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

Authors: Mahmoud El-Morshedy; Rashad M. El-Sagheer; Samah H. El-Essawy; Khaled M. Alqahtani; Mohamed El-Dawoody; Mohamed S. Eliwa
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

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.


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## Body

## 1. 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].The 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 Figure1). In the second type, the variable to be predicted comes from another independent future sample.Figure 1
Diagram of OSP under progT-II CS.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. The 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(1)fx;δ,β,θ=βδθxθβ−1e−δx/θβ,x>0;δ,β,θ>0,(2)Fx;δ,β,θ=1−e−δx/θβ,x>0;δ,β,θ>0,where δ and θ are the scale parameters, and β is the shape parameter. The reliability function St and hazard rate function ht of the NWPD take the forms, respectively, as(3)St=e−δt/θβ,t>0,(4)ht=βδθtθβ−1,t>0.It should be noted that the NWPD reduces to well-known distributions such as Weibull, Rayleigh, and exponential distributions as follows:(i)
Ifδ=θ=1, then NWPD reduces to Weibull β,1.(ii)
Ifδ=1, then NWPD reduces to Weibull β,θ.(iii)
Ifδ=1/2 and β=2, then NWPD reduces to Rayleigh θ.(iv)
Ifβ=1, then NWPD reduces to an exponential distribution with a mean equal θ/δ.It is clear that the shape ofht depends on the parameter β, and the following can be observed.(i)
Ifβ=1, the failure rate is constant and given by ht=δ/θ. This makes the NWPD suitable for modeling systems or components with a constant failure rate.(ii)
Ifβ>1, the hazard is an increasing function of t, which makes the NWPD suitable for modeling components that wear faster with time.(iii)
Ifβ<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.Figure 2
The hazard rate function of NWPD for different values ofδ, β, and θ.The 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 m3/s) of the Wheaton River near Carcross in Yukon Territory, Canada. In our paper, in the case of a one-sample 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. There 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].The remainder of the paper is organized as follows: the ML and Bayesian point estimates of the unknown parameters are discussed in Section2. In Section 3, the MLPI and BPI are explained in the case of OSP. The 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.


## 2. Maximum Likelihood and Bayesian Approaches

Suppose thatX1:m:nR1,…,Rm,X2:m:nR1,…,Rm,…,Xm:m:nR1,…,Rm be a ProgT-II C sample from the NWPD with a progressive censored scheme R=R1,R2,…,Rm. According to Balakrishnan and Aggarwala [19], the joint probability density function is given by(5)f1,2,…,mx1:m:nR1,…,Rm,x2:m:nR1,…,Rm,…,xm:m:nR1,…,Rm∝∏i=1mfxi:m:nR1,…,Rm1−Fxi:m:nR1,…,RmRi.Inserting (1) and (2) into (5), then the likelihood function can be written as(6)Lx¯;δ,β,θ∝βmδmθ−m∏i=1mxiθβ−1e−δ∑i=1mRi+1xi/θβ.Therefore, the log-likelihood functionℓx¯;δ,β,θ can be expressed as(7)ℓx¯;δ,β,θ=mlogβ+mlogδ−mlogθ+β−1∑i=1mlogxiθ−δ∑i=1mRi+1xiθβ.Upon differentiating (7) with respect to δ, β, and θ, respectively, and equating each result to zero, we obtain(8)mδ−∑i=1mRi+1xiθβ=0,(9)mβ+∑i=1mlogxiθ−δ∑i=1mRi+1xiθβlogxiθ=0,(10)mβθ−βδθ∑i=1mRi+1xiθβ=0.From (8), we get MLE of δ as(11)δ^=m∑i=1mRi+1xiθ^β^−1.Since (9) and (10) do not have closed-form solutions, the Newton-Raphson iteration method can be used to get the MLEs of β and θ. The reader can see the detailed steps of the Newton-Raphson algorithm in EL-Sagheer [20]. Now, we discuss how to obtain the Bayesian estimates for δ, β, and θ. Let the parameters δ, β, and θ be independent and follow the gamma prior distributions as(12)π1δ=η1γ1Γγ1δγ1−1e−η1δ,δ>0,γ1>0,η1>0,π2β=η2γ2Γγ2βγ2−1e−η2β,β>0,γ2>0,η2>0,π3θ=η3γ3Γγ3θγ3−1e−η3θ,θ>0,γ3>0,η3>0.,where the hyperparameters γi and ηi (where i=1,2,3) are reflected prior knowledge about δ, β, and θ. Note if γi=ηi=0, then the noninformative priors of δ, β, and θ are obtained.Hence, the joint prior function of the parametersδ, β, and θ is defined by(13)πδ,β,θ=η1γ1η2γ2η3γ3Γγ1Γγ2Γγ3δγ1−1βγ2−1θγ3−1e−η1δ−η2β−η3θ.From (6) and (13), the joint posterior density function can be given as follows:(14)π∗δ,β,θ|x¯=Lx¯;δ,β,θ×πδ,β,θ∫0∞∫0∞∫0∞Lx¯;δ,β,θ×πδ,β,θdδdβdθ∝βm+γ2−1δm+γ1−1θ−m+γ3−1∏i=1mxiθβ−1×e−η2β−η3θ−δη1+∑i=1mRi+1xi/θβ.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. The reader can see the detailed steps of the M-H technique with Gibbs sampling in the study of Mahmoud et al. [21].


## 3. One-Sample Prediction

OSP is a useful method to predict the failure lifetimes of the unobserved units (the removed surviving units) in the same sample generated by the ProgT-II C sampleX1:m:nR1,…,Rm, X2:m:nR1,…,Rm ,…, Xm:m:nR1,…,Rm with a progressive censoring scheme R1,R2,…,Rm. Suppose that Xi:Rl,i=1,2,…,Rl and l=1,2,…,m denote failure lifetimes of ith unobserved units, then the conditional pdf of Xi:Rl≡xi:Rl for a given value of δ,β, and θ defined as(15)g1xi:Rl|δ,β,θ,x¯=iRliFxi:Rl;δ,β,θ−Fx;δ,β,θi−11−Fxi:Rl;δ,β,θRl−i×fxi:Rl;δ,β,θ1−Fx;δ,β,θ−Rl,xi:Rl>x.Inserting (1) and (2) in (13), we get(16)g1xi:Rl|δ,β,θ,x¯=iRliβδθxi:Rlθβ−1eRlδx/θβe−Rl−i+1δxi:Rl/θβe−δx/θβ−e−δxi:Rl/θβi−1=iRliβδθxi:Rlθβ−1∑k=0i−1i−1k−1ke−Rl−i+k+1δxi:Rl/θβ×e−−Rl+i−k−1δx/θβ.The distribution function ofxi:Rl can be defined by(17)G1xi:Rl|δ,β,θ,x¯=∫xxi:Rlg1xi:Rl|δ,β,θ,x¯dxi:Rl=iRli∑k=0i−1i−1k−1ke−−Rl+i−k−1δx/θβ×e−Rl−i+k+1δx/θβ−e−Rl−i+k+1δxi:Rl/θβRl−i+k+1.


### 3.1. Maximum Likelihood Prediction

Due to ML prediction, the1−γ100% MLPI LB1,UB1 of xi:Rl can be written in the form(18)Prxi:Rl>LB1|x¯=1−γ2=1−G^1LB1|x¯⇒G^1LB1|x¯=γ2,(19)Prxi:Rl>UB1|x¯=γ2=1−G^1UB1|x¯⇒G^1UB1|x¯=1−γ2,where G^1xi:Rl|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).


### 3.2. Bayesian Prediction

Using (14) and (16), the predictive posterior density function of xi:Rl be given in the following form:(20)g1∗xi:Rl|x¯=∫0∞∫0∞∫0∞g1xi:Rl|δ,β,θ,x¯π∗δ,β,θ|x¯dδdβdθ.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 g1∗xi:Rl|x¯ as(21)g^1∗xi:Rl|x¯=1N−M∑j=M+1Ng1xi:Rl|δj,βj,θj,x¯.As in (17), we can approximate the distribution function of xi:Rl based on the generated MCMC samples as follows:(22)G^1∗xi:Rl|x¯=1N−M∑j=M+1NG1xi:Rl|δj,βj,θj,x¯.Then, the 1−γ100% BPI LB1,UB1 of xi:Rl takes the form as(23)Prxi:Rl>LB1|x¯=1−γ2=1−G^1∗LB1|x¯⇒G^1∗LB1|x¯=γ2,(24)Prxi:Rl>UB1|x¯=γ2=1−G^1∗UB1|x¯⇒G^1∗UB1|x¯=1−γ2.To solve (23) and (24), we employ the Newton-Raphson iteration method.


## 3.1. Maximum Likelihood Prediction

Due to ML prediction, the1−γ100% MLPI LB1,UB1 of xi:Rl can be written in the form(18)Prxi:Rl>LB1|x¯=1−γ2=1−G^1LB1|x¯⇒G^1LB1|x¯=γ2,(19)Prxi:Rl>UB1|x¯=γ2=1−G^1UB1|x¯⇒G^1UB1|x¯=1−γ2,where G^1xi:Rl|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).


## 3.2. Bayesian Prediction

Using (14) and (16), the predictive posterior density function of xi:Rl be given in the following form:(20)g1∗xi:Rl|x¯=∫0∞∫0∞∫0∞g1xi:Rl|δ,β,θ,x¯π∗δ,β,θ|x¯dδdβdθ.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 g1∗xi:Rl|x¯ as(21)g^1∗xi:Rl|x¯=1N−M∑j=M+1Ng1xi:Rl|δj,βj,θj,x¯.As in (17), we can approximate the distribution function of xi:Rl based on the generated MCMC samples as follows:(22)G^1∗xi:Rl|x¯=1N−M∑j=M+1NG1xi:Rl|δj,βj,θj,x¯.Then, the 1−γ100% BPI LB1,UB1 of xi:Rl takes the form as(23)Prxi:Rl>LB1|x¯=1−γ2=1−G^1∗LB1|x¯⇒G^1∗LB1|x¯=γ2,(24)Prxi:Rl>UB1|x¯=γ2=1−G^1∗UB1|x¯⇒G^1∗UB1|x¯=1−γ2.To solve (23) and (24), we employ the Newton-Raphson iteration method.


## 4. 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. The 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.


### 4.1. Prediction of Future-Order Statistics

Suppose that the available current sampleX1:m:nR1,…,Rm,X2:m:nR1,…,Rm,…,Xm:m:nR1,…,Rm be a ProgT-II C sample and let Y1,Y2,…,Yn1 be the FOS sample drawn from the same NWPD δ,β,θ. Our concern is to make predictions about the sth, 1≤s≤n1 FOS values. The conditional pdf of FOS Ys for a given values of δ,β, and θ is expressed in the formula, see David and Nagaraja [22].(25)g2ys|δ,β,θ,x¯=sn1s1−Fys;δ,β,θn1−sFys;δ,β,θs−1fys;δ,β,θ.Inserting (1) and (2) in (23), we get(26)g2ys;δ,β,θ=sn1sβδθysθβ−1e−n1−s+1δys/θβ1−e−δys/θβs−1=sn1sβδθysθβ−1∑k=0s−1s−1k−1ke−n1−s+k+1δys/θβ.The distribution function of Ys takes the form(27)G2ys|δ,β,θ,x¯=∫0ysg2ys|δ,β,θ,x¯dys=sn1s∑k=0s−1s−1k−1k1−e−n1−s+k+1δys/θβn1−s+k+1.


#### 4.1.1. Maximum Likelihood Prediction

Due to ML prediction, PI of FOSys can be computed by replacing the values of δ,β, and θ by their point estimates δ^,β^, and θ^. The 1−γ100% MLPI LB2,UB2 of FOS ys takes the form as(28)Prys>LB2|x¯=1−γ2=1−G^2LB2|x¯⇒G^2LB2|x¯=γ2,(29)Prys>UB2|x¯=γ2=1−G^2UB2|x¯⇒G^2UB2|x¯=1−γ2.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.


#### 4.1.2. Bayesian Prediction

The predictive posterior density function of FOSys can be written using (14) and (29) as follows:(30)g2∗ys|x¯=∫0∞∫0∞∫0∞g2ys|δ,β,θ,x¯π∗δ,β,θ|x¯dδdβdθ.The approximated solution of g2∗ys|x¯ and its distribution function can be obtained by applying the generated MCMC samples as follows:(31)g^2∗ys|x¯=1N−M∑j=M+1Ng1ys|δj,βj,θj,x¯,(32)G^2∗ys|x¯=1N−M∑j=M+1NG2ys|δj,βj,θj,x¯.Therefore, the 1−γ100% BPI LB2,UB2 of FOS ys is constructed.(33)Prys>LB2|x¯=1−γ2=1−G^2∗LB2|x¯⇒G^2∗LB2|x¯=γ2,(34)Prys>UB2|x¯=γ2=1−G^2∗UB2|x¯⇒G^2∗UB2|x¯=1−γ2.We need to apply some suitable numerical techniques such Newton-Raphson iteration method for solving (33) and (34).


### 4.2. Prediction of Future Upper Record Statistics

Suppose that the available current sampleX1:m:nR1,…,Rm,X2:m:nR1,…,Rm,…,Xm:m:nR1,…,Rm be ProgT-II C sample and let ZU1,ZU2,…,ZUn2 be the FURS sample drawn from the same NWPD (δ,β,θ). We want to make predictions about the sth, 1≤s≤n2 FURS values. The conditional pdf of FURS Zs for a given value of δ,β, and θ is given in the form; see Chandler [23].(35)g3zs|δ,β,θ,x¯=1s−1!−log1−Fzs;δ,β,θs−1fzs;δ,β,θ.Inserting (1) and (2) in (33), we get(36)g3zs|δ,β,θ,x¯=1s−1!βδθzsθβ−1−log1−1−e−δzs/θβs−1e−δzs/θβ.The distribution function of Zs defined as follows:(37)G3zs|δ,β,θ,x¯=∫0zsg3zs|δ,β,θ,x¯dzs=1s−1!βδθ∫0zszsθβ−1−log1−1−e−δzs/θβs−1e−δzs/θβdzs=1s−1!∫01−1−e−δzs/θβ−logφs−1dφ=1s−1!Γs−Γs,−log1−1−e−δzs/θβ.


#### 4.2.1. Maximum Likelihood Prediction

Due to ML prediction, PI ofzs can be computed by replacing the values of δ,β, and θ by their point estimates δ^,β^, and θ^. The 1−γ100% MLPI LB3,UB3 of FURS zs takes the form as(38)Przs>LB3|x¯=1−γ2=1−G^3LB3|x¯⇒G^3LB3|x¯=γ2,(39)Przs>UB3|x¯=γ2=1−G^3UB3|x¯⇒G^3UB3|x¯=1−γ2.For solving (38) and (39), we use the Newton-Raphson iteration method.


#### 4.2.2. Bayesian Prediction

The predictive posterior density function of FURSzs can be written using (14) and (36) as follows:(40)g3∗zs|x¯=∫0∞∫0∞∫0∞g3zs|δ,β,θ,x¯π∗δ,β,θ|x¯dδdβdθ.The approximated solution of g3∗zs|x¯ and its distribution function can be obtained by applying the generated MCMC samples as follows:(41)g^3∗zs|x¯=1N−M∑j=M+1Ng3zs|δj,βj,θj,x¯,(42)G^3∗zs|x¯=1N−M∑j=M+1NG3zs|δj,βj,θj,x¯.Therefore, the 1−γ100% BPI LB3,UB3 of FURS zs can be obtained in the following form:(43)Przs>LB3|x¯=1−γ2=1−G^3∗LB3|x¯⇒G^3∗LB3|x¯=γ2,(44)Przs>UB3|x¯=γ2=1−G^3∗UB3|x¯⇒G^3∗UB3|x¯=1−γ2.We need to apply some suitable numerical techniques such Newton-Raphson iteration method for solving (43) and (44).


## 4.1. Prediction of Future-Order Statistics

Suppose that the available current sampleX1:m:nR1,…,Rm,X2:m:nR1,…,Rm,…,Xm:m:nR1,…,Rm be a ProgT-II C sample and let Y1,Y2,…,Yn1 be the FOS sample drawn from the same NWPD δ,β,θ. Our concern is to make predictions about the sth, 1≤s≤n1 FOS values. The conditional pdf of FOS Ys for a given values of δ,β, and θ is expressed in the formula, see David and Nagaraja [22].(25)g2ys|δ,β,θ,x¯=sn1s1−Fys;δ,β,θn1−sFys;δ,β,θs−1fys;δ,β,θ.Inserting (1) and (2) in (23), we get(26)g2ys;δ,β,θ=sn1sβδθysθβ−1e−n1−s+1δys/θβ1−e−δys/θβs−1=sn1sβδθysθβ−1∑k=0s−1s−1k−1ke−n1−s+k+1δys/θβ.The distribution function of Ys takes the form(27)G2ys|δ,β,θ,x¯=∫0ysg2ys|δ,β,θ,x¯dys=sn1s∑k=0s−1s−1k−1k1−e−n1−s+k+1δys/θβn1−s+k+1.


### 4.1.1. Maximum Likelihood Prediction

Due to ML prediction, PI of FOSys can be computed by replacing the values of δ,β, and θ by their point estimates δ^,β^, and θ^. The 1−γ100% MLPI LB2,UB2 of FOS ys takes the form as(28)Prys>LB2|x¯=1−γ2=1−G^2LB2|x¯⇒G^2LB2|x¯=γ2,(29)Prys>UB2|x¯=γ2=1−G^2UB2|x¯⇒G^2UB2|x¯=1−γ2.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.


### 4.1.2. Bayesian Prediction

The predictive posterior density function of FOSys can be written using (14) and (29) as follows:(30)g2∗ys|x¯=∫0∞∫0∞∫0∞g2ys|δ,β,θ,x¯π∗δ,β,θ|x¯dδdβdθ.The approximated solution of g2∗ys|x¯ and its distribution function can be obtained by applying the generated MCMC samples as follows:(31)g^2∗ys|x¯=1N−M∑j=M+1Ng1ys|δj,βj,θj,x¯,(32)G^2∗ys|x¯=1N−M∑j=M+1NG2ys|δj,βj,θj,x¯.Therefore, the 1−γ100% BPI LB2,UB2 of FOS ys is constructed.(33)Prys>LB2|x¯=1−γ2=1−G^2∗LB2|x¯⇒G^2∗LB2|x¯=γ2,(34)Prys>UB2|x¯=γ2=1−G^2∗UB2|x¯⇒G^2∗UB2|x¯=1−γ2.We need to apply some suitable numerical techniques such Newton-Raphson iteration method for solving (33) and (34).


## 4.1.1. Maximum Likelihood Prediction

Due to ML prediction, PI of FOSys can be computed by replacing the values of δ,β, and θ by their point estimates δ^,β^, and θ^. The 1−γ100% MLPI LB2,UB2 of FOS ys takes the form as(28)Prys>LB2|x¯=1−γ2=1−G^2LB2|x¯⇒G^2LB2|x¯=γ2,(29)Prys>UB2|x¯=γ2=1−G^2UB2|x¯⇒G^2UB2|x¯=1−γ2.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.


## 4.1.2. Bayesian Prediction

The predictive posterior density function of FOSys can be written using (14) and (29) as follows:(30)g2∗ys|x¯=∫0∞∫0∞∫0∞g2ys|δ,β,θ,x¯π∗δ,β,θ|x¯dδdβdθ.The approximated solution of g2∗ys|x¯ and its distribution function can be obtained by applying the generated MCMC samples as follows:(31)g^2∗ys|x¯=1N−M∑j=M+1Ng1ys|δj,βj,θj,x¯,(32)G^2∗ys|x¯=1N−M∑j=M+1NG2ys|δj,βj,θj,x¯.Therefore, the 1−γ100% BPI LB2,UB2 of FOS ys is constructed.(33)Prys>LB2|x¯=1−γ2=1−G^2∗LB2|x¯⇒G^2∗LB2|x¯=γ2,(34)Prys>UB2|x¯=γ2=1−G^2∗UB2|x¯⇒G^2∗UB2|x¯=1−γ2.We need to apply some suitable numerical techniques such Newton-Raphson iteration method for solving (33) and (34).


## 4.2. Prediction of Future Upper Record Statistics

Suppose that the available current sampleX1:m:nR1,…,Rm,X2:m:nR1,…,Rm,…,Xm:m:nR1,…,Rm be ProgT-II C sample and let ZU1,ZU2,…,ZUn2 be the FURS sample drawn from the same NWPD (δ,β,θ). We want to make predictions about the sth, 1≤s≤n2 FURS values. The conditional pdf of FURS Zs for a given value of δ,β, and θ is given in the form; see Chandler [23].(35)g3zs|δ,β,θ,x¯=1s−1!−log1−Fzs;δ,β,θs−1fzs;δ,β,θ.Inserting (1) and (2) in (33), we get(36)g3zs|δ,β,θ,x¯=1s−1!βδθzsθβ−1−log1−1−e−δzs/θβs−1e−δzs/θβ.The distribution function of Zs defined as follows:(37)G3zs|δ,β,θ,x¯=∫0zsg3zs|δ,β,θ,x¯dzs=1s−1!βδθ∫0zszsθβ−1−log1−1−e−δzs/θβs−1e−δzs/θβdzs=1s−1!∫01−1−e−δzs/θβ−logφs−1dφ=1s−1!Γs−Γs,−log1−1−e−δzs/θβ.


### 4.2.1. Maximum Likelihood Prediction

Due to ML prediction, PI ofzs can be computed by replacing the values of δ,β, and θ by their point estimates δ^,β^, and θ^. The 1−γ100% MLPI LB3,UB3 of FURS zs takes the form as(38)Przs>LB3|x¯=1−γ2=1−G^3LB3|x¯⇒G^3LB3|x¯=γ2,(39)Przs>UB3|x¯=γ2=1−G^3UB3|x¯⇒G^3UB3|x¯=1−γ2.For solving (38) and (39), we use the Newton-Raphson iteration method.


### 4.2.2. Bayesian Prediction

The predictive posterior density function of FURSzs can be written using (14) and (36) as follows:(40)g3∗zs|x¯=∫0∞∫0∞∫0∞g3zs|δ,β,θ,x¯π∗δ,β,θ|x¯dδdβdθ.The approximated solution of g3∗zs|x¯ and its distribution function can be obtained by applying the generated MCMC samples as follows:(41)g^3∗zs|x¯=1N−M∑j=M+1Ng3zs|δj,βj,θj,x¯,(42)G^3∗zs|x¯=1N−M∑j=M+1NG3zs|δj,βj,θj,x¯.Therefore, the 1−γ100% BPI LB3,UB3 of FURS zs can be obtained in the following form:(43)Przs>LB3|x¯=1−γ2=1−G^3∗LB3|x¯⇒G^3∗LB3|x¯=γ2,(44)Przs>UB3|x¯=γ2=1−G^3∗UB3|x¯⇒G^3∗UB3|x¯=1−γ2.We need to apply some suitable numerical techniques such Newton-Raphson iteration method for solving (43) and (44).


## 4.2.1. Maximum Likelihood Prediction

Due to ML prediction, PI ofzs can be computed by replacing the values of δ,β, and θ by their point estimates δ^,β^, and θ^. The 1−γ100% MLPI LB3,UB3 of FURS zs takes the form as(38)Przs>LB3|x¯=1−γ2=1−G^3LB3|x¯⇒G^3LB3|x¯=γ2,(39)Przs>UB3|x¯=γ2=1−G^3UB3|x¯⇒G^3UB3|x¯=1−γ2.For solving (38) and (39), we use the Newton-Raphson iteration method.


## 4.2.2. Bayesian Prediction

The predictive posterior density function of FURSzs can be written using (14) and (36) as follows:(40)g3∗zs|x¯=∫0∞∫0∞∫0∞g3zs|δ,β,θ,x¯π∗δ,β,θ|x¯dδdβdθ.The approximated solution of g3∗zs|x¯ and its distribution function can be obtained by applying the generated MCMC samples as follows:(41)g^3∗zs|x¯=1N−M∑j=M+1Ng3zs|δj,βj,θj,x¯,(42)G^3∗zs|x¯=1N−M∑j=M+1NG3zs|δj,βj,θj,x¯.Therefore, the 1−γ100% BPI LB3,UB3 of FURS zs can be obtained in the following form:(43)Przs>LB3|x¯=1−γ2=1−G^3∗LB3|x¯⇒G^3∗LB3|x¯=γ2,(44)Przs>UB3|x¯=γ2=1−G^3∗UB3|x¯⇒G^3∗UB3|x¯=1−γ2.We need to apply some suitable numerical techniques such Newton-Raphson iteration method for solving (43) and (44).


## 5. 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 1.
(Simulated data). Based on the algorithm which is introduced by Balakrishnan and Sandhu [24], we generate a ProgT-II C sample from NWPD with parameters δ,β,θ=2.4,1.8,2.9 of size m=30, which is generated randomly of sample size n=50 with censoring scheme R = (2, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 1, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 2, 0, 2, 0, 0. 2). The ProgT-II C sample is given x = (0.205494, 0.274422, 0.360082, 0.416501, 0.527163, 0.58346, 0.614485, 0.665395, 0.666271, 0.693925, 0.697056, 0.893878, 0.920077, 0.929093, 0.956805, 0.978055, 1.11192, 1.27356, 1.3368, 1.35507, 1.38305, 1.59598, 1.63893, 1.86817, 1.90648, 2.01795, 2.02848, 2.2878, 2.37404, 2.51562). Based on the M-H technique within Gibbs sampling, we generate 32000 MCMC samples δj,βj,θj,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 γi and ηi=0, where i=1,2,3. The mean values of δ,β, and θ are given in Table 1. The results of 90% MLPI and BPI of xi:Rl are shown in Table 2. Also, the 95% MLPI and BPI of xi:Rl are summarized in Table 3. Table 4 shows the 90% MLPI and BPI of FOS ys. The 95% MLPI and BPI of FOS ys are listed in Table 5. The results of 90% MLPI and BPI of FURS zs are given in Table 6. Also, the 95% MLPI and BPI of FURS zs are obtained in Table 7.Table 1
Mean value ofδ, β, and θ.
ParameterMLBayesδ2.35882.3291β1.81571.8059θ2.87542.8586Table 2
90% MLPI and BPI of xi,Rl for simulated data.
PointMLPIBPILB1,UB1LengthLB1,UB1Lengthx1,10.3258,2.25521.92940.321,2.34572.0247x1,20.8763,3.68222.80590.8678,3.97463.1068x4,10.5625,3.32242.75990.5621,3.52392.9618x7,10.6729,2.35461.68180.6733,2.45871.7854x7,21.0765,3.74952.6731.0779,4.072.9922x10,10.7472,2.38241.63530.7478,2.48911.7413x10,21.1284,3.76862.64021.1303,4.09482.9645x12,10.9798,3.44712.46720.9813,3.67382.6925x15,10.9983,2.49111.49270.9991,2.60611.607x15,20.5527,3.84413.29140.5589,4.1893.6301x20,11.3866,2.69691.31031.3871,2.82361.4365x20,21.0821,3.99112.90911.0841,4.36323.2791x23,11.666,2.86811.20211.6664,3.00231.336x23,21.4116,4.11712.70541.4156,4.50713.0915x25,11.9304,3.04431.11391.9307,3.18511.2544x25,21.7086,4.24992.54131.7146,4.65572.9412x27,12.0513,3.12881.07762.0514,3.27231.2208x27,21.8413,4.31452.47331.8481,4.72722.8791x30,12.5348,3.48730.95262.5347,3.64031.1055x30,22.3606,4.59572.23512.3704,5.03292.6626Table 3
95% MLPI and BPI of xi,Rl for simulated data.
PointMLPIBPILB1,UB1LengthLB1,UB1Lengthx1,10.2704,2.52582.25540.267,2.66292.3959x1,20.7217,4.05113.32940.7095,4.49783.7883x4,10.493,3.71713.22410.4924,4.03773.5453x7,10.6438,2.61671.97280.644,2.7742.1299x7,20.9492,4.11343.16430.949,4.59743.6484x10,10.7206,2.64221.92160.7209,2.80352.0826x10,21.0064,4.13113.12471.0069,4.62283.6159x12,10.9371,3.83112.8940.9378,4.18773.2498x15,10.9775,2.74241.76490.9778,2.91671.9388x15,20.6822,4.20133.5190.6846,4.71834.0337x20,11.3707,2.93371.5631.371,3.12671.7557x20,21.1617,4.33843.17671.1629,4.89263.7297x23,11.6523,3.0941.44171.6525,3.29931.6468x23,21.4764,4.45622.97981.4791,5.03553.5565x25,11.9183,3.26021.34191.9184,3.47611.5577x25,21.7643,4.58092.81661.7684,5.18243.414x27,12.0397,3.34021.30052.0398,3.56061.5208x27,21.8938,4.64182.7481.8984,5.25293.3544x30,12.5251,3.68171.15662.525,3.91791.3928x30,22.4037,4.90752.50392.4104,5.55363.1433Table 4
90% MLPI and BPI of FOS ys for simulated data.
PointMLPIBPILB2,UB2LengthLB2,UB2Lengthy10.0982,0.92280.82460.0835,0.96090.8774y20.2937,1.22520.93150.2632,1.27621.013y30.4797,1.48091.00120.4426,1.54621.1036y40.659,1.72531.06620.6202,1.80941.1892y50.8383,1.97471.13650.8002,2.08471.2845y61.0238,2.24351.21960.9876,2.38941.4018y71.2234,2.55021.32681.1889,2.7471.5581y81.4488,2.92881.481.4144,3.21.7855y91.7232,3.46131.73811.6857,3.85142.1657y102.1158,4.47952.36362.0678,5.10973.0419Table 5
95% MLPI and BPI of FOS ys for simulated data.
PointMLPIBPILB2,UB2LengthLB2,UB2Lengthy10.0666,1.03490.96830.0536,1.08331.0297y20.2378,1.33891.10120.2054,1.40391.1986y30.4113,1.59841.1870.3688,1.68331.3145y40.5817,1.84791.26620.5352,1.95971.4245y50.7531,2.10421.35110.7062,2.2531.5468y60.9309,2.38231.45140.8854,2.58211.6967y71.1218,2.70241.58061.0783,2.97381.8955y81.3364,3.1021.76551.2941,3.47782.1837y91.5956,3.67382.07821.5516,4.21632.6647y101.9599,4.79982.83991.9072,5.68593.7788Table 6
90% MLPI and BPI of FURS zs for simulated data.
PointMLPIBPILB3,UB3LengthLB3,UB3Lengthz10.3491,3.282.93090.332,3.46233.1303z21.0138,4.2253.21121.0004,4.62823.6278z31.6044,4.93773.33331.5923,5.59434.0019z42.1286,5.53793.40932.103,6.46894.3659z52.604,6.0683.4642.5488,7.28764.7388z63.0422,6.54893.50672.9434,8.06675.1233z73.4511,6.99273.54173.2973,8.81535.518z83.836,7.40743.57133.6185,9.53925.9207z94.2011,7.79823.59713.9128,10.24226.3293z104.5492,8.16913.61984.185,10.9276.742Table 7
95% MLPI and BPI of FURS zs for simulated data.
PointMLPIBPILB3,UB3LengthLB3,UB3Lengthz10.2367,3.67843.44180.2183,3.97343.755z20.8209,4.61633.79550.8006,5.23484.4342z31.3759,5.32653.95061.3594,6.30074.9413z41.8794,5.92564.04621.8544,7.2775.4226z52.3407,6.45524.11452.2909,8.19825.9073z62.7685,6.93594.16752.6784,9.08066.4023z73.1691,7.37984.21073.0254,9.93326.9078z83.5474,7.79464.24723.3392,10.76157.4224z93.9069,8.18574.27873.6254,11.56957.9441z104.2502,8.55684.30663.8888,12.35998.4711Example 2.
(Real-life data): The 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 of20mm with sample sizes n=67. These data are reported by Badar and Priest [25] and used by Kundu and Raqab [26]. The 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. The data are as follows:(45)
The generated ProgT-II C sample from data set 1 with effective sample sizem=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, 0.803, 0.861, 0.865, 0.944, 0.958, 0.966, 0.997, 1.006, 1.021, 1.055, 1.063, 1.098, 1.14, 1.179, 1.224, 1.240, 1.253, 1.270, 1.272, 1.274, 2.128, 2.233, 2.433, 2.585, 2.585). Based on the M-H technique within Gibbs sampling, we generate 32000 MCMC samples δj,βj,θj,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 γi and ηi=0, where i=1,2,3. The mean values of δ,β, and θ are given in Table 8. The results of 90% MLPI and BPI of xi:Rl are shown in Table 9. Also, the 95% MLPI and BPI of xi:Rl are summarized in Table 10. Table 11 shows the 90% MLPI and BPI of FOS ys. The 95% MLPI and BPI of FOS ys are listed in Table 12. The results of 90% MLPI and BPI of FURS zs are shown in Table 13. Also, the 95% MLPI and BPI of FURS zs are obtained in Table 14.Figure 3
Empirical, Q-Q, and P-P plots of NWPD for real-life data.Table 8
Mean value ofδ, β, and θ for real-life data.
ParameterMLBayesδ2.35352.4621β1.98591.9742θ3.14723.2208Table 9
90% MLPI and BPI of xi,Rl for real-life data.
PointMLPIBPILB1,UB1LengthLB1,UB1Lengthx1,10.3735,1.61161.23820.3715,1.66111.2897x1,20.6524,2.1411.48850.6433,2.23241.5891x1,30.9836,2.66261.6790.9727,2.81961.8469x1,41.3579,3.30591.9481.3468,3.57382.227x1,51.8516,4.41452.56281.8357,4.91223.0765x4,10.5981,1.85431.25610.5981,1.91471.3167x4,20.855,2.47981.62480.8531,2.60431.7512x4,31.2227,3.17891.95621.2194,3.41252.1931x4,41.7267,4.3312.60421.7187,4.79443.0757x7,10.9014,2.22111.31970.902,2.30331.4014x7,20.5061,3.02762.52150.5167,3.21892.7022x7,31.6322,4.24142.60921.6304,4.66433.0339x10,10.9855,2.01481.02930.986,2.08051.0945x10,21.1604,2.60241.44211.1615,2.74131.5798x10,31.4536,3.27591.82221.4526,3.53172.0792x10,41.8979,4.40292.5051.8881,4.89493.0068x12,11.0321,2.27791.24581.0327,2.36281.3301x12,20.7123,3.06972.35750.7157,3.26712.5514x12,31.7084,4.27172.56331.706,4.70362.9976x15,11.0883,2.30421.21591.0889,2.39021.3013x15,20.7911,3.08942.29830.7934,3.28932.496x15,31.7432,4.28592.54271.7403,4.72162.9813x18,11.1709,2.34481.17381.1715,2.43231.2608x18,20.9009,3.11982.21890.9023,3.32362.4212x18,31.7962,4.3082.51181.7926,4.74932.9567x20,11.2671,2.79441.52731.2679,2.9231.6551x20,20.8101,4.12963.31950.813,4.49413.681x23,11.3116,2.81511.50351.3123,2.94491.6325x23,20.8777,4.14383.2660.8801,4.51083.6307x26,12.1533,3.29611.14282.1535,3.44491.2914x26,21.9179,4.4862.56811.925,4.89712.9721x28,12.4552,3.50191.04672.4552,3.65571.2005x28,22.2512,4.63992.38882.2601,5.06412.804x30,12.5955,3.13880.54342.5955,3.22080.6254x30,22.5164,3.54681.03032.5204,3.72971.2094x30,32.8091,4.06851.25942.7972,4.39761.6004x30,42.2466,5.02272.77612.2995,5.61783.3183Table 10
95% MLPI and BPI of xi,Rl for real-life data.
PointMLPIBPILB1,UB1LengthLB1,UB1Lengthx1,10.3437,1.78321.43950.3425,1.8511.5085x1,20.5664,2.31971.75330.5565,2.44831.8918x1,30.8684,2.85811.98970.8541,3.08162.2275x1,41.2182,3.5362.31781.203,3.9182.715x1,51.6771,4.74183.06471.6584,5.45443.796x4,10.5752,2.04151.46630.5752,2.12721.5521x4,20.7713,2.68231.9110.7687,2.86112.0924x4,31.0966,3.41532.31871.0912,3.75132.6601x4,41.5561,4.66383.10771.5462,5.33213.7859x7,10.8812,2.43071.54950.8814,2.55141.67x7,20.6232,3.27082.64760.6276,3.54992.9223x7,31.4789,4.58023.10121.4762,5.19583.7196x10,10.9717,2.18861.21690.9719,2.28371.3118x10,21.0999,2.79621.69641.1004,2.99561.8952x10,31.349,3.50582.15691.3471,3.87172.5246x10,41.7439,4.73072.98691.7336,5.43563.702x12,11.0145,2.48281.46831.0148,2.60791.5931x12,20.8001,3.30992.50980.8016,3.59752.7959x12,31.5626,4.60833.04571.5595,5.23573.6762x15,11.0716,2.5071.43541.0719,2.63391.562x15,20.8711,3.32812.4570.8721,3.61942.7473x15,31.6004,4.62143.0211.5971,5.25393.6568x18,11.1554,2.54431.3891.1556,2.67391.5183x18,20.9721,3.35642.38440.9727,3.6532.6803x18,31.658,4.6422.9841.6541,5.28183.6277x20,11.2454,3.04571.80021.2458,3.23971.9939x20,20.9409,4.47573.53480.942,5.01564.0736x23,11.2907,3.06471.7741.2911,3.26071.9696x23,20.9998,4.48883.48891.0008,5.03224.0314x26,12.1405,3.51211.37162.1406,3.73911.5985x26,21.9774,4.80682.82951.9821,5.41123.4291x28,12.444,3.7061.26212.444,3.94171.4977x28,22.3021,4.95092.64882.3082,5.57393.2657x30,12.5902,3.25370.66362.5902,3.37930.7891x30,22.5354,3.69191.15652.5381,3.95781.4197x30,32.7559,4.25641.50052.7454,4.72411.9787x30,42.2953,5.31293.01762.3393,6.15433.815Table 11
90% MLPI and BPI of FOS ys for real-life data.
PointMLPIBPILB2,UB2LengthLB2,UB2Lengthy10.1438,1.11460.97090.1273,1.15181.0245y20.3913,1.44431.0530.3618,1.49651.1347y30.6128,1.71761.10480.5796,1.78751.2079y40.8193,1.9751.15570.7862,2.06811.2818y51.0209,2.23461.21370.9895,2.35861.3691y61.2257,2.51111.28541.1959,2.67651.4807y71.4424,2.82331.38091.413,3.04511.6321y81.6835,3.20411.52051.652,3.50521.8532y91.9729,3.73281.75991.9349,4.15492.22y102.3801,4.72532.34512.3275,5.37783.0503Table 12
95% MLPI and BPI of FOS ys for real-life data.
PointMLPIBPILB2,UB2LengthLB2,UB2Lengthy10.1007,1.23781.13710.0855,1.28531.1998y20.3226,1.56641.24390.2901,1.6341.3439y30.5324,1.84181.30930.4931,1.93421.441y40.731,2.1031.3720.6902,2.22831.5381y50.9256,2.36821.44250.886,2.53731.6513y61.1235,2.65281.52921.0856,2.88011.7944y71.3325,2.97691.64441.2956,3.28271.9871y81.5638,3.37691.81311.5258,3.79262.2668y91.8389,3.94182.10291.7957,4.52542.7297y102.2192,5.03322.8142.1626,5.94293.7802Table 13
90% MLPI and BPI of FURS zs for real-life data.
PointMLPIBPILB3,UB3LengthLB3,UB3Lengthz10.4583,3.55373.09540.4426,3.75253.3099z21.2147,4.47923.26451.2056,4.90123.6956z31.8481,5.16533.31721.8378,5.8333.9952z42.3933,5.73653.34322.3651,6.66334.2982z52.8776,6.23653.35892.8162,7.43044.6143z63.3173,6.68693.36963.2101,8.15244.9423z73.7227,7.10013.37743.5601,8.83955.2795z84.1007,7.48413.38343.8753,9.49825.6229z94.4561,7.84433.38834.1625,10.13295.9704z104.7925,8.18473.39224.4269,10.74696.32Table 14
95% MLPI and BPI of FURS zs for real-life data.
PointMLPIBPILB3,UB3LengthLB3,UB3Lengthz10.3212,3.94643.62520.3031,4.26323.9601z21.0015,4.85713.85560.986,5.49564.5096z31.6059,5.5363.931.5927,6.51584.923z42.1358,6.10263.96682.1097,7.43555.3258z52.6104,6.59943.9892.555,8.29225.7372z63.0434,7.04744.0042.9444,9.10366.1592z73.4437,7.45864.0153.2896,9.87996.5903z83.8177,7.8414.02343.5995,10.62767.0281z94.1699,8.19994.033.8806,11.35127.4706z104.5037,8.53924.03554.1381,12.05417.916


## 6. Conclusion

In this paper, we have dealt with OSP and TSP problems for future observations having an NWPD under the ProgT-II C sample. The predictions of FOS and FURS samples are also studied. The construction of PI for future unobserved failures in all cases is obtained based on the invariant property of MLEs and the generated MCMC samples. N-RI is considered a suitable numerical method used in our paper to get the bounds of PI. A simulated data set and a real-life data set are performed to demonstrate the discussed methods. Summing up the results, it can be concluded that(i)
It is clear from all tables that the length of the MLPI is smaller than the length of the BPI.(ii)
For increasing the value of the survivor units in the same position ofxi,Rl in the case of OSP and s in the case of FOS or FURS, the length of the PI increases.(iii)
It can be seen that the length of90% PI is smaller than the length of 95% PI, which proved that when the significance level γ increases, then the interval length decreases.(iv)
In the TSP problem, the lengths of FOS are smaller than ones of FURS.(v)
Regarding the discussed problem, we can predict the exceedances of the future flood peaks based on the currently available data. Also, we can predict the unobserved value of the exceedances due to the recorded ones.(vi)
Finally, we can conclude that the proposed inference methods give consistent results.(vii)
Sometimes, the available data could be affected by uncertainties and/or inaccuracies. Therefore, strictly speaking, a prediction system based on soft computing techniques and, in particular, on the latest generation fuzzy techniques would be needed, see Cacciola et al. [27] as future work.



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*Source: 6416806-2022-05-12.xml*