Reliability Estimation in Stress Strength for Generalized Rayleigh Distribution Using a Lower Record Ranked Set Sampling Scheme

Abstract

This paper explores the likelihood and Bayesian estimation of the stress–strength reliability parameter ($\mathcal{R}$) based on a lower record ranked set sampling scheme from the generalized Rayleigh distribution. Maximum likelihood and Bayesian estimators as well as confidence intervals of $\mathcal{R}$ are derived and their properties are studied. Furthermore, two parametric bootstrap confidence intervals are introduced in the paper. A comparative simulation study is conducted to assess the effectiveness of these four confidence interval methodologies in estimating $\mathcal{R}$. The application of the methods is demonstrated using real data on fiber strength to showcase their practicability and relevance in the industry.

1. Introduction

Analysis of stress strength is a critical problem in the statistical literature. The parameter $\mathcal{R}$, is defined as $Pr (X<Y)$ and represents the probability occurring when the stress X surpasses the strength Y. In practical engineering, stress-strength modeling is widely used in material reliability analysis and design, and can also be used in product life testing. For example, in the aerospace field, life tests are required for components of aircraft to ensure the safety of the aircraft. When the product is used from the beginning to the end of the period, it will be affected by a variety of external stresses (such as pressure, wind, humidity, temperature, etc.). In addition to the influence of these external stresses, the product itself has a certain degree of strength to resist these external stresses. Currently, the stress-strength model is extensively applied across various domains, particularly in addressing issues related to reliability assessment.

Originally introduced by Birnbaum [1], the stress-strength model has since garnered significant interest from researchers in the field of reliability statistics. Bi et al. [2] studied the $\mathcal{R}$ estimation problem of the inverse Weibull distribution of the stress intensity. Hamad et al. [3] utilized maximum likelihood estimation and the least squares method to estimate $\mathcal{R}$ in Lomax distributed models. Beyond single-component models, multi-component stress-strength frameworks have increasingly attracted scholarly interest. Dey et al. [4] addressed the estimation of $\mathcal{R}$ in multi-component models with the Kumaraswamy distribution, applying both maximum likelihood and Bayesian estimation to derive asymptotic confidence and credible intervals. Additionally, Rasekhi et al. [5] explored both classical and Bayesian methods to estimate $\mathcal{R}$ in multi-component models under generalized logistic distribution.

Record-value data are already sampled in a random sampling process according to an established size order scheme, unlike order statistics. We refer to a sequence of records with an increasing trend as the upper records, and similarly a sequence of records with a decreasing trend as the lower records. McIntyre [6] first introduced ranked set sampling (RSS) when the variables under consideration are difficult to measure directly but easy to rank. This sampling method greatly increases the amount of information in the sample, making statistical analysis of the population more efficient. Takahasi and Wakimoto [7] demonstrated that the sample mean under RSS is more suitable than simple random sampling (SRS) to be used as an estimator when estimating the population mean. Dell [8] proved that RSS in the case of a wrong sort order does not affect the fact that it is more effective than simple random sampling for aggregate inference. Mahdizadeh [9] studied a goodness-of-fit test for the Rayleigh distribution using RSS and found that RSS-based tests were preferred through extensive Monte Carlo simulations.

Salehi and Ahmadi [10] proposed a novel sampling scheme termed record ranked set sampling (RRSS), an extension of traditional ranked set sampling. The final record from each sequence, denoted as $R_{i,i}$, is the only data item used for subsequent analysis. Therefore, the dataset $\mathbf{R}={\left(R_{1,1}, R_{2,2}, \dots , R_{m,m}\right)}^{\mathrm{T}}$ is a record ranked set sample of size m. The process can be visualized through the following schematic representation:

$$\begin{array}{ccc}1:R_{\left(1\right)1}\hfill & \hfill & \to R_{1,1}=R_{\left(1\right)1}\hfill \\ 2:R_{\left(1\right)2}{R}_{\left(2\right)2}\hfill & \hfill & \to R_{2,2}=R_{\left(2\right)2}\hfill \\ ⋮ ⋮ ⋮\ddots \hfill & ⋮\hfill & ⋮\hfill \\ n:R_{\left(1\right)m}{R}_{\left(2\right)m}\hfill & \cdots R_{\left(m\right)m}\hfill & \to R_{m,m}=R_{\left(m\right)m}\hfill \end{array}$$

(1)

Given that $F(.;\theta )$ represents the population’s cumulative distribution function ($cdf$) and $f(.;\theta )$ its probability density function ($pdf$), the joint density of $\mathbf{R}$ for lower RRSS of size m can be derived using the marginal density of ordinary records [11]:

$$f_{\mathbf{R}}\left(\mathbf{r}\right)=\prod _{i=1}^m\frac{{\left\{-logF\left(r_{i,i}\right)\right\}}^{i-1}}{(i-1)!}f\left(r_{i,i}\right),$$

(2)

where the vector $\mathbf{r}={\left(r_{1,1}, r_{2,2}, \dots , r_{m,m}\right)}^{\mathrm{T}}$ represents the observed outcomes for $\mathcal{R}$.

Salehi and Ahmadi [12] discussed the point and interval estimation of $\mathcal{R}$ for the one-parameter exponential distribution under RRSS. Srisodaphol et al. [13] discussed and tested, based on the median ranked set and RRSS samples, the interval estimates of $\mathcal{R}$ for single-parameter exponential distributions. Tahmasebi et al. [14] considered uncertain information for RRSS from an information-theoretic point of view. In practice, judgment ranking errors may occur due to subjectivity, inconsistent ranking criteria, weak correlation between variables, observation errors, intra-set homogeneity, and learning effects. Subjectivity arises due to visual inspection or sensory evaluation, leading to variations in criteria. Inconsistent ranking criteria may result from varying standards applied by observers. Weak correlations between variables can result in ranking inaccuracies, while observation errors exacerbate errors. Intra-set homogeneity makes accurate ranking challenging, especially if the auxiliary variable is weakly correlated. Lack of initial ranking experience can lead to errors, but accuracy improves over time.

The two-parameter generalized Rayleigh distribution with shape $\alpha >0$ and scale $\lambda >0$ parameters, say GR($\alpha$, $\lambda$), has the following cdf and pdf, respectively.

$$F(x;\alpha ,\lambda )={\left(1-e^{-\lambda x^2}\right)}^{\alpha }, x>0$$

(3)

$$f(x;\alpha ,\lambda )=2\alpha \lambda x{e}^{-\lambda x^2}{\left(1-e^{-\lambda x^2}\right)}^{\alpha -1}, x>0; \alpha , \lambda >0$$

(4)

The pdf and the hazard function of the GR distribution exhibit varying forms based on adjustments to the parameters $\alpha$ and $\lambda$, making it suitable for modeling diverse biased datasets. Specifically, for $\alpha \le 0.5$, the pdf is a decreasing function and the hazard function resembles a bathtub curve. Conversely, for $\alpha >0.5$, the pdf appears as a right-skewed, single-peaked curve with an increasing hazard function (refer to Figure 1). The GR distribution serves as an alternative to the Gamma or Weibull distributions, sharing many of their beneficial characteristics. However, it uniquely features a non-monotonic hazard function, advantageous in various practical settings. Notably, at $\alpha =1$, the GR distribution simplifies to the well-known Rayleigh distribution, a specific instance of the Weibull distribution, widely applied in communication engineering and electric vacuum technologies. Esemena and Gürlers [15] assessed the relative efficiencies of maximum likelihood estimators in RSS for this distribution. Additionally, Kotb and Raqab [16] explored Bayesian estimation of model parameters and prediction of unobserved data from the Rayleigh distribution using RSS. Furthermore, Ren et al. [17] examined entropy estimation for a GR distribution within progressively type-II censored samples. Rao [18] estimated $\mathcal{R}$ in a multi-component system using the maximum likelihood method of estimation in samples drawn from a generalized Rayleigh distribution. Ahmad [19] utilized maximum likelihood estimation and Bayesian approaches to examine two-parameter discrete models introduced as novel counterparts to the Rayleigh–Lindley mixture. Aljohani [20] demonstrated the supremacy of the proposed model parameter estimations of the unit generalized Rayleigh model under the RRS design.

It can be seen that the problem of $\mathcal{R}$ statistical inference has attracted a lot of attention. Salehi and Ahmadi [21] presented an interesting application of RRSS in the content of the reliability, considering the estimation of $\mathcal{R}$ using upper RRSS from the exponential distribution. In our study, we explore the use of the lower RRSS for estimating the stress–strength reliability instead of the upper one. In addition, we seek to know whether the results of Salehi and Ahmadi [10] are applicable to a more general distribution such as GR. Currently, there is a gap in the research regarding statistical inference of $\mathcal{R}$ under a GR distribution in more complex sampling schemes like RRSS. Therefore, this paper focuses on the statistical inference of $\mathcal{R}$ based on the lower RRSS sampling scheme. The paper is organized as follows: In Section 2, we obtain the maximum likelihood and Bayesian estimators for $\mathcal{R}$, and also monitor its performance. Section 3 presents the maximum likelihood confidence interval, Bayesian confidence interval, and two different parametric bootstrap confidence intervals (basic and percentile bootstrap) for $\mathcal{R}$. A simulation study described in Section 4 assesses and compares these confidence intervals. Section 5 analyses the real dataset of fiber strength. We conclude in Section 6.

2. Point Estimation

If $X\sim GR\left({\alpha }_1, {\lambda }_1\right)$ and $Y\sim GR\left({\alpha }_2, {\lambda }_2\right)$, the derivation of $\mathcal{R}$, which represents the probability $Pr(X<Y)$, is presented as (detail in the Appendix A):

$$\begin{array}{c}\hfill \mathcal{R}=1-{\int }_0^1{\alpha }_1{(1-t)}^{{\alpha }_1-1}{\left(1-t^{\frac{{\lambda }_2}{{\lambda }_1}}\right)}^{{\alpha }_2}\mathrm{d}t\end{array}$$

(5)

Consider the observation vectors $\mathbf{r}={(r_{1,1}, r_{2,2}, \dots , r_{m,m})}^{\top }$ and $\mathbf{s}={(s_{1,1}, s_{2,2}, \dots , s_{n,n})}^{\top }$, which are observed from the random vectors $\mathbf{R}={(R_{1,1}, R_{2,2}, \dots , R_{m,m})}^{\top }$ and $\mathbf{S}={(S_{1,1}, S_{2,2}, \dots , S_{n,n})}^{\top }$, respectively. $\mathbf{R}$ and $\mathbf{S}$ represent lower RRSS of size m and n from $GR\left({\alpha }_1, {\lambda }_1\right)$ and $GR\left({\alpha }_2, {\lambda }_2\right)$, respectively, and are assumed to be independent.

2.1. Maximum Likelihood Estimation

To estimate $\mathcal{R}$ using the maximum likelihood approach, we examine the following three scenarios:

$Scenario 1: {\alpha }_1\ne {\alpha }_2 \& {\lambda }_1\ne {\lambda }_2$

Integrating (3) and (4) into (2) yields the likelihood function as below:

$$\begin{array}{cc}\hfill L\left({\alpha }_1, {\alpha }_2, {\lambda }_1, {\lambda }_2\right)=& \prod _{i=1}^m\left\{\frac{{\left\{-log{\left(1-{\mathrm{e}}^{-{\lambda }_1{r}_{i,i}^2}\right)}^{{\alpha }_1}\right\}}^{i-1}2{\alpha }_1{\lambda }_1{r}_{i,i}{\mathrm{e}}^{-{\lambda }_1{r}_{i,i}^2}{\left(1-{\mathrm{e}}^{-{\lambda }_1{r}_{i,i}^2}\right)}^{{\alpha }_1-1}}{(i-1)!}\right\}\hfill \\ \hfill & \times \prod _{j=1}^n\left\{\frac{{\left\{-log{\left(1-{\mathrm{e}}^{-{\lambda }_2{s}_{j,j}^2}\right)}^{{\alpha }_2}\right\}}^{j-1}2{\alpha }_2{\lambda }_2{s}_{j,j}{\mathrm{e}}^{-{\lambda }_2{s}_{j,j}^2}{\left(1-{\mathrm{e}}^{-{\lambda }_2{s}_{j,j}^2}\right)}^{{\alpha }_2-1}}{(j-1)!}\right\}\hfill \end{array}$$

(6)

Consequently, the normal equations can be formulated as below:

$$\frac{\partial \ell \left({\alpha }_1, {\alpha }_2, {\lambda }_1, {\lambda }_2;\mathbf{r}, \mathbf{s}\right)}{\partial {\alpha }_1}=\sum _{i=1}^m\frac{(i-1)}{{\alpha }_1}+\frac{m}{{\alpha }_1}+\sum _{i=1}^{m}log\left(1-{\mathrm{e}}^{-{\lambda }_1{r}_{i,i}^2}\right)=0$$

(7)

$$\frac{\partial \ell \left({\alpha }_1, {\alpha }_2, {\lambda }_1, {\lambda }_2;\mathbf{r}, \mathbf{s}\right)}{\partial {\alpha }_2}=\sum _{j=1}^n\frac{(j-1)}{{\alpha }_2}+\frac{n}{{\alpha }_2}+\sum _{j=1}^{n}log\left(1-{\mathrm{e}}^{-{\lambda }_2{s}_{j,j}^2}\right)=0$$

(8)

$$\begin{array}{cc}\hfill \frac{\partial \ell \left({\alpha }_1, {\alpha }_2, {\lambda }_1, {\lambda }_2;\mathbf{r}, \mathbf{s}\right)}{\partial {\lambda }_1}=& \sum _{i=1}^m\frac{(i-1)r_{i,i}^2{\mathrm{e}}^{-{\lambda }_1{r}_{i,i}^2}}{\left(1-{\mathrm{e}}^{-{\lambda }_1{r}_{i,i}}\right)log\left(1-{\mathrm{e}}^{-{\lambda }_1{r}_{i,i}^2}\right)}-\sum _{i=1}^m{r}_{i,i}^2\hfill \\ \hfill & +\frac{m}{{\lambda }_1}+\sum _{i=1}^m\frac{\left({\alpha }_1-1\right)r_{i,i}^2{\mathrm{e}}^{-{\lambda }_1{r}_{i,i}^2}}{1-{\mathrm{e}}^{-{\lambda }_1{r}_{i,i}^2}}=0\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \frac{\partial \ell \left({\alpha }_1, {\alpha }_2, {\lambda }_1, {\lambda }_2;\mathbf{r}, \mathbf{s}\right)}{\partial {\lambda }_2}=& \sum _{j=1}^n\frac{(j-1)s_{j,j}^2{\mathrm{e}}^{-{\lambda }_2{s}_{j,j}^2}}{\left(1-{\mathrm{e}}^{-{\lambda }_2{s}_{j,j}}\right)log\left(1-{\mathrm{e}}^{-{\lambda }_2{s}_{j,j}^2}\right)}-\sum _{j=1}^n{s}_{j,j}^2\hfill \\ \hfill & +\frac{m}{{\lambda }_2}+\sum _{j=1}^n\frac{\left({\alpha }_2-1\right)s_{j,j}^2{\mathrm{e}}^{-{\lambda }_2{s}_{j,j}^2}}{1-{\mathrm{e}}^{-{\lambda }_2{s}_{j,j}^2}}=0\hfill \end{array}$$

(10)

Here, $\ell =logL$. Define M as $m(m+1)/2$ and N as $n(n+1)/2$. By resolving Equations (7) and (8), the solution is obtained as follows:

$${\widehat{\alpha }}_1\left({\lambda }_1\right)=\frac{-M}{{\sum }_{i=1}^{m}log\left(1-{\mathrm{e}}^{-{\lambda }_1{r}_{i,i}^2}\right)}$$

(11)

$${\widehat{\alpha }}_2\left({\lambda }_2\right)=\frac{-N}{{\sum }_{j=1}^{n}log\left(1-{\mathrm{e}}^{-{\lambda }_2{s}_{j,j}^2}\right)}$$

(12)

Substituting ${\widehat{\alpha }}_1\left({\lambda }_1\right)$ and ${\widehat{\alpha }}_2\left({\lambda }_2\right)$ into (9) and (10) and solving the resultant non-linear equations using a numerical method yields the maximum likelihood estimators of the parameters. Leveraging the invariance property of maximum likelihood estimation and employing Equation (5), we derive the maximum likelihood estimation for $\mathcal{R}$, denoted as ${\widehat{\mathcal{R}}}_{ML}$.

$Scenario 2: {\alpha }_1={\alpha }_2=\alpha \& {\lambda }_1\ne {\lambda }_2$

Following algebraic manipulations, the maximum likelihood estimators for the parameters can be determined by resolving the system described below:

$$\sum _{i=1}^m\frac{(i-1)r_{i,i}^2{\mathrm{e}}^{-{\lambda }_1{r}_{i,i}^2}}{\left(1-{\mathrm{e}}^{-{\lambda }_1{r}_{i,i}^2}\right)log\left(1-{\mathrm{e}}^{-{\lambda }_1{r}_{i,i}^2}\right)}-\sum _{i=1}^m{r}_{i,i}^2+\frac{m}{{\lambda }_1}+\sum _{i=1}^m\frac{\left(\widehat{\alpha }\left({\lambda }_1,{\lambda }_2\right)-1\right)r_{i,i}^2{\mathrm{e}}^{-{\lambda }_1{r}_{i,i}^2}}{1-{\mathrm{e}}^{-{\lambda }_1{r}_{i,i}^2}}=0$$

(13)

$$\sum _{j=1}^n\frac{(j-1)s_{j,j}^2{\mathrm{e}}^{-{\lambda }_2{s}_{j,j}^2}}{\left(1-{\mathrm{e}}^{-{\lambda }_2{s}_{j,j}^2}\right)log\left(1-{\mathrm{e}}^{-{\lambda }_2{s}_{j,j}^2}\right)}-\sum _{j=1}^n{s}_{j,j}^2+\frac{n}{{\lambda }_2}+\sum _{j=1}^n\frac{\left(\widehat{\alpha }\left({\lambda }_1,{\lambda }_2\right)-1\right)s_{j,j}^2{\mathrm{e}}^{-{\lambda }_2{s}_{j,j}^2}}{1-{\mathrm{e}}^{-{\lambda }_2{s}_{j,j}^2}}=0,$$

(14)

where

$$\widehat{\alpha }\left({\lambda }_1, {\lambda }_2\right)=\frac{-M-N}{{\sum }_{i=1}^{m}log\left(1-{\mathrm{e}}^{-{\lambda }_1{r}_{i,i}^2}\right)+{\sum }_{j=1}^{n}log\left(1-{\mathrm{e}}^{-{\lambda }_2{s}_{j,j}^2}\right)}$$

(15)

$Scenario 3: {\alpha }_1\ne {\alpha }_2 \& {\lambda }_1={\lambda }_2=\lambda$

In this scenario, it is not necessary to estimate $\lambda$, as (5) simplifies to

$$\mathcal{R}=\frac{{\alpha }_2}{{\alpha }_1+{\alpha }_2},$$

(16)

which does not depend on $\lambda$. We can assume, without loss of generality, that $\lambda =1$ in this scenario. Consequently, the maximum likelihood estimators for ${\alpha }_1$ and ${\alpha }_2$ are computed as follows:

$${\widehat{\alpha }}_1\left({\lambda }_1\right)=\frac{-M}{{\sum }_{i=1}^{m}log\left(1-{\mathrm{e}}^{-r_{i,i}^2}\right)}$$

(17)

$${\widehat{\alpha }}_2\left({\lambda }_2\right)=\frac{-N}{{\sum }_{j=1}^{n}log\left(1-{\mathrm{e}}^{-s_{j,j}^2}\right)}$$

(18)

As a result, ${\widehat{\mathcal{R}}}_{ML}$ is derived as follows:

$${\widehat{\mathcal{R}}}_{ML}=\frac{{\widehat{\alpha }}_2}{{\widehat{\alpha }}_1+{\widehat{\alpha }}_2}={\left[1+\frac{M}{N}\frac{T_2}{T_1}\right]}^{-1},$$

(19)

where

$$T_1=-\sum _{i=1}^{m}log\left(1-{\mathrm{e}}^{-R_{i,i}^2}\right)\sim \Gamma \left(M,{\alpha }_1\right)$$

(20)

and

$$T_2=-\sum _{j=1}^{n}log\left(1-{\mathrm{e}}^{-S_{j,j}^2}\right)\sim \Gamma \left(N,{\alpha }_2\right)$$

(21)

The pdf of ${\widehat{\mathcal{R}}}_{ML}$, as shown in (19), can be derived from (20) and (21). This is instrumental for calculating the moments of ${\widehat{\mathcal{R}}}_{ML}$.

Theorem 1.

The expectation of ${\widehat{\mathcal{R}}}_{ML}$ is calculated using a hyper-geometric function, as detailed below:

$$\begin{array}{cc}\hfill E\left({\widehat{\mathcal{R}}}_{ML}\right)=& M^{M+1}\Gamma (M+N){\left(\frac{1-\mathcal{R}}{N\mathcal{R}}\right)}^M\hfill \\ \hfill & \times {}_{2}F_1\left(1+M,M+N,1+M+N,1-\frac{M}{N}\frac{1-\mathcal{R}}{\mathcal{R}}\right),\hfill \end{array}$$

(22)

where ${}_{2}F_1$ is the hyper-geometric function [22].

Proof of Theorem 1.

Since (20) and (21), we obtain

$$2{\alpha }_1{T}_1\sim {\chi }_{2M}^2, 2{\alpha }_2{T}_2\sim {\chi }_{2N}^2$$

(23)

$$\frac{M{\alpha }_2{T}_2}{N{\alpha }_1{T}_1}\sim F(2N, 2M)$$

(24)

Utilizing simple transformation techniques and referring to (19), the pdf of ${\widehat{\mathcal{R}}}_{ML}$ is derived as follows:

$$f_{{\widehat{R}}_{ML}}\left(r\right)=\frac{1}{r^{2}B(N,M)}{\left(\frac{R}{1-R}\frac{N}{M}\right)}^N{\left(\frac{1-r}{r}\right)}^{N-1}{\left[1+\frac{R}{1-R}\frac{N}{M}\left(\frac{1}{r}-1\right)\right]}^{-M-N},$$

(25)

where $B(.,.)$ is a beta function.    □

According to Theorem 1, we plot the following images: Theorem 1 demonstrates that ${\widehat{\mathcal{R}}}_{ML}$ is the biased estimator of $\mathcal{R}$. The mean squared error (MSE) can be calculated as $MSE\left({\widehat{\mathcal{R}}}_{ML},\mathcal{R}\right)=Var\left({\widehat{\mathcal{R}}}_{ML}\right)+{\left(E\left({\widehat{\mathcal{R}}}_{ML}\right)-\mathcal{R}\right)}^2$. Figure 2 depicts the bias and MSE values of ${\widehat{\mathcal{R}}}_{ML}$ versus $\mathcal{R}$ for various combinations of M and N.

From the figure, we can see that

• The bias tends to increase as $\mathcal{R}$ approaches 0 or 1, and becomes minimal when $\mathcal{R}$ is around $0.5$.
• MSE peaks around $\mathcal{R}=0.5$ and diminishes towards the extremes of $\mathcal{R}$.

We simulated the maximum likelihood estimates of $\mathcal{R}$ under SRS and visualized the bias pattern. Upon comparison with the bias plot of RRSS, it was noted that the bias of the SRS distribution is more uniformly spread. Furthermore, RRSS demonstrated enhanced performance as $\mathcal{R}$ approaches 0.5, indicating its superior efficacy within this range.

2.2. Bayesian Estimation

This section will infer the reliability properties for unknown parameters from the Bayesian perspective. We consider the parameters ${\alpha }_1,{\alpha }_2,{\lambda }_1$, and ${\lambda }_2$ to be independent. Consequently, $\mathcal{R}$ is also treated as a random variable. Accordingly, we will present the Bayesian estimators for the same three distinct scenarios:

$Scenario 1: {\alpha }_1\ne {\alpha }_2 \& {\lambda }_1\ne {\lambda }_2$

A logical selection for the prior distributions of ${\alpha }_1, {\alpha }_2, {\lambda }_1$, and ${\lambda }_2$ would be

$${\alpha }_i\sim \Gamma \left(a_i,b_i\right), {\lambda }_i\sim \Gamma \left(c_i,d_i\right),$$

(26)

where $i=1, 2$ and $a_i, b_i, c_i, d_i(>0)$ are hyper-parameters.

Consider $x_i=1-{\mathrm{e}}^{{\lambda }_1{r}_{i,i}^2}$, $z_{{\lambda }_1}\left(\mathbf{r}\right)=-{\sum }_{i=1}^{m}log\left(1-{\mathrm{e}}^{-{\lambda }_1{r}_{i,i}^2}\right)$ and $y_j=1-{\mathrm{e}}^{{\lambda }_2{s}_{j,j}^2}$, $z_{{\lambda }_2}\left(\mathbf{s}\right)=-{\sum }_{j=1}^{n}log\left(1-{\mathrm{e}}^{-{\lambda }_2{s}_{j,j}^2}\right)$. Hence, combining the likelihood function (6) with the prior density (26) results in the joint posterior density of the parameters $({\alpha }_1,{\alpha }_2,{\lambda }_1,{\lambda }_2)$, which is proportional to

$$\begin{array}{cc}\hfill \pi \left({\alpha }_1, {\alpha }_2, {\lambda }_1, {\lambda }_2\mid \mathbf{r},\mathbf{s}\right)& \propto {\alpha }_1^{M+a_1+-1}{\mathrm{e}}^{-{\alpha }_1\left(b_1+z_{{\lambda }_1}\left(\mathbf{r}\right)\right)}\times {\alpha }_2^{N+a_2-1}{\mathrm{e}}^{-{\alpha }_2\left(b_2+z_{{\lambda }_2}\left(\mathbf{s}\right)\right)}\hfill \\ \hfill & \times {\lambda }_1^{m+c_1-1}{\mathrm{e}}^{\left(-{\lambda }_1\left[{\sum }_{i=1}^m{r}_{i,i}^2+d_1\right]+z_{{\lambda }_1}\left(\mathbf{r}\right)\right)}\prod _{i=1}^m{\left\{log{x}_i\right\}}^{i-1}\hfill \\ \hfill & \times {\lambda }_2^{n+c_2-1}{\mathrm{e}}^{\left(-{\lambda }_2\left[{\sum }_{j=1}^n{s}_{j,j}^2+d_2\right]+z_{{\lambda }_2}\left(\mathbf{s}\right)\right)}\prod _{j=1}^n{\left\{log{y}_i\right\}}^{j-1}\hfill \end{array}$$

(27)

Under the squared error loss function, the Bayesian estimator for $\mathcal{R}$, as indicated in (5), is equivalent to $E(\mathcal{R}\mid \mathbf{R},\mathbf{S})$ as below:

$${\widehat{\mathcal{R}}}_B={\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }\mathcal{R}\pi \left({\alpha }_1, {\alpha }_2, {\lambda }_1, {\lambda }_2\mid \mathbf{r},\mathbf{s}\right)\mathrm{d}{\alpha }_1 \mathrm{d}{\alpha }_2 \mathrm{d}{\lambda }_1 \mathrm{d}{\lambda }_2$$

(28)

Given that the integral (28) does not yield an explicit solution, employing a Monte Carlo simulation-based algorithm is a viable approach. This paper will provide a detailed description of this particular method.

The posterior density (27) can be reformulated as follows:

$$\begin{array}{c}\begin{array}{c}\pi \left({\alpha }_1, {\alpha }_2, {\lambda }_1, {\lambda }_2\mid \mathbf{r}, \mathbf{s}\right)\propto f_1\left({\alpha }_1\mid {\lambda }_1, \mathbf{r}\right)f_2\left({\lambda }_1\mid \mathbf{r}\right)h_1\left({\alpha }_1, {\lambda }_1\mid \mathbf{r}\right)g_1\left({\alpha }_2\mid {\lambda }_2,\mathbf{s}\right)h_2\left({\alpha }_2, {\lambda }_2\mid \mathbf{s}\right)g_2\left({\lambda }_2\mid \mathbf{s}\right)\hfill \end{array}\hfill \end{array}$$

(29)

So, we have

$$f_1\left({\alpha }_1\mid {\lambda }_1, \mathbf{r}\right)\propto {\alpha }_1^{a_1+M-1}{\mathrm{e}}^{\left(-{\alpha }_1\left[b_1+z_{{\lambda }_1}\left(\mathbf{r}\right)\right]\right)}$$

(30)

$$g_1\left({\alpha }_2\mid {\lambda }_2, \mathbf{s}\right)\propto {\alpha }_2^{a_2+N-1}{\mathrm{e}}^{\left(-{\alpha }_2\left[b_2+z_{{\lambda }_2}\left(\mathbf{s}\right)\right]\right)}$$

(31)

$$f_2\left({\lambda }_1\mid \mathbf{r}\right)\propto {\lambda }_1^{m+c_1-1}{\mathrm{e}}^{\left(-{\lambda }_1\left[{\sum }_{i=1}^m{r}_{i,i}^2+d_1\right]+z_{{\lambda }_1}\left(\mathbf{r}\right)\right)}$$

(32)

$$g_2\left({\lambda }_2\mid \mathbf{r}\right)\propto {\lambda }_2^{n+c_2-1}{\mathrm{e}}^{\left(-{\lambda }_2\left[{\sum }_{j=1}^n{s}_{j,j}^2+d_2\right]+z_{{\lambda }_2}\left(\mathbf{s}\right)\right)}$$

(33)

In addition,

$$h\left({\alpha }_1, {\alpha }_2, {\lambda }_1, {\lambda }_2\mid \mathbf{r}, \mathbf{s}\right)=\prod _{i=1}^m{\left\{log\left(1-{\mathrm{e}}^{-{\lambda }_1{r}_{i,i}^2}\right)\right\}}^{i-1}\times \prod _{j=1}^n{\left\{log\left(1-{\mathrm{e}}^{-{\lambda }_2{s}_{j,j}^2}\right)\right\}}^{j-1}$$

(34)

Ultimately, to approximate the integral detailed in (28), the importance sampling technique can be utilized according to Algorithm 1:

Algorithm 1 Importance sampling approximation algorithm

Step 1. Initiate the process by generating ${\lambda }_1$ and ${\lambda }_2$ from $f_2\left({\lambda }_1\mid \mathbf{r}\right)$ and $g_2\left({\lambda }_2\mid \mathbf{r}\right)$, respectively, using Markov chain Monte Carlo methods, such as the random walk.        Step 2. With ${\lambda }_i^{\prime }s$ in Step 1, simulate ${\alpha }_1$ from $\Gamma (M+a_1, z_{{\lambda }_1}\left(\mathbf{r}\right)+b_1)$ and ${\alpha }_2$ from $\Gamma (N+a_2, z_{{\lambda }_2}\left(\mathbf{s}\right)+b_2)$.           Step 3. Execute Steps 1 and 2 repetitively for B iterations to collect samples $\left({\alpha }_1^{\left(t\right)}, {\alpha }_2^{\left(t\right)}, {\lambda }_1^{\left(t\right)}, {\lambda }_2^{\left(t\right)}\right)$ for $t=1, \dots , B$, and subsequently compute ${\mathcal{R}}^{\left(t\right)}$ as specified in (5).   Step 4. The approximation of (28), denoted ${\widehat{\mathcal{R}}}_B$, is then calculated as the weighted mean of these observations: $${\widehat{\mathcal{R}}}_B=\sum _{t=1}^B{w}_t{\mathcal{R}}^{\left(t\right)},$$ (35) where $$w_t=\frac{h\left({\alpha }_1^{\left(t\right)}, {\alpha }_2^{\left(t\right)}, {\lambda }_1^{\left(t\right)}, {\lambda }_2^{\left(t\right)}\mid \mathbf{r}, \mathbf{s}\right)}{{\sum }_{t=1}^{B}h\left({\alpha }_1^{\left(t\right)}, {\alpha }_2^{\left(t\right)}, {\lambda }_1^{\left(t\right)}, {\lambda }_2^{\left(t\right)}\mid \mathbf{r}, \mathbf{s}\right)}$$ (36)

$Scenario 2: {\alpha }_1={\alpha }_2=\alpha \& {\lambda }_1\ne {\lambda }_2$

In this scenario, the prior distributions of the independent variables $\left(\alpha , {\lambda }_1, {\lambda }_2\right)$ would be as follows:

$$\alpha \sim \Gamma \left(a, b\right), {\lambda }_i\sim \Gamma \left(c_i, d_i\right),$$

(37)

where $i=1, 2$. Similarly, the joint posterior density of the variables $\left(\alpha , {\lambda }_1, {\lambda }_2\right)$ is derived as follows:

$$\begin{array}{cc}\hfill \pi \left(\alpha ,{\lambda }_1, {\lambda }_2\mid \mathbf{r}, \mathbf{s}\right)\propto & {\alpha }^{M+N+a-1}{\mathrm{e}}^{-\alpha \left(z_{{\lambda }_1}\left(\mathbf{r}\right)+z_{{\lambda }_2}\left(\mathbf{s}\right)+b\right)}\hfill \\ \hfill & \times {\lambda }_1^{c_1+m-1}{\mathrm{e}}^{\left(-{\lambda }_1\left[{\sum }_{i=1}^m{r}_{i,i}^2+d_1\right]+z_{{\lambda }_1}\left(\mathbf{r}\right)\right)}\prod _{i=1}^m{\left\{log{x}_i\right\}}^{i-1}\hfill \\ \hfill & \times {\lambda }_2^{c_2+n-1}{\mathrm{e}}^{\left(-{\lambda }_2\left[{\sum }_{j=1}^n{s}_{j,j}^2+d_2\right]+z_{{\lambda }_2}\left(\mathbf{s}\right)\right)}\prod _{j=1}^n{\left\{log{y}_j\right\}}^{j-1},\hfill \end{array}$$

(38)

where $z_{{\lambda }_1}\left(\mathbf{r}\right)$ and $z_{{\lambda }_2}\left(\mathbf{s}\right)$ are the same as in Scenario 1. Similarly, deriving ${\widehat{\mathcal{R}}}_B$ in a closed form is not feasible. Therefore, an analogous method can be employed to approximate ${\widehat{\mathcal{R}}}_B$.

$Scenario 3: {\alpha }_1\ne {\alpha }_2 \& {\lambda }_1={\lambda }_2=\lambda$

According to (16), $\mathcal{R}$ does not depend on $\lambda$. Consequently, we can assume $\lambda =1$ without loss of generality. Therefore,

$${\alpha }_1\mid r\sim \Gamma (M+a_1, t_1+b_1), {\alpha }_2\mid s\sim \Gamma (N+a_2, t_2+b_2),$$

(39)

where $t_i$, represents the observed value of the random variable $T_i$, $i=1, 2$.

Theorem 2.

The Bayesian estimator for $\mathcal{R}$ is formulated as follows:

$$\begin{array}{cc}\hfill {\widehat{\mathcal{R}}}_B=& \frac{\Gamma (1+N+a_2)\Gamma (M+a_1)}{B\left(M+a_1, N+a_2\right)}\hfill \\ \hfill & \times {}_{2}F_1\left(1, N+a_1, M+N+a_1+a_2+1, \frac{t_1-t_2+b_1-b_2}{T_1+b_1}\right).\hfill \end{array}$$

(40)

Proof of Theorem 2.

Utilizing basic transformation methods, the pdf of ${\widehat{\mathcal{R}}}_B$ under the squared error loss function, will be determined as follows:

$${\pi }_{\mathcal{R}\mid (\mathbf{R}, \mathbf{S})}\left(r\right)=\frac{r^{N+a_2-1}{(1-r)}^{M+a_1-1}{\left(t_1+b_1\right)}^{M+N+a_1}{\left(t_2+b_2\right)}^{N+a_2}}{B\left(M+a_1, N+a_2\right){\left[\left(t_1+b_1\right)(1-r)+\left(t_2+b_2\right)r\right]}^{M+N+a_1+a_2}}, 0<r<1,$$

(41)

where $B(.,.)$ is a beta function.    □

3. Interval Estimation

In this section, four confidence intervals for $\mathcal{R}$ will be developed for $Scenario 3: {\alpha }_1\ne {\alpha }_2 \& {\lambda }_1={\lambda }_2=1$.

3.1. Maximum Likelihood Interval

Theorem 3.

The $100(1-\alpha )\%$ maximum likelihood intervals of $\mathcal{R}$ will be derived as

$$\left\{{\left(1+\frac{1-{\widehat{\mathcal{R}}}_{ML}}{{\widehat{\mathcal{R}}}_{ML}}{F}_{1-\frac{\alpha }{2}}(2M, 2N)\right)}^{-1}, {\left(1+\frac{1-{\widehat{\mathcal{R}}}_{ML}}{{\widehat{\mathcal{R}}}_{ML}}{F}_{\frac{\alpha }{2}}(2M,2N)\right)}^{-1}\right\}$$

(42)

Proof of Theorem 3.

From (19)–(21), we have

$${\widehat{\mathcal{R}}}_{ML}\stackrel{d}{=}{\left(1+\frac{1-\mathcal{R}}{\mathcal{R}}F(2N,2M)\right)}^{-1},$$

(43)

where $\stackrel{d}{=}$ signifies equivalence in distribution. Utilizing (43), we construct the $100(1-\alpha )\%CI$ for $\mathcal{R}$.    □

3.2. Bayesian Interval

Theorem 4.

The $100(1-\alpha )\%$ Bayesian intervals of $\mathcal{R}$ will be derived as

$$\left\{{\left[1+\frac{\left(M+a_1\right)\left(T_2+b_2\right)}{\left(N+a_2\right)\left(T_1+b_1\right)}{F}_{1-\frac{\alpha }{2}}\left(2\left(M+a_1\right),2\left(N+a_2\right)\right)\right]}^{-1},{\left[1+\frac{\left(M+a_1\right)\left(T_2+b_2\right)}{\left(N+a_2\right)\left(T_1+b_1\right)}{F}_{\frac{\alpha }{2}}\left(2\left(M+a_1\right),2\left(N+a_2\right)\right)\right]}^{-1}\right\}$$

(44)

Proof of Theorem 4.

From (20), (21) and (39), we obtain

$$2\left(T_1+b_1\right){\alpha }_1\mid \mathbf{r}\sim {\chi }_{2\left(M+a_1\right)}^2 \mathrm{and} 2\left(T_2+b_2\right){\alpha }_2\mid \mathbf{s}\sim {\chi }_{2\left(N+a_2\right)}^2$$

(45)

Therefore, it follows that

$$\frac{{\alpha }_1(T_1+b_1)(a_2+N)}{{\alpha }_2(T_2+b_2)(a_1+M)}\mid \mathbf{r},\mathbf{s}\sim F(2(M+a_1),2(N+a_2))$$

(46)

   □

3.3. Parametric Bootstrap Confidence Intervals

This section outlines the development of two bootstrap confidence intervals for $Scenario 3$, where $\lambda$ is set to 1. Using the basic and percentile method, as developed by Efron, B. et al. [23], these intervals are established. The foundational calculations for all the procedures discussed earlier rely on $T_1=-{\sum }_{i=1}^{m}log\left(1-{\mathrm{e}}^{-R_{i,i}^2}\right)$ and $T_2=-{\sum }_{j=1}^{n}log\left(1-{\mathrm{e}}^{-S_{j,j}^2}\right)$. Given that $T_1\sim Gamma\left(M, {\alpha }_1\right)$ and $T_2\sim Gamma\left(N, {\alpha }_2\right)$, parametric bootstrap CIs can be utilized in place of non-parametric alternatives. The typical two-step process for bootstrap methods used to determine these intervals is as follows (Algorithm 2):

Algorithm 2 Parametric bootstrap algorithm

Step 1. Compute ${\widehat{\alpha }}_1, {\widehat{\alpha }}_2,$ and ${\widehat{\mathcal{R}}}_{ML}$ from (17), (18), and (19), respectively. This calculation should be based on independently observed samples $t_{i1}$ and $t_{j2}$, which are distributed according to $\Gamma (i, {\theta }_1), i=1, \dots , m,$ and $\Gamma (j, {\theta }_2),j=1, \dots , n$, respectively.        Step 2. Generate samples $t_{i1}^{*}$ and $t_{j2}^{*}$ from $\Gamma (i, {\widehat{\alpha }}_1)$ and $\Gamma (j, {\widehat{\alpha }}_2)$, respectively, for each $i=1, \dots , m$, and $j=1, \dots , n$. Use these samples to calculate ${\widehat{\alpha }}_1^{*}$ and ${\widehat{\alpha }}_2^{*}$, and subsequently, ${\widehat{\mathcal{R}}}_{ML}^{*}$. Repeat this step for $b=1, \dots , B,$ to obtain a series of estimators ${\widehat{\mathcal{R}}}_{\left(b\right)ML}^{*}$.

Basic bootstrap confidence interval

• Consider ${\widehat{\mathcal{R}}}_{ML}^{*}={({\widehat{\mathcal{R}}}_{\left(1\right)ML}^{*}, \dots , {\widehat{\mathcal{R}}}_{\left(B\right)ML}^{*})}^T$. Calculate the $100(1-\alpha )\%$ basic bootstrap confidence interval for $\mathcal{R}$ as $(-{\widehat{\mathcal{R}}}_{ML1-\frac{\alpha }{2}}^{*}, -{\widehat{\mathcal{R}}}_{ML\frac{\alpha }{2}}^{*})$.

Percentile bootstrap confidence interval

• Consider that F represents the empirical cumulative distribution function of ${\widehat{\mathcal{R}}}_{\left(b\right)ML}^{*}$ derived from its bootstrap samples. The $100(1-\alpha )\%$ percentile bootstrap confidence interval for $\mathcal{R}$ is then defined as $({\widehat{F}}_{\frac{\alpha }{2}}^{-1}, {\widehat{F}}_{1-\frac{\alpha }{2}}^{-1})$.

4. Simulation Study

To compare the confidence intervals proposed in Section 3, we conduct a simulation study thereafter. The simulation design involves examining every combination of m (the sample size of $\mathbf{r}$) $=5, 6, 7$ with n (the sample size of $\mathbf{s}$) $=5, 6, 7$. Additionally, we adopt $\mathcal{R}=0.4, 0.7$ and set ${\alpha }_2=0.4$. We standardize B (the sample size of the bootstrap) $=1000$ and for each combination, we generate 10,000 samples of $\mathbf{r}$ and $\mathbf{s}$ from $GR({\alpha }_1, 1)$ and $GR({\alpha }_2, 1)$, respectively. We select confidence levels $\alpha =0.05, 0.1$.

The hyper-parameters are set as $a_i=b_i=2$. As indicated above, the results of Bayesian estimation in scenario 3 (both point and interval estimation) are closely tied to the hyper-parameters $a_i, b_i, i=1,2$. To explore the impact on Bayesian estimation, we tested various combinations of hyper-parameters in the pre-simulation. Our findings revealed that different configurations of hyper-parameters lead to significant fluctuations in Bayesian estimation. After evaluating different parameter choices in our simulation, we settled on a specific combination of hyper-parameters based on their performance and generalizability, resulting in all four hyper-parameters being equal.

We made some modifications to the previous RRSS function by implementing the runif() function to generate a random number following uniform distribution to determine whether to randomly swap two data positions in each sequence. In this way, the data in each sequence are no longer sorted in perfect order.

The resulting confidence intervals’ coverage probability (CP) and expected length (EL) are compiled in

Table 1. The CP and EL values for the proposed 95% CIs.

Row	$\mathcal{R}$	m	n	CP.ML	EL.ML	CP.Be	EL.Be	CP.Be *	EL.Be *	CP.Basic	EL.Basic	CP.Perc	EL.Perc
1	0.40	5	5	0.992	0.644	0.947	0.313	0.946	0.313	0.889	0.326	0.946	0.326
2	0.40	5	6	0.996	0.646	0.892	0.298	0.886	0.298	0.896	0.302	0.947	0.302
3	0.40	5	7	0.997	0.656	0.782	0.286	0.792	0.286	0.901	0.287	0.948	0.287
4	0.40	6	5	0.974	0.526	0.936	0.285	0.935	0.285	0.898	0.306	0.946	0.306
5	0.40	6	6	0.992	0.525	0.940	0.271	0.937	0.271	0.903	0.279	0.943	0.279
6	0.40	6	7	0.997	0.528	0.885	0.260	0.879	0.260	0.916	0.261	0.950	0.261
7	0.40	7	5	0.943	0.451	0.870	0.262	0.874	0.262	0.906	0.291	0.946	0.291
8	0.40	7	6	0.977	0.447	0.928	0.249	0.926	0.249	0.912	0.263	0.947	0.263
9	0.40	7	7	0.992	0.445	0.929	0.238	0.927	0.238	0.917	0.244	0.949	0.244
10	0.70	5	5	0.999	1.103	0.934	0.289	0.929	0.289	0.892	0.292	0.947	0.292
11	0.70	5	6	1.000	1.069	0.940	0.255	0.937	0.256	0.896	0.274	0.944	0.274
12	0.70	5	7	0.999	0.857	0.901	0.230	0.900	0.230	0.903	0.262	0.944	0.262
13	0.70	6	5	0.997	0.775	0.851	0.281	0.852	0.281	0.894	0.269	0.944	0.269
14	0.70	6	6	0.999	0.745	0.922	0.248	0.922	0.248	0.912	0.249	0.948	0.249
15	0.70	6	7	0.999	0.724	0.936	0.222	0.931	0.222	0.904	0.234	0.945	0.234
16	0.70	7	5	0.992	0.682	0.720	0.275	0.731	0.275	0.901	0.254	0.944	0.254
17	0.70	7	6	0.997	0.660	0.845	0.243	0.840	0.243	0.909	0.231	0.945	0.231
18	0.70	7	7	0.999	0.638	0.913	0.217	0.908	0.217	0.916	0.216	0.950	0.216

Note: CI: confidence interval; ML: maximum likelihood; Be: Bayesian; Perc: percentile; The notation ‘*’ represents the estimate derived from the imperfect RRSS.

and

Table 2. The CP and EL values for the proposed 90% CIs.

Row	$\mathcal{R}$	m	n	CP.ML	EL.ML	CP.Be	EL.Be	CP.Be *	EL.Be *	CP.Basic	EL.Basic	CP.Perc	EL.Perc
1	0.40	5	5	0.978	0.524	0.894	0.264	0.890	0.264	0.840	0.276	0.897	0.276
2	0.40	5	6	0.990	0.527	0.818	0.251	0.820	0.252	0.844	0.256	0.893	0.256
3	0.40	5	7	0.983	0.534	0.684	0.241	0.690	0.241	0.856	0.242	0.898	0.242
4	0.40	6	5	0.950	0.434	0.878	0.240	0.878	0.240	0.846	0.258	0.891	0.258
5	0.40	6	6	0.978	0.431	0.876	0.228	0.875	0.228	0.856	0.236	0.896	0.236
6	0.40	6	7	0.985	0.435	0.801	0.219	0.808	0.219	0.855	0.221	0.891	0.221
7	0.40	7	5	0.900	0.373	0.800	0.211	0.799	0.221	0.855	0.246	0.893	0.246
8	0.40	7	6	0.951	0.369	0.871	0.210	0.871	0.210	0.864	0.222	0.897	0.222
9	0.40	7	7	0.976	0.368	0.863	0.201	0.867	0.201	0.870	0.206	0.900	0.206
10	0.70	5	5	0.999	0.899	0.874	0.244	0.866	0.243	0.849	0.246	0.893	0.246
11	0.70	5	6	0.999	0.871	0.894	0.216	0.884	0.216	0.852	0.230	0.896	0.230
12	0.70	5	7	0.999	1.048	0.840	0.194	0.831	0.194	0.856	0.220	0.894	0.220
13	0.70	6	5	0.999	0.946	0.777	0.237	0.766	0.237	0.851	0.227	0.897	0.227
14	0.70	6	6	1.000	0.905	0.861	0.209	0.856	0.209	0.863	0.209	0.899	0.209
15	0.70	6	7	0.999	0.880	0.873	0.187	0.877	0.187	0.858	0.197	0.891	0.197
16	0.70	7	5	0.998	0.840	0.618	0.232	0.616	0.232	0.851	0.214	0.894	0.214
17	0.70	7	6	0.999	0.797	0.751	0.202	0.752	0.204	0.867	0.195	0.899	0.195
18	0.70	7	7	0.999	0.772	0.844	0.182	0.854	0.182	0.870	0.182	0.901	0.182

Note: CI: confidence interval; ML: maximum likelihood; Be: Bayesian; Perc: percentile; The notation ‘*’ represents the estimate derived from the imperfect RRSS.

.

We arrive at

• Due to the extremely long EL.ML, the ML intervals are almost ineffective.
• CP.Be is closely related to the value of $a_i,b_i$, and both the CP and EL of Be decrease with an increase in sample size.
• As the sample size increases, the EL of the parametric bootstrap CI decreases. However, the CP of the parametric bootstrap CI remains stable compared to CP.Be.
• The basic bootstrap CI and percentile bootstrap CI have the same EL, with the percentile bootstrap CI performing better.
• An imperfect RRSS hardly affects the Bayesian estimation results.

5. Real Data Analysis

This section presents an analysis of real data on fiber strength. Xu et al. [24] mentioned in their paper that measuring interfacial bonding strength in fiber-reinforced soft composites becomes challenging because the large deformation capability of a soft matrix invalidates most of the current testing methods. Badar and Priest [25] published the first data on fiber strength. They provided measurements for the strength of individual carbon fibers and bundles, and impregnated tows consisting of 1000 carbon fibers. Individual fibers underwent tensile testing at various gauge lengths, including 1, 10, 20, and 50 mm, while impregnated fiber tows were tested at 20, 50, 150, and 300 mm lengths. The specific datasets discussed here include results from single fibers tested at 10 mm (referred to as Data I and set as X) and 20 mm (referred to as Data II and set as Y) gauge lengths, with the respective sample sizes being 63 and 69.

From the goodness-of-fit plots (Figure 3 and Figure 4), it can be observed that a generalized Rayleigh distribution fits the data. Further, we tested the model’s validity using the Kolmogorov–Smirnov (K-S) test for each dataset. The K-S p-values for datasets I and II were found to be 0.4426 and 0.5547. It can be concluded that a generalized Rayleigh distribution fits both datasets. The estimated parameters are ${\alpha }_1=17.4525984, {\alpha }_2=5.2951630, {\lambda }_1={\lambda }_2=0.3657918$. Following the previous assumptions, we set the sampling size to all combinations of 5, 6, and 7. The hyper-parameters are also held constant at $a_i=b_i=2$. Here, we use perfect RRSS. The maximum likelihood estimators, Bayesian estimators, maximum likelihood confidence intervals, Bayesian confidence intervals, basic bootstrap confidence intervals and percentile bootstrap confidence intervals for $\mathcal{R}$ are given in

Table 3. Results of data analysis.

Row	m	n	MLE	MLCI	BE	BCI	CI.Basic	CI.Perc
1	5	5	0.138	(0.066, 0.286)	0.491	(0.328, 0.657)	(0.088, 0.344)	(0.133, 0.388)
2	5	6	0.140	(0.073, 0.281)	0.565	(0.411, 0.713)	(0.104, 0.338)	(0.136, 0.370)
3	5	7	0.151	(0.082, 0.295)	0.627	(0.485, 0.759)	(0.113, 0.332)	(0.137, 0.357)
4	6	5	0.126	(0.063, 0.243)	0.417	(0.271, 0.571)	(0.097, 0.337)	(0.142, 0.382)
5	6	6	0.126	(0.068, 0.233)	0.490	(0.348, 0.633)	(0.112, 0.327)	(0.144, 0.359)
6	6	7	0.136	(0.077, 0.244)	0.554	(0.420, 0.685)	(0.124, 0.325)	(0.148, 0.349)
7	7	5	0.115	(0.059, 0.211)	0.354	(0.226, 0.494)	(0.103, 0.331)	(0.148, 0.377)
8	7	6	0.118	(0.065, 0.207)	0.424	(0.296, 0.558)	(0.119, 0.323)	(0.152, 0.356)
9	7	7	0.122	(0.071, 0.207)	0.488	(0.364, 0.614)	(0.131, 0.317)	(0.155, 0.342)

Note: MLE: maximum likelihood estimator; MLCI: maximum likelihood confidence interval; BE: Bayesian estimator; CI.Basic: basic bootstrap confidence interval; CI.Perc: percentile bootstrap confidence interval

.

As we can see, the estimation results obtained based on real data are basically in line with the conclusions obtained from the simulation study (Section 4). Surprisingly, the ML interval performs quite well in the real dataset, compared to the simulation study. Because the bootstrap the CI and ML interval start from the same facts, they are equally consistent numerically.

Our approach in real data tests involves following the hyper-parameter combination identified in the simulation study, rather than deriving the estimation of the the hyper-parameter theoretically. This distinction results in differences between Bayesian estimation and maximum likelihood estimation in real data tests. We do not explore the impact of various hyper-parameters on Bayesian estimation. Nor do we adjust the hyper-parameters for the simulation study. The relevant issues will be studied in the future.

Both the maximum likelihood point estimation and interval estimation were recalculated for the fiber strength dataset using the imperfect RRSS. It is evident that imperfect RRSS yields a significantly higher MLE.

6. Conclusions

In this paper, we explored the estimation of stress-strength reliability $\mathcal{R}$ using lower record ranked set samples from a generalized Rayleigh distribution. Both the maximum likelihood and Bayesian estimators were derived and analyzed. The effectiveness of Bayesian estimation proved to be highly dependent on the chosen reference prior. Various confidence intervals were evaluated through a simulation study based on the expected length and coverage probability. The results indicated that the Bayesian confidence interval outperformed the others, whereas the maximum likelihood confidence interval was relatively less effective. These findings were corroborated by a real data analysis, which also provided surprising results from the maximum likelihood confidence interval, reinforcing the conclusions from Section 2.