EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7126 ISSN 1307-5543 – ejpam.com Published by New York Business Global Statistical Inference of Accelerated Ishita Model Based on Type-I Generalized Hybrid Censoring Data with Applications Souha K. Badr1,∗ 1 Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah, Saudi Arabia Abstract. In this paper, we adopt the Ishita lifetime distribution to analyze biomedical science and engineering lifetime data under an accelerated life test (ALT) model. This data is exposed concerning the mechanism of a type-I generalized hybrid censoring scheme under a partially step- stress ALT model. The model parameters and the parameters of life (survival and hazard rate function) are estimated using maximum likelihood and Bayesian estimation. Also, the interval estimators are formulated with respect to the normal distribution of the maximum likelihood estimate, two parametric bootstrap confidence techniques, and Bayesian credible intervals. Two real data sets are analyzed to illustrate the proposed methods. Monte Carlo simulation is used to compare various methods. 2020 Mathematics Subject Classifications: 62F10, 62F15, 62F40 Key Words and Phrases: Bayesian estimation, bootstrap confidence interval, classical estima- tion, generalized hybrid censoring scheme, Ishita distribution, MCMC 1. Introduction In real-life applications such as engineering, medicine, insurance, and finance, the problem of modeling and analyzing real-life data is crucial. Exponential and Lindley dis- tributions are the most important one-parameter lifetime distributions popular for mod- eling biomedical science and engineering lifetime data, see Lindley [1]. Shanker et al. [2] observed that exponential and Lindley distributions are not suitable for many life- time data due to their nature of hazard rate functions, their shapes, and mean residual life. For searching for a better fit lifetime distribution than exponential and Lindley, Shanker [2] presented the Akash lifetime distribution. Shankar and Shukla [3] proposed a one-parameter model as a mixture of exponential(β) and gamma (3, β) distributions with mixing proportion β3 β3+2 . This model is known as the Ishita distribution (ID) and is ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7126 Email address: skbadr@uj.edu.sa (S. K. Badr) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. K. Badr / Eur. J. Pure Appl. Math, 18 (4) (2025), 7126 2 of 20 used for modeling the lifetime data in biomedical sciences and engineering. For model- ing lifetime data in reliability and in terms of its hazard rate shapes, this distribution is flexible than the exponential, Lindley, and Akash distributions. The ID has the advan- tage of increasing and decreasing the hazard rate function, making it more flexible than the exponential, Akash, and Lindley distributions. Also, ID is considered a model of the exponential family. Different author developed new distributions related to ID, such as the Power Ishita distribution by Shukla and Shankar [4], the truncated lifetime test for an Ishita distribution by Al-Nasser et al. [5] Poisson Ishita distribution by Anwar et al. [6], and new size-biased Ishita distribution by Al-Omari et al. [7]. The random variable X is called Ishita’s random variable if its probability density function (PDF) is given by f(x) = β3 β3 + 2 (β + x2) exp(−βx), x > 0, β > 0, (1) where β is scale parameter. The corresponding cumulative distribution function (CDF) and hazard failure rate function (h(.)) are given by F (x) = 1− ( 1 + βx(βx+ 2) β3 + 2 ) exp(−βx), (2) and h(t) = β4 + β3t2 β2t2 + 2βt+ β3 + 2 . (3) For a different choice of the parameter β the graph representation of The PDF and CDF of the Ishita distribution are shown in Figures 1 and 2. Figure 1: The graph of PDF of Ishita distribution. Commonly, in reliability analysis or medical study, the lifetime data is collected for some, but not all, population units under test, which is known as a censoring scheme (CS). Type-I and type-II censoring schemes (CSs) are commonly simple CSs. The test time in type-I CS is constant prior to, but a random number of data points. In type- II CS, constant prior number data and a random test time are used. When both the total test time τ and the number of data points m are considered, the hybrid censoring S. K. Badr / Eur. J. Pure Appl. Math, 18 (4) (2025), 7126 3 of 20 Figure 2: The graph of h(.) of Ishita distribution. scheme (HCS) was considered. The test is terminated at min(τ, Tm) at type-I HCS and terminated at max(τ, Tm) at type-II HCS, where Tm is m−th failure time and τ is the edial test time. For more information about type-I HCS see, Gupta and Kundu [8] and Kundu and Pradhan [9] and Childs et al. [10] and [11] for type-II HCS. All of these types of censoring schemes have the lack of memory that a small data size may be zero in type-I CS and type-I HCS. And, larger terminated test time in type-II CS and type-II HCS may be infinity. So that, authors can be avoid these schemes failure by considering the generalized hybrid censoring schemes (GHCSs), see [12]. In this paper, we adopted tyep-I GHCSs. Designing the experiment according to type-I GHCS provides us with the guarantee of saving the minimum number of failures needed for statistical inference. Therefore, Let a random sample of size n is selected from the population units to run under type-I GHCS. The minimum and ideal numbers of failure are prior propose to be k and m. Also, the ideal test time is denoted by τ . When the experiment is running the failure time is recorded until the k-th failure time Tk is observed. If, the failure time Tk larger than the time τ then, the test terimenated at Tk. But, if the failure time Tk is smaller than τ the test terimenated at min (τ, Tm), see [13] and [14]. The schematic diagram of type-I GHCS is presented in Figur3. Suppose that, the observed data under type-I GHCS is denoted by t = (t1 < t2 < ... < tr).The integer number r = k for τ ≤ tk, r = m for tm ≤ τ and k < r < m if tk < τ < tm. Figure 3: The schematic diagram of type-I GHCS. S. K. Badr / Eur. J. Pure Appl. Math, 18 (4) (2025), 7126 4 of 20 The problem of obtaining of enough failure time data under normal conditions is more difficult, in manufacturing industries. Therefore, the ALTs presented more suitable method to solve this problem in a short period of time. According to [15], different type of ALTs are defined in literature. The first type of ALTs was called constant-stress ALT. In this type of ALTs, we kept the stress throughout the test at a constant level. Several authors considered constant-stress ALT, see [16], [17] and [18]. If the stress level is changed under fixed time or number of failure then, step-stress ALT is defined, see [19] and [20]. The third type of ALTs is called progressive-stress ALT, the stress in this type is continual increasing through the test, see [21]. In several cases of ALTs, some units were tested under use conditions, and other units were tested under stress conditions, which is known by partially ALTs model, see [22]. Partially ALTs model defined as partially constant-stress ALT model and partially step-stress ALT model, see Almalki et al [23] and Almarashi and Abd-Elmougod [24]. In the partially constant-stress ALT model, some units are tested under normal condition and other units are tested under accelerated conditions. But, in partially step-stress ALT model, all units are tested under normal conditions until fixed prior time or number is observed, and hence are tested under accelerated conditions. Statistical inference of ID under partially step-stress ALT is developed when data is obtained with respect to type-I generalized HSC. The estimation results of the model parameters formulated with the resected MLE, bootstrap confidence interval, and Bayes method are constructed. The paper is described as follows: In Section 2, the essential assumptions and model formulation are presented. In Section 3, formulate the theoretical results of point estima- tion. In Section 4, formulate the theoretical results of interval estimation. The numerical study in the form of an analysis of two real data sets and a simulation study are formulated in Section 5. Finally, we conclude with some comments in Section 6. . 2. Assumptions and Model Formulation Table 1: List of Abbreviations. ID Ishita distribution. CDF Cumulative distribution function. PDF Probability density function ACI Approximate confidence interval CDF Cumulative distribution function. GHCS Generalized hybrid censoring scheme h(.) Hazrd fialure rate function. BTCI Bootstrap-t confidence interval. MH Metropolis–Hastings. CI Credible intervals ALT Accelerated life tests ME Mean estimate CI Credible intervals MSE Mean squared error. MIL Mean interval length. SEL Square error loss. PC Probability coverage. PBCI Percentile bootstrap confidence interval. Let a random sample of size n be selected from the population units to put under a life testing experiment. Also, suppose each of the lower and upper integer numbers of failures needed for statistical inference are prior proposed to be k and m. Without S. K. Badr / Eur. J. Pure Appl. Math, 18 (4) (2025), 7126 5 of 20 loss of generality, the stress time change η is selected to be smaller than the ideal test time τ . With respect to partially step-stress ALTs, firstly, run the experiment under normal stress conditions with all units until the time η is observed. Then, it runs under stress conditions. If the k-th failure time Xk < τ, we terminated the test at min(τ, Tm) otherwise the test terminated at Tk. Therefore, the data with respect to type-I GHCS can be described as X =(X1 < X2 < ...< XJ< η< XJ+1 < ... < Xr), where, k ≤ r ≤ m. The integer number J is observed under normal conditions, but r − J under accelerated conditions conditions. The objective of changing to a higher stress level is to shorten the test time. If we denote the total test time by T , which passes through two stages, normal and accelerated conditions. Then, T with respect to partially step-stress the ALT model is described as T = { X, X < η η + (X−η λ ), X > η, , (4) where, η is the stress change time, the acceleration factor is defined by λ and X is the lifetime at normal conditions. When, the lifetime of unit has a PDF defined by (1) with parameter β, the PDF of total lifetime T can be defined by f(t) =  f1(t), 0 < t ≤ η f2(t), t > η, 0, O.W. , (5) where f2(t) can be obtained from f1(t) in (1) by using transformation technique with respect to transformation (4) as f2(t) = β3λ β3 + 2 (β + (η + λ(t− η))2) exp(−β(η + λ(t− η))), t > 0, β > 0, λ ≥ 1, (6) The CDF, F (.) under accelerated conditions are given by F2(t)= { 1− ( 1 + β(η + λ(t− η))(β(η + λ(t− η)) + 2) β3 + 2 ) exp(−β(η + λ(t− η))) , (7) Remarks 1: (i) When r = k = J the accelerated type-I GHCS reduce to normal stress case only. (ii) When r > J the observed accelerated type-I GHC data t =(t1 < t2 < ...< tJ< τ < tJ+1 < ... < tr) and the joint likelihood function is given by L(t|λ, β) = n! (n− r)! [1− F2(tr)] n−r ( J∏ i=1 f1(ti) )( r∏ i=J+1 f2(ti) ) , (8) 3. Point Estimation In this section, for the given accelerated type-I GHC data the point ML estimation of the model parameters is discussed. Additionally, we discuss Bayes point estimation with gamma prior information for the ID parameter and non-informative prior information for the accelerated factor. S. K. Badr / Eur. J. Pure Appl. Math, 18 (4) (2025), 7126 6 of 20 3.1. Point ML estimation From (1) and (4), the joint likelihood function (8) for observed accelerated type-I GHC data t =(t1, t2, ..., tJ , tJ+1, ..., tr) is given by L (λ, β|t) ∝ ( β3 β3 + 2 )r λr−J ( J∏ i=1 (β + t2i ) )( r∏ i=J+1 (β + (η + λ(ti − η))2) ) × ( 1 + β(η + λ(tr − η))(β(η + λ(tr − η)) + 2) β3 + 2 )(n−r) × exp { −β J∑ i=1 ti − β r∑ i=J+1 (η + λ(ti − η)− β(n− r)(η + λ(tr − η)) } . (9) Taken the natural logarithm of the joint function (9) as ℓ (λ, β|t) = r log [ β3 β3 + 2 ] + (r − J) log λ+ r∑ i=J+1 log [ β + (η + λ(ti − η))2 ] + (n− r) ( log [ β3 + 2 + β(η + λ(tr − η))(β(η + λ(tr − η)) + 2) ] − log [ β3 + 2 ]) + J∑ i=1 log [ β + t2i ] − β J∑ i=1 ti − β r∑ i=J+1 (η + λ(ti − η)− β(n− r)(η + λ(ti − η)). (10) For simpilicty the log-likelihood function experssed as ℓ (λ, β|t) = r log [ β3 β3 + 2 ] + (r − J) log λ+ J∑ i=1 log [ β + t2i ] + r∑ i=J+1 log [ β + z2i ] − β J∑ i=1 ti −β r∑ i=J+1 zi − β(n− r)zr + (n− r) ( log [ β3 + 2 + βzr(βzr + 2) ] − log [ β3 + 2 ]) , (11) where, zi = η+λ(ti− η). The likelihood equations is obtained from (11) by taken the first partially derivatives respect to β and λ as follows ∂ℓ (λ, β|t) ∂β = 6r β (β3 + 2) + J∑ i=1 1 β + t2i + log r∑ i=J+1 1 β + z2i − J∑ i=1 ti − r∑ i=J+1 zi − (n− r)zr + (n− r) ( 3β2 + zr(βzr + 2) + β2z2r β3 + 2 + βzr(βzr + 2) − 3β2 β3 + 2 ) = 0, (12) and ∂ℓ (λ, β|t) ∂λ = r − J λ + r∑ i=J+1 2zi(ti − η) β + zi − β r∑ i=J+1 (ti − η)− β(n− r)(tr − η) S. K. Badr / Eur. J. Pure Appl. Math, 18 (4) (2025), 7126 7 of 20 + (n− r) ( β(tr − η)(βzr + 2) + β2(tr − η)zr β3 + 2 + βzr(βzr + 2) ) = 0, (13) Equations (12) and (13) have shown that, the ML estimate of the parameters λ and β is defined in the form of two non-linear equations. Therefore, the estimated λ̂ and β̂ can be obtained by applied any iteration method such as Newton Raphson iteration. Remarks (2): (i) The ML estimates of the parameter of life (Reliability and hazard rate function given by Ŝ(t) = 1− F̂ (t)|β=β̂ (14) ĥ(t) = h(t)|β=β̂ (15) (ii) When J = k = r the experiment run only at normal conditions and λ̂ = 0 (iii) The initial value of iteration method can be obtained from the joint profile log- likelihood function (11). 3.2. Point Bayesian estimation In this section, A challenging problem in statistical inference is selecting a prior dis- tribution that is suitable for the information about the parameters. In the problem at hand, the conjugate prior distribution does not exist. Due to several distributions, such as the chi-square distribution and the exponential being a special case of the gamma prior. Therefore, we adopt gamma prior distribution of the model parameters and non- informative prior information for the accelerated factor as follows. P1(β) ∝ βa−1 exp {−bβ} , a, b > 0, (16) and P2(λ) ∝ 1 λ . (17) Therefore, the joint prior information P (λ, β) ∝ λ−1βa−1 exp {−bβ} . (18) Generality, the joint posterior density function of λ and β compute from π(λ, β|t) = P1(β)P2(β)L (λ, β|t)∫∫ P1(β)P2(β)L (λ, β|t) dλdβ ∝ P1(β)P2(β)L (λ, β|t) . (19) Also, under squared error loss function the Bayes estimate of any function of λ and β say g(λ, β) is given by ĝ(λ, β) = ∫∫ g(λ, β)π(λ, β|t)dλdβ. (20) S. K. Badr / Eur. J. Pure Appl. Math, 18 (4) (2025), 7126 8 of 20 The closed form of posterior distribution (19) and the estimated value in (20) are gener- ally more complicated especially in a parameter vector with a large dimension. Therefore, the approximation method is the natural alternative method. Numerical integration, Lind- ley’s approximation, and MCMC methods can be applied. in this paper, we adopt the important one called the MCMC method as follows. Bayesian estimation using MCMC: From (19) the joint posterior distribution by using (18) and (9) reduce to π(λ, β|t) ∝ ( β3 β3 + 2 )r βa−1λr−J−1 ( J∏ i=1 (β + t2i ) )( r∏ i=J+1 (β + z2i ) )( 1 + βzr(βzr + 2) β3 + 2 )(n−r) × exp { −bβ − β J∑ i=1 ti − β r∑ i=J+1 zi − β(n− r)zr) } . (21) The joint posterior distribution (22) is reduce to two full conditional distributions given by π1(λ, |β, t) ∝ λr−J−1 ( r∏ i=J+1 (β + z2i ) )( 1 + βzr(βzr + 2) β3 + 2 )(n−r) × exp { −β r∑ i=J+1 (η + λ(t− η)− β(n− r)(η + λ(t− η)) } , (22) and π2(β|λ, t) ∝ ( β3 β3 + 2 )r βa−1 ( J∏ i=1 (β + t2i ) )( r∏ i=J+1 (β + z2i ) )( 1 + βzr(βzr + 2) β3 + 2 )(n−r) × exp { −bβ − β J∑ i=1 ti − β r∑ i=J+1 zi − β(n− r)zr) } . (23) The full conditional distributions (22) and (23) have shown that Metropolis-within- Gibbs samplers are more suitable algorithms. The flowering algorithm is used to generate from the posterior distribution (21) and hence obtain the empirical posterior distribution. Algorithm 1 (MCMC algorithm). (i) Begin with initial guess value λ(0) = λ̂ and β(0) = β̂. (ii) Put s = 1. (iii) Generate λ (s) 1 from (22) using (MH) algorithm. (iv) Generate β (s) 1 from (23) using (MH) algorithm. • Generate candidate sample points from normal distribution. S. K. Badr / Eur. J. Pure Appl. Math, 18 (4) (2025), 7126 9 of 20 • Compute the acceptance probability from Pl =min ( 1, πl(.|λ,t) πl(.|λ,t) ) , l =1,2. • Generate from uniform 0 and 1, Ul. • Accept the candidate sample points if Ul < Pl. Otherwise, reject the candidate point and repeat the last one. (v) For given t compute S(t, β (s) 1 ) and h(t, β (s) 1 ) (vi) Put s = s+ 1. (vii) Repeat Steps 3 to 4 N times to get (λ(1),β(1)), . . . ,(λ(N),β(N)). (viii) The Bayes estimates of λ and β with respect to the SEL function as are given by λ̂B = 1 N −N∗ N i=N∗+1 λ(i), (24) and β̂B = 1 N −N∗ N i=N∗+1 λ(i), (24) where N∗ s the number of iterations to reach the stationary distribution. (ix) Bayes estimates of the parameter of life (Reliability and hazard rate function) are given by ŜB(t) = 1 N −N∗ N i=N∗+1 S(t, β (i) 1 ) (25) ĥB(t) = 1 N −N∗ N i=N∗+1 h(t, β (i) 1 ) (26) 4. Interval Estimation In this section, we discuss the interval estimation of the ID parameter and the accel- erated factor with three different methods. The first one, confidence intervals depend on the asymptotic property of the ML estimate (Approximate ML confidence intervals), the second is the bootstrap confidence interval. Finally, we consider the probability credible intervals. 4.1. Approximate ML confidence intervals Interval estimation of the parameters depends on Fisher information matrix, which was defined as the minus expectation of the second partial derivative of the log-likelihood function as 𝟋(λ, β) = E ( −∂2ℓ (λ, β|t) ∂θi∂θj ) , i, j = 1, 2, θ1 = λ and θ2 = β, S. K. Badr / Eur. J. Pure Appl. Math, 18 (4) (2025), 7126 10 of 20 Generally, for models that have several parameters, the problem of computing expectation is more complicated. Therefore, replace the Fisher information matrix with the approxi- mate information matrix is given by Ψ(λ, β) = ( −∂2ℓ (λ, β|t) ∂θi∂θj ) |λ,β=λ̂,β̂ , i, j = 1, 2, θ1 = λ and θ2 = β. (23) From the log-likelihood function (10) the second partial derivative is obtained. Also, under bivariate normal distribution for the values of ML estimate of parameters λ̂ and β̂ with mean (λ, β) and variance as the diagonal of Ψ−1(λ̂, β̂). Therefore, we say( λ̂, β̂ ) −→ N ( (λ, β) ,Ψ−1(λ̂, β̂) ) . (24) Hence, the corresponding (1-2α)100% approximate confidence intervals of parameters are given by λ̂∓ ξα √ Ψ−1 1,1, and β̂ ∓ ξα √ Ψ−1 2,2 , (25) where, Ψ−1 i,j , i, j = 1, 2 are the element of diagonal for Ψ−1 i,j (λ̂, β̂) and the value ξα is a tabulated standard normal value with confidence level 2α. 4.2. Bootstrap confidence intervals Bootstrap techniques are commonly used methods in literature to formulate confidenca e interval of the parameters. In the problem at hand, paramteric bootstrap techniques is used to formulate boot-p and boot-t confidence interval, see [25], [26] and [27] as follows Algorithm 2 (Bootstrap confidence intervals): Step 1: From a real life population and the prior integer k, m and n as well as τ < η determine the original data set t =(t1, t2, ..., tJ , tJ+1, ..., tr). Step 2: The ML estimates λ̂ and β̂ are computed from the original data set t =(t1, t2, ..., tJ , tJ+1, ..., tr). Step 3: Put s = 1. Step 4: with the same prior integer k, m and n as well as τ < η generate a type-I GHC random from ID with parameters β̂. Step 5: Applied the transformation (4) for all data larger than η to get the accelerated type-I GHC bootstrap sample data t∗ =(t∗1, t∗2, ..., t∗J , t∗J+1, ..., t ∗ r). Step 6: Compute The ML estimates λ̂∗ and β̂∗ from bootstrap sample data t∗ =(t∗1, t∗2, ..., t∗J , t ∗ J+1, ..., t ∗ r). Step 7: Put s = s+ 1. S. K. Badr / Eur. J. Pure Appl. Math, 18 (4) (2025), 7126 11 of 20 Step 8: Repeat the steps from 3 to 7 NB-times, we obtain the bootstrap sample estimate θ ∗(1) l , θ ∗(2) l , ..., θ ∗(NB) l , l = 1,2, θ∗1 = λ∗ and θ∗2 = β∗ (26) Boot-p confidence intervals (PBCI) The ordered value of (26) is given by θ∗l(1), θ ∗ l(2), ..., θ ∗ l(NB), Suppose Γ(x) the emperical distribution of the ordered bootstrap sample estimated (27) then, F (x) = P (θl ⩽ x), i =1, 2 be a CDF of θl and the corresponding (1 − 2α)100% PBCIs are given by ( θ∗l(αNB), θ∗l((1-α)NB) ) , (28) where of θ∗l(.) = Γ−1(x). Boot-t confidence interval (BTCI) From the ordered sample θ∗l(1), θ ∗ l(2), ..., θ∗l(NB), we difene the statistics Φ∗ l(1) < Φ∗ l(2) < ... < Φ∗ l(MB) by Φ∗ l(j) = √ r θ∗l(j) − θ̂l√ var ( θ∗l(j) ) , l = 1, 2 . (29) Let Γ(x) = P (Φ∗ i ⩽ x) be the CDF of Φ∗ i . Therefore for given x, we define θ∗lboot-t = θ̂l + √ rVar(θ̂l)Γ−1(x), (30) The corresponding (1− 2α)100% PTCIs are given by( θ∗lboot-t(α), θ ∗ lboot-t(1-α) ) (31) 4.3. Bayesian credible interval From the Bayes estimate sample generated from MCMC method θ̂ (1) Bl , θ̂ (2) Bl , ..., θ̂ (N) Bl . (32) The ordred value θ̂Bl(1), θ̂Bl(2), ..., θ̂Bl(N), i = 1, 2. (33) The (1− 2α)100% equal two side credible intervals of the model parameters is defined by ( θ̂Bl(α 2 (N−N∗)), θ̂Bl((1−α 2 )(N−N∗)) ) (34) S. K. Badr / Eur. J. Pure Appl. Math, 18 (4) (2025), 7126 12 of 20 5. Numerical Results In this section, we adopt numerical computation in the form real data analysis and simulation study. For data analysis consider the data reported by Fuller et al. [28] of glass of the aircraft window and Bader and Priest [29] about the strength of single-carbon fibres in GPA. 5.1. Data analysis In this subsection, we consider two real data sets and the goodness of fit of these data to ID has been done by Shanker and Shukla [3]. Example 1: The following data describe strength data reported by Fuller et al. [28] of glass of the aircraft window Table 2: The orginal and the corresponding accelerated type-I GHC data sets Fuller et al. [28]. Orginal 18.83 20.800 21.657 23.030 23.230 24.050 24.321 25.500 25.520 data 25.800 26.69 26.770 26.780 27.050 27.670 29.900 31.110 33.200 33.730 33.760 33.890 34.760 35.750 35.910 36.980 37.080 37.090 39.580 44.045 45.290 45.381 accelerated 18.830 20.800 21.657 23.030 23.230 24.050 24.321 25.500 25.520 type-I GHC 25.800 26.690 26.770 26.780 27.017 27.223 27.967 28.370 29.067 data 29.243 29.253 29.297 29.587 Under consideration, the original data of size n =31 Fuller et al. [28], suppose m =25 k =15, τ =35.0 and the stress change time η =27 , the accelerated type-I GHC data can be obtained from the original data in Table 1 by considered λ =3.0. From accelerated type-I GHC data the integer numbers r =22 and J =13. The prior information about ID parameter is taken to be non-informative prior. With respect to MCMC method, chen iteration is running 11000 iterations and delete the first 1000 iterations as burn-in. Figures 4 and 5 show the empirical posterior distribution and its convergence under the Bayesian approach. The point estimate of the model parameters and the parameters of life when t =1.5 are reported in Table 3. The 95% interval estimation of the model parameters is reported in Table 3 Table 3: The results of point and interval estimate under 5% confidence level. Pa. (.)ML (.)B-MCMC 95% ACI 95% PBCI 95% BTCI 95% CI β 0.078268 0.078090 (0.0575, 0.0990) (0.0488, 0.0997) (0.0542, 0.0991) (1.1596, 4.6482) λ 2.524430 2.599440 (0.7772, 4.2716) (0.6542, 4.2744) (0.7723, 4.2722) (1.1596, 4.6482) S(.) 0.999726 0.999712 h(.) 0.000496 0.000520 S. K. Badr / Eur. J. Pure Appl. Math, 18 (4) (2025), 7126 13 of 20 Figure 4: Simulation number and Histogram of β and λ generated by MCMC method, respectively. Figure 5: Simulation number and Histogram of S and h generated by MCMC method, respectively. Example 2: The second data is the tensile strength, measured in GPa, of 69 carbon fibers represent tested under tension at gauge lengths of 20mm, Bader and Priest [29] The original Bader and Priest [29] data of size n =69 , suppose m =50 k =25, τ =2.0 and the stress change time η =1.0 , the accelerated type-I GHC data can be obtained from the original data in Table 2 by considered λ =3.0. The integer numbers r =25 and J =0 from accelerated type-I GHC data. With 11000 iterations and delete the first 1000 iterations as burn-in MCMC method is running . Figure 6 and 7 show the empirical posterior distribution and its convergence under Bayesian approach. The point estimate of the model parameters and the parameters of life when t =1.5 are reported in Table 5. The 95% interval estimation of the model parameters is reported in Table 5. 5.2. Simulation study In this section, the developed estimation results are discussed through a Monte Carlo simulation study to assess and compare the results. Through this study, we discuss the ef- fect of changing the ideal test time and the stress change time. Also, we test the results for the change in the values of the parameters and sample sizes. For the point estimate, check the ML estimate with the Bayes estimate for different prior information. In cases of in- terval estimation, compare the approximate confidence intervals, the bootstrap confidence S. K. Badr / Eur. J. Pure Appl. Math, 18 (4) (2025), 7126 14 of 20 Table 4: The orginal and the corresponding accelerated type-I GHC data sets Bader and Priest [28]. Orginal 1.312 1.314 1.479 1.552 1.700 1.803 1.861 1.865 1.944 1.958 data 1.966 1.997 2.006 2.021 2.027 2.055 2.063 2.098 2.140 2.179 2.224 2.240 2.253 2.270 2.272 2.274 2.301 2.301 2.359 2.382 2.382 2.426 2.434 2.435 2.478 2.490 2.511 2.514 2.535 2.554 2.566 2.570 2.586 2.629 2.633 2.642 2.648 2.684 2.697 2.726 2.770 2.773 2.800 2.809 2.818 2.821 2.848 2.880 2.954 3.012 3.067 3.084 3.090 3.096 3.128 3.233 3.433 3.585 3.585 accelerated 1.104 1.105 1.160 1.184 1.233 1.268 1.287 1.288 1.315 1.319 type-I GHC 1.322 1.332 1.335 1.340 1.342 1.352 1.354 1.366 1.380 1.393 data 1.408 1.413 1.418 1.423 1.424 Figure 6: Simulation number and Histogram of β and λ generated by MCMC method, respectively. interval, and the Bayesian credible intervals. The simulation results were formulated for 1000 different samples generated from the proposed model. The point estimate is tested under the mean estimate (ME) and mean squared error (MSE). In an interval estimate, we compute the mean interval length (MIL) and coverage percentage (CP). The following algorithms describe what happens in the simulation results Algorithm 3: (Monte Carlo simulation studying) Step 1: From ID with parameter β generate a random sample of size n . Step 2: The generate type-I GHC data is reported respected to k, m and τ . Step 3: Transform the type-I GHC data to accelerated type-I GHC data based on transfor- Table 5: The results of point and interval estimate under 5% confidence level. Pa. (.)ML (.)B-MCMC 95% ACI 95% PBCI 95% BTCI 95% CI β 0.51064 0.49666 (0.3178, 0.7035) (0.3001, 0.7457) (0.3124, 0.7011) (1.5527, 5.2164) λ 2.65934 2.91626 (1.0371, 4.2816) (1.0422, 4.2987) (1.0341, 4.2807) (1.5527, 5.2164) S(.) 0.92661 0.92657 h(.) 0.08645 0.08529 S. K. Badr / Eur. J. Pure Appl. Math, 18 (4) (2025), 7126 15 of 20 Figure 7: Simulation number and Histogram of S and h generated by MCMC method, respectively. mation (4) and accelerated factor λ. Step 4: Compute the point ML and Bayes estimate of λ and β. Step 5: Compute the interval ML confidence interval, bootstrap confidence intervals and Bayesian credible interval. Step 5: Repeated steps from 1 to 5 1000 times. Step 6: The values of MEs, MSEs, MILs and PCs are computed and results record in Tables (6) to (9). Table 6. Average ME and the corresponding MSEs when β =0.5 and λ =2.5. (k, m, n) (η, τ) MLE Bayes P0 Bayes P1 β λ β λ β λ (15,20,40) (2.5,6.5) ME 0.742 2.853 0.727 2.847 0.687 2.744 MSE 0.147 0.336 0.139 0.342 0.103 0.267 (15,20,40) (5,6.5) ME 0.719 2.871 0.705 2.869 0.661 2.761 MSE 0.136 0.341 0.125 0.359 0.099 0.271 (15,30,40) (2.5,6.5) ME 0.641 2.761 0.615 2.739 0.601 2.630 MSE 0.118 0.305 0.114 0.318 0.082 0.231 (15,30,30) (5,6.5) ME 0.630 2.782 0.672 2.800 0.598 2.670 MSE 0.112 0.315 0.101 0.331 0.075 0.242 (25,35,70) (2.5,6.5) ME 0.611 2.759 0.647 2.771 0.569 2.645 MSE 0.099 0.302 0.088 0.311 0.063 0.229 (25,35,70) (5,6.5) ME 0.601 2.769 0.631 2.789 0.551 2.653 MSE 0.092 0.308 0.081 0.313 0.060 0.234 (25,50,70) (2.5,6.5) ME 0.589 2.601 0.577 2.593 0.544 2.587 MSE 0.076 0.234 0.069 0.227 0.041 0.175 (25,50,70) (5,6.5) ME 0.565 2.612 0.564 2.597 0.528 2.591 MSE 0.071 0.239 0.063 0.232 0.035 0.181 S. K. Badr / Eur. J. Pure Appl. Math, 18 (4) (2025), 7126 16 of 20 Table 7. Average MIL and the corresponding PC when β =0.5 and λ =2.5. (k, m, n) (η, τ) MLE Boot-p Boot-t Bayes P1 β λ β λ β λ β λ (15,20,40) (2.5,6.5) MIL 0.642 4.245 0.666 4.542 0.626 4.011 0.603 3.897 PC 0.89 0.90 0.91 0.89 0.91 0.92 0.92 0.92 (15,20,40) (5.0,6.5) MIL 0.631 4.253 0.649 4.551 0.615 4.017 0.598 3.901 PC 0.91 0.90 0.92 0.90 0.91 0.93 0.94 0.91 (15,30,40) (2.5,6.5) MIL 0.601 4.225 0.618 4.527 0.589 4.001 0.579 3.891 PC 0.90 0.93 0.91 0.93 0.94 0.96 0.93 0.95 (15,30,30) (5.0,6.5) MIL 0.589 4.233 0.609 4.534 0.580 4.014 0.561 3.898 PC 0.91 0.93 0.93 0.94 0.95 0.92 0.96 0.94 (25,35,70) (2.5,6.5) MIL 0.566 4.211 0.575 4.508 0.550 3.980 0.542 3.859 PC 0.94 0.92 0.93 0.90 0.94 0.94 0.96 0.92 (25,35,70) (5.0,6.5) MIL 0.551 4.219 0.562 4.517 0.535 3.991 0.527 3.861 PC 0.93 0.93 0.93 0.96 0.92 0.94 0.95 0.95 (25,50,70) (2.5,6.5) MIL 0.515 4.170 0.531 4.466 0.505 3.938 0.511 3.815 PC 0.93 0.95 0.93 0.92 0.93 0.94 0.93 0.94 (25,50,70) (5.0,6.5) MIL 0.502 4.176 0.519 4.471 0.501 3.943 0.495 3.822 PC 0.92 0.92 0.96 0.94 0.93 0.95 0.95 0.91 Table 8. Average ME and the corresponding MSEs when β =1.2 and λ =1.5. (k, m, n) (η, τ) MLE Bayes P0 Bayes P1 β λ β λ β λ (15,20,40) (0.8,2.0) ME 1.454 2.147 1.404 2.103 1.389 2.001 MSE 0.284 0.314 0.275 0.303 0.197 0.287 (15,20,40) (1.2,2.0) ME 1.445 2.151 1.389 2.111 1.365 2.008 MSE 0.255 0.322 0.242 0.312 0.162 0.291 (15,30,40) (0.8,2.0) ME 1.392 2.118 1.379 1.893 1.314 1.872 MSE 0.225 0.289 0.222 0.275 0.135 0.214 (15,30,30) (1.2,2.0) ME 1.362 2.123 1.355 1.914 1.289 1.885 MSE 0.195 0.294 0.187 0.282 0.114 0.219 (25,35,70) (0.8,2.0) ME 1.278 1.874 1.282 1.971 1.244 1.754 MSE 0.135 0.166 0.127 0.159 0.094 0.114 (25,35,70) (1.2,2.0) ME 1.254 1.881 1.259 1.977 1.232 1.757 MSE 0.122 0.169 0.124 0.161 0.085 0.099 (25,50,70) (0.8,2.0) ME 1.247 1.866 1.244 1.974 1.225 1.761 MSE 0.115 0.151 0.127 0.153 0.079 0.084 (25,50,70) (1.2,2.0) ME 1.232 1.869 1.225 1.975 1.212 1.763 MSE 0.098 0.153 0.115 0.157 0.070 0.079 S. K. Badr / Eur. J. Pure Appl. Math, 18 (4) (2025), 7126 17 of 20 Table 9. Average MIL and the corresponding PC when β =1.2 and λ =1.5. (k, m, n) (η, τ) MLE Boot-p Boot-t Bayes P1 β λ β λ β λ β λ (15,20,40) (0.8,2.0) MIL 2.147 2.894 2.165 2.899 2.132 2.855 2.102 2.812 PC 0.90 0.90 0.91 0.90 0.91 0.92 0.92 0.90 (15,20,40) (1.2,2.0) MIL 2.122 2.899 2.142 2.915 2.112 2.866 2.055 2.817 PC 0.91 0.91 0.92 0.91 0.91 0.91 0.93 0.92 (15,30,40) (0.8,2.0) MIL 2.102 2.842 2.122 2.877 2.055 2.814 2.001 2.750 PC 0.93 0.93 0.90 0.93 0.96 0.93 0.94 0.94 (15,30,30) (1.2,2.0) MIL 2.072 2.851 2.100 2.884 2.033 2.821 1.982 2.754 PC 0.92 0.92 0.93 0.91 0.92 0.91 0.93 0.95 (25,35,70) (0.8,2.0) MIL 2.025 2.795 2.027 2.825 2.001 2.785 1.954 2.689 PC 0.93 0.90 0.94 0.94 0.96 0.93 0.94 0.93 (25,35,70) (1.2,2.0) MIL 2.002 2.797 2.001 2.831 1.975 2.789 1.932 2.691 PC 0.91 0.93 0.93 0.92 0.92 0.92 0.97 0.91 (25,50,70) (0.8,2.0) MIL 1.952 2.735 1.945 2.715 1.912 2.704 1.854 2.625 PC 0.94 0.92 0.92 0.94 0.94 0.95 0.93 0.94 (25,50,70) (1.2,2.0) MIL 1.924 2.741 1.931 2.722 1.900 2.713 1.832 2.631 PC 0.92 0.93 0.96 0.93 0.94 0.92 0.95 0.96 5.3. Numerical discussion From the numerical results presented in the data analysis and the Monte Carlo simu- lation study observe some points are reported as follows 1- In two real example, the results are formulated under a non-informative prior distri- bution. Therefore, the results of ML and Bayes estimates are closed. 2- Figures from (4) to (7) show the convergence happening in MCMC. 3- All results of point and interval estimation serve well for the larger ideal test time and larger sample size. 4- The results for the boot-t confidence interval estimate is best than ML and boot-p confidence interval. 5- The values of MSEs are reducing for increasing value of sample size and effective sample size. 6- For point estimate informative Bayes estimate is better than ML and the non-informative Bayes estimate. 7- For interval estimate, informative Bayes and bootstrap-t serve well than ML and boot- p. The results more suitable for large value of τ . S. K. Badr / Eur. J. Pure Appl. 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