# Analysis with Applications of the Generalized Type-II Progressive Hybrid Censoring Sample from Burr Type-XII Model

Authors: M. Nagy; M. E. Bakr; Adel Fahad Alrasheedi
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this article, based on the generalized Type-II progressive hybrid censoring sample from the Burr Type-XII distribution, maximum likelihood and Bayesian inference are discussed. Point and interval estimates of unknown parameters, reliability, and hazard functions are developed. We employed several loss functions, such as squared error, LINEX, and general entropy, as symmetric and asymmetric loss functions and various prior distributions as informative and non-informative priors for Bayesian inference of unknown parameters. Under a generalized Type-II progressive hybrid censoring sample, we also propose a Bayesian one-sample prediction for unobserved failures. We conduct simulation study using the MCMC algorithm for the Bayesian approach based on several prior distributions. Finally, we apply the results of the theoretical research to real data.


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## Body

## 1. Introduction

In reliability and survival analysis, the progressive Type-II censoring scheme is the most commonly used scheme. It is preferable to the classical Type-II censoring scheme. Progressive censoring is beneficial in various real-life applications such as industry, life research, and clinical applications. It allows the removal of surviving experimental units until the end of the test. Suppose that a life test is conducted withn units, and due to cost and time constraints, it is not desirable to capture all failures. Therefore, only a subset of the unit failures is observed. Such a sample is called a censored sample. Assume that one of the units was broken by accident after the test began, but before all of the units had burned out. If the experiment is still ongoing, this unit must be removed from the life test. The progressive censoring scheme gives a methodology for analyzing this type of data in this case. Some primary referred works are [1–5].The disadvantages of the Type-II progressive censoring method are that the experiment time can be quite long if the units are highly reliable. As a result, the works of [6, 7] address this issue by suggesting a new type of censoring in which the experiment’s stopping time is minXm:m:n,T, where T∈0,∞ is fixed before the starting of the life test. This type of censoring is known as progressive hybrid censoring (PHCS). Under PHCS, the total duration of the experiment will not exceed T. Several authors have looked into the PHCS, including [8, 9].However, the disadvantage of PHCS is that it cannot be implemented when so few failures can be detected beforeT. For this reason, Lee et al. [10] proposed a general type of censoring, called generalized Type-II PHCS, in which the lower number of failures is specified. The experiment of lifetime test would save time and cost of failures based on this censoring scheme. Moreover, the estimates of the statistical efficiency of the experiment are improved by more failures. In the following section, the comprehensive notion of generalized Type-II PHCS and its advantages are explained. For recent work on PHCS, see, for example, [11–14].The contribution in this paper is that we use the generalized Type-II PHCS data from Burr Type-XII distribution to discuss statistical inference for unknown distribution’s parameters and prediction for the removed units in each phase of the generalized Type-II PHCS.Using the exponential form, the Burr Type-XII distribution has the following density function (PDF), distribution function (CDF), and hazard rate function, respectively, given by(1)fx|λ,γ=λγxγ−11+xγe−λln1+xγ,(2)Fx|λ,γ=1−e−λln1+xγ,x>0,λ>0,γ>0.The survival and hazard rate functions are given, respectively, by(3)Sx|λ,γ=e−λln1+xγ,hx|λ,γ=λγxγ−11+xγ.The Burr Type-XII distribution has been extensively investigated by numerous researchers due to its applications in various disciplines such as biological, industrial, reliability and life testing, clinical studies, and so on (see [15, 16]). Recently, Nagy et al. in [17] have considered generalized Type-I PHCS with one specific time and two numbers of failures k,m. This is one of the methods’ drawbacks, especially when the units are highly reliable. In this paper, we employed a generalized Type-II PHCS to control the duration of the experiment which involves some additional complications.The rest of the article is structured as follows. Section2 gives a summary of the generalized Type-II PHCS. Section 3 calculates maximum likelihood (ML) estimates, while Section 4 calculates Bayesian estimates for unknown parameters and survival and hazard functions using three loss functions. The Bayesian prediction for the removed units in each phaseof the generalized Type-II PHCS is derived in Section 5. Simulation studies are carried out in Section 6 to compare the efficacy of the different inference methodologies. Real data are utilized to demonstrate the theoretical findings in Section 7. Finally, the article is concluded in Section 8.


## 2. The Model Explanation

Consider a life test in whichn identical items are put on test. The generalized Type-II PHCS may be described as follows. Let T1,T2∈0,∞ and integer m∈1,2,…,n be prefixed such that T1<T2 with R=R1,R2,…,Rm is also prefixed integer satisfying n=m+R1+⋯+Rm. At the time of first failure, R1 of the remaining units are randomly eliminated. Similarly at the time of the second failure,R2 of the remaining units are removed, and so on. This process repeats until the termination time T∗=maxT1,minXm:m:n,T2; at this time, all the remaining units are removed from the experiment. Let D1 and D2 denote the numbers of observed failures up to time T1 and T2, respectively. Also, let d1 and d2 be the observed values of D1 and D2, respectively. A schematic representation of the generalized Type-II PHCS can be found in Figure 1.Figure 1
Schematic representation of generalized Type-II progressive hybrid censoring scheme.Under the generalized Type-II PHCS described above, we have one of the following types of observations:(1)
If themth failure time occurs before T1, then instead of terminating the test by withdrawing the remaining Rm items after the mth failure, we continue to observe failures (without any further withdrawals) up to time T1. Therefore, the observed failure times are X1:m:n<…<Xm:m:n<…<Xd1:n.(2)
If themth failure occurs between T1 and T2, then the termination time is Xm:m:n, and the observed failure times are X1:m:n<…<Xd1:m:n<…<Xm:m:n.(3)
If themth failure occurs after T2, then the experiment terminates at T2 and the observed failure times are X1:m:n<…<Xd1:m:n<…<Xd2:m:n.Based on the generalized Type-II PHCS, the likelihood function is given by(4)Lθ|x¯=∏i=1d1∑j=imRj+1∏i=1d1fxi:m:nF¯xi:m:nRiF¯T1Rd1+1∗,d1=m,m+1,…,n,∏i=1d1∑j=imRj+1∏i=1mfxi:m:nF¯xi:m:nRi∗,d1=0,1,…,m−1,d2=m,∏i=1d2∑j=imRj+1∏i=1d2fxi:m:nF¯xi:m:nRiF¯T2Rd2+1∗,d2=0,1,…,m−1.Therefore, these cases can be combined and obtained as(5)Lθ|x¯=∏i=1d∗∑j=imRj+1∏i=1d∗fxi:m:nF¯xi:m:nRi∗F¯T1Rd1+1∗F¯T2Rd2+1∗,where(6)d∗=d1,ifXm:m:n<T1,m,ifT1<Xm:m:n<T2,d2,ifT2<Xm:m:n.R∗=R1,…,Rm−1,0,…,0,Rd1+1∗ifXm:m:n<T1,R1,…,Rm,ifT1<Xm:m:n<T2,R1,…,Rd2,Rd2+1∗,ifT2<Xm:m:n,where Rd1+1∗ and Rd2+1∗ represent the number of surviving units that are removed at T1 and T2 and are given by(7)Rd1+1∗=n−d1−∑j=1m−1Rj,ifXm:m:n<T1,0ifT1<Xm:m:n<T2,0ifT2<Xm:m:n,Rd2+1∗=0,ifXm:m:n<T1,0,ifT1<Xm:m:n<T2,n−d2−∑j=1d2Rj,ifT2<Xm:m:n.The likelihood function ofλ,γ under the generalized Type-II PHCS can be derived using (1) and (2) in (5), as(8)Lλ,γ|x¯=∏i=1d∗∑j=imRj+1λd∗γd∗∏i=1d∗xiγ−11+xiγ×exp−λ∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ=∏i=1d∗∑j=imRj+1λd∗γd∗exp∑i=1d∗lnxiγ−11+xiγ×exp−λ∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ,where xi=xi:d∗:n for simplicity of notation.


## 3. Estimation Using ML Technique

In this section, using the maximum likelihood technique, we compute the ML estimate of the unknown parameters of the Burr Type-XII model with generalized Type-II HPCS. Since the critical points of any functionfx are identical to the critical points of lnfx, (8) gives the corresponding log-likelihood function as(9)lnLλ,γ|x¯=const.+d∗lnλ+lnγ+γ−1∑i=1d∗lnxi−∑i=1d∗ln1+xiγ−λ∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ.To determine the critical points, set the first derivatives of (9) with respect to λ and γ equal to zero:(10)∂lnLλ,γ|x¯∂λ=d∗λ−∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ=0,(11)∂lnLλ,γ|x¯∂γ=d∗γ+∑i=1d∗lnxi−∑i=1d∗λRi∗+1+1xiγlnxi1+xiγ−λRd1+1∗T1γlnT1γ1+T1γ+Rd2+1∗T2γlnT2γ1+T2γ=0.Using numerical technique, we can compute ML estimates of the parametersλ and γ, are λ^ML and γ^ML respectively, by solving equations (9) and (11). Byusing the invariance property of the ML estimates, we can calculate the ML estimates of the survival and the hazard functions, respectively, as(12)S^MLt=1+tγ^ML−λ^ML,h^MLt=λ^MLγ^MLtγ^ML−11+tγ^ML−1.The conditions necessary for the existence and uniqueness of ML estimates of the parametersλ and γ are discussed by Nagy et al. [17].


### 3.1. Approximate Confidence Intervals

The1001−α% confidence intervals for any parameters θ are given by(13)θ^−zα/2SEθ^ML,θ^+zα/2SEθ^ML,where zα/2 is the right α/2 percentile of standard normal distribution and SEθ^ML=varθ^ML (standard deviation) is the square root of the estimated variance of θ^ML. Now, the estimated variances varλ^ML and varγ^ML of λ^ML and γ^ML are given by the first and the second diagonal elements of the Fisher information matrix of the parameters λ and γ. For large d∗, the observed Fisher information matrix of the parameters λ and γ is given by(14)FIλ^,γ^=d∗λ2,∑i=1d∗Ri∗+1xiγlnxi1+xiγ,∑i=1d∗Ri∗+1xiγlnxi1+xiγ,d∗γ2+∑i=1d∗λRi∗+1+1lnxi2xiγ1+xiγ2.λ^ML,γ^MLThe approximate estimates ofvarS^MLt and varh^MLt can be constructed by using the delta method (see [18, 19]) as(15)varS^t≃M1tI−1λ,γM1λ^ML,γ^ML,varh^t≃M2tI−1λ,γM2λ^ML,γ^ML,where Mt is the transpose of the matrix M, with(16)M1=∂St∂λ∂St∂γ,=−1+tγ−λln1+tγ−λtγ1+tγ−λ+1lnt,M2=∂ht∂λ∂ht∂γ,=γtγ−11+tγ−1λ1+tγtγ−1+γtγ−1lnt−γt2γ−1lnt1+tγ2.


## 3.1. Approximate Confidence Intervals

The1001−α% confidence intervals for any parameters θ are given by(13)θ^−zα/2SEθ^ML,θ^+zα/2SEθ^ML,where zα/2 is the right α/2 percentile of standard normal distribution and SEθ^ML=varθ^ML (standard deviation) is the square root of the estimated variance of θ^ML. Now, the estimated variances varλ^ML and varγ^ML of λ^ML and γ^ML are given by the first and the second diagonal elements of the Fisher information matrix of the parameters λ and γ. For large d∗, the observed Fisher information matrix of the parameters λ and γ is given by(14)FIλ^,γ^=d∗λ2,∑i=1d∗Ri∗+1xiγlnxi1+xiγ,∑i=1d∗Ri∗+1xiγlnxi1+xiγ,d∗γ2+∑i=1d∗λRi∗+1+1lnxi2xiγ1+xiγ2.λ^ML,γ^MLThe approximate estimates ofvarS^MLt and varh^MLt can be constructed by using the delta method (see [18, 19]) as(15)varS^t≃M1tI−1λ,γM1λ^ML,γ^ML,varh^t≃M2tI−1λ,γM2λ^ML,γ^ML,where Mt is the transpose of the matrix M, with(16)M1=∂St∂λ∂St∂γ,=−1+tγ−λln1+tγ−λtγ1+tγ−λ+1lnt,M2=∂ht∂λ∂ht∂γ,=γtγ−11+tγ−1λ1+tγtγ−1+γtγ−1lnt−γt2γ−1lnt1+tγ2.


## 4. Bayesian Estimations

In this study, we investigate three forms of loss functions for Bayesian estimation. The first is the squared error loss function (SELF), a symmetric function that in parameter estimation assigns equal weight to overestimates and underestimates. The second is the LINEX loss function (LLF), which is asymmetric and provides overestimation and underestimation distinct weights. The generalization of the entropy loss function is the third loss function (GELF). Assume that bothλ and γ are unknown and are separately distributed as πλa1,b1 and πγa2,b2 priors, respectively. The prior density function for joining is(17)πλ,γ∝λa1−1γa2−1exp−λb1exp−γb2,where a1,b1,a2,b2 are positive constants.The posterior density function ofλ,γ given by the generalized Type-II PHCS data is obtained as(18)π∗λ,γ|x¯=Lλ,γ|x¯πλ,γ∫0∞∫0∞Lλ,γ|x¯πλ,γdλdγ.From (8) and (17),(19)π∗λ,γ|x¯=I−1λd∗+a1−1γd∗+a2−1exp∑i=1d∗lnxiγ−11+xiγ−γb2×exp−λ∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ+b1,where the normalization constant I is given by(20)I=∫0∞∫0∞Lλ,γ|x¯πλ,γdλdγ=∫0∞∫0∞λd∗+a1−1γd∗+a2−1exp∑i=1d∗lnxiγ−11+xiγ−γb2×exp−λ∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ+b1dλdγ=Γd∗+a1∫0∞γd∗+a2−1exp∑i=1d∗lnxiγ−11+xiγ−γb2×∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ+b1−d∗+a1dγ.


### 4.1. Bayesian Estimations under SELF

A commonly used loss function is SELF, and the Bayesian estimateθ^BS relative to SELF is given by(21)θ^BS=Eπ∗θ.Hence, from (19) and (21), the Bayesian estimates of λ and γ under SELF are obtained, respectively, as(22)λ^BS=I−1Γd∗+a1+1∫0∞γd∗+a2−1exp∑i=1d∗lnxiγ−11+xiγ−γb2×∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ+b1−d∗+a1+1dγ,γ^BS=I−1Γd∗+a1∫0∞γd∗+a2exp∑i=1d∗lnxiγ−11+xiγ−γb2×∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ+b1−d∗+a1dγ.


### 4.2. Bayesian Estimations under LLF

Under the assumption that the minimal loss occurs atθ^=θ, LLF can be expressed as(23)LBLθ^,θ=expυθ^−θ−υθ^−θ−1,where υ≠0. The value θ^BL which minimizes Eπ∗LLθ^,θ is given by(24)θ^BL=−1υlnEπ∗exp−υθ,provided that the involved expectation Eπ∗exp−υθ is finite. The problem of choosing the value of the parameter v has been discussed by Calabria and Pulcini [20].From (19) and (24), the Bayesian estimator of λ and γ based on the LLF is as follows:(25)λ^BL=−1υlnI−1Γd∗+a1∫0∞γd∗+a2−1exp∑i=1d∗lnxiγ−11+xiγ−γb2×∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ+υ+b1−d∗+a1dγ,γ^BL=−1υlnI−1Γd∗+a1∫0∞γd∗+a2−1exp∑i=1d∗lnxiγ−11+xiγ−γb2+υ×∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ+b1−d∗+a1dγ.


### 4.3. Bayesian Estimations under GELF

Another commonly used asymmetric loss function is GELF which is given by(26)LBEθ^,θ∝θ^θκ−κlnθ^θ−1.Forκ>0, the Bayesian estimate θ^BE relative to the GELF is given by(27)θ^BE=Eπ∗θ−κ−1/κ,provided that the involved expectation Eπ∗θ−κ is finite. From (19) and (27), one obtains the Bayesian estimator of λ and γ using GELF as follows:(28)λ^BE=I−1Γd∗+a1−κ∫0∞γd∗+a2−1exp∑i=1d∗lnxiγ−11+xiγ−γb2×∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ+b1−d∗+a1−κdγ−1/κ,γ^BE=I−1Γd∗+a1∫0∞γd∗+a2−κ−1exp∑i=1d∗lnxiγ−11+xiγ−γb2×∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ+b1−d∗+a1dγ−1/κ.Because all of the above integrals in the Bayesian estimate of unknown parametersλ and γ cannot be computed analytically, the Markov chain Monte Carlo (MCMC) algorithm is employed to evaluate them. The conditional posterior distributions π1∗λ|γ;x¯ and π2∗γ|λ;x¯ of the parameters λ and γ can now be computed, respectively, based on the posterior distribution in (19).(29)π1∗λ|γ;x¯=Wγ;x¯+γd−d∗+a1Γd∗+a1λd∗+a1−1×exp−λ∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ+b1=Gd∗+a1,∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ+b1,(30)π2∗γ|λ;x¯=γd∗+a2−1exp∑i=1d∗lnxiγ−11+xiγ−γb2exp−λ∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ+b1.The Metropolis–Hastings sampler is employed to create samples ofγ inside the MCMC algorithm because the conditional distribution of γ in (30) is not a well-known distribution (for further details, see [21–23]). The MCMC algorithm (1) in [17] is used to create λ and γ samples from conditional posterior distributions, which will be utilized to approximate Bayes estimates.Assume thatgλ,γ is any function in λ and γ; then, the Bayesian estimates of g using the MCMC values are obtained as follows.Under SELF, the Bayesian estimate ofg is given by(31)gλ,γ^BS=1N−B∑i=1N−Bgλi,γi.Based on the LLF,(32)gλ,γ^BL=1υln1N−B∑i=1N−Beυgλi,γi.By using GELF,(33)gλ,γ^BE=1N−B∑i=1N−Bgλi,γi−κ−1/κ.The1001−α% Bayesian confidence interval or credible intervalL,Ufor any parameter θ, if(34)∫LUπ∗θ|x¯dθ=1−α.Since the integration in (34) cannot be solved analytically, the 1001−α MCMC approximate credible intervals for λ and γ using the N−B generated values after sorting it in an ascending order, λ1,λ2,…,λN−B and γ1,γ2,…,γN−B, are given as follows:(35)λN−Bα/2,λN−B1−α/2,γN−Bα/2,γN−B1−α/2,and the lengths of the credible intervals are the absolute difference between the upper and the lower bounds.


## 4.1. Bayesian Estimations under SELF

A commonly used loss function is SELF, and the Bayesian estimateθ^BS relative to SELF is given by(21)θ^BS=Eπ∗θ.Hence, from (19) and (21), the Bayesian estimates of λ and γ under SELF are obtained, respectively, as(22)λ^BS=I−1Γd∗+a1+1∫0∞γd∗+a2−1exp∑i=1d∗lnxiγ−11+xiγ−γb2×∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ+b1−d∗+a1+1dγ,γ^BS=I−1Γd∗+a1∫0∞γd∗+a2exp∑i=1d∗lnxiγ−11+xiγ−γb2×∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ+b1−d∗+a1dγ.


## 4.2. Bayesian Estimations under LLF

Under the assumption that the minimal loss occurs atθ^=θ, LLF can be expressed as(23)LBLθ^,θ=expυθ^−θ−υθ^−θ−1,where υ≠0. The value θ^BL which minimizes Eπ∗LLθ^,θ is given by(24)θ^BL=−1υlnEπ∗exp−υθ,provided that the involved expectation Eπ∗exp−υθ is finite. The problem of choosing the value of the parameter v has been discussed by Calabria and Pulcini [20].From (19) and (24), the Bayesian estimator of λ and γ based on the LLF is as follows:(25)λ^BL=−1υlnI−1Γd∗+a1∫0∞γd∗+a2−1exp∑i=1d∗lnxiγ−11+xiγ−γb2×∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ+υ+b1−d∗+a1dγ,γ^BL=−1υlnI−1Γd∗+a1∫0∞γd∗+a2−1exp∑i=1d∗lnxiγ−11+xiγ−γb2+υ×∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ+b1−d∗+a1dγ.


## 4.3. Bayesian Estimations under GELF

Another commonly used asymmetric loss function is GELF which is given by(26)LBEθ^,θ∝θ^θκ−κlnθ^θ−1.Forκ>0, the Bayesian estimate θ^BE relative to the GELF is given by(27)θ^BE=Eπ∗θ−κ−1/κ,provided that the involved expectation Eπ∗θ−κ is finite. From (19) and (27), one obtains the Bayesian estimator of λ and γ using GELF as follows:(28)λ^BE=I−1Γd∗+a1−κ∫0∞γd∗+a2−1exp∑i=1d∗lnxiγ−11+xiγ−γb2×∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ+b1−d∗+a1−κdγ−1/κ,γ^BE=I−1Γd∗+a1∫0∞γd∗+a2−κ−1exp∑i=1d∗lnxiγ−11+xiγ−γb2×∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ+b1−d∗+a1dγ−1/κ.Because all of the above integrals in the Bayesian estimate of unknown parametersλ and γ cannot be computed analytically, the Markov chain Monte Carlo (MCMC) algorithm is employed to evaluate them. The conditional posterior distributions π1∗λ|γ;x¯ and π2∗γ|λ;x¯ of the parameters λ and γ can now be computed, respectively, based on the posterior distribution in (19).(29)π1∗λ|γ;x¯=Wγ;x¯+γd−d∗+a1Γd∗+a1λd∗+a1−1×exp−λ∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ+b1=Gd∗+a1,∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ+b1,(30)π2∗γ|λ;x¯=γd∗+a2−1exp∑i=1d∗lnxiγ−11+xiγ−γb2exp−λ∑i=1d∗Ri∗+1ln1+xiγ+Rd1+1∗ln1+T1γ+Rd2+1∗ln1+T2γ+b1.The Metropolis–Hastings sampler is employed to create samples ofγ inside the MCMC algorithm because the conditional distribution of γ in (30) is not a well-known distribution (for further details, see [21–23]). The MCMC algorithm (1) in [17] is used to create λ and γ samples from conditional posterior distributions, which will be utilized to approximate Bayes estimates.Assume thatgλ,γ is any function in λ and γ; then, the Bayesian estimates of g using the MCMC values are obtained as follows.Under SELF, the Bayesian estimate ofg is given by(31)gλ,γ^BS=1N−B∑i=1N−Bgλi,γi.Based on the LLF,(32)gλ,γ^BL=1υln1N−B∑i=1N−Beυgλi,γi.By using GELF,(33)gλ,γ^BE=1N−B∑i=1N−Bgλi,γi−κ−1/κ.The1001−α% Bayesian confidence interval or credible intervalL,Ufor any parameter θ, if(34)∫LUπ∗θ|x¯dθ=1−α.Since the integration in (34) cannot be solved analytically, the 1001−α MCMC approximate credible intervals for λ and γ using the N−B generated values after sorting it in an ascending order, λ1,λ2,…,λN−B and γ1,γ2,…,γN−B, are given as follows:(35)λN−Bα/2,λN−B1−α/2,γN−Bα/2,γN−B1−α/2,and the lengths of the credible intervals are the absolute difference between the upper and the lower bounds.


## 5. One-Sample Bayesian Prediction

In this section, the Bayesian point and interval predictions are made for the unobserved or remote units in each phase given the generalized Type-II PHCS. The prediction interval is obtained by using equi-tailed (ET) or two-tailed interval and the highest posterior density (HPD) interval.LetYi:Rj∗ denote the ith order statistic out of Rj∗ removed units at stage j, (for i=1,2,…,Rj∗). Basak et al. in [24] calculated the conditional density function of Yi:Rj∗, given the observed generalized Type-II PHCS, by(36)fYi:Rj∗|x¯=fy|x¯=Rj∗!i−1!Rj∗−i!Fy−Fyji−11−FyRj∗−ify1−FyjRj∗,y>yj,where(37)j=1,…,d1,d1+1,ifXm:m:n<T1<T21,…,d1,…,m,ifT1<Xm:m:n<T2,1,…,d1,…,d2,d2+1,ifT1<T2<Xm:m:n.By compensation in (36) by using (1) and (2), the conditional density function of Yi:Rj∗ given generalized Type-II PHCS is then given as follows:(38)fy|x¯=∑k=0i−1Ckλγyγ−11+yγexp−λϖkln1+yγ1+yjγ,y>yj,where Ck=−1ki−1kRj∗!/i−1!Rj∗−i! and ϖk=k+Rj∗−i+1 for k=0,…,i−1; therefore, the Bayesian predictive density function of Yi:Rj∗, given generalized Type-II PHCS, is obtained as(39)f∗y|x¯=∫0∞∫0∞fy|x¯π∗λ,γ|x¯dλdγ=1N−B∑h=1N−B∑k=0i−1Ckλhγhyγh−11+yγhexp−λhϖkln1+yγh1+yjγh.The Bayesian predictive survival function ofYi:Rj∗, given generalized Type-II PHCS, is given as(40)F¯∗t|x¯=∫t∞f∗y|x¯dx=1N−B∑h=1N−B∑k=0i−1Ckϖk1+tγh1+yjγh−λhϖk.The Bayesian point predictor ofYi:Rj∗ under the squared error loss function is the mean of the predictive density, given by(41)Y^i:Rj∗=∫0∞xf∗y|x¯dx,=1N−B∑h=1N−B∑k=0i−1Ckλh1+yjγhλhϖkΓ1+1/γhΓλhϖk−1/γhΓ1+λhϖk.The1001−α% equi-tailed (ET) interval Bayesian predictive bounds for Yi:Rj∗ can be obtained by solving the following two equations:(42)F¯∗UET|x¯=1−α2,F¯∗LET|x¯=α2,but the 1001−α% highest posterior density (HPD) interval can be obtained by solving the following two equations:(43)f∗LHPD|x¯−f∗UHPD|x¯=0,F¯∗LHPD|x¯−F¯∗UHPD|x¯=1−α,where UET, UHPD, LET, and LHPD denote the upper and lower bounds of interval estimation for ET and HPD methods, respectively.


## 6. Simulation Study

For passing through and understanding statistical inferential approaches, and combination of theoretical and applied skills, sothe analytical technique was discussed in previous sections, and this section will focus on the numerical technique through a simulation study. We simulated random samples from the Burr Type-XII distribution in equation (2) and evaluated the coverage probabilities of the resulting CIs. We used parameter values λ=2 and γ=1, for sample of size n=50, with different values of m,T1, and T2. For the point estimate, we computed the ML and Bayesian estimates of λ, γ, St, and ht. For Bayesian estimates, we used SELF, LLF (with υ=0.5), and GELF (with κ=0.5) using informative (Prior1) and non-informative priors (Prior2); for each estimate, the mean square error (MSE) and estimated expected bias (EB) have been determined. We also construct the average confidence (ACL) and the coverage probabilities (CP) of the 90% and 95% asymptotic confidence intervals and Bayesian credible intervals for λ^, γ^, St^, and ht^ using 10,000 simulations. We used three different progressive Type-II censored sampling schemes, namely,(1)
Scheme 1:Rm=n−m,Ri=0 if i≠m.(2)
Scheme 2:R1=n−m,Ri=0 if i≠1.(3)
Scheme 3:R1=Rm=n−m/2,Ri=0 if i≠1 and m.For computing the Bayes estimates, we generate the MCMC algorithm suggested in Section4 with N=10,000 observations. The ML estimates for unknown parameters λ and γ have been used as initial values for running the MCMC algorithm. The generated sequences’ starting values may be different from converged sequences, so the first B=2,000 values are removed here to avoid the effects of the initial values. To check the convergence of MCMC samples and determine the B-burn-in, we provide the key diagnostic test, trace plots, and posterior density plots for different parameters and censoring schemes. Figures 2–5 show the trace plots of iterations for posterior densities of λ, γ, St, and ht using the three above schemes at m=40, T1=0.5, and T2=1 with Prior1 and Prior2. All censoring schemes are plotted in the same way, and it has been found that the trace plots of all censoring schemes converge very well. Furthermore, the approximate marginal posterior density with histograms for λ, γ, St=0.5, and ht=0.5 is shown in Figures 6–9, respectively.Figure 2
Trace plots of 10,000 iterations ofλ with Prior1 and Prior2. (a) Scheme I. (b) Scheme II. (c) Scheme III.
(a)(b)(c)Figure 3
Trace plots of 10,000 iterations ofγ with Prior1 and Prior2. (a) Scheme I. (b) Scheme II. (c) Scheme III.
(a)(b)(c)Figure 4
Trace plots of 10,000 iterations ofStwith Prior1 and Prior2. (a) Scheme I. (b) Scheme II. (c) Scheme III.
(a)(b)(c)Figure 5
Trace plots of 10,000 iterations ofht with Prior1 and Prior2. (a) Scheme I. (b) Scheme II. (c) Scheme III.
(a)(b)(c)Figure 6
The marginal posterior density with histograms forλ. (a) Scheme I. (b) Scheme II. (c) Scheme III.
(a)(b)(c)Figure 7
The marginal posterior density with histograms forγ. (a) Scheme I. (b) Scheme II. (c) Scheme III.
(a)(b)(c)Figure 8
The marginal posterior density with histograms forSt=0.5. (a) Scheme I. (b) Scheme II. (c) Scheme III.
(a)(b)(c)Figure 9
The marginal posterior density with histograms forht=0.5. (a) Scheme I. (b) Scheme II. (c) Scheme III.
(a)(b)(c)The trace plots, as seen in these diagrams, exhibit delicate mixing of the chains and converge to their distributions. Furthermore, in all situations, histograms and density plots are approximately symmetrical about their means. As a result, the MCMC-generated sample can be used to develop Bayesian estimates and approximate credible intervals, as well as to estimate parameters and their functions. The values ofMSE and EB of the ML and Bayesian estimates for λ, γ, St=0.5, and ht=0.5 are presented in Tables 1–4, respectively. Tables 5–8 show the ACL of 90% and 95% confidence intervals (CIs), as well as the associated coverage portability (CP).Table 1
MSE and EB of the estimates forλ.
Bayesianλ^BSλ^BLλ^BESch .T1,T2n,mλ^MLPrior2Prior2Prior2Prior2Prior2Prior2MSESch.10.5,1(50, 20)1.27850.01821.84650.01770.57100.01790.9019(50, 30)0.55710.02150.53400.02080.31400.02100.3705(50, 40)0.57600.02140.49920.02090.30080.02120.35631,1.5(50, 20)0.53010.02380.47780.02330.29810.02360.3538(50, 30)0.49400.02070.46180.02030.29150.02070.3327(50, 40)0.45440.02580.45540.02500.26320.02530.32761.5,2(50, 20)0.46680.02010.45550.01970.28130.01990.3221(50, 30)0.46490.02640.42270.02570.27890.02620.3095(50, 40)0.46490.02640.42270.02570.27890.02620.3095Sch.20.5,1(50, 20)1.33430.01911.66310.01880.55550.01900.8658(50, 30)0.43030.02580.42260.02500.26230.02530.3044(50, 40)0.46990.02010.42460.01970.26660.01990.30231,1.5(50, 20)0.45740.02570.42440.02500.26250.02550.2991(50, 30)0.37140.01820.34840.01780.21400.01790.2723(50, 40)0.34490.01720.32350.01690.21890.01710.24101.5,2(50, 20)0.32120.01600.29640.01560.20610.01580.2233(50, 30)0.30740.01700.29220.01670.20690.01690.2218(50, 40)0.31400.02680.26510.02590.19790.02620.2139Sch.30.5,1(50, 20)1.12230.01921.37390.01880.50060.01900.7552(50, 30)0.30320.02780.25510.02690.19610.02720.2067(50, 40)0.27720.01830.24010.01810.18280.01830.19671,1.5(50, 20)0.25910.02720.21330.02640.16910.02680.1795(50, 30)0.25590.02620.21370.02540.16680.02570.1759(50, 40)0.22750.01840.19520.01790.15370.01810.16291.5,2(50, 20)0.20420.01720.17080.01680.13870.01700.1435(50, 30)0.19170.01860.16700.01820.13260.01830.1390(50, 40)0.18990.01810.16060.01780.13170.01800.1367EBSch.10.5,1(50, 20)0.34740.00330.37860.00510.19210.00930.2113(50, 30)0.23040.01210.19810.00310.12260.00120.1163(50, 40)0.19880.00430.16380.00480.09460.00940.08681,1.5(50, 20)0.18750.00380.16220.00510.09450.00960.0854(50, 30)0.18190.00360.16120.00740.08560.01280.0715(50, 40)0.17350.00360.16120.00740.08560.01280.07151.5,2(50, 20)0.17660.00380.15860.00720.08200.01270.0684(50, 30)0.16340.00550.15600.00380.07200.00830.0572(50, 40)0.17840.01120.15600.00080.09790.00430.0912Sch.20.5,1(50, 20)0.32930.00180.34940.00660.18200.01080.1954(50, 30)0.16890.01130.14770.00090.09040.00420.0842(50, 40)0.15950.00060.14720.00830.06620.01280.05191,1.5(50, 20)0.17550.00100.14720.00810.08300.01250.0748(50, 30)0.15450.00010.14300.00720.09330.01080.0861(50, 40)0.14370.00230.13680.00480.08850.00850.08011.5,2(50, 20)0.14490.00410.13620.00310.08850.00670.0807(50, 30)0.14370.01090.11280.00010.07110.00540.0606(50, 40)0.13650.01150.10660.00070.06280.00460.0516Sch.30.5,1(50, 20)0.30350.00150.32920.00690.17150.01110.1814(50, 30)0.10970.00600.10140.00200.05190.00590.0402(50, 40)0.12060.00960.09200.00120.05290.00660.04131,1.5(50, 20)0.11110.00550.09000.00230.05980.00610.0517(50, 30)0.09970.00020.08940.00760.04340.01150.0303(50, 40)0.10840.00540.08480.00230.05470.00600.04581.5,2(50, 20)0.09910.00940.08010.00170.05090.00200.0423(50, 30)0.10020.00790.07490.00300.03480.00840.0219(50, 40)0.08860.00200.07070.00570.04150.00950.0322Table 2
MSE and EB of the estimates forγ.
Bayesianγ^BSγ^BLγ^BESch.T1,T2n,mλ^MLPrior2Prior2Prior2Prior2Prior2Prior2MSESch.10.5,1(50, 20)0.04160.00490.03380.00490.03140.00480.0296(50, 30)0.04220.00530.03360.00520.03140.00510.0297(50, 40)0.04130.00720.03270.00710.03060.00690.02911,1.5(50, 20)0.04130.00720.03270.00710.03060.00690.0291(50, 30)0.03850.00540.03150.00530.02950.00510.0279(50, 40)0.03690.00700.02890.00680.02700.00670.02561.5,2(50, 20)0.03600.00710.02830.00690.02660.00680.0254(50, 30)0.03150.00660.02430.00650.02290.00640.0218(50, 40)0.02790.00570.02170.00560.02070.00550.0202Sch.20.5,1(50, 20)0.02760.00540.02160.00540.02060.00530.0200(50, 30)0.02770.00720.02140.00710.02040.00700.0197(50, 40)0.02730.00560.02070.00550.01980.00540.01911,1.5(50, 20)0.02690.00630.02070.00620.01980.00610.0192(50, 30)0.02710.00720.02060.00710.01970.00700.0190(50, 40)0.02580.00700.01990.00690.01890.00670.01821.5,2(50, 20)0.02590.00730.01970.00710.01880.00700.0184(50, 30)0.02510.00550.01940.00550.01850.00550.0177(50, 40)0.02230.00450.01800.00450.01730.00440.0166Sch.30.5,1(50, 20)0.02280.00470.01790.00460.01710.00450.0164(50, 30)0.02300.00550.01760.00540.01680.00530.0161(50, 40)0.02130.00490.01710.00480.01650.00480.01621,1.5(50, 20)0.02090.00430.01660.00420.01590.00420.0153(50, 30)0.01930.00480.01520.00470.01470.00460.0144(50, 40)0.01680.00500.01300.00490.01260.00480.01241.5,2(50, 20)0.01660.00500.01290.00490.01250.00480.0122(50, 30)0.01590.00440.01230.00430.01190.00430.0115(50, 40)0.01440.00480.01120.00470.01090.00460.0107EBSch.10.5,1(50, 20)0.04160.00490.03380.00490.03140.00480.0296(50, 30)0.03850.00540.03150.00530.02950.00510.0279(50, 40)0.03690.00700.02890.00680.02700.00670.02561,1.5(50, 20)0.03600.00710.02830.00690.02660.00680.0254(50, 30)0.03150.00660.02430.00650.02290.00640.0218(50, 40)0.02790.00570.02170.00560.02070.00550.02021.5,2(50, 20)0.02760.00540.02160.00540.02060.00530.0200(50, 30)0.04220.00530.03360.00520.03140.00510.0297(50, 40)0.02770.00720.02140.00710.02040.00700.0197Sch.20.5,1(50, 20)0.04130.00720.03270.00710.03060.00690.0291(50, 30)0.02730.00560.02070.00550.01980.00540.0191(50, 40)0.02690.00630.02070.00620.01980.00610.01921,1.5(50, 20)0.02710.00720.02060.00710.01970.00700.0190(50, 30)0.02580.00700.01990.00690.01890.00670.0182(50, 40)0.02590.00730.01970.00710.01880.00700.01841.5,2(50, 20)0.02510.00550.01940.00550.01850.00550.0177(50, 30)0.02230.00450.01800.00450.01730.00440.0166(50, 40)0.02280.00470.01790.00460.01710.00450.0164Sch.30.5,1(50, 20)0.04130.00720.03270.00710.03060.00690.0291(50, 30)0.02300.00550.01760.00540.01680.00530.0161(50, 40)0.02130.00490.01710.00480.01650.00480.01621,1.5(50, 20)0.02090.00430.01660.00420.01590.00420.0153(50, 30)0.01930.00480.01520.00470.01470.00460.0144(50, 40)0.01680.00500.01300.00490.01260.00480.01241.5,2(50, 20)0.01660.00500.01290.00490.01250.00480.0122(50, 30)0.01590.00440.01230.00430.01190.00430.0115(50, 40)0.01440.00480.01120.00470.01090.00460.0107Table 3
MSE and EB of the estimates forSt.
BayesianSt^BSSt^BLSt^BESch.T1,T2n,mλ^MLPrior2Prior2Prior2Prior2Prior2Prior2MSESch−I0.5,1(50, 20)4.37×10−44.3×10−69.98×10−44.3×10−69.15×10−42.9×10−61.66×10−4(50, 30)7.65×10−44.3×10−60.0024.3×10−60.0021.4×10−62.50×10−4(50, 40)5.47×10−44.3×10−69.98×10−44.3×10−69.15×10−42.9×10−62.50×10−41,1.5(50, 20)4.37×10−45.7×10−67.49×10−45.7×10−66.66×10−42.9×10−61.66×10−4(50, 30)5.47×10−44.3×10−69.15×10−44.3×10−69.15×10−42.9×10−62.50×10−4(50, 40)2.19×10−45.7×10−64.16×10−45.7×10−64.16×10−42.9×10−61.66×10−41.5,2(50, 20)2.19×10−44.3×10−64.99×10−44.3×10−64.99×10−42.9×10−61.66×10−4(50, 30)4.37×10−45.7×10−67.49×10−45.7×10−67.49×10−42.9×10−62.50×10−4(50, 40)2.19×10−45.7×10−63.33×10−45.7×10−63.33×10−42.9×10−68.32×10−5Sch−II0.5,1(50, 20)5.47×10−44.3×10−60.0014.3×10−60.0012.9×10−62.50×10−4(50, 30)9.84×10−44.3×10−60.0024.3×10−60.0022.9×10−63.33×10−4(50, 40)6.56×10−44.3×10−60.0014.3×10−69.98×10−42.9×10−63.33×10−41,1.5(50, 20)4.37×10−45.7×10−66.66×10−45.7×10−66.66×10−42.9×10−62.50×10−4(50, 30)4.37×10−44.3×10−67.49×10−44.3×10−67.49×10−42.9×10−61.66×10−4(50, 40)2.19×10−45.7×10−63.33×10−45.7×10−63.33×10−42.9×10−68.32×10−51.5,2(50, 20)3.28×10−45.7×10−64.99×10−45.7×10−64.99×10−42.9×10−61.66×10−4(50, 30)3.28×10−44.3×10−65.82×10−44.3×10−65.82×10−42.9×10−61.66×10−4(50, 40)2.19×10−45.7×10−62.50×10−45.7×10−62.50×10−42.9×10−68.32×10−5Sch−III0.5,1(50, 20)5.47×10−44.3×10−60.0014.3×10−61.08E − 032.9×10−62.50×10−4(50, 30)5.47×10−42.9×10−60.0012.9×10−69.98×10−41.4×10−62.50×10−4(50, 40)3.28×10−44.3×10−64.99×10−44.3×10−64.99×10−42.9×10−61.66×10−41,1.5(50, 20)2.19×10−44.3×10−64.16×10−44.3×10−64.16×10−42.9×10−61.66×10−4(50, 30)5.47×10−44.3×10−60.0014.3×10−69.98×10−41.4×10−62.50×10−4(50, 40)2.19×10−44.3×10−64.16×10−44.3×10−64.16×10−42.9×10−68.32×10−51.5,2(50, 20)2.19×10−44.3×10−64.16×10−44.3×10−64.16×10−42.9×10−68.32×10−5(50, 30)6.56×10−44.3×10−60.0014.3×10−61.08E − 031.4×10−62.50×10−4(50, 40)2.19×10−44.3×10−64.16×10−44.3×10−64.16×10−42.9×10−68.32×10−5EBSch−I0.5,1(50, 20)0.0060.0020.0210.0020.0216.83×10−41.14E − 03(50, 30)0.0110.0020.0280.0020.0276.83×10−41.30×10−4(50, 40)0.0080.0010.0200.0010.0206.83×10−43.12×10−41,1.5(50, 20)0.0060.0020.0170.0010.0176.83×10−47.28×10−4(50, 30)0.0070.0010.0190.0010.0186.83×10−45.20×10−4(50, 40)0.0040.0020.0120.0010.0125.46×10−42.08×10−41.5,2(50, 20)0.0040.0010.0140.0010.0148.19×10−43.12×10−4(50, 30)0.0060.0020.0180.0020.0186.83×10−41.04×10−4(50, 40)0.0030.0010.0100.0010.0106.83×10−46.24×10−4Sch−II0.5,1(50, 20)0.0070.0020.0230.0020.0226.83×10−42.08×10−4(50, 30)0.0120.0020.0290.0020.0286.83×10−48.32×10−4(50, 40)0.0090.0020.0200.0020.0206.83×10−48.32×10−41,1.5(50, 20)0.0050.0010.0150.0010.0158.19×10−48.32×10−4(50, 30)0.0060.0010.0170.0010.0176.83×10−46.24×10−4(50, 40)0.0040.0010.0110.0010.0116.83×10−42.08×10−41.5,2(50, 20)0.0040.0010.0130.0010.0138.19×10−42.08×10−4(50, 30)0.0030.0010.0150.0010.0158.19×10−41.66E − 03(50, 40)0.0030.0010.0100.0010.0106.83×10−46.24×10−4Sch−III0.5,1(50, 20)0.0070.0020.0230.0020.0226.83×10−42.08×10−4(50, 30)0.0070.0010.0210.0010.0208.19×10−41.98E − 03(50, 40)0.0050.0010.0130.0010.0136.83×10−48.32×10−41,1.5(50, 20)0.0030.0010.0130.0010.0138.19×10−41.56E − 03(50, 30)0.0060.0010.0210.0010.0208.19×10−42.18E − 03(50, 40)0.0040.0010.0120.0010.0126.83×10−41.46E − 031.5,2(50, 20)0.0040.0010.0130.0010.0138.19×10−41.35E − 03(50, 30)0.0070.0020.0210.0010.0216.83×10−41.56E − 03(50, 40)0.0040.0010.0120.0010.0126.83×10−41.46E − 03Table 4
MSE and EB of the estimates forht.
Bayesianht^BSht^BLht^BESch.T1,T2n,mλ^MLPrior2Prior2Prior2Prior2Prior2Prior2MSESch−I0.5,1(50, 20)0.0234.4×10−50.0314.4×10−50.0264.4×10−50.016(50, 30)0.0204.4×10−50.0284.4×10−50.0244.4×10−50.015(50, 40)0.0225.9×10−50.0305.9×10−50.0215.9×10−50.0191,1.5(50, 20)0.0105.9×10−50.0115.9×10−50.0107.4×10−50.008(50, 30)0.0265.9×10−50.0325.9×10−50.0275.9×10−50.018(50, 40)0.0088.8×10−50.0098.8×10−50.0088.8×10−50.0061.5,2(50, 20)0.0117.4×10−50.0127.4×10−50.0117.4×10−50.009(50, 30)0.0225.9×10−50.0285.9×10−50.0245.9×10−50.017(50, 40)0.0097.4×10−50.0118.8×10−50.0108.8×10−50.008Sch−II0.5,1(50, 20)0.0244.4×10−50.0314.4×10−50.0275.9×10−50.017(50, 30)0.0204.4×10−50.0274.4×10−50.0244.4×10−50.015(50, 40)0.0115.9×10−50.0145.9×10−50.0125.9×10−50.0091,1.5(50, 20)0.0125.9×10−50.0145.9×10−50.0137.4×10−50.009(50, 30)0.0175.9×10−50.0235.9×10−50.0205.9×10−50.013(50, 40)0.0087.4×10−50.0097.4×10−50.0088.8×10−50.0061.5,2(50, 20)0.0127.4×10−50.0147.4×10−50.0137.4×10−50.009(50, 30)0.0244.4×10−50.0344.4×10−50.0285.9×10−50.018(50, 40)0.0077.4×10−50.0087.4×10−50.0087.4×10−50.006Sch−III0.5,1(50, 20)0.0244.4×10−50.0314.4×10−50.0275.9×10−50.017(50, 30)0.0674.4×10−50.1764.4×10−50.0904.4×10−50.054(50, 40)0.0145.9×10−50.0195.9×10−50.0175.9×10−50.0121,1.5(50, 20)0.0245.9×10−50.0345.9×10−50.0275.9×10−50.019(50, 30)0.0604.4×10−50.1164.4×10−50.0734.4×10−50.046(50, 40)0.0145.9×10−50.0185.9×10−50.0175.9×10−50.0111.5,2(50, 20)0.0235.9×10−50.0345.9×10−50.0275.9×10−50.018(50, 30)0.0784.4×10−50.1594.4×10−50.0824.4×10−50.056(50, 40)0.0155.9×10−50.0205.9×10−50.0185.9×10−50.012EBSch−I0.5,1(50, 20)0.0541.8×10−40.0713.4×10−40.0640.00280.026(50, 30)0.0463.8×10−40.0655.4×10−40.0590.00300.019(50, 40)0.0322.9×10−50.0431.2×10−40.0390.00250.0151,1.5(50, 20)0.0304.0×10−40.0325.4×10−40.0300.00300.009(50, 30)0.0551.0×10−40.0655.9×10−50.0590.00250.029(50, 40)0.0261.2×10−50.0301.6×10−40.0280.00250.0131.5,2(50, 20)0.0362.5×10−40.0381.0×10−40.0360.00220.016(50, 30)0.0491.6×10−40.0593.2×10−40.0550.00270.026(50, 40)0.0316.2×10−40.0354.7×10−40.0340.00180.018Sch−II0.5,1(50, 20)0.0573.5×10−40.0745.1×10−40.0670.00300.028(50, 30)0.0463.7×10−40.0625.3×10−40.0560.00300.018(50, 40)0.0301.5×10−50.0391.5×10−40.0360.00250.0121,1.5(50, 20)0.0394.0×10−40.0442.4×10−40.0410.00220.019(50, 30)0.0504.4×10−50.0621.2×10−40.0570.00250.027(50, 40)0.0305.1×10−40.0333.7×10−40.0310.00190.0161.5,2(50, 20)0.0412.6×10−40.0451.2×10−40.0430.00220.022(50, 30)0.0604.4×10−40.0742.9×10−40.0680.00210.036(50, 40)0.0261.3×10−40.0291.2×10−50.0280.00240.013Sch−III0.5,1(50, 20)0.0573.5×10−40.0745.1×10−40.0670.00300.028(50, 30)0.1003.5×10−40.1581.9×10−40.1290.00220.069(50, 40)0.0424.1×10−40.0562.5×10−40.0520.00210.0271,1.5(50, 20)0.0501.3×10−40.0621.2×10−50.0560.00250.029(50, 30)0.0883.1×10−40.1351.5×10−40.1150.00240.060(50, 40)0.0413.8×10−40.0552.2×10−40.0510.00220.0271.5,2(50, 20)0.0541.2×10−40.0664.4×10−50.0610.00250.033(50, 30)0.0968.8×10−50.1467.4×10−50.1210.00250.065(50, 40)0.0456.0×10−40.0584.4×10−40.0540.00190.029Table 5
The ACL of90% and 95% CIs and corresponding CP for λ^ML and λ^B.
λ^Bλ^MLPrior2Prior290%95%90%95%90%95%T1,T2n,mACLCPACLCPACLCPACLCPACLCPACLCPSch.10.5,1(50, 20)2.8700.9643.2470.9640.9570.9871.1070.9802.8170.9073.2230.942(50, 30)3.0150.9773.3990.9590.8700.9941.0120.9852.9500.9233.4050.939(50, 40)2.2420.9522.6120.9550.8070.9950.9440.9842.2030.9112.5770.9361,1.5(50, 20)1.7480.9171.9870.9570.7780.9810.9000.9801.6950.8931.9150.940(50, 30)2.1120.9702.4810.9490.7080.9920.8240.9792.0530.9212.4210.906(50, 40)1.4630.9221.6820.9600.6540.9780.7570.9841.4230.8981.6380.9321.5,2(50, 20)1.4910.9461.7370.9560.6910.9920.8010.9801.4460.9151.6770.941(50, 30)1.8350.9782.1220.9590.6320.9930.7270.9801.7920.9082.0690.921(50, 40)1.2680.9531.4990.9540.5820.9840.6730.9801.2320.9331.4570.927Sch.20.5,1(50, 20)2.8640.9543.2670.9540.9540.9771.1080.9802.7860.9013.2260.926(50, 30)3.0630.9663.3520.9510.8651.0000.9980.9803.0270.9113.3340.937(50, 40)2.2030.9442.5650.9440.8040.9930.9320.9832.1500.9092.5210.9201,1.5(50, 20)1.7300.9302.0350.9560.7750.9910.8990.9761.6720.9001.9800.931(50, 30)2.4450.9502.4820.9570.7070.9910.8160.9792.4110.8902.4280.927(50, 40)1.4360.9461.7040.9550.6470.9940.7540.9851.4010.9191.6540.9271.5,2(50, 20)1.5560.9231.7730.9490.6870.9810.8020.9741.5280.9001.7290.920(50, 30)1.9600.9852.2830.9650.6260.9950.7280.9811.9010.9072.2460.936(50, 40)1.2630.9451.4800.9600.5740.9920.6700.9801.2270.9251.4440.927Sch.30.5,1(50, 20)2.5460.9613.2580.9540.9310.9731.1070.9802.5020.9233.2180.926(50, 30)3.8590.9744.4440.9650.8370.9950.9700.9894.1290.8994.8710.926(50, 40)2.1870.9592.5160.9590.7790.9980.8980.9792.1600.8962.5230.9251,1.5(50, 20)1.8600.9652.1980.9630.7570.9870.8780.9811.8130.9102.1750.934(50, 30)3.2310.9703.4750.9660.6880.9890.7940.9833.4350.8863.7030.930(50, 40)1.7540.9592.0450.9630.6320.9960.7340.9821.7520.9022.0390.9311.5,2(50, 20)1.6210.9701.9690.9650.6760.9710.7820.9851.5880.9301.9600.921(50, 30)2.9330.9733.1840.9570.6090.9950.7030.9823.0560.8793.3830.909(50, 40)1.5930.9591.8400.9690.5610.9840.6500.9721.5770.8861.8390.931Table 6
The ACL of90% and 95% CIs and corresponding CP for γ^ML and γ^B.
Bayesianγ^MLPrior2Prior290%95%90%95%90%95%T1,T2n,mACLCPACLCPACLCPACLCPACLCPACLCPSch.10.5,1(50, 20)0.9800.9461.1300.9970.4781.0000.5551.0000.9550.9361.0840.975(50, 30)0.8740.9491.0050.9820.4130.9870.4781.0000.8450.9200.9630.964(50, 40)0.7450.9610.8590.9770.3840.9990.4391.0000.7300.9350.8270.9451,1.5(50, 20)0.7620.9630.8780.9760.4441.0000.5100.9990.7410.9420.8410.957(50, 30)0.7220.9620.8360.9740.3821.0000.4441.0000.7040.9400.8040.945(50, 40)0.5830.9390.6790.9590.3510.9860.4061.0000.5690.9260.6590.9411.5,2(50, 20)0.6770.9640.7860.9730.4070.9990.4690.9980.6640.9460.7590.953(50, 30)0.6440.9750.7540.9810.3481.0000.4030.9850.6240.9370.7260.944(50, 40)0.5250.9630.6150.9630.3211.0000.3731.0000.5120.9440.5940.941Sch.20.5,1(50, 20)0.9780.9531.1350.9870.4781.0000.5521.0000.9450.9371.0900.949(50, 30)0.8610.9500.9970.9850.4071.0000.4740.9980.8370.9260.9540.962(50, 40)0.7350.9640.8570.9900.3801.0000.4380.9970.7150.9500.8200.9631,1.5(50, 20)0.7540.9570.8810.9860.4371.0000.5071.0000.7380.9330.8560.966(50, 30)0.7950.9640.8390.9840.3761.0000.4341.0000.7720.9440.8080.970(50, 40)0.5830.9520.6790.9830.3511.0000.4020.9980.5720.9230.6540.9601.5,2(50, 20)0.6860.9540.7880.9800.4030.9990.4680.9990.6720.9180.7640.947(50, 30)0.6650.9520.7710.9850.3471.0000.3990.9950.6460.9180.7400.967(50, 40)0.5310.9350.6090.9850.3220.9860.3671.0000.5150.9160.5930.966Sch.30.5,1(50, 20)0.9610.9721.1350.9880.4690.9890.5511.0000.9350.9361.0890.949(50, 30)0.9640.9511.1090.9840.3950.9900.4571.0000.9280.9051.0580.951(50, 40)0.7280.9710.8530.9870.3691.0000.4281.0000.7090.9450.8220.9481,1.5(50, 20)0.8110.9680.9380.9950.4361.0000.4981.0000.7870.9470.9070.978(50, 30)0.8750.9711.0170.9890.3620.9970.4221.0000.8460.9200.9670.949(50, 40)0.6730.9650.7800.9820.3421.0000.3941.0000.6550.9350.7490.9611.5,2(50, 20)0.7390.9670.8640.9860.4020.9900.4581.0000.7200.9400.8300.961(50, 30)0.8120.9600.9360.9770.3341.0000.3870.9940.7760.9220.8830.937(50, 40)0.6180.9750.7210.9850.3110.9960.3631.0000.6020.9490.6950.955Table 7
The ACL of90% and 95% CIs and corresponding CP for St^ML and St^B.
St^BSt^MLPrior2Prior290%95%90%95%90%95%T1,T2n,mACLCPACLCPACLCPACLCPACLCPACLCPSch.10.5,1(50, 20)0.0920.7730.1120.8090.0221.0000.0271.0000.1450.9690.1890.964(50, 30)0.1180.8100.1440.8250.0231.0000.0271.0000.1720.9650.2210.963(50, 40)0.0950.8430.1130.8620.0221.0000.0260.9930.1290.9520.1620.9601,1.5(50, 20)0.0720.8280.0900.8680.0211.0000.0240.9780.1040.9480.1360.961(50, 30)0.0730.7630.0940.8100.0211.0000.0250.9630.1100.9720.1470.934(50, 40)0.0590.8520.0710.8880.0201.0000.0241.0000.0780.9600.1000.9441.5,2(50, 20)0.0610.8290.0700.8500.0191.0000.0220.9330.0880.9640.1100.954(50, 30)0.0690.7960.0850.8210.0191.0000.0231.0000.1030.9540.1350.949(50, 40)0.0510.8740.0600.8690.0191.0000.0220.9180.0690.9960.0850.944Sch.20.5,1(50, 20)0.0940.7800.1160.7810.0231.0000.0271.0000.1470.9680.1930.958(50, 30)0.1170.7910.1460.8140.0231.0000.0270.9780.1670.9490.2200.969(50, 40)0.0970.8550.1130.8720.0221.0000.0260.9630.1320.9680.1600.9491,1.5(50, 20)0.0720.8340.0820.8140.0210.9900.0240.9480.1030.9560.1270.959(50, 30)0.1100.7670.0910.8130.0210.9790.0250.9330.1570.9580.1440.955(50, 40)0.0580.8620.0680.8480.0201.0000.0241.0000.0780.9590.0960.9451.5,2(50, 20)0.0570.8030.0690.8050.0190.9900.0220.9030.0820.9500.1070.942(50, 30)0.0670.7720.0750.7920.0190.9790.0231.0000.1010.9400.1260.965(50, 40)0.0510.8690.0600.8970.0191.0000.0220.8880.0680.9590.0870.949Sch.30.5,1(50, 20)0.1060.8210.1160.7820.0221.0000.0271.0000.1510.9650.1930.958(50, 30)0.0940.7010.1160.7410.0220.9900.0270.9480.1380.9330.1840.940(50, 40)0.0730.8170.0870.8200.0220.9790.0260.9330.0970.9460.1220.9501,1.5(50, 20)0.0680.8040.0760.8270.0210.9670.0240.9180.0970.9530.1220.966(50, 30)0.0920.7180.1080.7690.0210.9560.0250.9020.1350.9360.1740.947(50, 40)0.0660.8430.0760.8430.0201.0000.0241.0000.0870.9510.1090.9521.5,2(50, 20)0.0600.8270.0710.8100.0190.9670.0220.8730.0890.9950.1110.956(50, 30)0.0820.7180.1020.7580.0190.9560.0230.9990.1220.9270.1610.941(50, 40)0.0600.8260.0700.8150.0191.0000.0220.9180.0810.9430.1010.953Table 8
The ACL of90% and 95% CIs and corresponding CP for ht^ML and ht^B.
ht^Bht^MLPrior2Prior290%95%90%95%90%95%T1,T2n,mACLCPACLCPACLCPACLCPACLCPACLCPSch.10.5,1(50, 20)0.4950.7520.5520.7800.0821.0000.0950.9820.4990.9910.5710.978(50, 30)0.4970.7880.5480.7960.0821.0000.0951.0000.4970.9790.5740.970(50, 40)0.3610.8200.4190.8310.0811.0000.0941.0000.3620.9680.4270.9731,1.5(50, 20)0.3250.8060.3640.8370.0781.0000.0901.0000.3150.9710.3520.967(50, 30)0.3610.7430.4280.7810.0711.0000.0820.9700.3550.9930.4280.940(50, 40)0.2690.8290.3090.8560.0761.0000.0881.0000.2620.9730.3020.9521.5,2(50, 20)0.3020.8070.3520.8200.0761.0000.0881.0000.2940.9770.3410.953(50, 30)0.3490.7740.4070.7910.0711.0000.0811.0000.3460.9760.4050.957(50, 40)0.2550.8500.3040.8380.0741.0000.0861.0000.2471.0000.2980.946Sch.20.5,1(50, 20)0.4930.7580.5610.7530.0820.9740.0950.9900.4890.9830.5760.963(50, 30)0.5090.7700.5420.7850.0821.0000.0951.0000.5180.9710.5620.969(50, 40)0.3500.8310.4090.8410.0811.0000.0941.0000.3450.9840.4110.9591,1.5(50, 20)0.3250.8110.3850.7850.0771.0000.0901.0000.3130.9720.3780.969(50, 30)0.4500.7460.4520.7840.0781.0000.0900.9740.4560.9820.4540.963(50, 40)0.2660.8380.3160.8180.0761.0000.0881.0000.2600.9730.3080.9551.5,2(50, 20)0.3300.7810.3670.7760.0761.0000.0871.0000.3290.9720.3590.950(50, 30)0.4060.7500.4670.7640.0760.9880.0880.9800.3990.9730.4730.975(50, 40)0.2600.8450.2980.8650.0741.0000.0861.0000.2510.9720.2930.956Sch.30.5,1(50, 20)0.4370.7990.5590.7540.0820.9740.0951.0150.4410.9820.5750.963(50, 30)0.6960.6820.8000.7140.0820.9240.0960.8910.8090.9520.9690.948(50, 40)0.3650.7940.4230.7910.0811.0220.0941.0150.3670.9630.4380.9621,1.5(50, 20)0.3800.7820.4410.7980.0781.0320.0901.0150.3760.9760.4480.977(50, 30)0.6380.6990.6840.7420.0780.9600.0910.9130.7300.9570.7800.960(50, 40)0.3480.8200.4050.8130.0771.0500.0901.0160.3550.9690.4160.9661.5,2(50, 20)0.3550.8040.4400.7810.0761.0090.0881.0150.3521.0000.4540.968(50, 30)0.6520.6990.7090.7310.0770.9450.0890.9130.7200.9480.8180.951(50, 40)0.3480.8030.4040.7860.0761.0160.0881.0150.3500.9690.4180.964


## 7. Numerical Example

For this example, we use the real data in ([25], p.105) which is consideredas breakdown times in minutes of an insulating fluid between electrodes at voltage 34 kV.Zimmer et al. in [26] indicated that the Burr Type-XII distribution fits these data and used it to obtain the estimates of the Burr Type-XII parameters. Table 9 shows the 19 breakdown times of an insulating fluid between electrodes. We will use these data to consider the following progressively censored schemes.Table 9
The observed breakdown times of an insulating fluid between electrodes.
i12345678910111213141516171819xi0.190.780.961.312.783.164.154.674.856.507.358.018.2712.0631.7532.5233.9136.7172.89For generating generalized Type-II progressive hybrid censored samples, supposem=15, R1=Rm=2; then, we would have the following progressive data: 0.19, 0.78, 1.31, 2.78, 3.16, 4.15, 4.67, 4.85, 6.50, 7.35, 8.01, 8.27, 31.75, 32.52, and 33.91, with 0.96, 12.06, 36.71, and 72.89 being randomly selected and removed from the given data. If we consider different values of T1 and T2, then we have different kinds of generalized Type-II PHCS, namely,(1)
Scheme I: supposeT1=50, T2=80; since X15:15:19<T1, then the experiment would have terminated at T1, with d∗=16,R1=2,Rm=0,Rd1+1′=1,Rd2+1′=0, and we would have the following data: 0.19, 0.78, 1.31, 2.78, 3.16, 4.15, 4.67, 4.85, 6.50, 7.35, 8.01, 8.27, 31.75, 32.52, 33.91, and 36.71.(2)
Scheme II: supposeT1=30, T2=40; since T1<X15:15:19<T2, then the experiment would have terminated at X15:15:19, with d∗=m=15,R1=Rm=2,Rd1+1′=Rd2+1′=0, and we would have the following data: 0.19, 0.78, 1.31, 2.78, 3.16, 4.15, 4.67, 4.85, 6.50, 7.35, 8.01, 8.27, 31.75, 32.52, and 33.91.(3)
Scheme III: supposeT1=20, T2=30; since T2<X15:15:19, then the experiment would have terminated at T2, with d∗=d2=12,R1=2,Rd1+1′=0,Rd2+1=4, and we would have the following data: 0.19, 0.78, 1.31, 2.78, 3.16, 4.15, 4.67, 4.85, 6.50, 7.35, 8.01, and 8.27, with 0.96, 12.06, 36.71, and 72.89 being removed from the data and 31.75, 32.52, and 33.91 being removed after T2.Table10 shows the ML and Bayesian estimates for the unknown parameters λ and γ and survival and hazard functions based on the generalized Type-II PHCS and two different selections Prior1 and Prior2. Table 11 also includes the asymptotic CIs of 90% and 95%, as well as the credible intervals. Table 12 shows the point predictors and 95% Bayesian prediction boundaries of Yi:Rk∗ for three different censoring schemes with Prior1 and Prior2.Table 10
The ML and Bayesian estimates ofλ^, mu^, St^, and ht^ at selected censoring schemes from the real dataset.
BayesianSELFLLFGELFEstimatorSchemeMLPrior1Prior2Prior1Prior2Prior1Prior2λ^10.2770.6770.4550.6700.4460.6450.39720.2660.6310.2660.6260.2630.6080.23330.2451.0390.7281.0210.6950.9890.603γ^11.6920.4400.4490.4390.4440.4310.42221.6770.7121.8660.7081.7560.6971.69131.6410.5851.4210.5821.3390.5681.248Rt=1^10.8250.6300.7360.6290.7330.6230.72620.8320.6490.8340.6480.8330.6440.83030.8440.4950.6220.4930.6170.4820.593ht=1^10.2350.1440.0930.1440.0930.1410.08820.2230.2190.2210.2190.2200.2140.20930.2010.2930.4380.2920.4340.2860.410Table 11
The CIs and ACL forλ^, γ^, St^, and ht^ at selected censoring schemes from real dataset.
BayesianAsymp.CIPrior1Prior2EstimatorSch.90%95%90%95%90%95%λ^1(0.090, 0.465)(0.055, 0.500)(0.424, 0.992)(0.388, 1.037)(0.196, 0.809)(0.184, 0.954)0.3750.4450.5680.6490.6130.7692(0.083, 0.448)(0.049, 0.483)(0.428, 0.880)(0.400, 0.944)(0.121, 0.482)(0.111, 0.547)0.3650.4340.4510.5440.3620.4363(0.067, 0.423)(0.033, 0.457)(0.657, 1.542)(0.602, 1.650)(0.276, 1.446)(0.210, 1.664)0.3570.4240.8851.0481.171.454γ^1(0.714, 2.669)(0.531, 2.852)(0.326, 0.571)(0.311, 0.596)(0.266, 0.686)(0.227, 0.700)1.9542.3220.2450.2850.420.4732(0.698, 2.656)(0.514, 2.840)(0.536, 0.924)(0.505, 0.959)(0.971, 3.294)(0.873, 3.385)1.9582.3260.3880.4542.3232.5123(0.651, 2.632)(0.465, 2.818)(0.406, 0.789)(0.387, 0.841)(0.645, 2.566)(0.554, 3.281)1.9812.3530.3830.4541.9212.727Rt=1^1(0.718, 0.932)(0.698, 0.952)(0.503, 0.745)(0.487, 0.764)(0.570, 0.873)(0.516, 0.880)0.2140.2550.2420.2770.3030.3642(0.726, 0.937)(0.707, 0.957)(0.543, 0.743)(0.520, 0.757)(0.716, 0.920)(0.684, 0.925)0.2110.250.20.2380.2040.2423(0.739, 0.948)(0.720, 0.968)(0.343, 0.634)(0.318, 0.658)(0.366, 0.826)(0.315, 0.864)0.2090.2480.2910.340.460.549ht=1^1(0.137, 0.332)(0.119, 0.350)(0.107, 0.188)(0.100, 0.194)(0.057, 0.136)(0.053, 0.147)0.1950.2310.0810.0940.0790.0942(0.127, 0.318)(0.109, 0.336)(0.162, 0.284)(0.152, 0.296)(0.136, 0.320)(0.122, 0.342)0.1910.2270.1230.1440.1840.223(0.105, 0.298)(0.087, 0.316)(0.213, 0.384)(0.202, 0.401)(0.249, 0.665)(0.223, 0.721)0.1930.2290.1710.1990.4160.498Table 12
Bayesian point predictor and 95% ET and HPD prediction intervals forYi:Rj∗.
Prior2Prior2Sch.jiY^i:Rj∗ET intervalHPD intervalY^i:Rj∗ET intervalHPD interval1115.958(1.462, 23.953)(1.336, 18.976)7.137(1.465, 25.126)(1.336, 19.782)22.49117.6423.66118.44629.780(5.901, 28.392)(5.775, 23.415)11.576(5.904, 29.565)(5.775, 24.220)22.49117.6423.66118.446d1+1113.267(8.339, 31.639)(7.319, 27.378)15.852(8.339, 34.000)(7.269, 28.999)23.29920.05825.66121.732118.914(6.880, 19.345)(6.533, 16.794)10.546(6.884, 20.413)(6.529, 17.515)12.46510.26113.5310.986216.993(7.195, 51.064)(5.917, 42.481)20.278(7.218, 53.677)(5.910, 44.332)43.86836.56446.45938.42215110.922(7.499, 25.069)(6.800, 21.527)12.995(7.503, 26.732)(6.678, 22.673)17.5714.72619.22915.995227.598(15.672, 94.192)(13.232, 60.735)33.467(15.717, 73.947)(13.176, 63.447)78.5247.50358.2350.2713119.945(5.907, 28.950)(5.775, 23.927)11.767(5.910, 30.059)(5.775, 24.706)23.04318.15224.14918.931213.583(2.829, 47.559)(1.492, 38.983)16.315(2.857, 49.985)(1.486, 40.743)44.7337.49147.12939.257d2+1112.867(2.659, 47.155)(1.451, 37.949)15.622(2.672, 50.932)(1.439, 40.491)44.49636.49848.2639.052220.884(13.339, 49.105)(13.315, 42.236)24.765(13.367, 51.445)(11.942, 43.930)35.76628.92138.07831.988332.324(10.856, 100.422)(7.273, 82.897)38.728(10.940, 105.534)(7.342, 87.613)89.56575.62494.59480.271440.942(6.162, 70.194)(5.718, 76.662)33.243(6.221, 99.726)(5.750, 80.644)64.03370.94493.50574.895


## 8. Conclusions

When the observed sample is a generalized Type-II PHCS sample, Bayesian and ML estimates of the unknown parameters, as well as the survival and hazard functions of the Burr Type-XII distribution, are produced. Squared error, LINEX, and general entropy loss functions based onPrior1 and Prior2 distributions are considered in the Bayesian method. The parameters, as well as the survival and hazard functions, are given asymptotic and credible CIs of 90% and 95%. In the generalized Type-II PHCS, Bayesian point and interval prediction of non-observed failures were also established. The following conclusions can be drawn from the numerical results:(1)
In most cases, the Bayesian estimates withPrior1 are better than the MLEs.(2)
The results of the estimates from ML that can be seen in Tables1–4 are similar to the Bayesian estimators using Prior2. Thus, when we have no prior knowledge about the unknown parameters, it is often easier to use the ML instead of the Bayesian estimators because the computation of the Bayesian estimators is more complicated.(3)
In most cases, the MSE increases asm decreases.(4)
The ACL of the CIs increases asT1 and T2 decrease.(5)
The credible intervals perform well compared to the asymptotic CIs.(6)
In all cases, the ACL of95% CIs is larger than that of 90% CIs.(7)
The HPD prediction intervals appear to be more accurate than the ET prediction intervals.



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*Source: 1241303-2022-02-18.xml*