# Bayesian Prediction Intervals Based on Type-I Hybrid Censored Data from the Lomax Distribution under Step-Stress Model

Authors: Abdalla Rabie; Abd EL-Baset A. Ahmad; Mohamad A. Fawzy; Tahani A. Aloafi
Journal: Journal of Mathematics
 (2022)

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## Abstract

The Bayesian prediction of future failures from Lomax distribution is the subject of this research. The observed data is censored using a Type-I hybrid censoring scheme under a step-stress partially accelerated life test. There are two types of sampling schemes considered: one-sample and two-sample. We create predictive intervals for failure observations in the future. Bayesian prediction intervals are constructed using MCMC algorithms. After all, two numerical examples, simulation study and a real-life example are provided for both one-sample and two-sample methods for the purpose of illustration.


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## Body

## 1. Introduction

In highly industrial products, it is difficult to obtain sufficient information about the product to test its validity, so it is appropriate to use an acceleration life test to obtain enough information with saving time and high cost. There are two types of accelerated life tests: the fully accelerated life test, in which it is assumed that the acceleration factor is known and there must be a mathematical model that determines the relationship between the life span and stress. Another one is the partially accelerated life test which is applied when the relationship between the life span and stress is unknown and is applied throughout step stress and constant stress. Many authors have written on this topic, see for example, Saeed, and Hanieh [1]; Lone et al. [2] and Mohammad and Mohammad [3] and Nagy et al. [4].It is important to use past data to predict future data in many experiments of life tests for units or products to improve product efficiency. One of the fascinating subjects in real-world reliability issues is prediction. Recently, there has been a lot of discussion about various prediction techniques.In multiple fields as healthcare, finance, technology, and education, Bayesian prediction of future data plays a critical role. The Bayesian approach, according to Geisser [5]; can be used to tackle the problem of prediction. See also AL-Hussaini et al. [6, 7], Ahmadi et al. [8], Ahmad [9], Ahmad et al. [10], Ateya [11], Balakrishnan and Shafay [12], Kundu and Howlader [13], and Singh et al. [14], Ahmad et al. [15] and Corcuera and Giummole [16].The most common type of prediction is predictive intervals. They are not the same as the commonly used confidence intervals and tolerance regions, which are intended to deal with unknown population parameters. A predictive interval is a range of values that uses the results of a previous sample to predict the outcomes of a future sample with a given probability.Prediction concerns can be divided into two categories:(1)
One-sample predictionLetX1:n,X2:n,…,Xr:n and Xr+1:n,Xr+2:n,…,Xn:n represent the known sample and a future sample, respectively. A one-sample prediction scheme entails predicting the future sample Xs for r<s≤n.(2)
Two-sample predictionLetX1:n,X2:n,…,Xn:n and Y1:m,Y2:m,…,Ym:m denote the informative sample of size n and a future sample of size m, respectively. The two samples are believed to be unrelated and drawn from the same distribution. The prediction of the future sample Y1:m,Y2:m,…,Ym:m is part of a two-sample prediction issue.For more details about one and two sample methods, one can refer to Prakash [17]; Wu and Gui [18]; and Mohie El-Din et al. [19].


### 1.1. The Model Description and Censoring Scheme

The Lomax distribution was introduced by Lomax [20] to analyse business failure data. Several authors have used Lomax distribution, for example, Howlader and Hossain [21], Abd-Ellah [22], Abd-Elfattah et al. [23], Hassan and Al-Ghamdi [24], among others. The cumulative distribution function (CDF) and the probability density function (PDF) of Lomax distribution are written, respectively, in the forms(1)Fx;α,θ=1−1+αx−θ,x>0,α,θ>0,fx;α,θ=αθ1+αx−θ+1,x>0,α,θ>0,where, θ and α are the shape and scale parameters, respectively.By mixing Type-I and Type-II censoring schemes, we obtain hybrid censoring scheme (HCS) in which the test is ended after obtaining a fixed numberr of failures from n items or after reaching a pre-determined time T. So, we can observe the following types of HCSs.(i)
Type-I HCS: Epstein [25] considered a HCS in which the life-testing experiment is terminated at a random time T∗=minXr:n,T, where 1≤r≤n and T>0 are known before.(ii)
Type-II HCS: In this type the experiment is finished at a random timeT∗=maxXr:n,T, this ensures at least r of failures are obtained.In this paper, we consider Type-I HCS which includes the following types of censored data:Case 1:X1:n<X2:n<⋯<Xr:n, if Xr:n<T, and pre-specified r number of failures occur before the censoring time T.Case 2:X1:n<X2:n<⋯<Xm:n, if Xm:n≤T<Xr:n, and only m number of failures occur before the pre-specified censoring time T.The rest of this paper is organized as follows: Section2 gives the test method and procedure. In Section 3, the likelihood function and the posterior PDF are stated. Section 4 is devoted for Bayesian one-sample prediction scheme and MCMC method. Bayesian two-sample prediction and MCMC technique are presented in Section 5. Section 6 presents numerical computations involving numerical examples and simulation study, as well as real-life example. The findings are displayed in four tables then concluding the results and methodology in Section 7.


## 1.1. The Model Description and Censoring Scheme

The Lomax distribution was introduced by Lomax [20] to analyse business failure data. Several authors have used Lomax distribution, for example, Howlader and Hossain [21], Abd-Ellah [22], Abd-Elfattah et al. [23], Hassan and Al-Ghamdi [24], among others. The cumulative distribution function (CDF) and the probability density function (PDF) of Lomax distribution are written, respectively, in the forms(1)Fx;α,θ=1−1+αx−θ,x>0,α,θ>0,fx;α,θ=αθ1+αx−θ+1,x>0,α,θ>0,where, θ and α are the shape and scale parameters, respectively.By mixing Type-I and Type-II censoring schemes, we obtain hybrid censoring scheme (HCS) in which the test is ended after obtaining a fixed numberr of failures from n items or after reaching a pre-determined time T. So, we can observe the following types of HCSs.(i)
Type-I HCS: Epstein [25] considered a HCS in which the life-testing experiment is terminated at a random time T∗=minXr:n,T, where 1≤r≤n and T>0 are known before.(ii)
Type-II HCS: In this type the experiment is finished at a random timeT∗=maxXr:n,T, this ensures at least r of failures are obtained.In this paper, we consider Type-I HCS which includes the following types of censored data:Case 1:X1:n<X2:n<⋯<Xr:n, if Xr:n<T, and pre-specified r number of failures occur before the censoring time T.Case 2:X1:n<X2:n<⋯<Xm:n, if Xm:n≤T<Xr:n, and only m number of failures occur before the pre-specified censoring time T.The rest of this paper is organized as follows: Section2 gives the test method and procedure. In Section 3, the likelihood function and the posterior PDF are stated. Section 4 is devoted for Bayesian one-sample prediction scheme and MCMC method. Bayesian two-sample prediction and MCMC technique are presented in Section 5. Section 6 presents numerical computations involving numerical examples and simulation study, as well as real-life example. The findings are displayed in four tables then concluding the results and methodology in Section 7.


## 2. The Test Method

In this section, we explain the fundamental assumptions and test procedures for the step-stress partially accelerated life test (SSPALT) model based on the Type-I hybrid censoring method.


### 2.1. Fundamental Presumptions

(1)
There are two stress levels:s1 and s2 (design and high).(2)
The life distribution of the test unit is Lomax distribution for any level of stress.(3)
An item’s entire lifetimeY is calculated as follows:(2)Y=T,T≤τ,τ+β−1T−τ,T>τ,where,T is an item lifetime at normal use condition and β>1, is the accelerating factor. This model discussed by Degroot and Goel [26] and referred to as tampered random variable (TRV).(4)
The lifetimesY1,…,Yn of the n test items are independent and identically distributed random variables.


### 2.2. Test Procedure

(1)
Alln test items are put through a normal environment.(2)
Ifr number failures does not fail by a predetermined time τ in normal use, it is switched to accelerated mode and run until it fails or the censorship time is reached.Based on the above assumptions, the CDF and PDF of a total lifetimeY of test item take the following forms, respectively,(3)Fy=1−1+αy−θ,0≤y≤τ,1−1+αδy−θ,y>τ,where, δy=τ+βy−τ and(4)fy=αθ1+αy−θ+1,0≤y≤τ,αθβ1+αδy−θ+1,y>τ.In Type-I HCS, the life-testing experiment is stopped according to the ruleT2=minyr:n,T1, that is the test is terminated at a predetermined time T1 if the rth failure obtained after T1, or as soon as the rth failure occurs, where 1≤r≤n. Thus,We may obtain a random number of failuresN∗ given by(5)N∗=Case1:r,ifyr:n≤τ≤T1,Case2:r−N1,ifτ<yr:n≤T1,Case3:N2,ifyr:n>T1.


## 2.1. Fundamental Presumptions

(1)
There are two stress levels:s1 and s2 (design and high).(2)
The life distribution of the test unit is Lomax distribution for any level of stress.(3)
An item’s entire lifetimeY is calculated as follows:(2)Y=T,T≤τ,τ+β−1T−τ,T>τ,where,T is an item lifetime at normal use condition and β>1, is the accelerating factor. This model discussed by Degroot and Goel [26] and referred to as tampered random variable (TRV).(4)
The lifetimesY1,…,Yn of the n test items are independent and identically distributed random variables.


## 2.2. Test Procedure

(1)
Alln test items are put through a normal environment.(2)
Ifr number failures does not fail by a predetermined time τ in normal use, it is switched to accelerated mode and run until it fails or the censorship time is reached.Based on the above assumptions, the CDF and PDF of a total lifetimeY of test item take the following forms, respectively,(3)Fy=1−1+αy−θ,0≤y≤τ,1−1+αδy−θ,y>τ,where, δy=τ+βy−τ and(4)fy=αθ1+αy−θ+1,0≤y≤τ,αθβ1+αδy−θ+1,y>τ.In Type-I HCS, the life-testing experiment is stopped according to the ruleT2=minyr:n,T1, that is the test is terminated at a predetermined time T1 if the rth failure obtained after T1, or as soon as the rth failure occurs, where 1≤r≤n. Thus,We may obtain a random number of failuresN∗ given by(5)N∗=Case1:r,ifyr:n≤τ≤T1,Case2:r−N1,ifτ<yr:n≤T1,Case3:N2,ifyr:n>T1.


## 3. The Likelihood Function and Posterior PDF

The likelihood function under Type-I HCS according to the previous assumptions can be written as:For Case 1(6)Lα,θ|y=n!n−r!1−Fyr:nn−r∏i=1rfyi:n=n!αθrn−r!1+αyr:nθr−n∏i=1r1+αyi:n−θ+1,y1:n<⋯<yr:n<τ<T1, in this case the prefixed number r out of n items is obtained at a normal use condition, so, this case will be discarded.And for Cases 2 and 3, the likelihood function under Type-I HCS can be combined and written as:(7)Lα,θ,β|y=n!n−r∗!∏i=1N1fyi:n∏i=N1+1r∗fyi:n1−FT2n−r∗=n!n−r∗!αθr∗βr∗−N11+αδT2θr∗−n×∏i=1N11+αyi:n−θ+1∏i=N1+1r∗1+αδyi:n−θ+1,where, 0<y1:n<⋯<yN1:n≤τ<yN1+1:n<⋯<yr∗:n≤T1, and r∗=N1+N∗.We consider the likelihood function in Cases 2 and 3 only, as it runs at accelerated use condition.We suggest the prior PDF for the parametersα and θ to be Γa,b and Γc,d, respectively; because the gamma prior is wealthy enough to cover the prior belief of the experimenter, so, many authors used it, see Pak and Mohammad [27]; Okasha [28]; Abd-Elfattah et al. [23]; which expressed as follows:(8)π1α∝αa−1exp−bα,a,b>0,(9)π2θ∝θc−1exp−dθ,c,d>0.It should be pointed out that, the empirical Bayes method can be used to estimate the hyper parameters if they are unknown using previous samples, see, for example, Maritz and Lwin [29]. As an alternative, the hierarchical Bayes approach, which employs a suitable prior for the hyper parameters, could be utilized, see Bernardo and Smith [30].The prior PDF for the acceleration factorβ is a non-informative prior given by:(10)π3β∝β−1,β>1.The joint prior PDF ofα,θ and β, can be written as follows:(11)πα,θ,β=π1απ2θπ3β∝1βαa−1θc−1e−bα+dθ,β>1,a,b,c,d,α,θ>0.The majority of published papers consider that the choice of prior distribution is the researcher’s belief and the assumption of independence between parameters is the best in terms of ease of calculations and good results, and there is no way to know independence or dependence between them. Also, they are specified as independent when you do not want to assume that they are prior informative about each other. That is, knowing the value of one would not change your mind about any of the others, before seeing any data, this assumption was chosen when we determined the prior distribution and this is more understandable. In addition, choosing another prior distribution or dependent parameters will increase the complexity and difficulty of mathematical equations. Furthermore, the independent gamma priors are relatively simple and concise, which may not yield much complex inferential and computational issues. Also, in many practical situations, although the dependent prior models seem more attractive, yet the dependent property between parameters cannot be justified from a statistical perspective due to historical information and expert experience where such prior data/information may be very rare. Therefore, independent priors are more popular in statistics under the Bayesian procedure for the sake of simplicity, see EL-Sagheer et al. [31]; and Nassar et al. [32].The joint posterior PDF ofα,θ and β, given y, can be written from (7) and (11), as(12)π∗α,θ,β|y=K−1αr∗+a−1θr∗+c−1βr∗−N1−11+αδT2θr∗−n×e−bα+dθ∏i=1N11+αyi:n−θ+1∏i=N1+1r∗1+αδyi:n−θ+1,where, a normalizing constant K is(13)K=∫1∞∫0∞∫0∞π∗α,θ,β|ydαdθdβ.The conditional posterior PDF ofα,θ and β, are, respectively, given by(14)π∗α|θ,β;y=αr∗+a−11+αδT2θr∗−ne−bα×∏i=1N11+αyi:n−θ+1∏i=N1+1r∗1+αδyi:n−θ+1,(15)π∗θ|α,β;y=θr∗+c−11+αδT2θr∗−ne−dθ×∏i=1N11+αyi:n−θ+1∏i=N1+1r∗1+αδyi:n−θ+1,(16)π∗β|α,θ;y=βr∗−N1−11+αδT2θr∗−n×∏i=N1+1r∗1+αδyi:n−θ+1.


## 4. Bayesian One-Sample Prediction

Let the firstr∗th order statistics, Y1:n,…,Yr∗:n, 1≤r∗<n, have been observed and we want to predict the future order statistics, Ys:n, s=r∗+1,r∗+2,…,n. The conditional PDF of the sth future order statistics given informative order statistics y=y1:n,y2:n,…,yr∗:n, can be written as:(17)f∗ys:n|y=n−r∗!n−s!s−r∗−1!Fys:n−FT2s−r∗−1×1−Fys:nn−s1−FT2−n−r∗fys:n,substituting (3) and (4) in (17), we get(18)f∗ys:n|α,θ,β;y=ℑαθβ1+αδT2θn−r∗1+αδT2δys:nθs−n−1−1×1+αδT2θ−1+αδys:nθs−r∗−1,where, ℑ=n−r∗!/n−s!s−r∗−1!.By multiplying (12) by (18) and then integrating with respect to α,θ,β, the predictive PDF of Ys:n, s=r∗+1,r∗+2,…,n given the informative order statistics y=y1:n,y2:n,…,yr∗:n, is given by(19)hys:n|α,θ,β;y=∫1∞∫0∞∫0∞π∗α,θ,β|yf∗ys:n|α,θ,β;ydαdθdβ,ys:n>yr∗:n.By substituting (12) and (18) in (19), we obtain(20)hys:n|α,θ,β;y=K−1ℑ∫1∞∫0∞∫0∞αr∗+aθr∗+cβr∗−N11+αδT2θn−r∗×1+αδT2θ−1+αδys:nθs−r∗−1×1+αδys:nθs−n−1−1e−bα+dθ∏i=1N11+αyi:n−θ+1×∏i=N1+1r∗1+αδyi:n−θ+1dαdθdβ.A1−γ100%, two-sided Bayesian predictive interval for Ys:n, is given by(21)PL<Ys:n<U=1−γ,where L and U are lower and upper bounds of Bayesian prediction for Ys, are obtained by solving the following two equations numerically(22)PYs>L|y=1−γ2,PYs>U|y=γ2..


### 4.1. Bayesian Prediction: One-Sample Scheme via MCMC

The predictive PDF (20), is approximated by using the MCMC method as follows:(23)hys:n|α,θ,β;y≈∑j=1Nf∗ys:n|αj,θj,βj;y∑j=1N∫yr∗:n∞f∗ys:n|αj,θj,βj;ydys:n,where, αj,θj,βj, j=1,2,3,…,N, are generated from the posterior PDF (12).A1−γ100%, two-sided Bayesian predictive interval, L,U, of observations Ys:n can be calculated from solving the following nonlinear equations for a given γ(24)∑j=1N∫L∞f∗ys:n|αj,θj,βj;ydys:n∑j=1N∫yr∗:n∞f∗ys:n|αj,θj,βj;ydys:n=1−γ2,∑j=1N∫U∞f∗ys:n|αj,θj,βj;ydys:n∑j=1N∫yr∗:n∞f∗ys:n|αj,θj,βj;ydys:n=γ2..


## 4.1. Bayesian Prediction: One-Sample Scheme via MCMC

The predictive PDF (20), is approximated by using the MCMC method as follows:(23)hys:n|α,θ,β;y≈∑j=1Nf∗ys:n|αj,θj,βj;y∑j=1N∫yr∗:n∞f∗ys:n|αj,θj,βj;ydys:n,where, αj,θj,βj, j=1,2,3,…,N, are generated from the posterior PDF (12).A1−γ100%, two-sided Bayesian predictive interval, L,U, of observations Ys:n can be calculated from solving the following nonlinear equations for a given γ(24)∑j=1N∫L∞f∗ys:n|αj,θj,βj;ydys:n∑j=1N∫yr∗:n∞f∗ys:n|αj,θj,βj;ydys:n=1−γ2,∑j=1N∫U∞f∗ys:n|αj,θj,βj;ydys:n∑j=1N∫yr∗:n∞f∗ys:n|αj,θj,βj;ydys:n=γ2..


## 5. Bayesian Two-Sample Prediction

LetX1:m≤X2:m≤⋯≤Xm:m be the order statistics from a future random sample of size m and independent of the informative sample Y1:n,Y2:n,…,Yr∗:n,1≤r∗<n. The future sample and the informative sample are following the Lomax SSPALT model under Type-I HCS. We wish to predict the first kth future order statistics of the future sample Xm:m,1≤k<m.The marginal PDF of thekth order statistics from Xk:m is expressed as:(25)h∗xk:m|α,θ,β=m!k−1!m−k!Fxk:mk−11−Fxk:mm−kfxk:m,xk:m>0=Ωαθβ1−1+αδxk:m−θk−1×1+αδxk:mθk−m−1−1,where, Ω=m!/k−1!m−k!.The predictive PDF ofXk:m, 1≤k<m say gxk:m|α,θ,β is given as:(26)gxk:m|α,θ,β=∫1∞∫0∞∫0∞π∗α,θ,β|yh∗xk:m|α,θ,βdαdθdβ,where, π∗α,θ,β|y is the joint posterior PDF of α,θ,β as given in (12). By substituting from (12) and (25) in (26), we get (27)gxk:m|θ,β=K−1Ω∫1∞∫0∞∫0∞αr∗+aθr∗+cβr∗−N11+αδT2θr∗−n×1−1+αδxk:m−θk−11+αδxk:mθk−m−1−1×e−bα+dθ∏i=1N11+αyi:n−θ+1∏i=N1+1r∗1+αδyi:n−θ+1dαdθdβ.A1−γ100%, two-sided Bayesian predictive interval for Xk:m, is written as:(28)PL<Xk:m<U=1−γ,where, bounds (L, U) of Xk:m, are computed by numerical solution of the two equations for a given γ(29)PXk:m>L=1−γ2,PXk:m>U=γ2..


### 5.1. Bayesian Two-Sample Prediction under MCMC

The predictive PDF (26), is approximated by applying the MCMC as follows:(30)gxk:m|α,θ,β≈∑j=1Nh∗xk:m|αj,θj,βj∑j=1N∫0∞h∗xk:m|αj,θj,βjdxk:m.A1−γ100%, two-sided Bayesian predictive interval, L,U, of the future Xk:m observations are obtained by solving the following nonlinear equations(31)∑j=1N∫L∞h∗xk:m|αj,θj,βjdxk:m∑j=1N∫0∞h∗xk:m|αj,θj,βjdxk:m=1−γ2,∑j=1N∫U∞h∗xk:m|αj,θj,βjdxk:m∑j=1N∫0∞h∗xk:m|αj,θj,βjdxk:m,=γ2..


## 5.1. Bayesian Two-Sample Prediction under MCMC

The predictive PDF (26), is approximated by applying the MCMC as follows:(30)gxk:m|α,θ,β≈∑j=1Nh∗xk:m|αj,θj,βj∑j=1N∫0∞h∗xk:m|αj,θj,βjdxk:m.A1−γ100%, two-sided Bayesian predictive interval, L,U, of the future Xk:m observations are obtained by solving the following nonlinear equations(31)∑j=1N∫L∞h∗xk:m|αj,θj,βjdxk:m∑j=1N∫0∞h∗xk:m|αj,θj,βjdxk:m=1−γ2,∑j=1N∫U∞h∗xk:m|αj,θj,βjdxk:m∑j=1N∫0∞h∗xk:m|αj,θj,βjdxk:m,=γ2..


## 6. Numerical Computations

Here, we present two examples to illustrate one-sample and two-sample techniques, as well as a simulation study and real-life data for the discussed methodology, is stated.


### 6.1. Numerical Examples

#### 6.1.1. One-Sample Scheme

(1)
For known values of of hyper parametersa=0.8;b=0.6;c=0.6;d=0.7, generate α=1.8;θ=2.5 and β=3.5 from equations (8), (9) and (10), respectively.(2)
By using generated values ofα;θ and β, a random sample of size 30 is simulated from (4). The first informative ordered failures r=20 are obtained and listed as follows:0.0113981, 0.0188695, 0.0638502, 0.06688, 0.0716792, 0.217725, 0.22276, 0.224412, 0.224614, 0.227522, 0.228146, 0.233232, 0.237106, 0.240437, 0.246286, 0.25205, 0.259451, 0.260386, 0.270926, 0.278369.(3)
Using these informative data in (25) with 0.95 CI, the lower and upper bounds of the next predicted failure Y21 are (0.27903, 0.400507) and for the last predicted failure Y30 are (0.279084, 0.402948).


#### 6.1.2. Two-Sample Scheme

(1)
For given values of of hyper parametersa=1.4;b=0.8;c=1.2;d=0.7, generate α=1.8;θ=1.4 and β=2.5 from equations (8), (9) and (10), respectively.(2)
By using generated values ofα;θ and β, a random sample of size 20 is simulated from (3). The first informative ordered failures r=15 are obtained and listed as follows:0.00167013, 0.157792, 0.193577, 0.19366, 0.207637, 0.219493, 0.226598, 0.242315, 0.271873, 0.294316, 0.334488, 0.603327, 0.651955, 0.70614, 0.743701.(3)
Using these informative data in (31) with 0.95 CI, the lower and upper bounds of the first four observations of a non-informative sample X1,X2,X3,X4 are (0.00129258, 0.191083), (0.00136062, 0.201295), (0.00143622, 0.21266) and (0.00152071, 0.225385), respectively.


### 6.2. Simulation Study

In this part of the paper, we explain the MCMC algorithm that used for computing Bayesian bounds for future samples in the case of one-sample and two-sample approaches. We determine values ofn,r and choose τ,T1. Using values of prior parameters a,b,c,d we generate initial values of the parameters α, θ and β from their prior PDF. Based on the generated α, θ and β, we generate a Lomax Type-I HC sample using SSPALT model with different sizes n=20;40;60;80 and censoring values r=15;30;45;60 based on the inverse function technique given by(32)X=1β1α1−U−1/θ−1−τ+τ,where, U denotes a number obtained from U0,1 randomly. From Metropolis algorithm, αj is generated from (14), θj from (15) and βj from (16) using normal distribution as a proposal distribution. Because the predictive PDF (21) cannot be produced in a closed form for one-sample prediction, it is approximated using the MCMC approach as described in (22). A 95% Bayesian predictive intervals for Ys::n, s=r∗ + 1, r∗ + 2, …, n are obtained by solving the two nonlinear Eqs.(23). For two-sample prediction, the predictive PDF (26), which cannot be produced in a closed form, is estimated using the MCMC approach as described in (28). A 95% Bayesian predictive intervals for Xk:m, k=1,2,…m are obtained by solving the two nonlinear equation (29).All results are derived by Mathematica 8 programming language software. The results of one-sample scheme with different values ofn,r,τ and T1 are listed in Table 1. The results of two-sample case with the same values of n,r,τ and T1 are obtained and displayed in Table 2. The numerical results are computed by MATHEMATICA 8 codes, such as: (FindRoot, NMaximize, NIntegrate and RandomReal).Table 1
95% Bayesian predictive intervals of future order statistics whenθ=2.5, α=1.8, β=3.5, a=0.8, b=0.6, c=0.6, and d=0.7 (one-sample scheme).
nrτ,T1YsLULength2015(0.3, 0.8)Y160.3949920.6699220.274931Y170.3949260.616210.221284Y180.3949310.6506720.255741(0.5, 1.6)Y160.5240270.6815950.157568Y170.5242050.7138180.189612Y180.5244620.7578830.2334224030(0.3, 0.8)Y310.374620.5456550.171035Y320.3745930.5349250.160332Y330.374570.5273260.152756(0.5, 1.6)Y310.5027750.6342170.131442Y320.5028390.6432970.140458Y330.5029120.6535770.1506656045(0.3, 0.8)Y460.3718940.4371510.0652573Y470.3719050.4384260.0665217Y480.3719160.4397310.0678151(0.5, 1.6)Y460.438030.4537870.0157567Y470.4380370.4548640.0168265Y480.4380460.4560980.0180528060(0.3, 0.8)Y610.3854210.4292080.0437874Y620.3854270.4300390.0446117Y630.3854340.4308950.0454615(0.5, 1.6)Y610.4408230.4794380.038615Y620.4408330.4812410.0404073Y630.4408450.4832090.0423641Table 2
95% Bayesian predictive intervals of future order statistics whenθ=1.4, α=1.8, and β=2.5a=1.4, b=0.8, c=1.2, and d=0.7 (two-sample scheme).
nrτ,T1XkLULength2015(0.3, 0.8)X10.001292580.1910830.189791X20.001360620.2012950.199935X30.001436220.212660.211224X40.001520710.2253850.223864(0.5, 1.6)X10.0011890.1780830.176894X20.001251590.1877310.186479X30.001321130.1984840.197163X40.001398870.2105430.2091454030(0.3, 0.8)X10.0008787210.1322370.131358X20.0009249810.1394370.138512X30.0009763810.1474650.146489X40.001033830.1564750.155441(0.5, 1.6)X10.0007399470.1090180.108278X20.0007788940.1148240.114045X30.000822170.1212820.12046X40.0008705370.1285110.127646045(0.3, 0.8)X10.0007985530.1180930.117294X20.0008405860.1244060.123566X30.0008872910.1314330.130546X40.0009394910.1393010.138361(0.5, 1.6)X10.0006211530.0915710.0909499X20.0006538480.09645060.0957967X30.0006901760.1018790.101189X40.0007307780.1079560.1072258060(0.3, 0.8)X10.0007533430.1111060.110352X20.0007929960.1170290.116236X30.0008370550.1236190.122782X40.0008862980.1309950.130109(0.5, 1.6)X10.0005779070.08546990.084892X20.0006083260.09003980.0894315X30.0006421260.09512590.0944837X40.0006799020.1008210.100141


### 6.3. Illustrative Example

In this section, we present a real life example, the dataset was initially considered by Chhikara and Folks [33]. It represents 46 repair times (in hours) for an airborne communication transceiver. The Lomax distribution has been fitted on this data set by Singh et al. [34]; and they stated that the Lomax distribution can be used for analyzing this data set. The data set is ordered and listed as follows:0.2, 0.3, 0.5, 0.5, 0.5, 0.5, 0.6, 0.6, 0.7, 0.7, 0.7, 0.8, 0.8, 1.0, 1.0, 1.0, 1.0, 1.1, 1.3, 1.5, 1.5, 1.5, 1.5, 2.0, 2.0, 2.2, 2.5, 2.7, 3.0, 3.0, 3.3, 3.3, 4.0, 4.0, 4.5, 4.7, 5.0, 5.4, 5.4, 7.0, 7.5, 8.8, 9.0, 10.3, 22.0, 24.5.By applying SSPALT on this data set, by considering this data represents lifetimes of units put on life testing, we can conclude the following:(1)
Atn=46, r=32, τ=1 and T1=3.8, we have 13 items are failed at τ=1 throughout normal usage. After changing mode to acceleration case, the test is terminated at T∗=minX32:46,T1=min3.3,3.8=3.3, that is, 32 items are failed at T∗=3.3.(2)
Atn=46, r=35, τ=1.5 and T1=4.6, there have been 20 items are failed at τ=1.5 throughout normal usage. After switching to acceleration case, the test is terminated at T∗=minX35:46,T1=min4.5,4.6=4.5, that is, 35 items are failed at T∗=4.5.Based on SSPALT model with Type-I HCS,95% one-sample Bayesian predictive intervals for order statistic Ys:n, s=r+1,r+2,…,n are reported in Table 3 and from the same sample 95% two-sample Bayesian predictive intervals for order statistic Xk:m, m=1;2;3;4, are reported in Table 4.Table 3
95% Bayesian predictive intervals of future order statistics whenβ=3, a=0.7, b=0.9, c=0.9, and d=0.6 (one-sample scheme).
nrτ,T1YsLULength4632(1, 3.8)Y333.334968.774665.4397Y344.5779913.89059.31248Y354.8462119.455714.609535(1.5, 4.6)Y364.514416.858742.34433Y374.800359.175454.3751Y385.0817412.46747.38569Table 4
95% Bayesian predictive intervals of future order statistics whenβ=3, a=0.7, b=0.9, c=0.9, and d=0.6 (two-sample scheme).
nrτ,T1XkLULength4632(1, 3.8)X10.004784180.7170010.712217X20.004943960.7386540.73371X30.005100120.7617390.756639X40.005219130.7861640.78094535(1.5, 4.6)X10.003363330.4969130.493549X20.00342890.5119130.508484X30.003507220.5278470.524339X40.003661620.5447670.541106


## 6.1. Numerical Examples

### 6.1.1. One-Sample Scheme

(1)
For known values of of hyper parametersa=0.8;b=0.6;c=0.6;d=0.7, generate α=1.8;θ=2.5 and β=3.5 from equations (8), (9) and (10), respectively.(2)
By using generated values ofα;θ and β, a random sample of size 30 is simulated from (4). The first informative ordered failures r=20 are obtained and listed as follows:0.0113981, 0.0188695, 0.0638502, 0.06688, 0.0716792, 0.217725, 0.22276, 0.224412, 0.224614, 0.227522, 0.228146, 0.233232, 0.237106, 0.240437, 0.246286, 0.25205, 0.259451, 0.260386, 0.270926, 0.278369.(3)
Using these informative data in (25) with 0.95 CI, the lower and upper bounds of the next predicted failure Y21 are (0.27903, 0.400507) and for the last predicted failure Y30 are (0.279084, 0.402948).


### 6.1.2. Two-Sample Scheme

(1)
For given values of of hyper parametersa=1.4;b=0.8;c=1.2;d=0.7, generate α=1.8;θ=1.4 and β=2.5 from equations (8), (9) and (10), respectively.(2)
By using generated values ofα;θ and β, a random sample of size 20 is simulated from (3). The first informative ordered failures r=15 are obtained and listed as follows:0.00167013, 0.157792, 0.193577, 0.19366, 0.207637, 0.219493, 0.226598, 0.242315, 0.271873, 0.294316, 0.334488, 0.603327, 0.651955, 0.70614, 0.743701.(3)
Using these informative data in (31) with 0.95 CI, the lower and upper bounds of the first four observations of a non-informative sample X1,X2,X3,X4 are (0.00129258, 0.191083), (0.00136062, 0.201295), (0.00143622, 0.21266) and (0.00152071, 0.225385), respectively.


## 6.1.1. One-Sample Scheme

(1)
For known values of of hyper parametersa=0.8;b=0.6;c=0.6;d=0.7, generate α=1.8;θ=2.5 and β=3.5 from equations (8), (9) and (10), respectively.(2)
By using generated values ofα;θ and β, a random sample of size 30 is simulated from (4). The first informative ordered failures r=20 are obtained and listed as follows:0.0113981, 0.0188695, 0.0638502, 0.06688, 0.0716792, 0.217725, 0.22276, 0.224412, 0.224614, 0.227522, 0.228146, 0.233232, 0.237106, 0.240437, 0.246286, 0.25205, 0.259451, 0.260386, 0.270926, 0.278369.(3)
Using these informative data in (25) with 0.95 CI, the lower and upper bounds of the next predicted failure Y21 are (0.27903, 0.400507) and for the last predicted failure Y30 are (0.279084, 0.402948).


## 6.1.2. Two-Sample Scheme

(1)
For given values of of hyper parametersa=1.4;b=0.8;c=1.2;d=0.7, generate α=1.8;θ=1.4 and β=2.5 from equations (8), (9) and (10), respectively.(2)
By using generated values ofα;θ and β, a random sample of size 20 is simulated from (3). The first informative ordered failures r=15 are obtained and listed as follows:0.00167013, 0.157792, 0.193577, 0.19366, 0.207637, 0.219493, 0.226598, 0.242315, 0.271873, 0.294316, 0.334488, 0.603327, 0.651955, 0.70614, 0.743701.(3)
Using these informative data in (31) with 0.95 CI, the lower and upper bounds of the first four observations of a non-informative sample X1,X2,X3,X4 are (0.00129258, 0.191083), (0.00136062, 0.201295), (0.00143622, 0.21266) and (0.00152071, 0.225385), respectively.


## 6.2. Simulation Study

In this part of the paper, we explain the MCMC algorithm that used for computing Bayesian bounds for future samples in the case of one-sample and two-sample approaches. We determine values ofn,r and choose τ,T1. Using values of prior parameters a,b,c,d we generate initial values of the parameters α, θ and β from their prior PDF. Based on the generated α, θ and β, we generate a Lomax Type-I HC sample using SSPALT model with different sizes n=20;40;60;80 and censoring values r=15;30;45;60 based on the inverse function technique given by(32)X=1β1α1−U−1/θ−1−τ+τ,where, U denotes a number obtained from U0,1 randomly. From Metropolis algorithm, αj is generated from (14), θj from (15) and βj from (16) using normal distribution as a proposal distribution. Because the predictive PDF (21) cannot be produced in a closed form for one-sample prediction, it is approximated using the MCMC approach as described in (22). A 95% Bayesian predictive intervals for Ys::n, s=r∗ + 1, r∗ + 2, …, n are obtained by solving the two nonlinear Eqs.(23). For two-sample prediction, the predictive PDF (26), which cannot be produced in a closed form, is estimated using the MCMC approach as described in (28). A 95% Bayesian predictive intervals for Xk:m, k=1,2,…m are obtained by solving the two nonlinear equation (29).All results are derived by Mathematica 8 programming language software. The results of one-sample scheme with different values ofn,r,τ and T1 are listed in Table 1. The results of two-sample case with the same values of n,r,τ and T1 are obtained and displayed in Table 2. The numerical results are computed by MATHEMATICA 8 codes, such as: (FindRoot, NMaximize, NIntegrate and RandomReal).Table 1
95% Bayesian predictive intervals of future order statistics whenθ=2.5, α=1.8, β=3.5, a=0.8, b=0.6, c=0.6, and d=0.7 (one-sample scheme).
nrτ,T1YsLULength2015(0.3, 0.8)Y160.3949920.6699220.274931Y170.3949260.616210.221284Y180.3949310.6506720.255741(0.5, 1.6)Y160.5240270.6815950.157568Y170.5242050.7138180.189612Y180.5244620.7578830.2334224030(0.3, 0.8)Y310.374620.5456550.171035Y320.3745930.5349250.160332Y330.374570.5273260.152756(0.5, 1.6)Y310.5027750.6342170.131442Y320.5028390.6432970.140458Y330.5029120.6535770.1506656045(0.3, 0.8)Y460.3718940.4371510.0652573Y470.3719050.4384260.0665217Y480.3719160.4397310.0678151(0.5, 1.6)Y460.438030.4537870.0157567Y470.4380370.4548640.0168265Y480.4380460.4560980.0180528060(0.3, 0.8)Y610.3854210.4292080.0437874Y620.3854270.4300390.0446117Y630.3854340.4308950.0454615(0.5, 1.6)Y610.4408230.4794380.038615Y620.4408330.4812410.0404073Y630.4408450.4832090.0423641Table 2
95% Bayesian predictive intervals of future order statistics whenθ=1.4, α=1.8, and β=2.5a=1.4, b=0.8, c=1.2, and d=0.7 (two-sample scheme).
nrτ,T1XkLULength2015(0.3, 0.8)X10.001292580.1910830.189791X20.001360620.2012950.199935X30.001436220.212660.211224X40.001520710.2253850.223864(0.5, 1.6)X10.0011890.1780830.176894X20.001251590.1877310.186479X30.001321130.1984840.197163X40.001398870.2105430.2091454030(0.3, 0.8)X10.0008787210.1322370.131358X20.0009249810.1394370.138512X30.0009763810.1474650.146489X40.001033830.1564750.155441(0.5, 1.6)X10.0007399470.1090180.108278X20.0007788940.1148240.114045X30.000822170.1212820.12046X40.0008705370.1285110.127646045(0.3, 0.8)X10.0007985530.1180930.117294X20.0008405860.1244060.123566X30.0008872910.1314330.130546X40.0009394910.1393010.138361(0.5, 1.6)X10.0006211530.0915710.0909499X20.0006538480.09645060.0957967X30.0006901760.1018790.101189X40.0007307780.1079560.1072258060(0.3, 0.8)X10.0007533430.1111060.110352X20.0007929960.1170290.116236X30.0008370550.1236190.122782X40.0008862980.1309950.130109(0.5, 1.6)X10.0005779070.08546990.084892X20.0006083260.09003980.0894315X30.0006421260.09512590.0944837X40.0006799020.1008210.100141


## 6.3. Illustrative Example

In this section, we present a real life example, the dataset was initially considered by Chhikara and Folks [33]. It represents 46 repair times (in hours) for an airborne communication transceiver. The Lomax distribution has been fitted on this data set by Singh et al. [34]; and they stated that the Lomax distribution can be used for analyzing this data set. The data set is ordered and listed as follows:0.2, 0.3, 0.5, 0.5, 0.5, 0.5, 0.6, 0.6, 0.7, 0.7, 0.7, 0.8, 0.8, 1.0, 1.0, 1.0, 1.0, 1.1, 1.3, 1.5, 1.5, 1.5, 1.5, 2.0, 2.0, 2.2, 2.5, 2.7, 3.0, 3.0, 3.3, 3.3, 4.0, 4.0, 4.5, 4.7, 5.0, 5.4, 5.4, 7.0, 7.5, 8.8, 9.0, 10.3, 22.0, 24.5.By applying SSPALT on this data set, by considering this data represents lifetimes of units put on life testing, we can conclude the following:(1)
Atn=46, r=32, τ=1 and T1=3.8, we have 13 items are failed at τ=1 throughout normal usage. After changing mode to acceleration case, the test is terminated at T∗=minX32:46,T1=min3.3,3.8=3.3, that is, 32 items are failed at T∗=3.3.(2)
Atn=46, r=35, τ=1.5 and T1=4.6, there have been 20 items are failed at τ=1.5 throughout normal usage. After switching to acceleration case, the test is terminated at T∗=minX35:46,T1=min4.5,4.6=4.5, that is, 35 items are failed at T∗=4.5.Based on SSPALT model with Type-I HCS,95% one-sample Bayesian predictive intervals for order statistic Ys:n, s=r+1,r+2,…,n are reported in Table 3 and from the same sample 95% two-sample Bayesian predictive intervals for order statistic Xk:m, m=1;2;3;4, are reported in Table 4.Table 3
95% Bayesian predictive intervals of future order statistics whenβ=3, a=0.7, b=0.9, c=0.9, and d=0.6 (one-sample scheme).
nrτ,T1YsLULength4632(1, 3.8)Y333.334968.774665.4397Y344.5779913.89059.31248Y354.8462119.455714.609535(1.5, 4.6)Y364.514416.858742.34433Y374.800359.175454.3751Y385.0817412.46747.38569Table 4
95% Bayesian predictive intervals of future order statistics whenβ=3, a=0.7, b=0.9, c=0.9, and d=0.6 (two-sample scheme).
nrτ,T1XkLULength4632(1, 3.8)X10.004784180.7170010.712217X20.004943960.7386540.73371X30.005100120.7617390.756639X40.005219130.7861640.78094535(1.5, 4.6)X10.003363330.4969130.493549X20.00342890.5119130.508484X30.003507220.5278470.524339X40.003661620.5447670.541106


## 7. Conclusion

Based on SSPALT under Type-I HCS of order statistics data from Lomax distribution, Bayesian prediction bounds for future observations are obtained by using one-sample and two-sample prediction techniques. The MCMC method is used to get Bayesian predictive intervals. We can state that using SSPALT with Type-I HCS, which has the advantage of having pre-determined experiment time, maximizes this advantage and speed up obtaining more information. Furthermore, one can see that SSPALT improves the disadvantage of this scheme, which is represented in the lack of failures in the specified time for the experiment. Finally, some numerical examples and simulation study are given to illustrate the results, and we observe the following remarks:From Tables1 and 2 (one-sample and two-sample cases), we noted that the Bayesian predictive intervals were affected by changing the values of n,r,τ and T1 as follows:(1)
When increasingτ and T1 with the same n,r, the length of Bayesian predictive intervals decreases.(2)
The length of Bayesian predictive intervals, for fixed values ofτ and T1, tends to be shorter when n and r increase, except for a few cases. This may be due to fluctuations in data.From Tables3 and 4 (one-sample and two-sample cases), we observed that:(3)
When values ofτ and T1 get larger the length of Bayesian predictive intervals becomes smaller.



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*Source: 2801582-2022-12-23.xml*