One- and Two-Sample Predictions Based on Progressively Type-II Censored Carbon Fibres Data Utilizing a Probability Model

New Weibull-Pareto distribution is a significant and practical continuous lifetime distribution, which plays an important role in reliability engineering and analysis of some physical properties of chemical compounds such as polymers and carbon fibres. In this paper, we construct the predictive interval of unobserved units in the same sample (one sample prediction) and the future sample based on the current sample (two-sample prediction). The used samples are generated from new Weibull-Pareto distribution due to a progressive type-II censoring scheme. Bayesian and maximum likelihood approaches are implemented to the prediction problems. In the Bayesian approach, it is not easy to simplify the predictive posterior density function in a closed form, so we use the generated Markov chain Monte Carlo samples from the Metropolis-Hastings technique with Gibbs sampling. Moreover, the predictive interval of future upper-order statistics is reported. Finally, to demonstrate the proposed methodology, both simulated data and real-life data of carbon fibres examples are considered to show the applicabilities of the proposed methods.


Introduction
Predictive analytics is used to reduce time, effort, and costs in forecasting business outcomes. A better decision will be supported when more data have been available. Moreover, organizations can solve their own problems and identify opportunities, by giving accurate and reliable insights. Using predictive analytics, we can analyse collective data to get new opportunities for customer attraction.
In the last few years, there has been growing interest in prediction which plays a vital role in many fields. For example, in industry, the experimenter wants to predict the lifetime of a future unobserved unit that relies on the information available from the current sample. So, the experimenter or the manufacturer introduces its products in the market and wants to make it on the place of desire and the focus of consumers by making their warranty limits more acceptable to them. For more information about applications of prediction, the reader can see the following researches: Ghafouri et al. [1], Pushpalatha et al. [2], Lee et al. [3], Burnaev [4], Sharma and Vijayakumar [5], and Asher et al. [6]. e future prediction problem can be separated into two types as follows: the first type is known as an OSP problem, and the other one is a TSP problem. In the OSP problem, the variable to be predicted comes from the same sequence of variables observed and is dependent on the current sample (see Figure 1). In the second type, the variable to be predicted comes from another independent future sample.
Suleman and Albert [7] suggested a new generalization form of Weibull-Pareto distribution denoted by NWPD, which is useful in modeling real-life situations and different scientific disciplines fields such as biological and marketing science in addition to reliability analysis and life testing. e probability density function (pdf ) and cumulative distribution function (cdf ) of a random variable X having an NWPD which is denoted by NWPD (δ, β, θ) are given, respectively, by where δ and θ are the scale parameters, and β is the shape parameter. e reliability function S(t) and hazard rate function h(t) of the NWPD take the forms, respectively, as It should be noted that the NWPD reduces to wellknown distributions such as Weibull, Rayleigh, and exponential distributions as follows: (i) If δ � θ � 1, then NWPD reduces to Weibull (β, 1). (ii) If δ � 1, then NWPD reduces to Weibull (β, θ). (iii) If δ � 1/2 and β � 2, then NWPD reduces to Rayleigh (θ). (iv) If β � 1, then NWPD reduces to an exponential distribution with a mean equal θ/δ.
It is clear that the shape of h(t) depends on the parameter β, and the following can be observed.
(i) If β � 1, the failure rate is constant and given by h(t) � δ/θ. is makes the NWPD suitable for modeling systems or components with a constant failure rate. (ii) If β > 1, the hazard is an increasing function of t, which makes the NWPD suitable for modeling components that wear faster with time.
(iii) If β < 1, the hazard is a decreasing function of t, which makes the NWPD suitable for modeling components that wear slower with time. For a quick illustration, see Figure 2.
e designed body of the paper is built to obtain the Bayesian and frequentist prediction under a ProgT-II C sample whose lifetime failures have NWPD. We study two popular techniques of the prediction problems known as OSP and TSP. As a vivid example of the applicability of the methodology used in our paper, the new Weibull-Pareto distribution was applied to model the exceedances of flood peaks (in m 3 /s) of the Wheaton River near Carcross in Yukon Territory, Canada. In our paper, in the case of a onesample prediction, it is possible to predict the values of the exceedances of the flood peaks that were not recorded for any reason while, in the case two-sample prediction, it is possible to predict the excesses of future flood peaks based on the available data. Accordingly, the necessary precautions can be taken to limit the destruction that may be caused by the flood. ere are several kinds of literature discussing the prediction problem under the ProgT-II CS for different distributions, for instance, Ghafouri et al. [8], Abdel-Hamid [9], AL-Hussaini et al. [10], Raqab et al. [11], Golparvar and Parsian [12] and Soliman et al. [13].
Also, many authors have focused on the problem of predicting either TSP or OSP and TSP together based on various types of censored data from different lifetime models, see, for example, Mahmoud et al. [14], EL-Sagheer [15], Ahmed [16], and Abushal and Al-Zaydi [17,18]. e remainder of the paper is organized as follows: the ML and Bayesian point estimates of the unknown parameters are discussed in Section 2. In Section 3, the MLPI and BPI are explained in the case of OSP. e MLPI and BPI of the FOS sample are outlined in Section 4. In the same section, the MLPI and BPI for the FURS sample are also obtained. Section 5 is devoted to analyse two real-life examples. Conclusion remarks and the results of this work are reported in Section 6.

Maximum Likelihood and Bayesian Approaches
Suppose that X erefore, the log-likelihood function ℓ(x; δ, β, θ) can be expressed as Unobserved Failures:  Upon differentiating (7) with respect to δ, β, and θ, respectively, and equating each result to zero, we obtain It is clear that (14) cannot be obtained in a closed form. So, we apply the M-H technique with Gibbs sampling to generate MCMC samples and obtain the Bayes estimates of δ, β, and θ. e reader can see the detailed steps of the M-H technique with Gibbs sampling in the study of Mahmoud et al. [21]. with a progressive censoring scheme (R 1 , R 2 , . . . , R m ). Suppose that X i: R l , i � 1, 2, . . . , R l and l � 1, 2, . . . , m denote failure lifetimes of i th unobserved units, then the conditional pdf of X i: R l ≡ x i: R l for a given value of δ, β, and θ defined as

One-Sample Prediction
Inserting (1) and (2) in (13), we get 4 Computational Intelligence and Neuroscience e distribution function of x i: R l can be defined by where G 1 (x i: R l | x) can be obtained after replacing the values of δ, β, and θ by their point estimates δ, β, and θ as in (17). Newton-Raphson iteration method is employed to get the approximated solutions of (18) and (19).

Bayesian Prediction.
Using (14) and (16), the predictive posterior density function of x i: R l be given in the following form: It is so hard to simplify (20) in a closed formula. So, MCMC samples generated by applying the M-H technique within Gibbs sampling can be used to approximate the g * 1 (x i: R l | x) as As in (17), we can approximate the distribution function of x i: R l based on the generated MCMC samples as follows: Computational Intelligence and Neuroscience 5 To solve (23) and (24), we employ the Newton-Raphson iteration method.

Two-Sample Prediction
TSP is a useful method to predict the failure lifetimes in the future sample based on the available current sample which was drawn from the same population. In this section, we discuss two cases of TSP. e first one is the TSP for FOS, and the other is the TSP for FURS. Also, the construction of PI based on ML and Bayesian predictions in the two cases of TSP is discussed. be a ProgT-II C sample and let Y 1 , Y 2 , . . . , Y n 1 be the FOS sample drawn from the same NWPD (δ, β, θ). Our concern is to make predictions about the s th , 1 ≤ s ≤ n 1 FOS values. e conditional pdf of FOS Y s for a given values of δ, β, and θ is expressed in the formula, see David and Nagaraja [22].

Prediction of
Inserting (1) and (2) in (23), we get e distribution function of Y s takes the form It is evident that (28) and (29) do not have an analytical solution; therefore, the Newton-Raphson iteration method is applied to get the approximated solutions.

Bayesian Prediction.
e predictive posterior density function of FOS y s can be written using (14) and (29) as follows: e approximated solution of g * 2 (y s | x) and its distribution function can be obtained by applying the generated MCMC samples as follows: 6 Computational Intelligence and Neuroscience Pr We need to apply some suitable numerical techniques such Newton-Raphson iteration method for solving (33) and (34).

Prediction of Future Upper Record Statistics.
Suppose that the available current sample X  (2) , . . . , Z U(n 2 ) be the FURS sample drawn from the same NWPD (δ, β, θ). We want to make predictions about the s th , 1 ≤ s ≤ n 2 FURS values. e conditional pdf of FURS Z s for a given value of δ, β, and θ is given in the form; see Chandler [23].
Inserting (1) and (2) in (33), we get e distribution function of Z s defined as follows:

Computational Intelligence and Neuroscience
For solving (38) and (39), we use the Newton-Raphson iteration method.

Bayesian Prediction.
e predictive posterior density function of FURS z s can be written using (14) and (36) as follows: e approximated solution of g * 3 (z s | x) and its distribution function can be obtained by applying the generated MCMC samples as follows: erefore, the (1 − c)100% BPI (LB 3 , UB 3 ) of FURS z s can be obtained in the following form: We need to apply some suitable numerical techniques such Newton-Raphson iteration method for solving (43) and (44).

Numerical Computations
To illustrate the proposed methods discussed in the previous sections, we consider two examples, the first one is a simulated data set, and the other is a real data set.
Example 2. (Real-life data): e data are represented by the strength data measured in GPA, for single carbon fibres, and impregnated 1000 carbon fibre tows. For analyzed purposes, we consider single fibres of 20 mm with sample sizes n � 67. ese data are reported by Badar and Priest [25] and used by Kundu and Raqab [26]. e distance between the empirical and the fitted distribution functions as computed by using Kolmogorov-Smirnov (K-S) is 0.046121, and the corresponding p value is 0.9988 Since the p value is quite high, we cannot reject the null hypothesis that the data are coming from the NWPD. Empirical, Q − Q, and P − P plots are shown in Figure 3, which clear that the NWPD fits the data very well. e data are as follows: (45) e generated ProgT-II C sample from data set 1 with effective sample size m � 30 and censoring scheme R � (5, 0, 0, 4, 0, 0, 3, 0, 0, 4, 0, 3, 0, 0, 3, 0, 0, 3, 0, 2, 0, 0, 2, 0, 0, 2, 0, 2, 0, 4) is given as follows: x � (0.312, 0.314, 0.479, 0.552, 0.70, 0. , β j , θ j ), j � 1, 2, . . . , 32000 and discard the first 2000 values as "burn-in" periods under the consideration of the noninformative prior gamma functions of δ, β, and θ with hyperparameters c i and η i � 0, where i � 1, 2, 3. e mean values of δ, β, and θ are given in Table 8. e results of 90% MLPI and BPI of x i: R l are shown in Table 9. Also, the 95% MLPI and BPI of x i: R l are summarized in Table 10. Table 11 shows the 90% MLPI and BPI of FOS y s . e 95% MLPI and BPI of FOS y s are listed in Table 12. e results of 90% MLPI and BPI of FURS z s are shown in Table 13. Also, the 95% MLPI and BPI of FURS z s are obtained in Table 14. 8 Computational Intelligence and Neuroscience