Estimation and Prediction for Alpha-Power Weibull Distribution Based on Hybrid Censoring

: This work discusses the issues of estimation and prediction when lifespan data following alpha-power Weibull distribution are observed under Type II hybrid censoring. We calculate point and related interval estimates for both issues using both non-Bayesian and Bayesian methods. Using the Newton–Raphson technique under the classical approach, we compute maximum likelihood estimates for point estimates in the estimation problem. Under the Bayesian approach, we compute Bayes estimates under informative and non-informative priors using the symmetric loss function. Using the Fisher information matrix under classical and Bayesian techniques, the corresponding interval estimates are derived. Additionally, using the best unbiased and conditional median predictors under the classical approach, as well as Bayesian predictive and associated Bayesian predictive interval estimates in the prediction approach, the predictive point estimates and associated predictive interval estimates are computed. We compare several suggested approaches of estimation and prediction using real data sets and Monte Carlo simulation studies. A conclusion is provided.


Introduction
Using predictive analytics to foresee business outcomes may save time, money, and effort. More information will be accessible to help make better judgments. Organizations may also uncover possibilities and address their own challenges by providing precise and trustworthy insights. Aggregate data may be evaluated, using predictive analytics, to find new prospects for consumer acquisition. Prediction, which is important in many domains, has drawn greater attention in recent years. For instance, in business, an experimenter may wish to estimate the lifespan of an unseen future unit using data from a present sample. If so, the producer or the experimenter may release their goods into the market with the intention of capturing customers' attention and increasing demand by lowering the thresholds for their warranties. See [1][2][3] for additional details regarding the applicability of predictions in business and industry.
In cases involving parametric estimating, we usually assume the availability of a group of samples selected at random from a preset number of units (n). Furthermore, it is anticipated that random observations will fit a certain distribution. In practice, however, it is conceivable that not all of the unit observations are accessible, as some unit observations are either not captured or are lost during intermediary transmission. These data are referred to as censored samples. In the literature, there are techniques for estimating sample instances that have already been censored. Notably, from a theoretical standpoint, censoring may h(x) = logα and rh(x) = logα The formula for the mean time to failure (MTF) is Despite there being a wealth of research on estimates under hybrid censoring and the APW distribution being superior to many competing distributions, including the Weibull, alpha power exponential, McDonald Weibull, beta Weibull, transmuted Weibull, gamma Lomax, Zografos-Balakrishnan log-logistic, exponentiated Weibull, and exponentiated Weibull distributions, there is no body of work discussing the prediction of the future ordered statistics based on hybrid censored samples for the APW model. In addition, there is not much information available regarding research achieving classical and Bayesian estimates of the unknown parameters, reliability, hazard rate functions, and the MTF for APW distribution under hybrid Type II censoring. All of these gaps in the current literature prompted us to produce this work, which has three major purposes. The first goal is to examine the issue of estimating the unknown parameters, reliability, hazard rate functions, and the MTF of APW distribution using classical and Bayesian methods of estimation under Type II hybrid censoring. We also calculate the corresponding interval estimates for the model parameter. The second goal is to obtain one-and two-sample Bayesian prediction problems. Also, the one-and two-sample Bayesian prediction interval limits are constructed based on the Type II hybrid censored data. The third goal is to perform a simulation study to ensure the theoretical results of the paper and that an actual data set is given to indicate the proposed methodology in real life.
The rest of the article is organized as follows: Both classical and Bayesian approaches are used to estimate the parameters, reliability, hazard rate, and the MTF under Type II hybrid censoring, and the associated interval limits of the unknown parameters are determined in Section 2. In Section 3, it is discussed how to solve for one-and twosample Bayesian predictive posterior density. Also built in this section are the prediction boundaries for next samples. The results of the Monte Carlo simulation are studied in Section 4 to allow for the comparison of the performance of the suggested estimators. In Section 5, an actual data set is examined as an example. Concluding remarks are provided in Section 6 at the end of this essay.

Maximum Likelihood
Assume that n identical units are put through a life test, and that they all have lives that follow the APW distribution specified in the pdf Equation (1). The life test is ended at max(X R:n , T), where X R:n stands for the R th order statistic, or at the later of a predetermined time (say, T) and a predetermined number of failures (say, R ≤ n). The following scenarios would result in the random observations being obtained under this sort of censoring system (also called the Type II hybrid censoring method): Case I : If T < X R:n , then(X 1:n , X 2:n , . . . , X R:n ). Case II : (X 1:n , X 2:n , . . . , X k:n ) if X r:n ≤ T, R < k < n.
According to the Type II hybrid censoring method, the likelihood function can be expressed as follows: The log-likelihood function, is provided by The following log-likelihood equations can be simultaneously solved to give the maximum likelihood estimates (MLEs),α,β andλ.
and ∂ ∂λ As can be seen, the aforementioned equations cannot be solved explicitly and must instead be solved numerically using an iterative process. It is impossible to explicitly obtain the precise distribution of MLEs. Nonetheless, it is possible to build the confidence intervals for the parameters using the asymptotic characteristics of MLEs. The MLEs of the reliability (rf), hrf, and MTF may be directly calculated, respectively, by applying the invariance property of the maximum likelihood function, as shown below: and The MLEs α,θ,λ roughly follow a bivariate normal distribution under some regularity criteria, with mean ψ= (α, θ, λ) and variance matrix σ = I −1 α,θ,λ , where I α,θ,λ is the observed Fisher's information matrix, which is given by The asymptotic variances for the parameters ψ = (α, θ, λ), are provided by the diagonal elements of I −1 α,θ,λ , respectively. One may obtain the two-sided (1 − φ)100% normal approximation confidence interval aŝ where Z is the common normal variate.

Bayesian Estimation
On the basis of a Type II hybrid censored sample, we suggested Bayes estimators of the APW distribution's unknown parameters in this section. We must define the prior distributions for the parameters in Bayesian estimation. The authors of [19] proposed a number of methods for earlier definitions. Assuming the functional form for the prior density, eliciting its parameter(s), also known as the hyper-parameter(s), is the simplest method for prior specifications. The gamma distribution contains closed-form equations for moments and is highly versatile and able to accept different density forms. In some situations, it also offers conjugacy and mathematical simplicity. The gamma distribution has been extensively used in Bayesian estimation as a prior distribution for the parameters of the other lifespan distributions. Moreover, we assume that ψ = (α, θ, λ) are independent gamma random variables in this study. For more details, see [20][21][22]. The prior distribution of ψ = (α, θ, λ) are provided by where it is assumed that the hyper-parameters ν 1 , ν 2 , ν 3 , ν 4 , ν 5 , and ν 6 are known and are selected to represent the previous assumption about the unknown parameters. These can be acquired if we can predict the parameter's anticipated values using the prior mean (M) and confidence in the predicted value using the prior variance (V). Consequently, by resolving the following previous moments equations for ν 1 , ν 2 , ν 3 , ν 4 , ν 5 , and ν 6 may be obtained: , M = ν 6 ν 5 , and V = ν 6 ν 2

5
. The prior densities become non-informative inappropriate prior distributions as soon as The prior distribution mentioned above occasionally matches the scale-invariant Jeffrey prior (see [23]). You can find the joint posterior density of ψ = (α, θ, λ) as It is generally known that the expected value of any parametric function using the posterior distribution, or E π [ψ(α, θ, λ)], is the Bayes estimator of that function, ψ = (α, θ, λ), under squared error loss. Consequently, the Bayes estimates of α, θ, and λ are the means of their posteriors. Similar to this, it is simple to construct the Bayes estimates of the reliability and hazard rate under squared error loss aŝ similar to how the MTF's Bayes estimate may be determined by It is clear that the aforementioned expressions, Equations (23)- (25), cannot be resolved in a pleasing closed form. To take a random sample from the joint posterior and enable sample-based inference, we proposed using Markov chain Monte Carlo (MCMC) techniques, such as the Gibbs sampler (see [24]) and Metropolis Hastings algorithm (see [25,26]).

Predictive Posterior Density
In this part, we have used a Type II hybrid censored sample to determine the density of future order statistics from the APW distribution. Predictive density is a term used to describe the pdf of the future sample. There are two categories for the prediction problems, with two sample predictions being covered in the following two successive subsections: 1-One-sample prediction; 2-Two-sample prediction.

One Sample Prediction
Out of all the units/items tested, only a small number of them were noticed in the Type II hybrid censoring scheme. In such cases, the investigator could be curious to learn how long the removed surviving units had lived based on the sample at hand.
Let r 0 < n and X s:n (r 0 < s ≤ n) be the prediction. Given the Type II hybrid censored sample x, the conditional distribution of sth order statistics should be as follows: Case I: so where (R < s ≤ n), and x = {x 1:n , x 2:n , . . . , x R:n }x s:n > x R:n . By adding Equations (1) and (2) to Equation (27), we have Case II: where f ( X s:n = x s:n , X K:n = x k: So, where x = {x 1:n , x 2:n , . . . , x k:n }; k < s ≤ n, x s:n > T; and K is a random variable. The value T denotes the maximum number of x ' i s. Then, Substituting Equations (1) and (2) into Equation (30), we obtain The following formula defines the one-sample predictive posterior density of future observables: Case I: Case II: Numerical approaches are thus needed to investigate the posteriors' characteristics as the aforementioned predictive posteriors cannot be reduced to any standard distribution. Using the method described in the previous section, we can utilize the M-H algorithm to extract a sample from the predicted posteriors and provide estimates and forecast intervals for upcoming data. By resolving the following equations, we can also obtain the two-sided 100(1 − φ)% prediction intervals (L s , U s ) for y (s) .
As the aforementioned equations cannot be solved directly, confidence intervals can be calculated by utilizing any acceptable iterative approach.

Two-Sample Prediction
In another circumstance, we may be interested in the failure time of the kth ordered observation from a future sample of size N from the same lifespan distribution, i.e., P y (s) ψ, x = P(y(s)|ψ). The two sample prediction issues are the result of this. The sth order statistic's pdf is provided by And when we add Equations (1) and (2) to Equation (34), we obtain As stated below, the two-sample predictive posterior density of a future observation is As the density in Equation (34) is independent of Type II censoring, it is the same for both instances (Case I and Case II). We can also obtain the predictive density for Case I and Case II by replacing the posterior π( ψ|x) of the corresponding case. Using the MCMC techniques covered in the preceding section, it is possible to quantitatively investigate the aforementioned posterior density. By resolving the following two equations, we can also obtain the two-sided 100 As the aforementioned equations cannot be solved directly, confidence intervals can be calculated by utilizing any acceptable iterative approach.

Simulation
In this section, we give some experimental findings, mostly for the purpose of seeing how the various approaches behave for various sample sizes and time censoring techniques. We use the MLE and Bayes estimators developed by employing the MCMC approach to estimate the unknown parameters. We contrast how well each estimator performs in terms of relative absolute bias (RAB) and MSE. Additionally, as the aim of this section, Bayes prediction has been obtained from the s-th observation, where r 0 < s ≤ n for a one-sample and two-sample prediction problem, and is associated with the inference based on the available data, namely . Specifically, we wish to provide an estimate of the posterior density function of x (s) given the data and also construct a 100(1 − φ)% predictive interval of x (s) . We consider these two cases separately.
The estimates of the parameters in the Bayes technique are generated based on informative priors in order to evaluate the type of prior. In the case of informative priors, the hyper-parameters are selected by elective hyper-parameters utilizing MLE information to display the outcomes of estimated parameters with the help of the MLE's asymptotic distribution and Gibbs samples. For confidence intervals, we compare the average intervals (lower and upper) of the asymptotic confidence intervals for the MLE and HPD for the Bayesian estimators based on their coverage percentages of 95%. Also, we have calculated approximate 95% confidence intervals for the unknown parameters. The procedure is repeated 1000 times, and the average confidence/credibility is reported. Tables 1-4 present the complete results. The acquired estimates were calculated using the "maxLik" and "coda" packages of the R programming language because the theoretical findings of d and q obtained using the suggested estimation methods cannot be represented in closed form. For these estimates, the "cmaxLik" package was used for MLE, while the "coda" package was used to obtain the Bayesian estimation for MCMC based on MH algorithms.
The simulation results concluded the following findings, which we highlight: • As the sample size increases for the parameters of APW α, θ, and λ, the minimum RAB and MSE decline for the estimated parameters of MLE and Bayes estimates; • The Bayes estimates consistently outperform the MLE in terms of RAB, MSE, and interval values; • In most cases, the HPD intervals are shorter than ACI; • When the censored sample size r increases while keeping the hybrid censored sample's sample size n and time constant, and the performance improves.
When sample size n and censored sample size r are kept constant, performance improves as the time of the hybrid censored sample lengthens.   Table 3. MLE and Bayesian α = 0.5, θ = 0.6, λ = 3.

MLE Bayesian Prediction
One Two

Application
Reference [18] used two real-life data sets and confirmed that APW distribution is better than many competitive distributions such as: Weibull, alpha power exponential, Mc-Donald Weibull, beta Weibull, transmuted Weibull, gamma Lomax, Zografos-Balakrishnan log-logistic, exponentiated Weibull, and exponentiated Weibull distributions. The APW distribution has more different applications, such as inferences and engineering applications based on progressive Type II censoring (see [27]); the optimal test plan of step-stress models under progressively Type II censored samples (see [28]); and discrete APW and its applications (see [29]). In this section, we consider two real-life data sets, which are discussed by [18], and illustrate the methods proposed in the previous sections.
First, the data set is from [18] for the application to the APW distribution, and it represents the survival times (in days) of 109 successive coal-mining disasters in Great Britain, within the period of 1875-1951. The statistical summarized measures of this data are given as follows: minimum value is 1; first quartile is 54; median is 145; mean is 233.3; third quartile is 312; and maximum value is 1630. Before moving on, we want to look at the data set using a scaled Total Time on Test (TTT) plot, a strip plot, a violin plot, and an empirical hazard function of the observed data. Figure 1 gives a clear picture of the distribution's hazard function's shape, which has a decreasing shape and a TTT line under half line. In Figures 1 and 2, the APW is good model to describe these data. Figure 2 illustrates the APW distribution's theoretical and empirical pdf, QQ, CDF, and P-P plot using the data set of the survival times data, and it can be seen that the APW is suitable and reliable for fitting the survival times data set.  Reference [18] obtained the MLE parameters of APW as = 0.01956 , = 1.04985, = 0.00106, the KSD is 0.0592, and its P-value is 0.8383. Table 5 discussed MLE and Bayesian estimation to estimate parameters of APW based on hybrid censoring with different size of samples, which different measures have been obtained as the MLE of the parameters with standard error (SE), lower, upper for confidence intervals with 95%, reliability, and hazard value with . Figure 3 discussed the existence and uniqueness plot of maximum likelihood estimates for survival times data to check the existence and uniqueness estimators. The maximum likelihood values for survival time data set, where r = 60 and T = 110 for the estimated parameter values that coincide with the MLE estimates in Table 5, are shown in Figure 3, which also supports the MLE estimates.   Reference [18] obtained the MLE parameters of APW as = 0.01956 , = 1.04985, = 0.00106, the KSD is 0.0592, and its P-value is 0.8383. Table 5 discussed MLE and Bayesian estimation to estimate parameters of APW based on hybrid censoring with different size of samples, which different measures have been obtained as the MLE of the parameters with standard error (SE), lower, upper for confidence intervals with 95%, reliability, and hazard value with . Figure 3 discussed the existence and uniqueness plot of maximum likelihood estimates for survival times data to check the existence and uniqueness estimators. The maximum likelihood values for survival time data set, where r = 60 and T = 110 for the estimated parameter values that coincide with the MLE estimates in Table 5, are shown in Figure 3, which also supports the MLE estimates.  Reference [18] obtained the MLE parameters of APW as α =0.01956, θ = 1.04985, and λ = 0.00106, the KSD is 0.0592, and its P-value is 0.8383. Table 5 discussed MLE and Bayesian estimation to estimate parameters of APW based on hybrid censoring with different size of samples, which different measures have been obtained as the MLE of the parameters with standard error (SE), lower, upper for confidence intervals with 95%, reliability, and hazard value with t o . Figure 3 discussed the existence and uniqueness plot of maximum likelihood estimates for survival times data to check the existence and uniqueness estimators. The maximum likelihood values for survival time data set, where r = 60 and T = 110 for the estimated parameter values that coincide with the MLE estimates in Table 5, are shown in Figure 3, which also supports the MLE estimates.   The MCMC results are normal, exhibit symmetric posterior density histograms, and have convergence measures by using the Brooks-Gelman-Rubin (BGR) statistic, as shown in Figures 4 and 5. According to Figure 6, the values for the MCMC series, which began with zero and ended with one, do not exhibit any auto-correlation.  The MCMC results are normal, exhibit symmetric posterior density histograms, and have convergence measures by using the Brooks-Gelman-Rubin (BGR) statistic, as shown in Figures 4 and 5. According to Figure 6, the values for the MCMC series, which began with zero and ended with one, do not exhibit any auto-correlation.  The MCMC results are normal, exhibit symmetric posterior density histograms, and have convergence measures by using the Brooks-Gelman-Rubin (BGR) statistic, as shown in Figures 4 and 5. According to Figure 6, the values for the MCMC series, which began with zero and ended with one, do not exhibit any auto-correlation.   Now let us look at the sample prediction problems for one and two. Based on the observed sample, we present in Figure 7 the predictive point for one sample and two sample, and the predictive point of the s-st order statistic. Accordingly, the s-st failure will occur between r and 109 days based on the observed sample. Based on the results in Table  6 and Figure 7, we note the closeness of the results of the two methods for predicting values, but to determine the best method, the key is the KS test for two independent samples. It has been noted that the first method is the best because it contains the least distance between the data and also has the largest p-value of KS test.   Now let us look at the sample prediction problems for one and two. Based on the observed sample, we present in Figure 7 the predictive point for one sample and two sample, and the predictive point of the s-st order statistic. Accordingly, the s-st failure will occur between r and 109 days based on the observed sample. Based on the results in Table  6 and Figure 7, we note the closeness of the results of the two methods for predicting values, but to determine the best method, the key is the KS test for two independent samples. It has been noted that the first method is the best because it contains the least distance between the data and also has the largest p-value of KS test.  Now let us look at the sample prediction problems for one and two. Based on the observed sample, we present in Figure 7 the predictive point for one sample and two sample, and the predictive point of the s-st order statistic. Accordingly, the s-st failure will occur between r and 109 days based on the observed sample. Based on the results in Table 6 and Figure 7, we note the closeness of the results of the two methods for predicting values, but to determine the best method, the key is the KS test for two independent samples. It has been noted that the first method is the best because it contains the least distance between the data and also has the largest p-value of KS test. The authors of [30] provided the second data set. The statistics are 1.5 cm glass fiber strengths, as determined by the National Physical Laboratory in England. The statistically summarized measures of these data are given as follows: minimum value is 0.55; first quartile is 1.375; median is 1.590; mean is 1.507; third quartile is 1.685; and maximum value is 2.240. Reference [18] obtained the MLE parameters of APW for glass fiber data as = 10.8558, = 4.48362, = 0.194777, the KSD is 0.10661, and its p-value is 0.47107. Figure 8 gives a clear picture of the distribution's hazard function's shape, which has a decreasing shape, and the TTT line is under half line. Based on Figures 8 and 9, the APW is a good model to describe these data. Figure 9 illustrates the APW distribution's theoretical and empirical pdf, QQ, CDF, and P-P plot using the glass fiber data, and it can be seen  The authors of [30] provided the second data set. The statistics are 1.5 cm glass fiber strengths, as determined by the National Physical Laboratory in England. The statistically summarized measures of these data are given as follows: minimum value is 0.55; first quartile is 1.375; median is 1.590; mean is 1.507; third quartile is 1.685; and maximum value is 2.240. Reference [18] obtained the MLE parameters of APW for glass fiber data as α = 10.8558, θ = 4.48362, and λ = 0.194777, the KSD is 0.10661, and its p-value is 0.47107. Figure 8 gives a clear picture of the distribution's hazard function's shape, which has a decreasing shape, and the TTT line is under half line. Based on Figures 8 and 9, the APW is a good model to describe these data. Figure 9 illustrates the APW distribution's theoretical and empirical pdf, QQ, CDF, and P-P plot using the glass fiber data, and it can be seen that the APW is suitable and reliable for the fitting glass fiber data. The authors of [30] provided the second data set. The statistics are 1.5 cm glass fiber strengths, as determined by the National Physical Laboratory in England. The statistically summarized measures of these data are given as follows: minimum value is 0.55; first quartile is 1.375; median is 1.590; mean is 1.507; third quartile is 1.685; and maximum value is 2.240. Reference [18] obtained the MLE parameters of APW for glass fiber data as = 10.8558, = 4.48362, = 0.194777, the KSD is 0.10661, and its p-value is 0.47107. Figure 8 gives a clear picture of the distribution's hazard function's shape, which has a decreasing shape, and the TTT line is under half line. Based on Figures 8 and 9, the APW is a good model to describe these data. Figure 9 illustrates the APW distribution's theoretical and empirical pdf, QQ, CDF, and P-P plot using the glass fiber data, and it can be seen that the APW is suitable and reliable for the fitting glass fiber data.    Figure 10 discussed the existence and uniqueness plot of maximum likelihood estimates for glass fiber data to check for existence and uniqueness estimators. The maximum likelihood values for the glass fiber data set, where r = 40 and T = 1.5 for the estimated parameter values that The authors of [30] provided the second data set. The statistics are 1.5 cm glass fiber strengths, as determined by the National Physical Laboratory in England. The statistically summarized measures of these data are given as follows: minimum value is 0.55; first quartile is 1.375; median is 1.590; mean is 1.507; third quartile is 1.685; and maximum value is 2.240. Reference [18] obtained the MLE parameters of APW for glass fiber data as = 10.8558, = 4.48362, = 0.194777, the KSD is 0.10661, and its p-value is 0.47107. Figure 8 gives a clear picture of the distribution's hazard function's shape, which has a decreasing shape, and the TTT line is under half line. Based on Figures 8 and 9, the APW is a good model to describe these data. Figure 9 illustrates the APW distribution's theoretical and empirical pdf, QQ, CDF, and P-P plot using the glass fiber data, and it can be seen that the APW is suitable and reliable for the fitting glass fiber data.    Figure 10 discussed the existence and uniqueness plot of maximum likelihood estimates for glass fiber data to check for existence and uniqueness estimators. The maximum likelihood values for the glass fiber data set, where r = 40 and T = 1.5 for the estimated parameter values that   Figure 10 discussed the existence and uniqueness plot of maximum likelihood estimates for glass fiber data to check for existence and uniqueness estimators. The maximum likelihood values for the glass fiber data set, where r = 40 and T = 1.5 for the estimated parameter values that coincide with the MLE estimates in Table 7, are shown in Figure 10, which also supports the MLE estimates. coincide with the MLE estimates in Table 7, are shown in Figure 10, which also supports the MLE estimates.  The MCMC results are normal, exhibit symmetric posterior density histograms, and have convergence measures for APW parameters with the glass fiber data, as shown in Figures 11 and 12. According to Figure 13, the values for the MCMC series, which began with zero and ended with one, do not exhibit any auto-correlation for APW parameters with glass fibers data. The MCMC results are normal, exhibit symmetric posterior density histograms, and have convergence measures for APW parameters with the glass fiber data, as shown in Figures 11 and 12. According to Figure 13, the values for the MCMC series, which began with zero and ended with one, do not exhibit any auto-correlation for APW parameters with glass fibers data.   Now let us look at the sample prediction problems for one and two. Based on the observed sample, we present in Figure 14 the predictive point for the one-sample and twosample prediction of the s-st order statistic for glass fiber data. Accordingly, the s-st failure will occur between the r and 63rd observation based on the observed sample. Based on the results in Table 8 and Figure 7, we note the closeness of the results of the two methods for predicting values, but to suggest the best method, the key is the KS test for two independent samples. It has been noted that the first method is the best because it contains the least distance between the data and also has the largest p-value of KS test.    Now let us look at the sample prediction problems for one and two. Based on the observed sample, we present in Figure 14 the predictive point for the one-sample and twosample prediction of the s-st order statistic for glass fiber data. Accordingly, the s-st failure will occur between the r and 63rd observation based on the observed sample. Based on the results in Table 8 and Figure 7, we note the closeness of the results of the two methods for predicting values, but to suggest the best method, the key is the KS test for two independent samples. It has been noted that the first method is the best because it contains the least distance between the data and also has the largest p-value of KS test.    Now let us look at the sample prediction problems for one and two. Based on the observed sample, we present in Figure 14 the predictive point for the one-sample and twosample prediction of the s-st order statistic for glass fiber data. Accordingly, the s-st failure will occur between the r and 63rd observation based on the observed sample. Based on the results in Table 8 and Figure 7, we note the closeness of the results of the two methods for predicting values, but to suggest the best method, the key is the KS test for two independent samples. It has been noted that the first method is the best because it contains the least distance between the data and also has the largest p-value of KS test.  Now let us look at the sample prediction problems for one and two. Based on the observed sample, we present in Figure 14 the predictive point for the one-sample and two-sample prediction of the s-st order statistic for glass fiber data. Accordingly, the s-st failure will occur between the r and 63rd observation based on the observed sample. Based on the results in Table 8 and Figure 7, we note the closeness of the results of the two methods for predicting values, but to suggest the best method, the key is the KS test for two independent samples. It has been noted that the first method is the best because it contains the least distance between the data and also has the largest p-value of KS test.  Figure 14. Bayes prediction by one-sample prediction and two-sample prediction: glass fiber data.

Concluding Remarks
Under the hybrid Type II censoring schemes, the maximum likelihood estimation and Bayesian estimation techniques for the alpha power Weibull distribution are devised. Both paradigms take into account parameter estimates as well as reliability, hazard rate functions and mean failure time. On the basis of simulated samples produced from the distribution for various hybrid Type II censoring techniques, the long-term performances of the suggested estimators are compared. Under the symmetric loss function, Bayes estimators are built. In reality, Bayes estimators give accurate estimates of the parameters because, as predicted, they have mean squared errors that are lower than those of MLEs. Additionally, the issue of predicted future observables was also covered, and one-and two-sample predictive posteriors of the future order statistics were obtained. The posteriors become quite complex and are impossible to simplify into any clear forms. The samples are extracted from the posteriors using the Metropolis-Hastings method, and a sample-based summary of the predicted posteriors is then provided. Because of reasonable results of the APW model in application and its flexibility in this study, in the future, researchers should discuss several new issues for the APW model. For instance, the Bayesian estimation of one-and two-sample prediction based on different censored samples and constant/step stress for APW will be studied. Also, the classical two-sample prediction of APW distribution based on different censored samples can be obtained. Finally, we think that academics, reliability experts, and scientists in the engineering and manufacturing industries will find the approaches outlined in this work to be of great use.

Concluding Remarks
Under the hybrid Type II censoring schemes, the maximum likelihood estimation and Bayesian estimation techniques for the alpha power Weibull distribution are devised. Both paradigms take into account parameter estimates as well as reliability, hazard rate functions and mean failure time. On the basis of simulated samples produced from the distribution for various hybrid Type II censoring techniques, the long-term performances of the suggested estimators are compared. Under the symmetric loss function, Bayes estimators are built. In reality, Bayes estimators give accurate estimates of the parameters because, as predicted, they have mean squared errors that are lower than those of MLEs. Additionally, the issue of predicted future observables was also covered, and one-and two-sample predictive posteriors of the future order statistics were obtained. The posteriors become quite complex and are impossible to simplify into any clear forms. The samples are extracted from the posteriors using the Metropolis-Hastings method, and a sample-based summary of the predicted posteriors is then provided. Because of reasonable results of the APW model in application and its flexibility in this study, in the future, researchers should discuss several new issues for the APW model. For instance, the Bayesian estimation of oneand two-sample prediction based on different censored samples and constant/step stress for APW will be studied. Also, the classical two-sample prediction of APW distribution based on different censored samples can be obtained. Finally, we think that academics, reliability experts, and scientists in the engineering and manufacturing industries will find the approaches outlined in this work to be of great use.