	(11.5407)	(0.5199)	(0.5160)
		70:100	0.915	0.78	0.944	0.944	1	0.96	0.951
			(0.4969)	(0.4864)	(0.5394)	(0.5397)	(15.6492)	(0.5217)	(0.5177)
		100::100	0.95	0.839	0.946	0.947	1	0.961	0.957
			(0.4542)	(0.4608)	(0.4590)	(0.4598)	(15.6280)	(0.4412)	(0.4384)
		100::200	0.891	0.782	0.929	0.932	1	0.954	0.948
			(0.4003)	(0.4112)	(0.4401)	(0.4406)	(15.5066)	(0.4409)	(0.4382)
		200:200	0.946	0.831	0.965	0.966	1	0.955	0.952
			(0.3199)	(0.3213)	(0.3224)	(0.3227)	(15.1855)	(0.3157)	(0.3143)
	0.1:0.3	10:10	0.928	0.861	0.968	0.964	1	0.917	0.872
			(3.1737)	(3.2835)	(4.0275)	(4.2649)	(20.2615)	(1.5550)	(1.4894)
		10:40	0.758	0.642	0.921	0.913	0.999	0.915	0.862
			(1.4608)	(1.4689)	(2.9838)	(3.1197)	(21.7976)	(1.5491)	(1.4843)
		10:70	0.723	0.591	0.938	0.922	0.98	0.91	0.855
			(1.3487)	(1.3488)	(2.7793)	(1.9569)	(8.6714)	(1.5371)	(1.4726)
		10:100	0.682	0.557	0.93	0.924	0.998	0.905	0.858
			(1.3203)	(1.2898)	(2.6547)	(3.0178)	(19.5596)	(1.5338)	(1.4701)
		40:40	0.945	0.835	0.947	0.949	1	0.931	0.903
			(1.2159)	(1.2074)	(1.2579)	(1.2603)	(21.9385)	(1.0053)	(0.9869)
		40:70	0.777	0.662	0.922	0.926	1	0.941	0.924
			(0.8288)	(0.8317)	(1.1345)	(1.1360)	(21.4322)	(1.0251)	(1.0061)
		40:100	0.583	0.488	0.923	0.925	1	0.933	0.913
			(0.7344)	(0.7434)	(1.0792)	(1.0805)	(20.8017)	(1.0109)	(0.9922)
		70:70	0.939	0.834	0.949	0.952	1	0.932	0.911
			(0.8877)	(0.8867)	(0.9089)	(0.9104)	(10.5684)	(0.7968)	(0.7871)
		70:100	0.789	0.663	0.938	0.937	1	0.932	0.913
			(0.6689)	(0.6598)	(0.8432)	(0.8437)	(20.3708)	(0.7967)	(0.7872)
		100::100	0.943	0.839	0.942	0.941	1	0.949	0.94
			(0.7475)	(0.7501)	(0.7463)	(0.7468)	(20.6395)	(0.6889)	(0.6822)
		100::200	0.362	0.324	0.922	0.924	1	0.943	0.934
			(0.4874)	(0.4766)	(0.6602)	(0.6607)	(10.3036)	(0.6805)	(0.6739)
		200:200	0.941	0.821	0.946	0.946	1	0.946	0.942
			(0.5159)	(0.5122)	(0.5163)	(0.5169)	(19.4086)	(0.4978)	(0.4946)
	0.3:0.3	10:10	0.915	0.852	0.975	0.971	0.996	0.996	0.984
			(2.7894)	(2.8138)	(3.6066)	(3.4667)	(14.7114)	(1.5104)	(1.4414)
		10:40	0.804	0.706	0.942	0.938	0.996	0.995	0.989
			(1.3841)	(1.4159)	(2.6966)	(2.7398)	(14.1320)	(1.5072)	(1.4393)
		10:70	0.774	0.659	0.932	0.922	0.99	0.992	0.986
			(1.3176)	(1.3145)	(2.6355)	(1.9825)	(8.0045)	(1.5082)	(1.4400)
		10:100	0.751	0.644	0.934	0.925	0.998	0.994	0.982
			(1.3037)	(1.2707)	(2.6254)	(2.4449)	(15.8105)	(1.5153)	(1.4462)
		40:40	0.919	0.845	0.948	0.947	1	0.953	0.946
			(1.0908)	(1.1033)	(1.1263)	(1.1291)	(16.4584)	(0.9067)	(0.8904)
		40:70	0.785	0.664	0.93	0.93	1	0.961	0.95
			(0.7711)	(0.7715)	(1.0362)	(1.0373)	(15.9752)	(0.9020)	(0.8858)
		40:100	0.678	0.588	0.926	0.93	1	0.977	0.967
			(0.7086)	(0.7100)	(0.9920)	(0.9938)	(16.8030)	(0.9185)	(0.9022)
		70:70	0.942	0.828	0.955	0.955	1	0.957	0.953
			(0.8090)	(0.7951)	(0.8183)	(0.8194)	(9.0969)	(0.7176)	(0.7091)
70:100		0.841	0.705	0.937	0.94	1	0.962	0.953
		(0.6355)	(0.6339)	(0.7835)	(0.7844)	(16.0980)	(0.7229)	(0.7144)
100::100		0.941	0.857	0.957	0.956	1	0.953	0.95
		(0.6726)	(0.6765)	(0.6714)	(0.6715)	(17.2198)	(0.6128)	(0.6072)
100::200		0.557	0.503	0.946	0.945	1	0.958	0.959
		(0.4715)	(0.4701)	(0.6186)	(0.6183)	(15.8078)	(0.6156)	(0.6099)
200:200		0.949	0.844	0.947	0.945	1	0.95	0.952
		(0.4668)	(0.4674)	(0.4685)	(0.4689)	(15.1548)	(0.4413)	(0.4385)

Values in bold correspond to the shortest expected lengths among those whose coverage probabilities are equal to or greater than the nominal confidence level.

and

Table 10. CP and EL for 95% confidence intervals for the ratio of two means for zero-inflated two-parameter Rayleigh distribution when location parameter $\left({\mu }_1,{\mu }_2\right)=\left(0.1,0.1\right)$ and scale parameter $\left({\lambda }_1,{\lambda }_2\right)=\left(1,1\right)$.

Scale	Prob	Coverage Probability
		Samplesize	PB	BS	GCI	MOVER	AN	MCMC	HPD
1:1	0.1:0.1	10:10	0.895	0.824	0.958	0.958	0.999	0.999	0.999
			(1.1517)	(1.1724)	(1.3634)	(1.3731)	(12.9024)	(1.0028)	(0.9708)
		10:40	0.906	0.793	0.961	0.956	1	1	0.999
			(0.8830)	(0.8741)	(1.1836)	(1.1880)	(13.0147)	(1.0081)	(0.9750)
		10:70	0.887	0.766	0.965	0.965	1	1	1
			(0.8706)	(0.8618)	(1.1830)	(1.1876)	(13.7601)	(1.0013)	(0.9693)
		10:100	0.88	0.786	0.948	0.947	1	1	1
			(0.8598)	(0.8562)	(1.1741)	(1.1939)	(14.0276)	(1.0004)	(0.9685)
		40:40	0.939	0.829	0.959	0.959	1	0.984	0.984
			(0.5176)	(0.5144)	(0.5427)	(0.5440)	(13.6094)	(0.5087)	(0.5033)
		40:70	0.926	0.807	0.949	0.948	1	0.975	0.979
			(0.4664)	(0.4584)	(0.5253)	(0.5264)	(13.2948)	(0.5154)	(0.5095)
		40:100	0.917	0.79	0.942	0.943	1	0.984	0.982
			(0.4471)	(0.4494)	(0.5088)	(0.5095)	(13.4411)	(0.5134)	(0.5078)
		70:70	0.934	0.806	0.956	0.954	1	0.961	0.962
			(0.3917)	(0.3754)	(0.3996)	(0.4004)	(12.7013)	(0.3861)	(0.3832)
		70:100	0.931	0.818	0.956	0.958	1	0.975	0.974
			(0.3586)	(0.3575)	(0.3898)	(0.3906)	(12.7383)	(0.3856)	(0.3827)
		100::100	0.942	0.825	0.947	0.948	1	0.957	0.956
			(0.3271)	(0.3214)	(0.3319)	(0.3325)	(12.6701)	(0.3248)	(0.3228)
		100::200	0.861	0.762	0.933	0.935	1	0.965	0.97
			(0.2864)	(0.2878)	(0.3173)	(0.3178)	(12.6259)	(0.3231)	(0.3211)
		200:200	0.948	0.824	0.952	0.949	1	0.958	0.956
			(0.2305)	(0.2307)	(0.2318)	(0.2320)	(12.4838)	(0.2293)	(0.2283)
	0.1:0.3	10:10	0.919	0.835	0.964	0.958	1	1	1
			(2.0215)	(2.0125)	(2.9968)	(3.0930)	(12.9967)	(1.2418)	(1.1852)
		10:40	0.774	0.631	0.934	0.926	1	0.991	0.977
			(1.0617)	(1.0552)	(2.1676)	(2.0984)	(18.0072)	(1.2800)	(1.2246)
		10:70	0.744	0.638	0.927	0.912	1	0.982	0.966
			(1.0040)	(1.0091)	(2.0504)	(1.6644)	(16.9154)	(1.2650)	(1.2112)
		10:100	0.673	0.555	0.917	0.906	0.998	0.981	0.964
			(0.9587)	(0.9535)	(2.1455)	(2.0073)	(17.3471)	(1.2546)	(1.2006)
		40:40	0.937	0.826	0.945	0.944	1	0.957	0.947
			(0.9069)	(0.9183)	(0.9165)	(0.9180)	(18.2027)	(0.7690)	(0.7548)
		40:70	0.764	0.641	0.929	0.926	1	0.966	0.947
			(0.5926)	(0.5841)	(0.8076)	(0.8084)	(17.7230)	(0.7603)	(0.7464)
		40:100	0.564	0.495	0.929	0.929	1	0.971	0.948
			(0.5257)	(0.5223)	(0.7658)	(0.7660)	(17.5106)	(0.7584)	(0.7444)
		70:70	0.951	0.847	0.955	0.957	1	0.969	0.963
			(0.6605)	(0.6570)	(0.6592)	(0.6598)	(16.7957)	(0.6029)	(0.5954)
		70:100	0.784	0.683	0.939	0.94	1	0.951	0.94
			(0.4861)	(0.4896)	(0.6210)	(0.6215)	(17.0425)	(0.5937)	(0.5865)
		100::100	0.946	0.841	0.944	0.944	1	0.951	0.949
			(0.5395)	(0.5430)	(0.5433)	(0.5440)	(16.7848)	(0.5069)	(0.5021)
		100::200	0.373	0.364	0.93	0.929	1	0.947	0.945
			(0.3534)	(0.3547)	(0.4803)	(0.4805)	(10.2865)	(0.5034)	(0.4986)
		200:200	0.943	0.807	0.942	0.949	1	0.947	0.944
			(0.3751)	(0.3670)	(0.3751)	(0.3755)	(16.1312)	(0.3647)	(0.3625)
	0.3:0.3	10:10	0.923	0.866	0.98	0.979	1	1	1
			(2.0376)	(2.1342)	(2.4464)	(2.5956)	(14.0034)	(1.2531)	(1.1947)
		10:40	0.782	0.682	0.944	0.942	0.999	1	0.999
			(0.9988)	(0.9956)	(1.9536)	(2.1447)	(12.2856)	(1.2556)	(1.1974)
		10:70	0.775	0.678	0.941	0.936	0.995	0.999	0.999
			(0.9522)	(0.9504)	(1.8719)	(1.8464)	(14.2481)	(1.2573)	(1.1993)
		10:100	0.767	0.655	0.924	0.921	0.998	1	1
			(0.9263)	(0.9351)	(1.8612)	(1.8936)	(12.2530)	(1.2540)	(1.1957)
		40:40	0.943	0.816	0.947	0.947	1	0.978	0.978
			(0.7979)	(0.7975)	(0.8180)	(0.8193)	(13.6500)	(0.6934)	(0.6810)
		40:70	0.821	0.699	0.932	0.931	1	0.987	0.983
			(0.5697)	(0.5726)	(0.7508)	(0.7517)	(13.7719)	(0.7092)	(0.6963)
		40:100	0.664	0.585	0.927	0.931	1	0.982	0.982
			(0.5138)	(0.5068)	(0.7262)	(0.7262)	(13.6826)	(0.6906)	(0.6784)
		70:70	0.935	0.849	0.956	0.954	1	0.969	0.958
			(0.5862)	(0.5976)	(0.5894)	(0.5905)	(13.5250)	(0.5375)	(0.5310)
70:100		0.836	0.73	0.943	0.948	1	0.97	0.969
		(0.4611)	(0.4643)	(0.5678)	(0.5685)	(13.0638)	(0.5404)	(0.5339)
100::100		0.938	0.827	0.959	0.955	1	0.956	0.957
		(0.4858)	(0.4822)	(0.4883)	(0.4891)	(12.8257)	(0.4526)	(0.4485)
100::200		0.51	0.468	0.923	0.923	1	0.958	0.96
		(0.3390)	(0.3321)	(0.4468)	(0.4477)	(12.8912)	(0.4501)	(0.4460)
200:200		0.952	0.828	0.949	0.95	1	0.955	0.957
		(0.3431)	(0.3393)	(0.3396)	(0.3396)	(9.6765)	(0.3258)	(0.3238)

Values in bold correspond to the shortest expected lengths among those whose coverage probabilities are equal to or greater than the nominal confidence level.

and Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10, respectively, with the figures illustrating only a selected set of representative scenarios for clarity. The investigation of confidence interval estimation for both measures revealed consistent patterns. When the sample size was small, i.e., $\left(n_1,n_2\right)=\left(10,10\right)$, the CP values of GCI, MOVER, NA, MCMC, and HPD were close to or greater than the nominal confidence level of 95%. In terms of expected length, the HPD credible interval produced the shortest EL among all methods. However, in certain parameter settings, for example $\left({\mu }_1,{\mu }_2\right)=\left(0.1,0.1\right)$, $\left({\lambda }_1,{\lambda }_2\right)=\left(0.1,0.1\right)$, and $\left({\delta }_1,{\delta }_2\right)=\left(0.1,0.3\right)$, some methods did not perform well, as their CP values deviated noticeably from the nominal level. This behavior is mainly attributable to model error, which arises when the extremely small sample size amplifies the zero-inflation effect and increases the variability of the estimators. Even under such conditions, the GCI method continued to demonstrate good performance.

For other small sample scenarios with $n_1=10$ (i.e., $\left(n_1,n_2\right)=\left(10,40\right)$, $\left(10,70\right)$, and $\left(10,100\right)$), the results were broadly similar. Based on coverage probability, the most appropriate methods remained GCI, MOVER, AN, MCMC, and HPD, with HPD consistently producing the shortest expected length.

At $\left(n_1,n_2\right)=\left(40,40\right)$, the PB method began to show improved performance, although its CP values were still below the nominal level. For this sample size, the most suitable methods were again GCI, MOVER, AN, MCMC, and HPD, with HPD yielding the shortest EL. Similar behavior was observed for $\left(n_1,n_2\right)=\left(40,70\right)$ and $\left(40,100\right)$.

For larger sample sizes specifically $\left(70,70\right)$, $\left(70,100\right)$, $\left(100,100\right)$, $\left(100,200\right)$, and $\left(200,200\right)$ the CP values of PB, GCI, MOVER, AN, MCMC, and HPD were close to or exceeded the nominal 95% level. With respect to expected length, the HPD credible interval continued to provide the shortest EL among all competing methods.

We present the simulation results for constructing confidence intervals of the difference and the ratio of means in

Table 11. Summary of the best-performing methods across all sample sizes for the difference between means sceanrio.

Parameter		Sample Size
${\lambda }_1$,${\lambda }_2$	${\delta }_1$,${\delta }_2$	10:10	10:40	10:70	10:100	40:40	40:70	40:100	70:70	70:100	100:100	100:200	200:200
(0.1,0.1)	(0.1,0.1)	HPD	GCI	HPD	GCI	GCI	HPD	HPD	HPD	AN	AN	AN	MOVER
	(0.1,0.3)	GCI	AN	AN	AN	GCI	AN	AN	GCI	AN	GCI	AN	HPD
	(0.3,0.3)	MOVER	HPD	HPD	HPD	AN	AN	HPD	GCI	AN	AN	AN	GCI
(0.1,0.5)	(0.1,0.1)	GCI	GCI	GCI	GCI	GCI	GCI	HPD	HPD	AN	AN	AN	MOVER
	(0.1,0.3)	GCI	GCI	AN	AN	MOVER	AN	AN	HPD	GCI	GCI	HPD	HPD
	(0.3,0.3)	HPD	HPD	HPD	HPD	HPD	HPD	HPD	AN	HPD	HPD	AN	HPD
(0.5,0.5)	(0.1,0.1)	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	AN	HPD
	(0.1,0.3)	HPD	HPD	HPD	HPD	AN	HPD	HPD	HPD	AN	AN	AN	MCMC
	(0.3,0.3)	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	MCMC	HPD	HPD
(0.5,1)	(0.1,0.1)	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD
	(0.1,0.3)	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	AN	AN	HPD	HPD
	(0.3,0.3)	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD
(1,1)	(0.1,0.1)	GCI	GCI	GCI	HPD	GCI	HPD	HPD	HPD	HPD	HPD	HPD	HPD
	(0.1,0.3)	GCI	HPD	HPD	HPD	HPD	HPD	HPD	GCI	HPD	HPD	HPD	HPD
	(0.3,0.3)	MOVER	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	MOVER

and

Table 12. Summary of the best-performing methods across all sample sizes for the ratio means sceanrio.

Parameter		Sample Size
${\lambda }_1$,${\lambda }_2$	${\delta }_1$,${\delta }_2$	10:10	10:40	10:70	10:100	40:40	40:70	40:100	70:70	70:100	100:100	100:200	200:200
(0.1,0.1)	(0.1,0.1)	GCI	GCI	GCI	GCI	GCI	HPD	AN	HPD	AN	AN	AN	MOVER
	(0.1,0.3)	GCI	AN	AN	AN	GCI	AN	AN	MOVER	AN	AN	AN	GCI
	(0.3,0.3)	MOVER	HPD	HPD	HPD	HPD	AN	HPD	GCI	AN	AN	AN	HPD
(0.1,0.5)	(0.1,0.1)	GCI	GCI	MCMC	GCI	GCI	MCMC	MCMC	HPD	AN	PB	AN	MOVER
	(0.1,0.3)	GCI	AN	AN	-	GCI	AN	AN	PB	AN	GCI	AN	MCMC
	(0.3,0.3)	MCMC	AN	AN	AN	MCMC	MOVER	MCMC	GCI	MCMC	MCMC	AN	HPD
(0.5,0.5)	(0.1,0.1)	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD
	(0.1,0.3)	MCMC	MCMC	MCMC	MCMC	AN	MCMC	MCMC	AN	AN	AN	AN	GCI
	(0.3,0.3)	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	MCMC	HPD	HPD
(0.5,1)	(0.1,0.1)	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	MCMC	HPD
	(0.1,0.3)	GCI	AN	AN	AN	AN	AN	AN	MOVER	AN	AN	AN	AN
	(0.3,0.3)	HPD	HPD	HPD	HPD	MCMC	HPD	HPD	HPD	HPD	HPD	HPD	HPD
(1,1)	(0.1,0.1)	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD
	(0.1,0.3)	HPD	HPD	HPD	HPD	MCMC	MCMC	MCMC	HPD	MCMC	MCMC	AN	AN
	(0.3,0.3)	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD	HPD

, respectively. These tables identify the best-performing methods based on two criteria: (i) coverage probability (CP) exceeding the nominal level, and (ii) the shortest expected length (EL). After considering both criteria, we found that for the difference of means, the HPD method delivers the best overall performance because it consistently attains the nominal 0.95 coverage probability while producing the shortest interval length. The second-best method is GCI, followed by MOVER. When none of these methods achieve adequate coverage, the AN method becomes the next option; however, its considerably wider interval makes it less desirable in practice. Methods PB and BS are not recommended, as the simulation results indicate that they never outperform the other methods in terms of both CP and EL in any scenario. For the ratio of means, the HPD method also demonstrates strong performance across a wide range of scenarios and consistently provides shorter expected lengths than the other approaches. The next best-performing methods are GCI, MCMC, and AN, respectively. Similar to the results for the difference between means, the AN method yields confidence intervals that are substantially wider than those produced by the other methods. Although its coverage probability is acceptable, the excessive interval width makes the AN method less desirable and therefore not recommended.

In summary, the simulation study showed that the Bayesian HPD credible interval is suitable for constructing confidence intervals for both the difference and the ratio of means for the zero-inflated two-parameter Rayleigh distribution, as the coverage probabilities remained consistently close to the nominal level in nearly all scenarios.

4. Application

The Songkran festival in Thailand is a period associated with increased travel and high-risk road behavior, leading the government to designate it as the “Seven Dangerous Days.” Comparing the average number of road traffic fatalities between two periods is therefore crucial for assessing yearly changes in risk and determining whether safety conditions have improved or deteriorated. Such comparisons also provide practical insight into the effectiveness of implemented road-safety interventions. In this study, we analyzed road traffic fatality data during the Songkran festival in southern Thailand for 2024 and 2025. The estimated difference and ratio means between the two years were interpreted to reflect changes in the actual fatality burden, allowing us to quantify both the magnitude and direction of risk variation across years. This interpretation helps contextualize the statistical findings in terms of public safety implications. The data were obtained from the Thailand Road Safety Collaboration (TRSC) [25]. The dataset consists of 14 observations for each year. The recorded values for 2024 are 3, 3, 6, 7, 4, 2, 1, 6, 2, 3, 2, 7, 0, and 6, while the corresponding observations for 2025 are 2, 3, 6, 7, 3, 2, 4, 8, 0, 2, 0, 4, 0, and 6. Observations with zero counts were modeled using a binomial distribution, whereas positive counts were analyzed using the distribution selected based on the lowest Akaike information criterion (AIC) and Bayesian information criterion (BIC), as reported in

Table 13. AIC and BIC values for road traffic fatalities during the Seven Dangerous Days of the Songkran festival in Thailand (2024).

Distribution	Two-Parameter Rayleigh	Normal	Weibull	Lognormal	Exponential	Gamma
AIC	57.80345	59.40485	57.96072	58.94112	63.04365	58.22936
BIC	50.13734	60.53475	59.09061	60.07102	64.6086	59.35926

Values in bold indicate the distributions with the lowest AIC and BIC.

and

Table 14. AIC and BIC values for road traffic fatalities during the Seven Dangerous Days of the Songkran festival in Thailand (2025).

Distribution	Two-Parameter Rayleigh	Normal	Weibull	Lognormal	Exponential	Gamma
AIC	49.00744	50.99825	49.63039	49.0718	54.94955	49.23649
BIC	49.80323	51.79404	50.42618	49.8676	56.34745	50.03228

Values in bold indicate the distributions with the lowest AIC and BIC.

. A summary of the parameter estimates for the dataset is provided, along with the resulting 95% confidence intervals for the difference and ratio of the two means, presented in

Table 15. Confidence intervals of difference means by Various Methods under the Zero-inflated Two-parameter Rayleigh Distribution Using Road Traffic Accident Data in Southern Thailand during the Songkran Festival in 2024 and 2025.

Method	Lower	Upper	Length
PB	−1.42825	2.07826	3.506514
BS	0.269991	2.38972	2.119724
GCI	−1.697773	2.19267	3.890439
MOVER	−1.644535	2.23543	3.879967
AN	−2.816778	3.45609	6.272865
Bayes	−1.473315	1.61163	3.08494
HPD	−1.480229	1.59935	3.079581

and

Table 16. Confidence intervals of ratio means by Various Methods under the Zero-inflated Two-parameter Rayleigh Distribution Using Road Traffic Accident Data in Southern Thailand during the Songkran Festival in 2024 and 2025.

Method	Lower	Upper	Length
PB	0.684892	1.92286	1.237972
BS	1.248869	1.80943	0.560564
GCI	0.621492	1.96538	1.343889
MOVER	0.637013	1.96949	1.332481
AN	0.158052	2.02809	1.870038
Bayes	0.664939	1.59267	0.92773
HPD	0.620266	1.51781	0.89754

.

The results indicate that the BS method produces the shortest expected length; however, simulation findings show that its coverage probability falls below the nominal level. For small sample sizes, methods that yield coverage probabilities close to or above the 0.95 nominal confidence level—namely GCI, MOVER, NA, MCMC, and HPD—perform more reliably. Based on the simulation evidence, the HPD method is identified as the most appropriate approach for constructing confidence intervals for both the difference and ratio of means under the zero-inflated two-parameter Rayleigh distribution.

5. Discussion

Kijsason et al. [26] proposed credible and confidence intervals for the single mean of the zero-inflated two-parameter Rayleigh distribution and applied the method to total COVID-19 deaths in Singapore. Building on this concept, we further developed estimation procedures for constructing confidence intervals for the difference and ratio of means under the zero-inflated two-parameter Rayleigh distribution.

In this study, the HPD method outperformed the other approaches namely PB, BS, GCI, MOVER, NA, and MCMC in constructing confidence intervals for both the difference and ratio of means. However, the HPD method still exhibited limitations in certain scenarios, where the resulting confidence intervals did not achieve the desired confidence level. Alternative methods such as GCI, MOVER, NA, and MCMC also demonstrated satisfactory performance across various sample sizes, although they generally produced wider confidence intervals compared to the HPD method.

6. Conclusions

This paper develops confidence intervals for the difference and ratio of means under the zero-inflated two-parameter Rayleigh distribution using seven estimation methods. The intervals are constructed based on the percentile bootstrap, bootstrap with standard error, generalized confidence interval (GCI), method of variance estimates recovery (MOVER), normal approximation via the delta method, Bayesian Markov Chain Monte Carlo (MCMC), and Bayesian highest posterior density (HPD). Their performance is assessed through coverage probabilities and expected interval lengths. For both parameters of interest, the simulation study indicates that the GCI, normal approximation, Bayesian MCMC, and Bayesian HPD methods perform well across all sample sizes, while the percentile bootstrap improves as the sample size increases. Among all approaches, the Bayesian HPD interval provides the most reliable results, consistently maintaining coverage probabilities close to the nominal 0.95 level and producing comparatively shorter intervals. The Bayesian MCMC and GCI methods rank second and third, respectively.

The performance of the proposed methods was evaluated using actual road traffic fatality data during the Songkran festival in southern Thailand for the years 2024 and 2025. Both the simulation and empirical findings indicate that Bayesian HPD credible interval is the most suitable method for constructing confidence intervals for the difference and ratio of means under the zero-inflated two-parameter Rayleigh distribution, as it consistently achieves coverage probabilities at or near the nominal 0.95 level.