**8. Simulation Study**

A simulation study is carried out to illustrate the relative efficiency of multi-stress– strength reliability under the progressive first failure based on different censored schemes and to evaluate it as a function of changing factors of a parameter. For a better understanding of this model, we use the following procedure to produce samples from the progressive first failure based on different censored schemes for APE distribution described in Section 3.

A large number N = 1000 of progressively first-failure censored samples for a true value of parameters α, β1,β2, and β3 different combinations of n (number of groups), m (progressively first-failure-censoring data), and k (number of items within each group) are generated from the APE by using the algorithm described in Balakrishnan and Sandhu (1995). In each case, the MLE and Bayesian of the multi-stress–strength reliability are computed. The asymptotic CIs and two parametric bootstrap CIs are used for MLE computation purposes. The HPD CIs are used for Bayesian computation purposes. The MSE and Bias values are used to compare different estimators. The average lengths are also used to compare the performances of the two-sided 95% asymptotic CI/HPD credible intervals, where the length of asymptotic CI is (LACI), length of bootstrap-p CI is (LBPCI), length of bootstrap-t CI is (LBTCI), and length of credible CI is (LCCI). Comparison between censoring schemes is made with respect to their optimum criteria measures; see Table 1, where we consider the various sampling schemes listed as follows:

Scheme I: R m = n − m and Ri = 0; i = 1, . . . , m − 1, Scheme II: R1= n − m and Ri= 0; i = 2, . . . , m.

The simulation study was conducted with various values of (k, n, m), such as n = 20, 50, and k = 2 and 4 for each group size. When the number of failed participants reaches or exceeds a specified value m, the test is over, where m =12 and 18 when n = 20, and m = 35 and 45 when n = 50. The joint posterior distribution of the unknown four parameters is proportional to the likelihood function based on the non-informative priors of hyperparameters ai, bi for I = 1, ... , 4. As a result, we employed an informative prior of, and using elective hyper-parameters, the values of hyper-parameters are chosen to satisfy the prior mean, resulting in the expected value of the corresponding parameter; see Refs. [56,57]. The Bayesian estimation based on 12,000 MCMC samples and discarding the first 2000 values as "burn-in" are generated using the M-H sampler technique introduced in Section 3.

The progressive first failure of censored samples was generated from APE distribution for four sets of parametric values:

In Table 2: α = 0.8, β1 = 1.8, β2 = 0.8, β3 = 0.2 and α = 2, β1 = 1.8, β2 = 0.8, β3 = 0.2. In Table 3: α = 0.8, β1 = 3, β2 = 2, β3 = 1.5 and α = 2, β1 = 3, β2 = 2, β3 = 1.5.

In computational analysis, extensive computations were carried out using the R statistical programming language software, with the "coda" package proposed by Ref. [58], and the "maxLik" package proposed by Henningsen and Toomet (2011), which uses the Newton–Raphson method of maximizing the computations. The average results of MLE and Bayesian for multi-stress–strength reliability are presented in Tables 2 and 3.


**Table 3.** MLE and Bayesian point and interval estimations for multi-stress–strength reliability with optimality measures when β1 = 1.8, β2 = 0.8, β3 = 0.2.

Tables 3 and 4 show that APE based on the multi-stress–strength model MLE and Bayesian of multi-stress–strength reliability is excellent in terms of MSE, Bias, and CI length (LCI). The MSE, Bias, and LCI drop as n and m rise, as expected. Furthermore, the MSE, Bias, and LCI drop as group size k grows. In terms of MSE, Bias, and LCI, Bayesian estimation utilizing gamma informative prior is also superior to MLE because it includes prior knowledge. In terms of the length of CI values, HPD credible intervals outperform asymptotic CI for interval estimation. As a result, we recommend using the M-H approach to estimate multi-stress–strength reliability using Bayesian point and interval estimates. Furthermore, when comparing Scheme I and Scheme II, it is obvious that the MLE optimum criteria measures for Scheme II are higher than for Scheme I.


**Table 4.** MLE and Bayesian point and interval estimations for multi-stress–strength reliability with optimality measures when β1 = 3, β2 = 2, β3 = 1.5.

### **9. Application of Real Data**

The analysis of real data is presented in this part for demonstration reasons. We look at data from three distinct voltages of 36, 34, and 32 KV that show times to breakdown of an insulating fluid between electrodes. This information is taken from page 105 of [59].

Data set 1: Times to breakdown of an insulated fluid at 32 KV (Z): 0.27, 0.40, 0.69, 0.79, 0.75, 2.75, 3.91, 9.88, 13.95, 15.93, 27.80, 53.24, 82.85, 89.29, 100.58, 215.10.

Data set 2: Times to breakdown of an insulated fluid at 34 KV (Y): 0.19, 0.78, 0.96, 1.31, 2.78, 3.16, 4.15, 4.67, 4.85, 6.50,7.35, 8.01, 8.27, 12.06, 31.75, 32.52, 33.91, 36.71, 72.89.

Data set 3: Times to breakdown of an insulated fluid at 36 KV (X): 0.35, 0.59, 0.96, 0.99, 1.69, 1.97, 2.07, 2.58, 2.71, 2.90,3.67, 3.99, 5.35, 13.77, 25.50.

Ref. [60] discusses the estimation of R = P[Y < X < Z] of the Weibull distribution. Table 5 discusses parameter estimation with stander error (SE) for this model and R = P[Y < X < Z] by the MLE method.


**Table 5.** MLE with SE and R = P[Y < X < Z] for the Weibull model.

First, we check the fitting of APE distribution to this data; see Table 6. Distance of Kolmogorov–Smirnov (DKS) with p values (PVKS) for three distinct voltages data. The values of KSD statistics are found to be 0.2598, 0.1612, and 0.1427 with corresponding PVKS 0.2214, 0.6492, and 0.8786. The *p* values indicate that the APE distribution with the above-mentioned parameters is a suitable model for modeling these three data sets. The plots of the estimated PDF, CDF, and PP plot of the three data sets in Figures 3–5 also confirm the same.

**Table 6.** MLE with SE, KSD, and different measures for three distinct voltages data.


**Figure 3.** Plots of the estimated PDF, CDF, and PP of APE distribution in data set I.

**Figure 4.** Plots of the estimated PDF, CDF and PP plot of APE distribution in data set II.

**Figure 5.** Plots of the estimated PDF, CDF, and PP plot of APE distribution in data set III.

Based on the complete data, the MLE and Bayesian estimate for the APE model of R = P[Y < X < Z] are found to be 0.4523 and 0.4570, respectively, as shown in Table 7. Here, it is to be noted in the Bayesian estimation of parameters that we use informative priors, as gamma prior is available regarding the model parameters. From the results of Table 7, we show that the Bayesian estimation is the best estimation of this model where the multi-stress–strength reliability R = P[Y < X < Z] is larger than MLE. In addition, the SE of Bayesian is smaller than MLE. Figure 6 shows the contour plot of the log-likelihood function of this model with different values of parameters to check the unique and global values of these parameters. Figure 7 discusses the MCMC trace, convergence, and plot of the posterior distribution of this model.


**Table 7.** MLE and Bayesian estimation for the parameters and R = P[Y < X < Z] for the APE model.

**Figure 6.** Contour plot of log-likelihood function with different values of parameters; complete sample.

**Figure 7.** MCMC trace, convergence and plot of posterior distribution; complete sample.
