EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6337 ISSN 1307-5543 – ejpam.com Published by New York Business Global Advanced Censoring Schemes for Statistical Inference of Reliability in Engineering Contexts Amal S. Hassan1, Gaber Sallam Salem Abdalla2, Ehab M. Almetwally3, Mohammed Elgarhy4,∗, Manal M. Yousef5 1 Faculty of Graduate Studies for Statistical Research, Cairo University, Giza, 12613, Egypt 2 Department of Insurance and Risk Management, Faculty of Business, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia 3 Department of Mathematics and Statistics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Riyadh, Saudi Arabia 4 Department of Basic Sciences, Higher Institute of Administrative Sciences, Belbeis, AlSharkia, Egypt 5 Department of Mathematics, Faculty of Science, New Valley University, El-Khargah 72511, Egypt Abstract. In practice, systems that are exposed to rigorous operational environments frequently fail. The critical concept that system failure often occurs when these extremely strict operating constraints are reached has not yet been thoroughly investigated by researchers. This study address this gap in an analysis of the multi-stress-strength model Υ = P (U < W < V ), in which stresses (U &V ) and strength (W ) are defined using the exponentiated Weibull distribution. In this work, point and interval estimators for Υ under a generalized progressive hybrid censoring scheme are derived. Using symmetric and asymmetric loss functions, we derive Bayesian estimators and maximum likelihood estimators. We employ Markov chain Monte Carlo techniques because the Bayesian estimators are computationally complex. Furthermore, to assess estimator performance, we create percentile bootstrap intervals, bootstrap-t intervals, and Bayesian credible intervals. A simulation study was conducted to evaluate the efficacy of the proposed estimates. Numerical results lead us to the conclusion that the Bayesian estimates based on informative priors outperform classical estimates in terms of biases and mean squared errors. The highest posterior density credible intervals offer a more favorable average length than asymptotic confidence intervals when performing Bayesian estimation. Real progressively censored engineering data applications of real data are presented to demonstrate the effectiveness of the proposed estimators. 2020 Mathematics Subject Classifications: 62N01, 62F10, 62F15, 62P30 Key Words and Phrases: Stress-strength model, Exponentiated Weibull, Generalized progres- sive hybrid censoring, Maximum likelihood method, Bayesian inference ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6337 Email addresses: amal52_soliman@cu.edu.eg (A. S. Hassan), jssabdullah@imamu.edu.sa (G. S. S. Abdalla), emalmetwally@imamu.edu.sa (E. M. Almetwally), dr.moelgarhy@gmail.com (M. Elgarhy), manal.mansour@sci.nvu.edu.eg (M. M. Yousef) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. S. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6337 2 of 26 1. Introduction A significant issue in statistical literature is the analysis of the stress-strength model (SSM). The probability that the stress X will exceed the strength Y is represented by the parameter R, defined as R = Pr(X < Y ). The SSM is commonly utilized in practical engineering for product life testing, material reliability analysis, and design. For instance, life tests are essential for aircraft components in the aerospace industry to ensure safety. Throughout its lifecycle, a product faces various external challenges such as temperature, humidity, wind, and pressure. Alongside these external factors, the product’s inherent strength allows it to withstand these stresses. The SSM is now widely used across many fields, especially for tackling reliability assessment challenges. An insightful monograph discussing various SSMs was authored by Kotz et al. [1]. The lifespan of a component with strength W under two stresses U and V is dic- tated by the SSM. A component can survive if its strength W is greater than stress U and smaller than stress V . The quantity Υ = P (U < W < V ) quantifies the reliability of the device within the framework of the mechanical reliability of this model. Systolic and diastolic blood pressure, for instance, have two limits that a person’s blood pressure should fall within. Consequently, the quantity Υ, despite being termed the SSM, possesses applications that extend beyond the assessment of actual SSM. It is applicable in a variety of fields, including mechanical design, information engineering, quality control, reliability analysis, and materials science. Chandra and Owen [2] first introduced the methodol- ogy’s fundamental concept of Υ. In the literature, various studies regarding the model Υ = P (U < W < V ) have been conducted by different authors, employing a range of sam- pling designs and distributions. Ivshin [3] investigated the maximum likelihood estimator (MLE) and the minimal variance unbiased estimator of Υ when both stress and strength are uniform or exponential random variables with an unknown location parameter. With a Weibull distribution, Hassan et al. [4] concentrated on estimating the parameter Υ in the presence of k outliers. The case of nonparametric inference of Υ was covered by Guangming et al. [5]. The estimate of Υ, assuming that the stresses and strength are independent vari- ables that adhere to the inverse Kumaraswamy distribution, was examined, respectively, by Hameed et al. [6] in a complete sample and Hassen et al. [7] in ranked set sampling. In the work of Abd Elfattah and Taha [8], the reliability estimator of Υ based on the inverse Rayleigh distribution was analyzed, taking into account data outliers. Raheem et al. [9] looked into traditional estimation methods, assuming an inverse Rayleigh distribution for both stress and strength random variables. Attia and Karam[10] explored the Bayesian estimation of Υ in the context of a Dagum distribution. Choudhary et al. [11] performed a statistical estimation of Υ within a Weibull distribution framework, utilizing progres- sively censored data. Yousef and Elmetwally [12] investigated the reliability estimator of Υ based on progressive first failure. For recent studies, the reader can refer to Yousef et al. [13], Hassan et al. [14], Alotaibi et al. [15], Moheb et al. [16] and Hassan and Mogran [17]. Cho et al. [18] presented a novel censoring scheme called the generalized progressive hybrid censoring (GPHC). This plan ensures a sufficient number of failures, which can A. S. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6337 3 of 26 increase the effectiveness of statistical inference, in addition to ending the experiment within the predetermined testing period. The following is a description of the GPHC. Assume that our research group consists of n independent units with the same lifetime distribution, where Z1, Z2, ..., Zn represent the corresponding lifetime. The integers k and s, k < s, as well as R1, R2, ..., Rs, which can satisfy the equation ∑s i=1Ri+s = n function as preplanned integers and have been under predetermination between zero and n. On the arrival of the first failure Z1, we withdraw R1 units. When the second failure, Z2, occurs, we remove R2 units at random from the n−2−R1 survivors. With the rest of the survival units removed, the process is repeated and ended at τ∗ = max(min(τ, Zs), Zk). It vastly improved prior approaches by allowing us to choose whether or not to continue the exper- iment if the sample size is insufficient at the predetermined cut-off time τ . Researchers would prefer to obtain s failures under the GPHC scheme, but they can alternatively choose k failures, which are considered the bare minimum. The GPHC scheme is referred to as R1, R2, ..., Rs. Let D be the observed failure times before arriving at the predefined time τ . The GPHC scheme can be classified into the following categories: Case 1: Z1, ..., Zd, ..., Zk for τ < Zk < Zs, Case 2: Z1, ..., Zk, ..., Zd for Zk < τ < Zs, Case 3: Z1, ..., Zk, ..., Zs for Zk < Zs < τ . The likelihood function of a random sample with cumulative distribution function (CDF) F (z) under GPHC is as follows. l(γ;Z) =  A1 k−1∏ j=1 f(zj:s:n)(1− F (zj:s:n)) Rjf(zk:s:n)(1− F (zk:s:n)) R∗ k Case 1, A2 d∏ j=1 f(zj:s:n)(1− F (zj:s:n)) Rj (1− F (τ))R ∗ d+1 Case 2, A3 s∏ j=1 f(zj:s:n)(1− F (zj:s:n)) Rj Case 3, (1) where γ is the vector of parameters for the lifetime distribution, A1 = k∏ j=1 ∑s k=j(Rk + 1), A2 = d∏ j=1 ∑s k=j(Rk + 1), A3 = s∏ j=1 ∑s k=j(Rk + 1), R∗ k = n − k − ∑k−1 i=1 Ri and R∗ d+1 = n− d− ∑d i=1Ri. Tu and Gui [19] considered the estimation of unknown parameters featured by the Ku- maraswamy distribution based on GPHC. Alotaibi et al. [20] discussed reliability analysis of Kavya Manoharan Kumaraswamy distribution under GPHC. Nagy et al. [21] recently employed a GPHC sample from the Burr XII distribution to estimate the unknown pa- rameters, reliability, and hazard functions. The Runge-Kutta technique was employed by Maswadah [22] to enhance the maximum likelihood estimation method. Based on GPHC, Liu and Gui. [23] derived the point and interval estimators for the unknown parame- ters, reliability, and hazard rate functions of the bathtub model. Abdelwahab et al. [24] A. S. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6337 4 of 26 discussed classical and Bayesian inference for the Kavya–Manoharan generalized exponen- tial distribution based on GPHC. Wang et al. [25] discussed a competing risk model for bivariate Kumaraswamy distributed based on GPHC. Hassan et al. [26], [27], and [28] implemented a GPHC for an inverted Topp-Leone distribution, a generalized inverse ex- ponential distribution, and a generalized Lomax distribution, respectively. For more see [29], [30] and [31]. To the best of our knowledge, no prior work has attempted to estimate Υ using an exponentiated Weibull distribution (EWD) based on the GPHC scheme, which is the motivation behind this paper. The importance of the EWD and its extensive use in numerous fields motivate us to address this issue. Furthermore, this study is thought to be a generalization of the Yousef et al. [13] research. Thus, the main driving force behind this can be summed up as follows: • Establish reliability inferences for Υ = P (U < W < V ), assuming that strength (W ) and two stresses (U and V ) have independent EWD with the same scale parameter. • Derive the MLE as well as the Bayesian estimator of Υ under symmetric and asym- metric loss functions. • Construct asymptotic confidence intervals with the help of the delta method and provide Bayesian credible intervals. • Create percentile bootstrap (Boot-P) and bootstrap-t (Boot-T) intervals. • Markov chain Monte Carlo (MCMC) methods are used to tackle the intricate inte- grals found in the Bayesian analysis of the posterior distribution. • Analyze the data and conduct a simulation study to see how various estimates be- have. We proceed with the reliability formulation calculation in Section 2 of this paper. Section 3 includes the derivation of the MLEs, approximate confidence intervals, Boot- P, and Boot-T intervals of Υ. The Bayesian estimators with varying loss functions are computed in Section 4. In addition, the MCMC algorithm is employed to derive Bayesian estimators and establish the highest posterior density (HPD) intervals in Section 5. Then, in Section 6, real data applications are used to demonstrate data analysis. Conclusions are organized in Section 7. 2. Reliability Formulation A very adaptable class of probability distribution functions is the three-parameter EWD, which Mudholkar and Srivastava [32] introduced as an extension of the Weibull family. The significance of this distribution arises from the fact that the function of the survival rate takes various forms, making it suitable for studying and covering many problems of time reconciliation and validity. Many probability distributions are included A. S. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6337 5 of 26 as special cases. Mudholkar et al. [33] provided examples of how the EWD is used in reliability and survival studies. The probability density function (PDF) of the EWD is defined as f(x; η, λ, θ) = ηλθxθ−1e−ηx θ (1− e−ηx θ )λ−1; x > 0, η, λ, θ > 0, (2) and the corresponding CDF is as follows: F (x; η, λ, θ) = (1− e−ηx θ )λ, (3) where λ > 0 and θ > 0 are the shape parameters and η > 0 is the scale parameter. Assuming that η is known parameter (η = 1), then we write X ∼ EWD(λ, θ). The hazard function has various shapes depending on parameter values, such as mono- tone increasing (θ ≥ 1, θλ ≥ 1), monotone decreasing (θ ≤ 1, θλ ≤ 1), or unimodal (θ < 1, θλ > 1), see Figure 1 for illustration. These different shapes gave the distribution more flexibility in fitting different real data. When λ = 1, the distribution is referred to as a Weibull distribution, and when θ = 1, it reduces to the exponentiated exponential distribution (EED). If θ = λ = 1, the distribution has an exponential constant hazard function. For additional results and applications, one can see Almalki and Nadarajah [34], Wu and Lee [35], Ahmad et al. [36], Cheema et al. [37], Rahman et al. [38], Xie [39] and Ishag et al. [40]. 0 1 2 3 4 0 1 2 3 4 5 x h (x ) 0 1 2 3 4 0 1 2 3 4 5 0 1 2 3 4 0 1 2 3 4 5 0 1 2 3 4 0 1 2 3 4 5 0 1 2 3 4 0 1 2 3 4 5 λ = 1.5 θ = 1.3 λ = 10 θ = 3 λ = 0.8 θ = 2 λ = 0.7 θ = 0.4 λ = 5 θ = 1.6 Figure 1: hazard function with various parameters values Let the random variables U ∼ EWD(λ1, θ), W ∼ EWD(λ2, θ,), V ∼ EWD(λ3, θ) be independent. The reliability formula of the SSM that the probability of a component strength falling in between two stresses is given by: Υ = ∫ ∞ 0 ∫ ∞ u ∫ v u f(u;λ1, θ)f(w;λ2, θ)f(v;λ3, θ)dwdvdu, A. S. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6337 6 of 26 which gets the following formula: Υ = λ2λ3 (λ1 + λ2)(λ1 + λ2 + λ3) . (4) The reliability Υ of the SSM is shown in Figure 2, and it is clear that the reliability value increases as the value of the parameter λ3 increases. Also, it is observed that as the parameter values change, the reliability rating does so as well. In most cases, this rating is high and covers the majority of the values. 2 4 6 1 2 3 4 5 6 Y 0.2 0.4 0.6 0.8 λ3 = 20 0.2 0.4 0.6 0.8 λ2 0.2 0.4 0.6 0.8 λ2 λ1 0.2 0.4 0.6 0.8 2 4 6 1 2 3 4 5 6 Y 0.2 0.4 0.6 0.8 λ3 = 15 0.2 0.4 0.6 0.8 λ2 0.2 0.4 0.6 0.8 λ2 λ1 0.2 0.4 0.6 0.8 2 4 6 1 2 3 4 5 6 Y 0.2 0.4 0.6 0.8 λ3 = 4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 λ2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 λ2 λ1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Figure 2: 3D plot of Υ reliability in SSM 3. Maximum Likelihood Estimation The maximum likelihood (ML) procedure is a popular and effective strategy used by statisticians when dealing with reliability issues and survival analysis. The unknown A. S. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6337 7 of 26 parameters λ1, λ2, λ3, and θ will are estimated using this method to obtain Υ. In order to start we have one of the following types of observations for each identical EWD component stress U = (u1:s1:n1 , u2:s1:n1 , ..., uD1:s1:n1), V = (v1:s3:n3 , v2:s3:n3 , ..., vD32:s3:n3) and strength W = (w1:s2:n2 , w2:s2:n2 , ..., wD21:s2:n2) sample under GPHC scheme. Throughout the paper, denote U = (u1, ..., uD1), where uj = uj:s1:n1 , j = 1, ..., D1, similarly for vj , and wj . In our situation, the likelihood function of the observed stress sample U can be obtained by replacing its CDF and PDF in (1) as shown below. l(λ1, θ;U) = A∗ 1λ D1 1 θD1 D1∏ j=1 uθ−1 j e−u θ j (1− e−u θ j )λ1−1(1− (1− e−u θ j )λ1)Rj1(1− (1− e−τ θ 1 )λ1)R ∗ d1+1 , where D1 =  k1 − 1 Case 1, d1 Case 2, s1 Case 3. Let Q = (U, V,W ) represents the stress and strength samples, and let γ = (λ1, λ2, λ3, θ) be the vector of parameters. The likelihood function of the observed data can be expressed as follows based on the observations of the given data under GPHC scheme. l(γ;Q) = A∗ 1A ∗ 2A ∗ 3λ D1 1 λD2 2 λD3 3 θD1+D2+D3 × D1∏ j=1 uθ−1 j e−u θ jψ1(uj , θ) λ1−1(1− ψ1(uj , θ) λ1)Rj1(1− ψ1(τ1, θ) λ1)R ∗ d1+1 × D2∏ j=1 wθ−1 j e−w θ jψ2(wj , θ) λ2−1(1− ψ2(wj , θ) λ2)Rj2(1− ψ2(τ2, θ) λ2)R ∗ d2+1 × D3∏ j=1 vθ−1 j e−v θ jψ3(vj , θ) λ3−1(1− ψ3(vj , θ) λ3)Rj3(1− ψ3(τ3, θ) λ3)R ∗ d3+1 , (5) where ψ1(uj , θ) = 1− e−u θ j , ψ2(wj , θ) = 1− e−w θ j , ψ3(vj , θ) = 1− e−v θ j , (6) and ψ1(τ1, θ) is as given by Equation (6) with uj = τ1. Similarly for ψ2(τ2, θ) and ψ3(τ3, θ). By taking the logarithm of (5), say L, we can get the log-likelihood function as: L = lnA∗ 1A ∗ 2A ∗ 3 +D1 lnλ1 +D2 lnλ2 +D3 lnλ3 + (D1 +D2 +D3) ln θ − (θ − 1)[ D1∑ j=1 lnuj + D2∑ j=1 lnwj + D3∑ j=1 ln vj ]− [ D1∑ j=1 uθj + D2∑ j=1 wθj + D3∑ j=1 vθj ] + (λ1 − 1) D1∑ j=1 lnψ1(uj , θ) + (λ2 − 1) D2∑ j=1 lnψ2(wj , θ) + (λ3 − 1) D3∑ j=1 lnψ3(vj , θ) A. S. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6337 8 of 26 + D1∑ j=1 Rj1 ln(1− (ψ1(uj , θ)) λ1) + D2∑ j=1 Rj2 ln(1− (ψ2(wj , θ)) λ2) + D3∑ j=1 Rj3 ln(1− (ψ3(vj , θ)) λ3) +R∗ d1+1 ln(1− (ψ1(τ1, θ)) λ1) +R∗ d2+1 ln(1− (ψ2(τ2, θ)) λ2) +R∗ d3+1 ln(1− (ψ3(τ3, θ)) λ3). (7) We take the partial derivatives of (7) for λ1, λ2, λ3, and θ respectively, and get a set of likelihood equations as follows: ∂L ∂λ1 = D1 λ1 + D1∑ j=1 [lnψ1(uj , θ)−Rj1ϑ1(ψ1(uj , θ), λ1)]−R∗ d1+1ϑ1(ψ1(τ1, θ), λ1), ∂L ∂λ2 = D2 λ2 + D2∑ j=1 [lnψ2(wj , θ)−Rj2ϑ2(ψ2(wj , θ), λ2)]−R∗ d2+1ϑ2(ψ2(τ2, θ), λ2), ∂L ∂λ3 = D3 λ3 + D3∑ j=1 [lnψ3(vj , θ)−Rj3ϑ3(ψ3(vj , θ), λ3)]−R∗ d3+1ϑ3(ψ3(τ3, θ), λ3), ∂L ∂θ = D1 +D2 +D3 θ − [ D1∑ j=1 (uθj + 1) lnuj + D2∑ j=1 (wθj + 1) lnwj + D3∑ j=1 (vθj + 1) ln vj ] + (λ1 − 1) D1∑ j=1 ϕ1(ψ1(uj , θ), θ) + (λ2 − 1) D2∑ j=1 ϕ2(ψ2(wj , θ), θ) + (λ3 − 1) D3∑ j=1 ϕ3(ψ3(vj , θ), θ)− D1∑ j=1 Rj1$1(ψ1(uj , θ), θ)− D2∑ j=1 Rj2$2(ψ2(wj , θ), θ) − D3∑ j=1 Rj3$3(ψ3(vj , θ), θ)−R∗ d1+1$1(ψ1(τ1, θ), θ)−R∗ d2+1$2(ψ2(τ2, θ), θ) −R∗ d3+1$3(ψ3(τ3, θ), θ),  (8) where ϑi(ψi(qj , θ), λi) = (ψi(qj , θ)) λi 1− (ψi(qj , θ))λi lnψi(qj , θ), ϑi(ψi(τi, θ), λi) = (ψi(τi, θ)) λi 1− (ψi(τi, θ))λi lnψi(τi, θ), ϕi(ψi(qj , θ), θ) = qθj e −qθj ψi(qj , θ) ln qj , $i(ψi(qj , θ), θ) = λi(ψi(qj , θ)) λi 1− (ψi(qj , θ))λi ϕi(ψi(qj , θ), θ), for q = u, v, w, and i = 1, 2, 3. A. S. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6337 9 of 26 Since the equations in (8) are nonlinear, it is obvious that simplifying and obtaining closed-form solutions is difficult. In this situation, Newton’s iteration method can be applied to get the MLEs of λ1, λ2, λ3, and θ, and then substitute these estimates into Equation (4) to gain the MLE of Υ as: Υ̂ = λ̂2λ̂3 (λ̂1 + λ̂2)(λ̂1 + λ̂2 + λ̂3) . (9) 3.1. Asymptotic Confidence Intervals The asymptotic confidence interval (CI) of Υ is determined in this subsection using the asymptotic distribution of Υ̂. A 100(1− ζ)% asymptotic CI of Υ can be constructed as ( Υ̂− z 1− ζ 2 √ V ar(Υ̂), Υ̂ + z 1− ζ 2 √ V ar(Υ̂) ) , where z 1− ζ 2 is the standard normal variate’s upper pth percentile. V ar(Υ̂) can be obtained by applying a result of Rao [41] (pp. 387), which gives us V ar(Υ̂) = 4∑ i=1 ( ∂L ∂γi )2 γ̂i + 2 2∑ i=1 ( ∂L ∂γi ) γ̂i ( ∂L ∂γj ) γ̂j Cov(γ̂i, γ̂j). The observed Fisher information matrix can be used to determine the variance and co- variance of λ̂1, λ̂2, λ̂3, and θ̂. Therefore, the observed Fisher information matrix is given by: I(γ) = [Iij ] = −E [ ∂2L ∂γi∂γj ] , i, j = 1, 2, 3, 4, where γ is the parameter vector (λ1, λ2, λ3, θ) with γ1 = λ1, γ2 = λ2, γ3 = λ3, γ4 = θ. Unfortunately, the exact mathematical expressions for the above expectation are very difficult to obtain, so it is obtained by dropping the expectation on an operation (see Cohen [42, 43]). The elements of I(γ) are given as follows I11 = − D1 λ21 + D1∑ j=1 Rj1 ∂ϑ1(ψ1(uj , θ), λ1) ∂λ1 +R∗ d1+1 ∂ϑ1(ψ1(τ1, θ), λ1) ∂λ1  , I22 = − D2 λ22 + D2∑ j=1 Rj2 ∂ϑ2(ψ2(wj , θ), λ2) ∂λ2 +R∗ d2+1 ∂ϑ2(ψ2(τ2, θ), λ2) ∂λ2  , I33 = − D3 λ23 + D3∑ j=1 Rj3 ∂ϑ3(ψ3(vj , θ), λ3) ∂λ3 +R∗ d3+1 ∂ϑ3(ψ3(τ3, θ), λ3) ∂λ3  , A. S. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6337 10 of 26 I14 = D1∑ j=1 [ ϕ1(ψ1(uj , θ)−Rj1 ∂ϑ1(ψ1(uj , θ), λ1) ∂θ ] −R∗ d1+1 ∂ϑ1(ψ1(τ1, θ), λ1) ∂θ , I24 = D2∑ j=1 [ ϕ2(ψ2(wj , θ)−Rj2 ∂ϑ2(ψ2(wj , θ), λ2) ∂θ ] −R∗ d2+1 ∂ϑ2(ψ2(τ2, θ), λ2) ∂θ , I34 = D3∑ j=1 [ ϕ3(ψ3(vj , θ)−Rj3 ∂ϑ3(ψ3(vj , θ), λ3) ∂θ ] −R∗ d3+1 ∂ϑ3(ψ3(τ3, θ), λ3) ∂θ , I44 = −D1 +D2 +D3 θ2 − [ D1∑ j=1 uθj + D2∑ j=1 wθj + D3∑ j=1 vθj ](ln 2 θ + 1 θ ) + (λ1 − 1) D1∑ j=1 ∂ϕ1(ψ1(uj , θ), θ) ∂θ + (λ2 − 1) D2∑ j=1 ∂ϕ2(ψ2(wj , θ), θ) ∂θ + (λ3 − 1) D3∑ j=1 ∂ϕ3(ψ3(vj , θ), θ) ∂θ − D1∑ j=1 Rj1 ∂$1(ψ1(uj , θ), θ) ∂θ − D2∑ j=1 Rj2 ∂$2(ψ2(wj , θ), θ) ∂θ − D3∑ j=1 Rj3 ∂$3(ψ3(vj , θ), θ) ∂θ −R∗ d1+1 ∂$1(ψ1(τ1, θ), θ) ∂θ −R∗ d2+1 ∂$2(ψ2(τ2, θ), θ) ∂θ −R∗ d3+1 ∂$3(ψ3(τ3, θ), θ) ∂θ , I12 = I21 = I13 = I31 = I32 = I23 = 0, where ∂ϑi(ψi(qj , θ), λi) ∂λi = (ϑi(ψi(qj , θ), λi)) 2 (ψi(qj , θ))λi , ∂ϑi(ψi(τi, θ), λi) ∂λi = (ϑi(ψi(τi, θ), λi)) 2 (ψi(τi, θ))λi , ∂ϑi(ψi(qj , θ), λi) ∂θ = 2ψi(ψi(qj , θ), θ) 1− (ψi(qj , θ))λi , ∂ϕi(ψi(qj , θ), θ) ∂θ = ϕi(ψi(qj , θ), θ)(ln qj − eq θ j ), ∂$i(ψi(qj , θ), θ) ∂θ = ∂$i(ψi(qj , θ), θ) ∂θ = λiϕ ′ i(ψi(qj , θ), θ)[ (ψi(qj , θ)) −λi − 1 ] + λ2iϕi(ψi(qj , θ), θ) (ψi(qj , θ)) −λi−1[ (ψi(qj , θ)) −λi − 1 ]2 , for q = u, v, w, i = 1, 2, 3 and ϕ′ i(ψi(qj , θ), θ) = (ln qj) 2qθj e −qθj [ ψi(qj ,θ)(1−qθj )+qθj e −qθj ] (ψi(qj ,θ)) 2 . 3.2. Bootstrap Methods A resampling technique, the bootstrap method for constructing more widely used CIs is discussed in this subsection. Algorithms 1 and 2 present the algorithms for the Boot-P and Boot-T methods, respectively. • Boot-P method: A. S. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6337 11 of 26 Algorithm 1 The algorithm of Boot-P method. Step 1: Generate three random samples from EWD(λ1, θ), EWD(λ2, θ), and EWD(λ3, θ), and use the GPHC scheme to obtain the observed data. Step 2: Calculate the ML estimates (for example, λ̂1, λ̂2, λ̂3). Step 3: Utilize (λ̂1, λ̂2, λ̂3), to generate three new samples, respectively. Step 4: Obtain new ML estimates (λ̂∗(k)1 , λ̂ ∗(k) 2 , λ̂ ∗(k) 3 ). Step 5: Compute the bootstrap estimate of Υ in Step 4 and indicate it by Υ̂∗. Step 6: Repeat Steps 3 and 5 B times. Step 7: Obtain the results ((Υ̂∗ 1, Υ̂ ∗ 2, ..., Υ̂ ∗ B)). Step 8: Define ~(x) = P (Υ̂∗ ≤ x) be the CDF of Υ̂∗. Let Υ̂Boot−P (x) = ~−1(x) for a given x. Step 9: The 100(1− ζ)% symmetric Boot-P CI for Υ is (Υ̂Boot−P (ζ/2), Υ̂Boot−P (1− ζ/2)). • Boot-T method: Algorithm 2 The algorithm of Boot-T method. Steps 1 to 5 are the same as those in Algorithm 1. Step 6: Define statistic T ∗ Υ = (Υ̂∗−Υ̂) σ∗ Υ . Step 7: Repeat Steps 3 and 6 B times, then we have T ∗(1) Υ , T ∗(2) Υ , ..., T ∗(B) Υ . Step 8: Define ℵ(x) = P (T ∗ Υ ≤ x) be the CDF of T ∗ Υ. Let Υ̂Boot−T = Υ̂+ℵ−1(x)σΥ. Step 9: The 100(1− ζ)% symmetric Boot-T CI for Υ is (Υ̂Boot−T (ζ/2), Υ̂Boot−T (1− ζ/2)). 4. Bayesian Estimation In this section, Bayesian estimation of Υ is obtained when data are observed using GPHC based on a squared error loss function (SELF) and a linear exponential (LINEX) loss function, which are defined respectively by L1 = (ρ, ρ̌) = (ρ̌− ρ)2, L2 = (ρ, ρ̌) = ec(ρ̌−ρ) − c(ρ̌− ρ)− 1, where ρ̌ is an estimator of ρ. Denote the prior and posterior distributions of ρ by π(ρ) and π∗(ρ | q), respectively. Under the SELF and LINEX loss functions, the Bayesian estimation of any function B(ρ) of ρ is given by ṽSELF = E[B(ρ) | q] = ∫ ∞ 0 B(ρ)π∗(ρ | q)dρ, ṽLINEX = −1 c ln ( E(e−cB(ρ)) ) = −1 c ln (∫ ∞ 0 e−cB(ρ)π∗(ρ | q)dρ ) . A. S. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6337 12 of 26 The prior distribution is important for the development of Bayes estimators. Under the assumption of gamma prior distributions, we investigate this estimation problem. Therefore, it is assumed here that λ1, λ2, λ3, and θ follow independent gamma distributions with λ1 ∼ G(a1, b1), λ2 ∼ G(a2, b2), λ3 ∼ G(a3, b3), and θ ∼ G(a4, b4) with probability densities given by, respectively, π(λi) = λai−1 i Γ(ai)b ai i e −λi bi , π(θ) = θa4−1 Γ(a4)b a4 4 e − θ b4 , λi > 0, ai, bi > 0, i = 1, 2, 3. (10) Using the informative prior (10) and the likelihood function (5), the joint posterior density can be derived as follows: π∗(γ) = 3∏ i=1 A∗ i λDi+ai−1 i Γ(ai)b ai i θa4−1+ ∑3 j=1Dj Γ(a4)b a4 4 e − θ b4 × e − ∑D1 j=1 [ (1−θ) lnuj+uθj+ λ1 b1 −(λ1−1) lnψ1(uj ,θ)−Rj1 ln(1−(ψ1(uj ,θ)) λ1 ) ] +R∗ d1+1 ln(1−(ψ1(τ1,θ))λ1 ) × e − ∑D2 j=1 [ (1−θ) lnwj+w θ j+ λ2 b2 −(λ2−1) lnψ2(wj ,θ)−Rj2 ln(1−(ψ2(wj ,θ)) λ2 ) ] +R∗ d2+1 ln(1−(ψ2(τ2,θ))λ2 ) × e − ∑D3 j=1 [ (1−θ) ln vj+vθj+ λ3 b3 −(λ3−1) lnψ3(vj ,θ)−Rj3 ln(1−(ψ3(vj ,θ)) λ3 ) ] +R∗ d3+1 ln(1−(ψ3(τ3,θ))λ3 ). The marginal posterior densities of the parameters λ1, λ2, λ3, and θ can be derived as π∗(λ1) ∝ λD1+a1−1 1 e − ∑D1 j=1 [ λ1( 1 b1 −lnψ1(uj ,θ))−Rj1 ln(1−(ψ1(uj ,θ)) λ1 ) ] +R∗ d1+1 ln(1−(ψ1(τ1,θ))λ1 ), π∗(λ2) ∝ λD2+a2−1 2 e − ∑D2 j=1 [ λ2( 1 b2 −lnψ2(wj ,θ))−Rj2 ln(1−(ψ2(wj ,θ)) λ2 ) ] +R∗ d2+1 ln(1−(ψ2(τ2,θ))λ2 ), π∗(λ3) ∝ λD3+a3−1 3 e − ∑D3 j=1 [ λ3( 1 b3 −lnψ3(vj ,θ))−Rj3 ln(1−(ψ3(vj ,θ)) λ3 ) ] +R∗ d3+1 ln(1−(ψ3(τ3,θ))λ3 ), π∗(θ) ∝ θa4−1+ ∑3 i=1Die − θ b4 × e − ∑D1 j=1 [ uθj−θ lnuj−(λ1−1) lnψ1(uj ,θ)−Rj1 ln(1−(ψ1(uj ,θ)) λ1 ) ] +R∗ d1+1 ln(1−(ψ1(τ1,θ))λ1 ) × e − ∑D2 j=1 [ wθ j−θ lnwj−(λ2−1) lnψ2(wj ,θ)−Rj2 ln(1−(ψ2(wj ,θ)) λ2 ) ] +R∗ d2+1 ln(1−(ψ2(τ2,θ))λ2 ) × e − ∑D3 j=1 [ vθj−θ ln vj−(λ3−1) lnψ3(vj ,θ)−Rj3 ln(1−(ψ3(vj ,θ)) λ3 ) ] +R∗ d3+1 ln(1−(ψ3(τ3,θ))λ3 ).  (11) The marginal posterior densities in (11) are not well-known distributions, so we will use the Metropolis-Hastings (MH) sampler to generate the values of λ1, λ2, λ3, and θ with a normal proposal distribution to generate samples from it in (11). Furthermore, the approach of Chen and Shao [44] is extensively used to construct HPD intervals with unknown benefit distribution parameters for Bayesian estimates. For example, a 95% HPD interval can be created using two endpoints from the MCMC sample outputs: 2.5% and 97.5% percentiles, respectively. The Θ parameters’ Bayes, trustworthy intervals are calculated as follows: (i) Sorted parameters as λ̃[1]l < λ̃ [2] l < ... < λ̃ [N ] l ; l = 1, 2, 3, θ̃[1] < θ̃[2] < ... < θ̃[N ], and Υ[1] < Υ[2] < ... < Υ[N ], and N is the length of MCMC generated. A. S. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6337 13 of 26 (ii) The 95% symmetric credible intervals of λ̃1, λ̃2, λ̃3, θ̃ and Υ̃ become ( λ̃ L 25 1000 l , λ̃ L 975 1000 l ) ,( θ̃L 25 1000 , θ̃L 975 1000 ) and ( Υ̃L 25 1000 , Υ̃L 975 1000 ) . 5. Simulation Study This section presents some simulation findings that demonstrate how the various tactics discussed in this study perform in real-world situations. • Various circumstances were used as: Case I: λ1 = 0.5, λ2 = 4, λ3 = 20, θ = 0.5 and τ1 = 14, τ2 = 6, τ3 = 18. Case II:λ1 = 0.5, λ2 = 1.2, λ3 = 10, θ = 3 and τ1 = 1, τ2 = 1.2, τ3 = 1.6. Case III:λ1 = 0.3, λ2 = 0.6, λ3 = 5, θ = 0.8 and τ1 = 2, τ2 = 1.8, τ3 = 3. Case IV: λ1 = 1.3, λ2 = 5, λ3 = 15, θ = 1.5 and τ1 = 2, τ2 = 1.8, τ3 = 3 and τ1 = 2.5, τ2 = 2.5, τ3 = 4. • A variety of sample sizes were selected, n1 = 20, n2 = 25, n3 = 15 with different effective sample sizes for each sample as s1 = 15, s2 = 18, s3 = 11, and different k values for each sample as k1 = 12, k2 = 16, k3 = 10. • Big sample sizes were selected as n1 = 30, n2 = 40, n3 = 30 with different effective sample sizes for each sample as s1 = 17, s2 = 22, s3 = 13, and different k values for each sample as k1 = 15, k2 = 20, k3 = 11. • Two different progressive censoring schemes, namely, Scheme-IR1 = (n1−s1, rep(0, s1− 1)), R2 = (n2− s2, rep(0, s2− 1)), R3 = (n3− s3, rep(0, s3− 1)) and Scheme-II R1 = (rep(0, s1 − 1), n1 − s1), R2 = (rep(0, s2 − 1), n1 − s1), R3 = (rep(0, s3 − 1), n3 − s3). • Calculate the point estimates based on both estimation methods. Also, calculate the credible and approximate 95% CIs for both loss functions in each instance. • The procedures are carried out 5000 times, and the results are presented for the bias and mean squared errors (MSE) for all estimates. • The average lengths of CI (LCI) with related coverage percentages (CP), average lengths of Boot-P (LCIBP), average lengths of Boot-T (LCIBT), and average lengths of HPD credible CI (LCCI) where LCCI1 for SELF, LCCI2 for LINEX when c= 0.5 and LCCI3 for LINEX when c=1.5. Elective hyperparameters based on the mean and variance of the gamma prior distri- bution are used to choose the hyper-parameters for prior distribution. Using the likelihood method’s estimate and variance-covariance matrix, we may learn how to elicit hyperpa- rameters of the independent joint prior. The resulting hyperparameters can be represented as the mean and variance of gamma priors. A. S. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6337 14 of 26 aj = [ 1 N ∑N i=1 ρ̂ i j ]2 1 N−1 ∑N i=1 [ ρ̂ij − 1 N ∑N i=1 ρ̂ i j ]2 ; j = 1, 2, ..., p− 1, bj = 1 N ∑N i=1 ρ̂ i j 1 N−1 ∑N i=1 [ ρ̂ij − 1 N ∑N i=1 ρ̂ i j ]2 ; j = 1, 2, ..., p− 1, where N is the number of iterations (for more details see Dey et al. [45]). We replicate the MH algorithm process 10,000 times for MCMC approaches. The following conclusions can be drawn from the results shown in Tables”1, 2, 3, and 4. • Based on the average (AvE), where they approach to the actual values, the estimates of population parameters for ML and Bayesian methods are pretty good (see Tables 1, 2, 3, and 4). • The MSE reduces with sample size, as is expected for the ML and Bayesian estima- tion techniques as seen in Tables 1, 2, 3, and 4. • As s increases for a given sample size, the MSE likewise gets worse. • The MSEs fall as the tolerable absolute minimum of failures, k, rises, if n and s are held constant as seen in Table 1 to Table 4. • Based on the evidence presented in Tables 1- 4, Scheme II demonstrates clear supe- riority over Scheme I in terms of bias, MSE, length of CI. • The Bayesian estimates perform better than ML estimates in terms of bias, MSE, and length of CI because they take into account prior information based on a gamma informative prior (see Tables 1-4). • Bayes estimates derived using asymmetric loss functions are more accurate than those employing symmetric loss functions, as demonstrated in Tables 1-4. • For Bayesian estimation, the average length of the HPD credible CIs is preferable to the average length of an asymptotic CI. The shortest CI is the average bootstrapping time (see Tables 1-4). 6. Data Analysis In order to illustrate the approaches suggested in this study, this section looks at a real data set. The EWD based on the GPHC technique is also displayed using this dataset. Chapter Three of Nelson’s book [46] contains the results of a stress-strength life test of transformer insulation. The test included three levels of voltage, which are 35:4kv, 42:4kv, and 46:7kv, respectively, with a normal voltage of 14:4kv. A. S. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6337 15 of 26 Table 1: AvE and MSE for Estimation methods: Case I λ1 = 0.5, λ2 = 4, λ3 = 20, θ = 0.5 τ1 = 14, τ2 = 6, τ3 = 18 ML SELF LINEX c=0.5 LINEX c=1.5 n1, n2, n3 Scheme s1, s2, s3 k1, k2, k3 Bias MSE Bias MSE Bias MSE Bias MSE LACI CP LCIBP LCIBT LCCI1 LCCI2 LCCI3 20,15,15 1 15,18,11 12,16,10 λ1 0.0288 0.0173 0.0085 0.0087 0.0067 0.0086 0.0032 0.0085 0.5032 96% 0.0225 0.0224 0.3396 0.3381 0.3351 λ2 0.1576 0.4243 -0.0038 0.0193 -0.0065 0.0193 -0.0119 0.0193 2.4799 94% 0.1146 0.1148 0.5392 0.5405 0.5382 λ3 0.0455 0.0314 -0.0114 0.0218 -0.0142 0.0220 -0.0197 0.0223 0.6724 95% 0.0319 0.0319 0.5938 0.5880 0.5829 θ 0.0048 0.0012 0.0024 0.0011 0.0020 0.0011 0.0011 0.0010 0.1327 95% 0.0055 0.0055 0.1297 0.1295 0.1291 Υ -0.0072 0.0006 -0.0014 0.0003 -0.0010 0.0003 -0.0004 0.0003 0.0956 96% 0.0044 0.0045 0.0651 0.0649 0.0648 17,22,13 15,20,11 λ1 0.0356 0.0163 0.0025 0.0053 0.0017 0.0053 0.0002 0.0053 0.4812 95% 0.0227 0.0228 0.2787 0.2783 0.2783 λ2 0.1404 0.4167 0.0039 0.0081 0.0029 0.0081 0.0010 0.0081 2.4722 94% 0.1128 0.1126 0.3422 0.3417 0.3443 λ3 0.0425 0.0301 0.0068 0.0081 0.0058 0.0080 0.0038 0.0079 0.6284 99% 0.0308 0.0306 0.3472 0.3450 0.3413 θ 0.0064 0.0011 0.0048 0.0010 0.0045 0.0010 0.0038 0.0009 0.1278 95% 0.0055 0.0053 0.1334 0.1329 0.1325 Υ -0.0086 0.0006 -0.0003 0.0002 -0.0001 0.0002 0.0002 0.0002 0.0915 95% 0.0041 0.0040 0.0534 0.0532 0.0527 2 15,18,11 12,16,10 λ1 0.0551 0.0214 0.0124 0.0095 0.0106 0.0093 0.0071 0.0091 0.5320 96% 0.0228 0.0229 0.3717 0.3692 0.3641 λ2 0.2231 0.7048 -0.0037 0.0197 -0.0063 0.0198 -0.0115 0.0199 3.1757 95% 0.1536 0.1519 0.5060 0.5048 0.5059 λ3 0.0623 0.5909 0.0051 0.0216 0.0024 0.0215 -0.0031 0.0215 3.0064 95% 0.1394 0.1416 0.5883 0.5866 0.5928 θ 0.0067 0.0014 0.0027 0.0013 0.0022 0.0013 0.0011 0.0013 0.1440 95% 0.0067 0.0067 0.1298 0.1296 0.1293 Υ -0.0134 0.0009 -0.0020 0.0003 -0.0016 0.0003 -0.0009 0.0003 0.1040 96% 0.0049 0.0049 0.0712 0.0708 0.0702 17,22,13 15,20,11 λ1 0.0363 0.0182 0.0024 0.0055 0.0016 0.0055 0.0000 0.0054 0.5102 96% 0.0228 0.0219 0.2805 0.2798 0.2789 λ2 0.1437 0.3905 0.0022 0.0083 0.0012 0.0083 -0.0008 0.0084 2.3865 95% 0.1113 0.1090 0.3542 0.3541 0.3490 λ3 0.0144 0.1303 -0.0016 0.0087 -0.0026 0.0088 -0.0047 0.0088 1.4152 99% 0.0601 0.0637 0.3829 0.3823 0.3800 θ 0.0045 0.0013 0.0037 0.0012 0.0033 0.0012 0.0026 0.0012 0.1384 96% 0.0059 0.0062 0.1371 0.1373 0.1370 Υ -0.0088 0.0007 -0.0003 0.0002 -0.0002 0.0002 0.0002 0.0002 0.0958 96% 0.0044 0.0043 0.0534 0.0532 0.0530 30,40,30 1 25,30,22 20,24,20 λ1 0.0282 0.0105 0.0120 0.0070 0.0104 0.0069 0.0072 0.0067 0.3872 97% 0.0186 0.0185 0.3222 0.3217 0.3200 λ2 0.1331 0.0243 0.0102 0.0190 0.0075 0.0199 0.0023 0.0199 1.8612 94% 0.0786 0.0764 0.5319 0.5279 0.5310 λ3 0.0068 0.0011 -0.0106 0.0216 -0.0132 0.0217 -0.0184 0.0218 1.4103 98% 0.0662 0.0806 0.5735 0.5757 0.5758 θ 0.0009 0.0005 0.0000 0.0005 -0.0003 0.0005 -0.0008 0.0005 0.0905 95% 0.0043 0.0043 0.0897 0.0897 0.0896 Υ -0.0066 0.0004 -0.0022 0.0003 -0.0019 0.0002 -0.0013 0.0002 0.0731 96% 0.0034 0.0035 0.0613 0.0612 0.0608 27,34,26 23,30,23 λ1 0.0234 0.0091 0.0060 0.0048 0.0052 0.0048 0.0037 0.0048 0.4103 96% 0.0193 0.0194 0.2629 0.2622 0.2610 λ2 0.0326 0.0235 -0.0042 0.0076 -0.0052 0.0076 -0.0071 0.0077 1.8923 95% 0.0851 0.0690 0.3244 0.3225 0.3220 λ3 0.0053 0.0010 -0.0002 0.0008 -0.0012 0.0084 -0.0031 0.0084 0.8675 97% 0.0380 0.0381 0.3506 0.3513 0.3511 θ 0.0024 0.0005 0.0024 0.0005 0.0022 0.0006 0.0017 0.0004 0.0903 95% 0.0039 0.0038 0.0950 0.0950 0.0950 Υ -0.0081 0.0004 -0.0009 0.0002 -0.0008 0.0002 -0.0005 0.0002 0.0761 95% 0.0032 0.0032 0.0506 0.0507 0.0507 2 25,30,22 20,24,20 λ1 0.0374 0.0117 0.0143 0.0080 0.0128 0.0079 0.0098 0.0077 0.3974 95% 0.0169 0.0170 0.3298 0.3274 0.3228 λ2 0.1411 0.3371 0.0129 0.0194 0.0103 0.0193 0.0051 0.0191 2.2098 95% 0.0976 0.0985 0.5626 0.5598 0.5562 λ3 0.0702 0.0009 -0.0027 0.0008 -0.0053 0.0007 -0.0106 0.0006 1.0712 97% 0.0523 0.0534 0.5576 0.5555 0.5481 θ 0.0015 0.0007 0.0009 0.0007 0.0006 0.0007 0.0000 0.0007 0.1032 95% 0.0048 0.0048 0.1056 0.1057 0.1053 Υ -0.0089 0.0004 -0.0026 0.0003 -0.0023 0.0003 -0.0018 0.0003 0.0742 95% 0.0033 0.0034 0.0616 0.0614 0.0612 27,34,26 23,30,23 λ1 0.0302 0.0113 0.0057 0.0054 0.0050 0.0054 0.0035 0.0053 0.3999 94% 0.0159 0.0169 0.2829 0.2816 0.2793 λ2 0.1151 0.1259 -0.0072 0.0076 -0.0082 0.0077 -0.0102 0.0077 1.9472 96% 0.0885 0.0878 0.3480 0.3489 0.3468 λ3 0.0118 0.0008 0.0004 0.0007 -0.0007 0.0007 -0.0027 0.0006 0.9266 99% 0.0410 0.0412 0.3375 0.3362 0.3380 θ 0.0033 0.0007 0.0028 0.0008 0.0025 0.0006 0.0021 0.0005 0.1023 95% 0.0045 0.0045 0.1047 0.1047 0.1049 Υ -0.0069 0.0004 -0.0008 0.0002 -0.0007 0.0002 -0.0004 0.0002 0.0715 94% 0.0032 0.0032 0.0530 0.0529 0.0526 At 42:4kv, the dataset is 0.6, 13.4, 15.2, 19.9, 25.0, 30.2, 32.8, 44.4, 56.2. At 46:7kv, the dataset is 3.1, 8.3, 8.9, 9.0, 13.6, 14.9, 16.1, 16.9, 21.3,48.1. At 35:4kv, the dataset is 40.1, 59.4, 71.2, 166.5, 204.7, 229.7, 308.3, and 537.9. We wish to determine the SSM P (U < W < V ) dependability. First, Table 5 presents estimated values for several measures as: ’Kolmogorov Smirnov distance (KS) statistic along with its P-value (P-V), Akaike information criterion (AIC), Bayesian information criterion (BIC), corrected AIC (CAIC), and Hannan-Quinn information criterion (HQIC)’ for the EWD. The EWD fits each data set according to the KS test. The ML estimates via a complete sample are listed in Table 6 for EWD and EED with stress-strength Υ. According to the value of different measures and P-V of Υ, it is noted that the EWD is better than the EED. Also, the SE for the ML estimate is smaller than that of the other estimates. In light of the GPHC, the Bayesian estimation method represents the most accurate for the EWD. The Bayesian estimating method’s reliability is higher than the ML process’s, supporting the stated result. Figures 3, 4, and 5, prove the fitting of each data set fitted by EWD, according to empirical CDF, histogram, and probability- probability (P-P) behavior plots. A. S. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6337 16 of 26 Table 2: Bias, MSE, and LCI for Estimation methods: Case II λ1 = 0.5, λ2 = 1.2, λ3 = 10, θ = 3 τ1 = 1, τ2 = 1.2, τ3 = 1.6 ML SELF LINEX c=0.5 LINEX c=1.5 ML Bayesian n1, n2, n3 Scheme s1, s2, s3 k1, k2, k3 Bias MSE Bias MSE Bias MSE Bias MSE LACI CP LCIBP LCIBT LCCI1 LCCI2 LCCI3 20,25,15 1 15,18,11 12,16,10 λ1 0.0193 0.0155 0.0128 0.0095 0.0111 0.0094 0.0079 0.0092 0.4826 95% 0.0231 0.0230 0.3574 0.3550 0.3515 λ2 0.0498 0.0708 -0.0011 0.0188 -0.0036 0.0187 -0.0087 0.0187 1.0260 96% 0.0476 0.0480 0.5286 0.5272 0.5268 λ3 1.1425 7.8730 -0.0019 0.0238 -0.0047 0.0238 -0.0105 0.0239 10.0560 96% 0.4620 0.4721 0.5673 0.5671 0.5658 θ 0.1315 0.1374 0.0015 0.0137 -0.0006 0.0137 -0.0047 0.0138 1.3597 95% 0.0637 0.0630 0.4445 0.4448 0.4449 Υ 0.0003 0.0029 -0.0049 0.0017 -0.0045 0.0017 -0.0038 0.0017 0.2119 95% 0.0095 0.0097 0.1533 0.1535 0.1541 17,22,13 15,20,11 λ1 0.0170 0.0149 0.0059 0.0053 0.0051 0.0052 0.0034 0.0052 0.4509 95% 0.0226 0.0226 0.2665 0.2674 0.2695 λ2 0.0430 0.0690 0.0003 0.0073 -0.0007 0.0073 -0.0025 0.0073 1.0168 96% 0.0489 0.0474 0.3184 0.3180 0.3178 λ3 1.0793 4.7472 0.0066 0.0080 0.0056 0.0080 0.0037 0.0080 10.0673 96% 0.4718 0.4477 0.3657 0.3658 0.3634 θ 0.0811 0.1010 -0.0009 0.0067 -0.0019 0.0067 -0.0037 0.0067 1.2054 95% 0.0572 0.0570 0.3174 0.3172 0.3150 Υ -0.0003 0.0028 -0.0017 0.0009 -0.0015 0.0009 -0.0010 0.0009 0.2114 96% 0.0099 0.0093 0.1136 0.1141 0.1149 2 15,18,11 12,16,10 λ1 0.0299 0.0181 0.0099 0.0098 0.0082 0.0097 0.0047 0.0095 0.5152 97% 0.0247 0.0246 0.3533 0.3525 0.3527 λ2 0.0642 0.0853 -0.0010 0.0180 -0.0035 0.0180 -0.0086 0.0182 1.1182 95% 0.0512 0.0504 0.5110 0.5130 0.5080 λ3 0.7819 6.4649 0.0077 0.0235 0.0048 0.0233 -0.0011 0.0231 9.4937 96% 0.4457 0.4583 0.6016 0.6044 0.6026 θ 0.0913 0.1337 0.0034 0.0154 0.0011 0.0154 -0.0036 0.0153 1.3895 94% 0.0621 0.0624 0.4865 0.4826 0.4787 Υ -0.0054 0.0033 -0.0034 0.0017 -0.0030 0.0017 -0.0022 0.0017 0.2254 96% 0.0110 0.0108 0.1481 0.1481 0.1474 17,22,13 15,20,11 λ1 0.0230 0.0160 -0.0013 0.0052 -0.0021 0.0052 -0.0037 0.0052 0.4888 95% 0.0220 0.0215 0.2630 0.2625 0.2628 λ2 0.0750 0.0843 0.0031 0.0081 0.0022 0.0081 0.0002 0.0081 1.1006 94% 0.0489 0.0503 0.3410 0.3388 0.3393 λ3 0.9118 3.0464 -0.0007 0.0073 -0.0017 0.0073 -0.0038 0.0073 7.2292 96% 0.4381 0.4355 0.3225 0.3224 0.3244 θ 0.0914 0.1008 0.0011 0.0075 0.0002 0.0075 -0.0017 0.0075 1.1930 95% 0.0519 0.0530 0.3267 0.3266 0.3260 Υ 0.0006 0.0028 0.0013 0.0010 0.0015 0.0010 0.0019 0.0010 0.2070 95% 0.0103 0.0101 0.1155 0.1155 0.1147 30,40,30 1 25,30,22 20,24,20 λ1 0.0131 0.0087 0.0166 0.0081 0.0150 0.0079 0.0116 0.0076 0.3624 95% 0.0157 0.0157 0.3287 0.3232 0.3182 λ2 0.0421 0.0437 0.0080 0.0159 0.0056 0.0158 0.0009 0.0156 0.8036 95% 0.0364 0.0361 0.4913 0.4876 0.4838 λ3 0.9674 4.0388 -0.0012 0.0228 -0.0041 0.0229 -0.0099 0.0232 6.9122 96% 0.3122 0.3125 0.6000 0.6053 0.6134 θ 0.0804 0.0559 0.0097 0.0124 0.0077 0.0123 0.0037 0.0123 0.8725 96% 0.0397 0.0396 0.4200 0.4196 0.4182 Υ 0.0031 0.0016 -0.0056 0.0014 -0.0052 0.0013 -0.0044 0.0013 0.1582 95% 0.0067 0.0067 0.1334 0.1325 0.1313 27,34,26 23,30,23 λ1 0.0159 0.0079 0.0069 0.0047 0.0062 0.0047 0.0047 0.0046 0.3442 95% 0.0143 0.0144 0.2593 0.2584 0.2555 λ2 0.0450 0.0391 0.0013 0.0066 0.0004 0.0066 -0.0014 0.0066 0.7553 96% 0.0339 0.0341 0.3140 0.3159 0.3185 λ3 0.8746 3.3364 0.0064 0.0080 0.0053 0.0080 0.0033 0.0080 4.2210 94% 0.3042 0.3041 0.3275 0.3269 0.3252 θ 0.0767 0.0519 0.0068 0.0066 0.0059 0.0066 0.0042 0.0066 0.8190 95% 0.0342 0.0314 0.3236 0.3232 0.3205 Υ 0.0026 0.0016 -0.0021 0.0008 -0.0019 0.0008 -0.0015 0.0008 0.1565 95% 0.0065 0.0064 0.1084 0.1084 0.1084 2 25,30,22 20,24,20 λ1 0.0255 0.0116 0.0096 0.0069 0.0079 0.0068 0.0047 0.0066 0.4099 95% 0.0191 0.0191 0.2993 0.2989 0.2972 λ2 0.0562 0.0617 0.0117 0.0191 0.0092 0.0190 0.0044 0.0188 0.9491 95% 0.0430 0.0434 0.5359 0.5350 0.5368 λ3 0.8660 4.2109 -0.0091 0.0225 -0.0121 0.0226 -0.0180 0.0229 7.3000 95% 0.3315 0.3315 0.6000 0.5992 0.6038 θ 0.0775 0.0794 0.0026 0.0130 0.0004 0.0129 -0.0038 0.0129 1.0628 94% 0.0490 0.0487 0.4419 0.4416 0.4449 Υ -0.0020 0.0022 -0.0030 0.0012 -0.0026 0.0012 -0.0019 0.0012 0.1832 96% 0.0080 0.0078 0.1322 0.1317 0.1309 27,34,26 23,30,23 λ1 0.0194 0.0114 0.0042 0.0054 0.0034 0.0054 0.0018 0.0053 0.3541 95% 0.0152 0.0182 0.2849 0.2851 0.2843 λ2 0.0584 0.0510 -0.0008 0.0080 -0.0018 0.0080 -0.0038 0.0080 0.8556 97% 0.0384 0.0382 0.3434 0.3462 0.3461 λ3 0.6843 1.6573 0.0079 0.0081 0.0068 0.0080 0.0046 0.0080 7.0074 95% 0.2970 0.2985 0.3455 0.3456 0.3460 θ 0.0612 0.0579 0.0058 0.0070 0.0049 0.0070 0.0031 0.0069 0.9127 95% 0.0389 0.0389 0.3182 0.3170 0.3151 Υ 0.0001 0.0020 -0.0012 0.0010 -0.0010 0.0010 -0.0006 0.0010 0.1762 95% 0.0078 0.0068 0.1205 0.1207 0.1212 10 20 30 40 0.0 0.2 0.4 0.6 0.8 1.0 10 20 30 40 0.0 0.2 0.4 0.6 0.8 1.0 x F(x ) Empirical CDF Estimated CDF Estimated PDF x f(x ) 0 10 20 30 40 50 0.0 0 0.0 1 0.0 2 0.0 3 0.0 4 0.0 5 0.0 6 0.04 0.04 0.01 0 0.01 PP probability(x) F(x ) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Figure 3: Empirical CDF, histogram, and P-P plots for the EWD for data set 1 Figure 6 discussed the profile likelihood of the EWD parameters, which index the estimators’ maximum log-likelihood values. Figure 7 discussed count-our plots of log- likelihood values with EWD parameters which are indicated by the estimators’ uniqueness. The posterior distribution for the MCMC estimate of the SSM for the EWD based on the GPHC is shown in Figure 8 along with its trace and normal curve. Figure 10 presents the MCMC samples as a pairs plot, which shows the pairwise correlation between parameters in the top plot, correlation coefficients in the bottom plot, and marginal fre- quency distribution for each parameter on the diagonal. Additionally, as seen in Figure 9, A. S. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6337 17 of 26 Table 3: Bias, MSE, and LCI for Estimation methods: Case III λ1 = 0.3, λ2 = 0.6, λ3 = 5, θ = 0.8 τ1 = 2, τ2 = 1.8, τ3 = 3 ML SELF LINEX c=0.5 LINEX c=1.5 ML Bayesian n1, n2, n3 Scheme s1, s2, s3 k1, k2, k3 Bias MSE Bias MSE Bias MSE Bias MSE LACI CP LCIBP LCIBT LCCI1 LCCI2 LCCI3 20,25,15 1 15,18,11 12,16,10 λ1 0.0082 0.0078 0.0196 0.0062 0.0183 0.0060 0.0156 0.0058 0.3461 95% 0.0148 0.0145 0.2770 0.2757 0.2731 λ2 0.0091 0.0254 0.0076 0.0120 0.0055 0.0119 0.0015 0.0117 0.6240 95% 0.0268 0.0269 0.4186 0.4188 0.4189 λ3 0.3314 1.8832 -0.0060 0.0209 -0.0086 0.0211 -0.0138 0.0213 5.2253 96% 0.2362 0.2366 0.5804 0.5804 0.5847 θ 0.0484 0.0256 0.0063 0.0081 0.0047 0.0081 0.0014 0.0080 0.5990 94% 0.0264 0.0263 0.3331 0.3323 0.3296 Υ -0.0048 0.0040 -0.0126 0.0033 -0.0123 0.0033 -0.0115 0.0032 0.2477 94% 0.0110 0.0111 0.2177 0.2171 0.2177 17,22,13 15,20,11 λ1 0.0112 0.0068 -0.0002 0.0038 -0.0009 0.0038 -0.0022 0.0038 0.3202 95% 0.0144 0.0142 0.2323 0.2315 0.2283 λ2 0.0142 0.0248 -0.0034 0.0058 -0.0043 0.0058 -0.0060 0.0058 0.6148 96% 0.0285 0.0283 0.2935 0.2928 0.2915 λ3 0.2902 1.6877 0.0059 0.0076 0.0049 0.0076 0.0030 0.0076 4.9689 97% 0.2380 0.2384 0.3390 0.3395 0.3378 θ 0.0359 0.0206 0.0048 0.0048 0.0040 0.0047 0.0025 0.0047 0.5454 94% 0.0255 0.0257 0.2535 0.2537 0.2544 Υ -0.0067 0.0038 0.0002 0.0023 0.0005 0.0023 0.0010 0.0023 0.2417 95% 0.0112 0.0113 0.1803 0.1801 0.1801 2 15,18,11 12,16,10 λ1 0.0081 0.0084 0.0134 0.0065 0.0121 0.0063 0.0093 0.0060 0.3578 95% 0.0153 0.0154 0.2818 0.2793 0.2762 λ2 0.0085 0.0278 -0.0062 0.0139 -0.0084 0.0138 -0.0126 0.0136 0.6536 95% 0.0285 0.0282 0.4266 0.4253 0.4255 λ3 0.3314 1.9484 -0.0040 0.0230 -0.0068 0.0231 -0.0124 0.0233 5.3205 96% 0.2465 0.2366 0.5550 0.5529 0.5541 θ 0.0454 0.0211 0.0122 0.0097 0.0104 0.0096 0.0067 0.0094 0.5413 96% 0.0236 0.0229 0.3715 0.3714 0.3683 Υ -0.0059 0.0043 -0.0113 0.0034 -0.0110 0.0034 -0.0102 0.0033 0.2573 96% 0.0111 0.0111 0.2102 0.2107 0.2077 17,22,13 15,20,11 λ1 0.0181 0.0082 0.0041 0.0041 0.0035 0.0041 0.0021 0.0040 0.3483 95% 0.0147 0.0148 0.2394 0.2391 0.2378 λ2 0.0287 0.0265 0.0004 0.0069 -0.0004 0.0069 -0.0022 0.0068 0.6289 97% 0.0272 0.0272 0.3203 0.3194 0.3150 λ3 0.3534 1.7370 -0.0012 0.0081 -0.0022 0.0081 -0.0043 0.0081 4.9821 96% 0.2196 0.2212 0.3555 0.3566 0.3577 θ 0.0253 0.0184 0.0047 0.0056 0.0040 0.0056 0.0024 0.0056 0.5223 94% 0.0224 0.0214 0.2910 0.2913 0.2905 Υ -0.0052 0.0042 -0.0023 0.0025 -0.0021 0.0025 -0.0015 0.0025 0.2468 95% 0.0110 0.0103 0.1845 0.1851 0.1866 30,40,30 1 28,30,28 18,22,22 λ1 0.0081 0.0042 0.0128 0.0037 0.0118 0.0037 0.0098 0.0035 0.2508 95% 0.0113 0.0112 0.2233 0.2218 0.2178 λ2 0.0179 0.0138 0.0095 0.0092 0.0078 0.0090 0.0044 0.0089 0.4561 95% 0.0190 0.0190 0.3643 0.3640 0.3636 λ3 0.3321 0.6837 0.0020 0.0211 -0.0008 0.0211 -0.0063 0.0211 2.9713 95% 0.1332 0.1349 0.5546 0.5509 0.5515 θ 0.0278 0.0085 0.0081 0.0052 0.0069 0.0052 0.0044 0.0051 0.3439 96% 0.0153 0.0153 0.2760 0.2757 0.2782 Υ 0.0006 0.0022 -0.0077 0.0021 -0.0074 0.0021 -0.0068 0.0021 0.1834 97% 0.0079 0.0079 0.1712 0.1709 0.1709 25,35,35 22,30,30 λ1 0.0057 0.0039 0.0083 0.0032 0.0077 0.0031 0.0064 0.0030 0.2440 95% 0.0110 0.0106 0.2069 0.2060 0.2022 λ2 0.0135 0.0115 0.0072 0.0050 0.0065 0.0050 0.0049 0.0050 0.4173 96% 0.0187 0.0191 0.2734 0.2718 0.2716 λ3 0.2902 0.6492 0.0060 0.0079 0.0050 0.0079 0.0030 0.0078 3.0593 96% 0.1241 0.1421 0.3382 0.3369 0.3343 θ 0.0306 0.0077 0.0049 0.0036 0.0043 0.0036 0.0030 0.0036 0.3221 96% 0.0150 0.0156 0.2283 0.2288 0.2295 Υ 0.0016 0.0021 -0.0038 0.0017 -0.0035 0.0017 -0.0030 0.0016 0.1800 95% 0.0077 0.0077 0.1514 0.1512 0.1495 2 28,30,28 18,22,22 λ1 0.0161 0.0060 0.0180 0.0052 0.0168 0.0051 0.0144 0.0049 0.2979 96% 0.0139 0.0140 0.2627 0.2620 0.2612 λ2 0.0230 0.0150 0.0122 0.0110 0.0104 0.0109 0.0068 0.0106 0.4720 94% 0.0223 0.0221 0.3958 0.3897 0.3834 λ3 0.3658 1.0361 -0.0004 0.0222 -0.0034 0.0223 -0.0093 0.0223 3.7274 95% 0.1684 0.1759 0.5571 0.5613 0.5589 θ 0.0272 0.0098 0.0105 0.0062 0.0090 0.0061 0.0061 0.0060 0.3737 95% 0.0173 0.0170 0.3030 0.3012 0.3014 Υ -0.0025 0.0027 -0.0102 0.0026 -0.0099 0.0025 -0.0092 0.0025 0.2030 96% 0.0086 0.0086 0.1905 0.1900 0.1899 25,35,35 22,30,30 λ1 0.0095 0.0042 0.0105 0.0031 0.0099 0.0031 0.0086 0.0030 0.2527 95% 0.0109 0.0109 0.2035 0.2019 0.2014 λ2 0.0255 0.0152 -0.0007 0.0050 -0.0015 0.0050 -0.0030 0.0049 0.4736 95% 0.0225 0.0225 0.2685 0.2683 0.2683 λ3 0.3813 0.9327 -0.0045 0.0075 -0.0055 0.0075 -0.0075 0.0075 3.4817 95% 0.1526 0.1498 0.3194 0.3201 0.3195 θ 0.0283 0.0081 0.0084 0.0040 0.0077 0.0039 0.0064 0.0039 0.3356 95% 0.0149 0.0147 0.2323 0.2319 0.2320 Υ 0.0016 0.0022 -0.0072 0.0017 -0.0070 0.0017 -0.0065 0.0017 0.1855 95% 0.0078 0.0079 0.1558 0.1558 0.1551 0 10 20 30 40 50 0.0 0.2 0.4 0.6 0.8 1.0 0 10 20 30 40 50 0.0 0.2 0.4 0.6 0.8 1.0 y F(y ) Empirical CDF Estimated CDF Estimated PDF y f(y ) 0 10 20 30 40 50 60 0.0 00 0.0 05 0.0 10 0.0 15 0.0 20 0.0 25 0.0 30 0.009 0.027 0.009 0.018 0.009 0.027 PP probability(y) F(y ) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Figure 4: Empirical CDF, histogram, and P-P plots for the EWD for data set 2 convergence starts at 2000 iterations or less for the SSM estimate for EWD based on the entire sample. Figures 13 and 12 show the MCMC samples as a pairs plot that represents the pairwise relationship between parameters as independent with the scatter plot matrix in the top plot, correlation coefficients in the bottom plot, and marginal frequency distri- bution for each parameter on the diagonal. In this picture, the parameters p3 and p4 are shown to be medially related, where p1 is a λ1, p2 is a λ2, p3 is a λ3, and p4 is a θ. For each component of this model, we suggested using the following GPHC sample as follows: A. S. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6337 18 of 26 Table 4: Bias, MSE, and LCI for Estimation methods: Case IV λ1 = 1.3, λ2 = 5, λ3 = 15, θ = 1.5 n1 = 30, n2 = 40, n3 = 40 ML SELF LINEX c=0.5 LINEX c=1.5 ML Bayesian τ1, τ2, τ3 Scheme s1, s2, s3 k1, k2, k3 Bias MSE Bias MSE Bias MSE Bias MSE LACI CP LCIBP LCIBT LCCI1 LCCI2 LCCI3 2,1.8,3 1 20,28,28 18,22,22 λ1 0.0436 0.0673 0.0268 0.0132 0.0169 0.0124 -0.0025 0.0114 1.0034 95% 0.0455 0.0460 0.4201 0.4178 0.4044 λ2 0.1569 0.7586 0.0122 0.0787 -0.0201 0.0779 -0.0825 0.0842 3.3618 96% 0.1516 0.1512 1.0791 1.0632 1.0881 λ3 0.1916 1.8396 -0.0229 0.5429 -0.0932 0.5571 -0.2204 0.6113 5.2688 94% 0.2449 0.2454 2.8574 2.7490 2.6732 θ 0.0118 0.0062 0.0025 0.0044 0.0003 0.0044 -0.0040 0.0044 0.3049 95% 0.0141 0.0142 0.2574 0.2570 0.2553 Υ -0.0055 0.0012 -0.0037 0.0003 -0.0033 0.0003 -0.0023 0.0002 0.1336 95% 0.0062 0.0062 0.0584 0.0569 0.0560 25,35,35 22,30,30 λ1 0.0588 0.0617 0.0288 0.0121 0.0210 0.0120 0.0056 0.0109 0.9804 96% 0.0448 0.0472 0.5470 0.5410 0.5312 λ2 0.1264 0.7014 -0.0015 0.0749 -0.0166 0.0729 -0.0462 0.0817 3.2487 95% 0.1430 0.1370 1.1184 1.1026 1.0797 λ3 0.1802 1.2887 0.0000 0.1828 -0.0226 0.1841 -0.0664 0.1901 4.8852 94% 0.2367 0.2357 1.6691 1.6675 1.6510 θ 0.0074 0.0051 0.0015 0.0037 -0.0003 0.0037 -0.0038 0.0037 0.2794 96% 0.0124 0.0124 0.2292 0.2278 0.2265 Υ -0.0068 0.0011 -0.0033 0.0003 -0.0027 0.0003 -0.0013 0.0003 0.1372 96% 0.0058 0.0058 0.0625 0.0616 0.0619 2 20,28,28 18,22,22 λ1 0.0583 0.0914 0.0257 0.0127 0.0147 0.0119 -0.0067 0.0109 1.1637 96% 0.0501 0.0501 0.4224 0.4134 0.3968 λ2 0.1521 0.8848 0.0049 0.0707 -0.0277 0.0703 -0.0903 0.0774 3.6424 97% 0.1535 0.1535 1.0520 1.0518 1.0699 λ3 0.2225 2.9436 0.0244 0.5724 -0.0503 0.5641 -0.1839 0.5905 6.6753 95% 0.2892 0.2902 2.9323 2.8524 2.7197 θ 0.0128 0.0078 0.0046 0.0049 0.0022 0.0049 -0.0026 0.0048 0.3419 95% 0.0151 0.0152 0.2682 0.2675 0.2660 Υ -0.0071 0.0014 -0.0030 0.0002 -0.0025 0.0002 -0.0013 0.0002 0.1437 94% 0.0068 0.0070 0.0564 0.0563 0.0559 25,35,35 22,30,30 λ1 0.0552 0.0900 0.0261 0.0129 0.0180 0.0109 0.0021 0.0108 1.1571 96% 0.0495 0.0495 0.4945 0.5197 0.5101 λ2 0.2528 0.8691 0.0063 0.0681 -0.0087 0.0681 -0.0382 0.0618 3.6131 94% 0.1508 0.1529 1.0891 1.0761 1.0816 λ3 0.1178 1.8800 0.0102 0.1480 -0.0109 0.1484 -0.0521 0.1533 5.3603 95% 0.2305 0.2290 1.5367 1.5377 1.4936 θ 0.0087 0.0053 0.0019 0.0040 0.0001 0.0040 -0.0037 0.0040 0.2840 96% 0.0133 0.0130 0.2419 0.2420 0.2424 Υ -0.0076 0.0013 -0.0029 0.0002 -0.0021 0.0003 -0.0007 0.0002 0.1390 96% 0.0055 0.0057 0.0598 0.0598 0.0612 2.5,2.5,4 1 20,28,28 18,22,22 λ1 0.0416 0.0659 0.0117 0.0132 0.0095 0.0131 0.0050 0.0130 1.0038 95% 0.0421 0.0432 0.4289 0.4290 0.4309 λ2 0.1320 0.7271 -0.0161 0.0220 -0.0188 0.0222 -0.0241 0.0227 3.4061 95% 0.1578 0.1571 0.5741 0.5771 0.5809 λ3 0.1546 1.6025 -0.0066 0.0230 -0.0095 0.0232 -0.0153 0.0235 5.5506 95% 0.2547 0.2547 0.5874 0.5837 0.5918 θ 0.0178 0.0061 0.0095 0.0044 0.0084 0.0044 0.0060 0.0043 0.2992 95% 0.0136 0.0133 0.2610 0.2612 0.2603 Υ -0.0080 0.0011 -0.0012 0.0002 -0.0010 0.0002 -0.0005 0.0002 0.1261 95% 0.0058 0.0058 0.0498 0.0498 0.0506 25,35,35 22,30,30 λ1 0.0409 0.0624 -0.0006 0.0069 -0.0015 0.0069 -0.0033 0.0069 0.9773 95% 0.0420 0.0424 0.3138 0.3137 0.3120 λ2 0.1143 0.6581 0.0070 0.0081 0.0060 0.0081 0.0040 0.0080 3.1516 95% 0.1392 0.1445 0.3528 0.3523 0.3511 λ3 0.1512 1.1030 0.0018 0.0085 0.0008 0.0084 -0.0011 0.0084 5.5025 95% 0.2440 0.2438 0.3480 0.3471 0.3481 θ 0.0097 0.0044 0.0022 0.0033 0.0016 0.0033 0.0004 0.0033 0.2589 95% 0.0130 0.0128 0.2213 0.2208 0.2199 Υ -0.0040 0.0010 0.0002 0.0001 0.0003 0.0001 0.0005 0.0001 0.1258 95% 0.0056 0.0056 0.0357 0.0357 0.0356 2 20,28,28 18,22,22 λ1 0.0315 0.0862 -0.0093 0.0126 -0.0115 0.0157 -0.0159 0.0158 1.1455 95% 0.0523 0.0523 0.4861 0.4858 0.4857 λ2 0.1702 0.8386 -0.0208 0.0219 -0.0235 0.0221 -0.0290 0.0226 3.5308 95% 0.1509 0.1506 0.5741 0.5742 0.5848 λ3 0.0818 2.0260 -0.0086 0.0253 -0.0117 0.0254 -0.0180 0.0258 5.5759 95% 0.2527 0.2540 0.6038 0.6118 0.6167 θ 0.0136 0.0061 0.0027 0.0045 0.0014 0.0054 -0.0012 0.0054 0.3021 96% 0.0133 0.0133 0.2822 0.2813 0.2818 Υ -0.0049 0.0014 0.0012 0.0002 0.0015 0.0002 0.0019 0.0002 0.1430 95% 0.0063 0.0061 0.0567 0.0560 0.0561 25,35,35 22,30,30 λ1 0.0524 0.0754 0.0008 0.0070 -0.0001 0.0070 -0.0020 0.0070 1.0576 97% 0.0473 0.0472 0.3323 0.3330 0.3302 λ2 0.1255 0.8078 0.0048 0.0079 0.0038 0.0079 0.0018 0.0079 2.6264 96% 0.1464 0.1473 0.3296 0.3299 0.3308 λ3 0.0900 1.9416 0.0088 0.0093 0.0077 0.0092 0.0056 0.0091 4.0894 93% 0.2170 0.2171 0.3673 0.3665 0.3669 θ 0.0091 0.0049 0.0018 0.0035 0.0012 0.0035 0.0000 0.0035 0.2729 94% 0.0124 0.0124 0.2251 0.2252 0.2247 Υ -0.0080 0.0014 0.0001 0.0001 0.0002 0.0001 0.0004 0.0001 0.1408 96% 0.0064 0.0063 0.0389 0.0388 0.0389 Table 5: ML estimates for parameters of EWD with different measures Estimates SE KS P-V AIC CAIC BIC HQIC u EWD λ1 20.6859 9.2567 0.1680 0.8978 75.6779 77.3921 76.2830 75.0140 θ 0.4841 0.0557 w EWD λ2 10.2683 3.5525 0.2689 0.4039 106.5047 108.0047 107.3005 106.0030 θ 0.3303 0.0398 v EWD λ3 39.2764 21.1171 0.1479 0.9810 141.3349 143.0492 141.9401 140.6710 θ 0.2623 0.0283 V = (40.1, 59.4, 71.2, 166.5, 204.7, 229.7, 308.3), U= (3.1, 8.3, 8.9, 9.0, 13.6, 14.9, 16.1), W= (0.6, 13.4, 15.2, 19.9, 25.0, 30.2, 32.8, 44.4), R2 = (0, 0, 0, 0, 0, 0, 0, 1, 0), R1 =(0, 0, 0, 0, 0, 0, 0, 0, 1), R3=(0, 0, 0, 0, 0, 0, 0, 2). Table 7 discusses the ML and Bayesian estimates of the EWD parameters based on the GPHC sample of the SSM. Figure 11 displays the posterior distribution’s trace and A. S. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6337 19 of 26 200 400 600 800 1000 0.0 0.2 0.4 0.6 0.8 1.0 200 400 600 800 1000 0.0 0.2 0.4 0.6 0.8 1.0 z F(z ) Empirical CDF Estimated CDF Estimated PDF z f(z ) 0 200 400 600 800 1000 1200 0.0 00 0 0.0 01 0 0.0 02 0 0.0 03 0 0.002 0.002 0 0 0 0.001 P−P plot probability(z) F(z ) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Figure 5: Empirical CDF, histogram, and P-P plots for the EWD for Data set 3 Table 6: ML and Bayesian estimates for reliability in SSM based on complete sample for EWD and EED ML Bayesian Estimates SE Lower Upper Estimates SE Lower Upper EWD λ1 7.6867 2.5593 2.6704 12.7030 2.5194 0.7976 1.0926 4.0157 λ2 9.1172 2.8817 3.4691 14.7654 3.4328 1.1334 1.4230 5.6384 λ3 75.7190 34.8958 7.3232 144.1147 72.1292 25.4282 23.7217 122.9561 θ 0.3036 0.0204 0.2636 0.3437 0.0620 0.0064 0.0526 0.0726 Υ 0.3744 0.3910 EED λ1 0.6170 0.2216 0.1827 1.0513 0.6506 0.1647 0.2896 1.0329 λ2 0.5031 0.1781 0.1540 0.8523 0.5416 0.1716 0.2189 0.8249 λ3 3.6845 1.7466 0.2612 7.1077 4.0167 1.6591 1.0943 7.0266 Υ 0.3445 0.3503 normal curve for the SSM estimation for EWD based on the GPHC. Figure 12, which dis- plays the pairwise correlation between parameters in the top plot, correlation coefficients in the bottom plot, and the marginal frequency distribution for each parameter on the diagonal, presents the MCMC samples as a pairs plot. Additionally, as shown in Figure 13, convergence starts at 2000 iterations or less for the reliability estimate of the SSM for EWD based on the censored sample. Table 7: ML and Bayesian estimates for reliability of SSM ML Bayesian τ1, τ2, τ3 Estimates SE Lower Upper Estimates SE Lower Upper 20, 50, 500 λ1 7.7896 2.8023 2.2970 13.2821 8.1878 2.7952 3.3206 14.2400 λ2 8.7026 3.0212 2.7810 14.6241 9.3778 2.9498 3.9458 15.1873 λ3 68.6337 39.8506 1.4735 146.7409 85.1238 38.3644 15.9283 172.4221 θ 0.3108 0.0281 0.2557 0.3660 0.3155 0.0250 0.2681 0.3648 Υ 0.4254 0.4425 25, 60, 550 λ1 7.8717 2.8090 2.3660 13.3773 8.3905 2.7092 2.7466 14.6373 λ2 8.6463 2.9969 2.7725 14.5202 8.9148 2.8299 3.9046 14.4624 λ3 74.3011 41.5613 2.1590 155.7611 84.1920 36.4978 21.1738 154.9815 θ 0.3156 0.0263 0.2641 0.3670 0.3180 0.0230 0.2749 0.3650 Υ 0.4282 0.4293 A. S. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6337 20 of 26 6.5 7.5 8.5 − 16 1. 54 − 16 1. 50 − 16 1. 46 − 16 1. 42 λ1 l(γ ) 8.5 9.0 9.5 10.0 − 16 1. 46 − 16 1. 44 − 16 1. 42 λ2 l(γ ) 74.0 75.0 76.0− 16 1. 41 40 − 16 1. 41 30 − 16 1. 41 20 λ3 l(γ ) 0.1 0.3 0.5 − 40 0 − 35 0 − 30 0 − 25 0 − 20 0 θ l(γ ) Figure 6: Profile likelihood of EWD parameters 6.5 7.0 7.5 8.0 8.5 8. 6 8. 8 9. 0 9. 2 9. 4 9. 6 9. 8 10 .0 λ1 λ 2 −161.5451 −161.5451 −161.4914 −161.4687 −161.4687 −161.4534 −161.439 −161.4272 −161.4153 6.5 7.0 7.5 8.0 8.5 74 .0 74 .5 75 .0 75 .5 76 .0 76 .5 λ1 λ 3 −161.5288 −161.4762 −161.4762 −161.4536 −161.4536 −161.4371 −161.4371 −161.4249 −161.4249 −161.4167 −161.4167 −161.4123 8.6 9.0 9.4 9.8 74 .0 74 .5 75 .0 75 .5 76 .0 76 .5 λ2 λ 3 −161.4548 −161.4376 −161.4376 −161.429 −161.429 −161.4226 −161.4226 −161.4169 −161.4169 −161.4138 −161.4118 Figure 7: Count-our plots of log-likelihood values with EWD parameters 0 4000 10000 1 2 3 4 5 6 7 Iterations λ 1 0 4000 10000 2 4 6 8 10 Iterations λ 2 0 4000 10000 50 10 0 15 0 Iterations λ 3 0 4000 10000 0.0 4 0.0 5 0.0 6 0.0 7 0.0 8 0.0 9 Iterations θ λ1 Fre qu en cy 0 2 4 6 0.0 0.1 0.2 0.3 0.4 0.5 λ2 Fre qu en cy 2 4 6 8 10 0.0 0.1 0.2 0.3 λ3 Fre qu en cy 0 50 150 0.0 00 0.0 04 0.0 08 0.0 12 θ Fre qu en cy 0.04 0.06 0.08 0 10 20 30 40 50 60 Figure 8: Trace and normal curve of posterior distribution for MCMC estimation of SSM for EWD based on complete sample 7. Summary and Conclusion In this paper, it is explained how to draw the statistical conclusion that Υ = P (U < W < V ) for a component with a strength that is independent of opposite lower and upper bound stresses when the stresses and strength both follow EWD. We presume that the random variables for stresses and strength are both independent and have an EWD with a shared scale parameter. Due to ML and Bayesian methods, various point and interval es- A. S. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6337 21 of 26 0 2000 4000 6000 8000 10000 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 Bayes estimate of expression(Υ) time Υ Point estimate Lower limit Upper limit Υ F re q u e n c y 0.2 0.4 0.6 0.8 0 1 2 3 4 0 2000 4000 6000 8000 10000 0 .5 0 0 .5 5 0 .6 0 time Υ Figure 9: Trace and normal curve of the posterior distribution for MCMC estimation of SSM for EWD based on complete sample p1 2 4 6 8 10 0.071 0.19 1 2 3 4 5 6 7 0. 05 0. 10 0. 15 0. 20 0.3 2 4 6 8 10 p2 0.11 0.27 p3 50 100 150 0.29 0.05 0.10 0.15 0.20 1 2 3 4 5 6 7 50 10 0 15 0 p4 Figure 10: Pairs plot of the MCMC samples for parameter estimates of SSM for EWD based on complete sample timates for the reliability model Υ are derived using a GPHC design. The MCMC method and the MH algorithm, which are both based on the SELF and LINEX loss functions and are all carried out in the context of informative priors, both produce Bayesian estimators. Asymptotic distribution theory and the construction of Bayes credible intervals are used to derive CIs. Boot-T is preferable to Boot-P, according to discussions on bootstrap CIs. In order to compare the usefulness of the suggested estimates using several metrics, in- cluding average values, mean squared error, and length of CIs, the Monte Carlo simulation is carried out. The study’s findings show that, for four parameters and SSM based on the GPHC scheme, the Bayes estimates produce lower MSE. For illustrative reasons, an actual data set that has been gradually suppressed is given. This study’s main limitations in- volve using ML estimation and MCMC techniques for Υ = P (U < W < V ) computation. A. S. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6337 22 of 26 0 2000 4000 6000 8000 10000 5 1 0 1 5 2 0 Iterations λ 1 0 2000 4000 6000 8000 10000 5 1 0 1 5 2 0 Iterations λ 2 0 2000 4000 6000 8000 10000 0 5 0 1 0 0 1 5 0 2 0 0 Iterations λ 3 0 2000 4000 6000 8000 10000 0 .2 5 0 .3 0 0 .3 5 0 .4 0 Iterations θ λ1 F re q u e n cy 5 10 15 20 0 .0 0 0 .0 5 0 .1 0 0 .1 5 λ2 F re q u e n cy 5 10 15 20 0 .0 0 0 .0 4 0 .0 8 0 .1 2 λ3 F re q u e n cy 0 50 100 150 200 0 .0 0 0 0 .0 0 4 0 .0 0 8 0 .0 1 2 θ F re q u e n cy 0.25 0.30 0.35 0.40 0 5 1 0 1 5 Figure 11: Convergence plots of MCMC for parameter estimates of the EWD for censored sample p1 5 1 0 1 5 2 0 0.05 0.048 5 10 15 0 .2 5 0 .3 0 0 .3 5 0 .4 0 0.11 5 10 15 20 p2 −0.07 0.055 p3 0 50 100 150 200 250 0.58 0.25 0.30 0.35 0.40 5 1 0 1 5 0 5 0 1 0 0 2 0 0 p4 Figure 12: Pairs plots of the MCMC results for parameter estimates of stress-strength for EWD based on complete sample 0 2000 4000 6000 8000 10000 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 Bayes estimate of expression(Υ) time Υ Point estimate Lower limit Upper limit Υ F re q u e n cy 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 1 2 3 0 2000 4000 6000 8000 10000 0 .3 4 0 .3 6 0 .3 8 0 .4 0 0 .4 2 0 .4 4 time Υ Figure 13: Trace and normal curve of posterior distribution for MCMC estimation of reliability stress-strength for EWD based on censored sample A. 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