# Classical and Bayesian Inference Using Type-II Unified Progressive Hybrid Censored Samples for Pareto Model

Authors: M. Nagy; Adel Fahad Alrasheedi
Journal: Applied Bionics and Biomechanics
 (2022)

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## Abstract

In the lifetime and reliability experiments, the censored samples play a fundamental and important role in order to control time and cost. The researchers developed the censored sample schemes to solve the problems that arise by applying the previous methods. Recently, Górny and Cramer (2018) proposed a new general type of censored sample called Type-II unified progressive hybrid censored sample. In this paper, we present an overview of the Type-II unified progressive hybrid censored sample. We used this censored sample to compute the maximum likelihood estimates of unknown parameters from the Pareto distribution, as well as Bayesian estimates for unknown parameters under three different error loss functions. The point and interval Bayesian predictions one- and two-sample Bayesian predictions from the Pareto distribution are shown. Simulation studies are carried out to compare the efficacy of the various inference approaches. Finally, real data sets are examined to determine the applicability of the proposed model and various estimating approaches.


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## Body

## 1. Introduction

In order to time and expense constraints, experiments in reliability analysis frequently end before all units in the test have failed. In such circumstances, failure information is only accessible for a portion of the sample, and only limited information is given on all units that have not failed. Data that has been censored is referred to as censored data. There are several different censoring schemes such as Type-I and Type-II. Since Epstein [1] presented Type-I hybrid censoring, various hybrid censoring modifications have been developed to address the model’s flaws. Due to the fact that Type-I hybrid censoring does not guarantee the observation of at least one of the failures, Childs et al. [2] developed Type-II hybrid censoring, which ensures the observation of at least m failures from the n units put on the life test. However, the main disadvantage of this censoring system is that the experimenter has not controlled the test time. The disadvantages of both Type-I and Type-II hybrid censoring are mitigated by Chandrasekar et al. [3]. In addition, the unified hybrid censoring methods are even more flexible than hybrid censoring techniques (see, e.g., Balakrishnan et al. [4]; Huang and Yang [5]; Park and Balakrishnan [6]). In unified hybrid censoring method, consider, n identical units are placed on a life-testing device. Fix the integers k, m∈1,2,⋯,n, and T1 and T2∈0,∞ such that k<m and T1<T2. The experiment is stopped at minmaxT1,Ym:n,T2 if the kth failure occurs before time T1. Otherwise, the experiment is stopped at minmaxYk:n,T2,Ym:n%. We can guarantee that the experiment will be completed at most in time T2 with at least k failures, and if not, we can guarantee exactly k failures under this censoring strategy.If one of these units is inadvertently broken but the experiment has not yet been terminated, this unit must be removed from the life test, and the progressive censoring methodology is the best method for this case. Complete failures ofm units will be observed in Type-II progressive censoring methods. When the first failure occurs, R1 of the n−1 remaining units is chosen at random and removed from the lifetime test. R2 of the n−R1−2 surviving units is randomly selected and eliminated at the second observed failure. Finally, Rm surviving units are removed after the mth failure, and the experiment comes to an end. We will denote the m ordered failure times thus observed by Y1:m:n,⋯,Ym:m:n. It is evident that n=m+∑k=1mRk.The downsides of the Type-II progressive censoring system are that if the units are highly reliable, the experiment can take a long time. Therefore, Kundu and Joarder [7] and Childs et al. [8] proposed a progressive hybrid censoring scheme (PHCS) in which the life-testing experiment is ended at time minYm:m:n,T, with T∈0,∞. For more details, we refer our readers to Tomer and Panwar [9], Panahi [10], Almarashi et al. [11], and Moihe El-Din et al. [12, 13]. On the other hand, the disadvantage of the PHCS is that it cannot be applied when only a few failures are likely to occur before time T. For this reason, Cho et al. [14] proposed a Type-I generalized PHCS in which the life-testing experiment is terminated at the time minmaxT,Yk:m:n,Ym:m:n for prefixed k<m1,2,⋯,n. Moreover, Lee et al. [15] proposed Type-II generalized PHCS, in which the life-testing experiment is terminated at time minmaxT1,Ym:m:n,T2 for prefixed T1<T20,∞. For recent work on this topic, see, for example, Moihe El-Din and Nagy [16], Nagy et al. [17, 18], and Nagy and Alrasheedi [19].While generalized PHCS are superior to Type-I and Type-II PHSC, they do have significant disadvantages. Therefore, Górny and Cramer [20] developed a general type of generalized PHCS, called Type-II unified PHCS to address some of the shortcomings of these schemes. Under Type-II unified PHCS, we can guarantee that the lifetime experiment will be completed at no later than T2 with at least k number of unit failures; this ensures that the statistical inference is carried out with more efficiency. For recent work on the Type-II unified PHCS, see, for example, Górny and Cramer in [21] and Kim and Lee in [22].The following is how the rest of the article is structured: Section2 provides an overview of the Type-II unified PHCS. Section 3 determines the maximum likelihood estimates (ML) of unknown parameters, while Section 4 derives the Bayesian estimates for the unknown parameters with three loss functions. Sections 5 and 6 calculate the point and interval Bayesian predictions for one- and two-sample Bayesian predictions, respectively. Simulation studies are carried out in Section 7 to compare the efficacy of the offered inference methodologies. A real data is utilized to demonstrate the theoretical findings in Section 8. Finally, the paper is concluded in Section 9.


## 2. The Type-II Unified PHCS and Likelihood Function

Consider a life test in whichn identical items are put on test. Then, the Type-II unified PHCS may be described as follows. Let T1,T2∈0,∞ and integer k,m∈1,2,⋯,n are prefixed such that T1<T2 and k<m with R=R1,R2,⋯,Rm is also prefixed integers satisfying n=m+R1+⋯+Rm. At the time of first failure, R1 of the remaining units are randomly removed. Similarly, at the time of the second failure R2, of the remaining units are removed and so on. If the kth failure occurs before time T1, the experiment is terminated at minmaxYm:m:n,T1,T2. If the kth failure occurs between T1 and T2, the experiment is terminated at minYrm:m:n,T2 and if the kth failure occurs after time T2, the experiment is terminated at Yk:n. Under this censoring scheme, we can guarantee that the experiment would be completed at most in time T2 with at least k failure and if not, we can guarantee exactly k failures. Let D1 and D2 denote the numbers of observed failures up to time T1 and T2, respectively. In addition, d1 and d2 are the observed values of D1 and D2, respectively.Under the UPHCS described above, we have one of the following types of observations:(1)
If thekth failure occurs before time T1, the experiment is terminated at minmaxYm:m:n,T1,T2 and then we have the following three subcases:(a)
If themth failure occurs before T1, i.e., 0<Yk:m:n<Ym:m:n<T1<T2, 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 the experiment end at time T∗=T1. Therefore, the observed failure times are Y1:m:n<⋯<Yk:m:n<⋯<Ym:m:n<⋯<Yd1:n(b)
If themth failure occurs between T1 and T2, i.e., 0<Yk:m:n<T1<Ym:m:n<T2, then the experiment will end at T∗=Ym:m:n and the observed failure times are Y1:m:n<⋯<Yk:m:n<⋯<Yd1:m:n<⋯<Ym:m:n(c)
If themth failure occurs after T2, i.e., 0<Yk:m:n<T1<T2<Ym:m:n, then the experiment will end at T∗=T2 and the observed failure times are Y1:m:n<⋯<Yk:m:n<⋯<Yd1:m:n<⋯<Yd2:m:n(2)
If the timeT1 pass before the kth, then the experiment will end at minmaxYk:m:n,T2,Ym:m:n and then we have the following three subcases:
(a)
IfT2 passes before the kth failure occurs, i.e., 0<T1<T2<Yk:m:n<Ym:m:n, then the experiment will end at T∗=Yk:m:n and we will observe Y1:m:n<⋯<Yd1:m:n<⋯<Yd2:m:n<⋯<Yk:m:n(b)
If themth failure occurs before T2, i.e., 0<T1<Yk:m:n<Ym:m:n<T2, then the experiment will end at T∗=Ym:m:n and we will observe Y1:m:n<⋯<Yd1:m:n<⋯<Yk:m:n<⋯<Ym:m:n(c)
If the timeT2 between Yk:m:n and Ym:m:n, i.e., 0<T1<Yk:m:n<T2<Ym:m:n, then the experiment will end at T∗=T2 and the observed failure times are Y1:m:n<⋯<Yd1:m:n<⋯<Yk:m:n<⋯<Yd2:m:nLetY¯ be the Type-II unified progressive hybrid censored sample from distribution with the probability density function (PDF) gy, and the cumulative distribution function (CDF) Gy, then, based on the Type-II unified PHCS, the likelihood function is given by
(1)LY¯Y¯=∏i=1d1∑j=1mR~j+1∏i=1d1gyi:m:nG¯yi:m:nR~iG¯T1R~t1inCase1a,∏i=1m∑j=1mR~j+1∏i=1mgyi:m:nG¯yi:m:nR~iinCase1b,∏i=1d2∑j=1mR~j+1∏i=1d2gyi:m:nG¯yi:m:nR~iG¯T2R~t2inCase1c,∏i=1k∑j=1mR~j+1∏i=1kgyi:m:nG¯yi:m:nR~iinCase2a,∏i=1m∑j=1mR~j+1∏i=1mgyi:m:nG¯yi:m:nR~iinCase2b,∏i=1d2∑j=1mR~j+1∏i=1d2gyi:m:nG¯yi:m:nR~iG¯T2R~t2inCase2c,Therefore, these cases can be combined and obtained as
(2)Lθ∣Y¯∏i=1d∗∑j=1mR~j+1∏i=1d∗gyi:m:nG¯yi:m:nR~iG¯T1R~t1G¯T2R~t2,where G¯=1−G and
(3)Y¯=y1:m:n,⋯,yk:m:n,⋯,ym−1:m:n,ym:m:n,⋯,yd1:ninCase1a,y1:m:n,⋯,yk:m:n,⋯,yd1:m:n,⋯,ym:m:ninCase1b,y1:m:n,⋯,yk:m:n,⋯,yd1:m:n,⋯,yd2:m:ninCases1c,y1:m:n,⋯,yd1:m:n,⋯,yd2:m:n,⋯,yk:m:ninCase2a,y1:m:n,⋯,yd1,⋯,yk:m:n,⋯,ym:m:ninCase2b,y1:m:n,⋯,yd1:m:n,⋯,yk:m:n,⋯,yd2:m:ninCases2c,d∗=d1inCase1a,minCases1band2b,d2inCases1cand2c,kinCase2a,R~=R1,⋯,Rk,⋯,Rm−1,0,⋯,0,Rt1inCase1a,R1,⋯,Rk,⋯,Rd1,⋯,RminCase1b,R1,⋯,Rk,⋯,Rd1,⋯,Rt2inCases1c,R1,⋯,Rd1,⋯,Rd2,⋯,Rk∗inCase2a,R1,⋯,Rd1,⋯,Rk,⋯,RminCase2b,R1,⋯,Rd1,⋯,Rk,⋯,Rt2inCases2c,with R~k∗=n−k−∑j=1k−1R~j,R~t1 is the number of surviving units that are eliminated at T1, given by
(4)R~t1=n−d1−∑j=1m−1R~jinCase1a,0inallothercases,and R~t2 is the number of surviving units that are eliminated at T2, given by
(5)R~t2=n−d2−∑j=1d2R~jinCases1cand2c,0inallothercases.Special cases: The Type-II unified PHCS is a generalization of many censoring schemes, for example:(1)
IfRi=0 for all i<m and Rm=n−m, the Type-II unified PHCS becomes unified HCS(2)
IfT2=∞, the Type-II unified PHCS becomes generalized Type-I PHCS(3)
Ifk=m, the Type-II unified PHCS becomes generalized Type-II PHCS(4)
IfT1=0 and k=m, the Type-II unified PHCS becomes Type-I PHCS(5)
IfT2=∞ and k=0, the Type-II unified PHCS becomes Type-II PHCSNote: In order for the experiment to be terminated at time T1, Rm must be not equal to zero; if Rm is equal to zero and the mth failure occurs before T1, then the experiment is terminated at Ym:m:n.


## 3. The ML Estimation

In this section, we derive the ML inference of the unknown parametersλ and θ for the Pareto distribution which was introduced by Pareto [23] as a model for the distribution of income, based on the Type-II unified PHCS. Using the exponential form, Pareto distribution has the following density function (PDF) and distribution function (CDF), respectively, given by
(6)gy∣λ,θ=λyexp−λlnyθ,λ,θ>0,y≥θ,(7)Gy∣λ,θ=1−exp−λlny%θ,λ,θ>0,y≥θ.%.From (7), (6), and (2), the likelihood function of λ,θ under the Type-II unified PHCS can be derived as
(8)Lλ,θ∣Y¯=∏i=1d∗∑j=imR~j+1λd∗∏d∗i=11yiexp−ληy¯+R~t1lnT1+R~t2lnT2−nlnθ,where ηy¯=∑i=1d∗R~i+1lnyi, and yi=yi:d∗:n for simplicity of notation.Since the likelihood function (8) is an increasing function in θ, but θ is the lower bound of yi for all yi∈Y¯, so its maximum value will be attained at the maximum value y1 of θ. From (8), the log-likelihood function of λ,θ is given by
(9)lnLλ,θ∣Y¯∝d∗lnλ−λη%y¯+R~t1lnT1+R~t2lnT2−nlnθ.To maximize relative toλ, differentiate (9) with respect to λ and solve the equation
(10)∂lnLλ,θ∣Y¯∂λ=0,so the ML estimator λ^ML of λ is obtained as
(11)λ^ML=d∗ηy¯+R~t1lnT1+R~t2lnT2−nlny1:n.


### 3.1. Approximate Confidence Intervals forλ and θ

For larged∗, the observed Fisher information matrix of the parameters λ and θ is given by
(12)Iλ^,θ^=−∂2lnLλ,θ∣Y¯∂λ2−∂2lnLλ,θ∣Y¯∂λ∂θ−∂2lnLλ,θ∣Y¯∂θ∂λ−∂2lnLλ,θ∣Y¯∂θ2λ^ML,θ^ML,where
(13)∂2lnLλ,θ∣Y¯∂λ2=−d∗λ2,∂2lnLλ,θ∣Y¯∂θ2=−nλθ2,∂2lnLλ,θ∣Y¯∂λ∂θ=−nθ,and a 1001−α% two-sided approximate confidence intervals for the parameters λ and θ are then
(14)λ^−zα/2Vλ^,λ^+zα/2Vλ^,θ^−zα/2Vθ^,θ^+zα/2Vθ^,respectively, where Vλ^ and Vθ^ are the estimated variances of λ^ML and θ^ML, which are given by the first and the second diagonal element of I−1λ^,θ^, and zα/2 is the upper α/2 percentile of the standard normal distribution.


## 3.1. Approximate Confidence Intervals forλ and θ

For larged∗, the observed Fisher information matrix of the parameters λ and θ is given by
(12)Iλ^,θ^=−∂2lnLλ,θ∣Y¯∂λ2−∂2lnLλ,θ∣Y¯∂λ∂θ−∂2lnLλ,θ∣Y¯∂θ∂λ−∂2lnLλ,θ∣Y¯∂θ2λ^ML,θ^ML,where
(13)∂2lnLλ,θ∣Y¯∂λ2=−d∗λ2,∂2lnLλ,θ∣Y¯∂θ2=−nλθ2,∂2lnLλ,θ∣Y¯∂λ∂θ=−nθ,and a 1001−α% two-sided approximate confidence intervals for the parameters λ and θ are then
(14)λ^−zα/2Vλ^,λ^+zα/2Vλ^,θ^−zα/2Vθ^,θ^+zα/2Vθ^,respectively, where Vλ^ and Vθ^ are the estimated variances of λ^ML and θ^ML, which are given by the first and the second diagonal element of I−1λ^,θ^, and zα/2 is the upper α/2 percentile of the standard normal distribution.


## 4. Bayesian Estimation

In this study, we investigate three forms of loss functions for Bayesian estimation. The first is the squared error loss function (SELF), which is a symmetric function that values overestimation and underestimation equally when estimating parameters. The LINEX loss function (LLF), which is asymmetric and offers different weights due to overestimation and underestimation, is the second option. The generalization of the entropy loss function is the third loss function (GELF).Under the assumption that both parametersλ and θ are unknown, we can use the joint prior density function of λ and θ proposed by Lwin [24] and generalized by Arnold and Press [25] for Bayesian Estimations. The generalized Lwin prior is given by
(15)πλ,θ∝λa1θexp−λlna2−b1lnθ,λ>0,0<θ<d,where a1,b1,a2,b2 are positive constants and b2b1<a2.Upon combining (8) and (15), given UPHCS, the posterior density function of λ,θ is obtained as
(16)π∗λ,θ∣Y¯=Lλ,θ∣Y¯πλ,θ∫0∞Lλ,θ∣Y¯πλ,θdλdθ=I−1λd∗+a1θ−1exp−ληy¯+R~t1lnT1+R~t2lnT2−n+b1lnθ+lna2,where
(17)I=∫0δ∫0∞λd∗+a1θ−1exp−ληy¯+R~t1lnT1+R~t2lnT2−n+b1lnθ+lna2dλdθ=Γd∗+a1n+b1ηy¯+R~t1lnT1+R~t2lnT2−n+b1lnδ+lna2−d∗+a1,withδ=miny1:n,b2.


### 4.1. The Bayesian Estimation under SELF

A commonly used loss function is the squared error loss function (SELF) defined as follows:(18)LBSβ^,β∝β∧−β2The Bayesian estimateβ^BS for the unknown parameter β, relative to the squared error loss function, is given by
(19)β^BS=Eπ∗βBy using (16), the Bayesian estimator of λ under the squared error loss function is the mean of the posterior density function, given by
(20)λ^BS=∫0δ∫0∞λπ∗λ,θ∣Y¯dλdθ.Hence, the Bayesian estimator ofλ under the squared error loss function is obtained as
(21)λ^BS=d∗+a1ηy¯+R~t1lnT1+R~t2lnT2−n+b1lnδ+lna2,and the Bayesian estimator of θ under the squared error loss function is obtained as
(22)θ^BS=∫0δ∫0∞θπ∗λ,θ∣Y¯dλdθ=I−1δ∫0∞λd∗+a1λn+b1+1exp−ληy¯+R~t1lnT1+R~t2lnT2−n+b1lnδ+lna2dλ=I−1δn+b1ηy¯+R~t1lnT1+R~t2lnT2−n+b1lnδ+lna2−d∗+a1×∫0∞td∗+a1e−tt+ηy¯+R~t1lnT1+R~t2lnT2−n+b1lnδ+lna2/n+b1dt=δΓd∗+a1Φd∗+a1,ηy¯+R~t1lnT1+R~t2lnT2−n+b1lnδ+lna2n+b1,where
(23)Φy,y=∫0∞tye−tt+ydt.A partial tabulation ofψy,y=y/ΓyΦy−1,y has been provided by Arnold and Press in [25].


### 4.2. The Bayesian Estimation under GELF

Another commonly used asymmetric loss function is the general entropy (GE) loss function given by(24)LBEβ^,β∝β∧βω−ωlnβ^β−1.Forω>0, a positive error has a more serious effect than a negative error, and for ω<0, a negative error has a more serious effect than a positive error. In this case, the Bayesian estimate β^BE relative to the GE loss function is given by
(25)θ^BE=Eπ∗β−ω−1ω,provided that the involved expectation Eπ∗β−ω is finite. It can be shown that, when ω=1, the Bayesian estimate in Eq. (25) coincides with the Bayesian estimate under the weighted squared error loss function. Similarly, when ω=−1, the Bayesian estimate in Eq. (25) coincides with the Bayesian estimate under the SE loss function.By using (16), the Bayesian estimator of λ under GELF is given by
(26)λ^BE=∫0δ∫0∞λ−ωπ∗λ,θ∣Y¯dλdθ−1ω=Γd∗+a1−ωηy¯+R~t1lnT1+R~t2lnT2−n+b1lnδ+lna2d∗+a1Γd∗+a1ηy¯+R~t1lnT1+R~t2lnT2−n+b1lnδ+lna2+εd∗+a1−ω−1ωand the Bayesian estimator of θ under GEF is obtained as
(27)θ^BE=∫0δ∫0∞θ−ωπ∗λ,θ∣Y¯dλdθ−1ω=I−1∫0δΓd∗+a1+1θ1−ωηy¯+R~t1lnT1+R~t2lnT2−n+b1lnθ+lna2d∗+a1+1dθ.


### 4.3. The Bayesian Estimation under LLF

Under the assumption that the minimal loss occurs atβ^=β, the LINEX loss function can be expressed as
(28)LBLβ^,β=expεβ^−β−εβ^−β−1where ε≠0. The sign and magnitude of the shape parameter v represent the direction and degree of asymmetry, respectively. It is easily seen the (unique) Bayesian estimator of θ, denoted by θ^Lunder the LINEX loss function, and the valueβ^Lwhich minimizesEπ∗LLβ^,βis given by
(29)β^BL=−1εlnEπ∗exp−υβ,provided that the involved expectation Eπ∗exp−υβ is finite.By using (16), the Bayesian estimator of λ under LLF is given by
(30)λ^BL=−1εln∫0δ∫0∞exp−υλπ∗λ,θ∣Y¯dλdθ=−1εlnηy¯+R~t1lnT1+R~t2lnT2−n+b1lnδ+lna2d∗+a1ηy¯+R~t1lnT1+R~t2lnT2−n+b1lnδ+lna2+εd∗+a1,and the Bayesian estimator of θ under LLF is obtained as
(31)θ^BL=−1εln∫0δ∫0∞exp−υθπ∗λ,θ∣Y¯dλdθ=−1εlnI−1∫0δΓd∗+a1+1θexp−υθ×ηy¯+R~t1lnT1+R~t2lnT2−n+b1lnθ+lna2d∗+a1+1dθ.


## 4.1. The Bayesian Estimation under SELF

A commonly used loss function is the squared error loss function (SELF) defined as follows:(18)LBSβ^,β∝β∧−β2The Bayesian estimateβ^BS for the unknown parameter β, relative to the squared error loss function, is given by
(19)β^BS=Eπ∗βBy using (16), the Bayesian estimator of λ under the squared error loss function is the mean of the posterior density function, given by
(20)λ^BS=∫0δ∫0∞λπ∗λ,θ∣Y¯dλdθ.Hence, the Bayesian estimator ofλ under the squared error loss function is obtained as
(21)λ^BS=d∗+a1ηy¯+R~t1lnT1+R~t2lnT2−n+b1lnδ+lna2,and the Bayesian estimator of θ under the squared error loss function is obtained as
(22)θ^BS=∫0δ∫0∞θπ∗λ,θ∣Y¯dλdθ=I−1δ∫0∞λd∗+a1λn+b1+1exp−ληy¯+R~t1lnT1+R~t2lnT2−n+b1lnδ+lna2dλ=I−1δn+b1ηy¯+R~t1lnT1+R~t2lnT2−n+b1lnδ+lna2−d∗+a1×∫0∞td∗+a1e−tt+ηy¯+R~t1lnT1+R~t2lnT2−n+b1lnδ+lna2/n+b1dt=δΓd∗+a1Φd∗+a1,ηy¯+R~t1lnT1+R~t2lnT2−n+b1lnδ+lna2n+b1,where
(23)Φy,y=∫0∞tye−tt+ydt.A partial tabulation ofψy,y=y/ΓyΦy−1,y has been provided by Arnold and Press in [25].


## 4.2. The Bayesian Estimation under GELF

Another commonly used asymmetric loss function is the general entropy (GE) loss function given by(24)LBEβ^,β∝β∧βω−ωlnβ^β−1.Forω>0, a positive error has a more serious effect than a negative error, and for ω<0, a negative error has a more serious effect than a positive error. In this case, the Bayesian estimate β^BE relative to the GE loss function is given by
(25)θ^BE=Eπ∗β−ω−1ω,provided that the involved expectation Eπ∗β−ω is finite. It can be shown that, when ω=1, the Bayesian estimate in Eq. (25) coincides with the Bayesian estimate under the weighted squared error loss function. Similarly, when ω=−1, the Bayesian estimate in Eq. (25) coincides with the Bayesian estimate under the SE loss function.By using (16), the Bayesian estimator of λ under GELF is given by
(26)λ^BE=∫0δ∫0∞λ−ωπ∗λ,θ∣Y¯dλdθ−1ω=Γd∗+a1−ωηy¯+R~t1lnT1+R~t2lnT2−n+b1lnδ+lna2d∗+a1Γd∗+a1ηy¯+R~t1lnT1+R~t2lnT2−n+b1lnδ+lna2+εd∗+a1−ω−1ωand the Bayesian estimator of θ under GEF is obtained as
(27)θ^BE=∫0δ∫0∞θ−ωπ∗λ,θ∣Y¯dλdθ−1ω=I−1∫0δΓd∗+a1+1θ1−ωηy¯+R~t1lnT1+R~t2lnT2−n+b1lnθ+lna2d∗+a1+1dθ.


## 4.3. The Bayesian Estimation under LLF

Under the assumption that the minimal loss occurs atβ^=β, the LINEX loss function can be expressed as
(28)LBLβ^,β=expεβ^−β−εβ^−β−1where ε≠0. The sign and magnitude of the shape parameter v represent the direction and degree of asymmetry, respectively. It is easily seen the (unique) Bayesian estimator of θ, denoted by θ^Lunder the LINEX loss function, and the valueβ^Lwhich minimizesEπ∗LLβ^,βis given by
(29)β^BL=−1εlnEπ∗exp−υβ,provided that the involved expectation Eπ∗exp−υβ is finite.By using (16), the Bayesian estimator of λ under LLF is given by
(30)λ^BL=−1εln∫0δ∫0∞exp−υλπ∗λ,θ∣Y¯dλdθ=−1εlnηy¯+R~t1lnT1+R~t2lnT2−n+b1lnδ+lna2d∗+a1ηy¯+R~t1lnT1+R~t2lnT2−n+b1lnδ+lna2+εd∗+a1,and the Bayesian estimator of θ under LLF is obtained as
(31)θ^BL=−1εln∫0δ∫0∞exp−υθπ∗λ,θ∣Y¯dλdθ=−1εlnI−1∫0δΓd∗+a1+1θexp−υθ×ηy¯+R~t1lnT1+R~t2lnT2−n+b1lnθ+lna2d∗+a1+1dθ.


## 5. One-Sample Bayesian Prediction

Forq=1,2,⋯,R~j, let Yq:R~j denote the qth order statistic out of R~j removed units at stage j. Then, the conditional density function of Yq:R~j, given the observed Type-II unified PHCS, is given, see Basak et al. [26], by
(32)gYq:R~j∣Y¯=gy∣Y¯=R~j!q−1!R~j−q!Gy−Gyjq−11−GyR~j−qgy1−GyjR~j,y>yj,where
(33)j=1,⋯,k,⋯,m−1,t1inCase1a,1,⋯,k,⋯,d1,⋯,minCase1b,1,⋯,k,⋯,d1,⋯,t2inCases1c,1,⋯,d1,⋯,d1,⋯,k∗inCase2a,1,⋯,d1,⋯,k,⋯,minCase2b,1,⋯,d1,⋯,k,⋯,t2inCases2c,with yt1=T1 and yt2=T2.By using (6) and (7) in (32), given Type-II unified PHCS, the conditional density function of Yq:R~j is then given as follows:
(34)gy∣Y¯=∑h=0q−1Chλyexp−λϖhlny−lnyj,y>yj,where Ch=−1hq−1hR~j!/q−1!R~j−q! and ϖh=h+R~j−q+1 for h=0,⋯,q−1..Upon combining (16) and (34), the Bayesian predictive density function of Yq:R~j, given UPHCS, is obtained as
(35)y∣Y¯=I−1∑h=0q−1Ch∫0δ∫0∞λd∗+a1+1θyexp−ληy¯+R~t1lnT1+R~t2lnT2−n+b1lnθ+lna2×exp−λϖhlny−lnyjdλdθ=I−1Γd∗+a1+1n+b1∑h=0q−1Chyηy¯+R~t1lnT1+R~t2lnT2−n+b1lnδ+lna2+ϖhlny−lnyj−d∗+a1+1.The Bayesian predictive survival function ofYq:R~j, given Type-II unified PHCS, is given as
(36)G¯∗t∣Y¯=∫t∞g∗y∣Y¯dy=I−1Γd∗+a1n+b1∑h=0q−1Chϖhηy¯+R~t1lnT1+R~t2lnT2−n+b1lnδ+lna2+ϖhlnt−lnyj−d∗+a1.The Bayesian point predictor ofY under the squared error loss function is the mean of the predictive density, given by
(37)Y^q:R~j=∫0∞yf∗y∣Y¯dy,where g∗y∣Y¯ is given as in (35). The Bayesian predictive bounds of 1001−α% two-sided equi-tailed (ET) interval for Ys:n can be obtained by solving the following two equations:
(38)G¯∗LET∣Y¯=α2andG¯∗UET∣Y¯=1−α2,where G¯∗t∣Y¯ is given as in (36), and LET and UET denote the lower and upper bounds, respectively.


## 6. Two-Sample Bayesian Prediction

LetY1:ℓ:m≤Y2:ℓ:N≤⋯≤Yℓ:ℓ:N be a future independent progressive Type-II censored sample from the same population with censoring scheme S¯=S1,⋯,Sℓ. In this section, we develop a general procedure for deriving the point and interval predictions for Ys:ℓ:N, 1≤s≤ℓ, based on the observed UPHCS. The marginal density function of Ys:ℓ:N is given by Balakrishnan et al. [27] as
(39)gYs:ℓ:Nys∣θ=CN,s∑h=0s−1ch,s−11−GysWh,s−1gys,where 1≤s≤q,CN,s=NN−S1−1⋯N−S1⋯−Ss−1+1,Wh,s=N−S1−⋯−Ss−h−1−s+h+1,and ch,s−1=−1h∏u=1h∑%ε=s−hs−h+u−1Sε+1∏u=1s−h−1∑ε=us−h−1Sε+1−1..Upon substituting (7) and (6) in (39), the marginal density function of Ys:ℓ:N is then obtained as
(40)gYs:ℓ:Nys∣θ=CN,s∑h=0s−1ch,s−1λysexp−λWh,slnys%θ,ys>0.Upon combining (16) and (39), given UPHCS, the Bayesian predictive density function of Ys:ℓ:N is obtained as
(41)gYs:ℓ:N∗ys∣Y¯=g1Ys:ℓ:N∗ys∣Y¯,0<ys≤δ,g2Ys:ℓ:N∗ys∣Y¯,ys>δ,where
(42)g1Ys:ℓ:N∗ys∣Y¯=∫0ys∫0∞gYs:ℓ:Nys∣Y¯π∗λ,θ∣Y¯dλdθ=I−1Γd∗+a1+1CN,s∑h=0s−1ch,s−1%n+b1+Wh,sys×ηy¯+R~t1lnT1+R~t2lnT2−n+b1lnys+lna2−d∗+a1+1,g2Ys:ℓ:N∗ys∣Y¯=∫0δ∫0∞gYs:ℓ:Nys∣Y¯π∗λ,θ∣Y¯dλdθ=I−1Γd∗+a1+1CN,s∑h=0q−1ch,s−1n+b1+Wh,sys×ηy¯+R~t1lnT1+R~t2lnT2−n+b1+Wh,slnδ+Wh,slnys+lna2−d∗+a1+1.From (41), we simply obtain the predictive survival function of Ys:ℓ:N, given UPHCS, as
(43)G¯Ys:ℓ:N∗t∣Y¯=∫t∞g∗ys∣Y¯dys=G¯1Ys:ℓ:N∗t∣Y¯,0<t≤δ,G¯2Ys:ℓ:N∗t∣Y¯,t>δ,where
(44)G¯1Ys:ℓ:N∗t∣Y¯=∫0tδ∞g1Ys:ℓ:N∗ys∣Y¯dys+∫0δ∞∞g2Ys:ℓ:N∗ys∣Y¯dys=I−1Γd∗+a1CN,s∑h=0s−1ch,s−1n+b1n+b1+Wh,sWh,s×n+b1+Wh,sηy¯+R~t1lnT1+R~t2lnT2−n+b1lnδ+lna2−d∗+a1−Wh,sηy¯+R~t1lnT1+R~t2lnT2−n+b1lnt+lna2−d∗+a1G¯2Ys:ℓ:N∗t∣Y¯=∫0t∞∞g2Ys:ℓ:N∗ys∣Y¯dys=I−1Γd∗+a1CN,s∑h=0s−1ch,s−1Wh,sn+b1+Wh,s×ηy¯+R~t1lnT1+R~t2lnT2−n+b1+Wh,slnδ+Wh,slnt+lna2−d∗+a1.The Bayesian point predictor ofYs:ℓ:N, 1≤s≤m, under the squared error loss function is the mean of the predictive density, given by
(45)Y^s:ℓ:N=∫0∞ysgYsℓ:N∗ys∣Y¯dys,where gYs:ℓ:N∗ys∣Y¯ is given as in (41).The Bayesian predictive bounds of1001−α% ET interval for Ys:ℓ:N, 1≤s≤m, can be obtained by solving the following two equations:
(46)G¯Ys:ℓ:N∗LET∣Y¯=α2andG¯Ys:ℓ:N∗UET∣Y¯=1−α2,where G¯Ys:ℓ:N∗t∣Y¯ is given as in (43), and LET and UET denote the lower and upper bounds, respectively.


## 7. Simulation Study

In this section, we present a simulation study to compare the performance of the classical ML and Bayesian estimation procedures under different Type-II unified PHCS. Extensive computations were performed using the statistical software maple.Firstly, we show how we generate Type-II unified PHC data from Pareto distribution. For given values ofn, m, T1, T2, and R=R1,⋯,Rm. We will use the transformation which was suggested by Balakrishnan and Aggarwala in [28] to generate Type-II progressive censored data from Pareto distribution. Let the generated Type-II PC data is y1,m,n,y2,m,n,⋯,ym,m,n, if ym,mn<T1, we set Rm=0 and use the transformation which was suggested by Ng et al. in [29] to generate Rm order statistics from left truncated Pareto distribution with truncated value ym,m,n. Now, we m Type-II progressive censored data and Rm order statistics as the following y1,m,n,y2,m,n,⋯,ym,m,n,ym+1,n,ym+Rm,n. Then, we determined the termination time of the experiment and the corresponding observed Type-II unified PHC data as shown in Section 2.We simulate Type-II unified PHCS for different combinations for a sample of sizen=50, with different values of m=2k, and T2=2T1 from the Pareto distribution. For convenience, we consider the true values of the unknown parameters as λ=1 and θ=3.For the point estimate, we computed the ML estimate and Bayesian estimates ofλ and θ, under SELF, LLF (with ε=0.5), and GELF (with ω=0.5) using informative prior (IP) and non-informative priors (NIP) values for the mean square error (MSE) and the estimated bias (EB) for each estimate. We construct also the average confidence length (ACL) and the coverage probabilities (CP) of the 90% and 95% asymptotic confidence intervals and Bayesian credible intervals for λb and θb, using 1,000 simulations.We take the different censoring schemes as follows:(1)
Scheme 1Rm=Rk=n−m/2,Ri=0 for all i≠k,m.(2)
Scheme 2R1=Rm=n−m/2,Ri=0 for all i≠1,m..The average estimates, MSE and EB for ML and Bayesian estimates ofλ and θ, have been reported in Tables 1 and 2, respectively, also, Tables 3 and 4 are present the ACL of 90% and 95% confidence intervals with corresponding CP for λ^ and θ^, respectively.Table 1
The values of MSE and EB of ML and Bayesian estimates forλ based on the different Type-II unified PHCSs.
n,m,kSch.T1,T2λ^MLλ^Bλ^BSλ^BEλ^BLIPNIPIPNIPIPNIPMSE(50,20,10)1(5,10)0.26370.20750.23410.19540.21910.19990.224720.28230.22150.25160.20830.23530.21330.2414(50,30,15)10.21920.18910.20530.18370.19890.18510.200620.21930.18990.20640.18540.20100.18610.2020(50,40,20)10.20090.17940.19200.17680.18890.17680.189020.19960.17820.19040.17510.18690.17550.1873(50,20,10)1(10,20)0.23820.18680.21070.17590.19720.17990.202220.19490.21580.23330.18750.21180.19200.2406(50,30,15)10.15030.13310.14200.13260.14120.13140.140020.16720.14520.15550.14190.15160.14250.1523(50,40,20)10.16720.14520.15550.14190.15160.14250.152320.18770.16760.17790.16330.17320.16470.1747(50,20,10)1(15,30)0.21440.23000.24740.25310.27240.23560.253420.17540.19420.21000.21860.23630.20050.2165(50,30,15)10.12080.13200.13960.11930.12710.11830.126020.10980.11730.12410.13110.13870.12080.1278(50,40,20)10.15050.13070.14000.12770.13640.12830.137120.17700.15810.16740.15390.16270.15540.1643EB(50,20,10)1(5,10)0.11020.04590.05470.00860.01310.03260.039720.11780.05160.06180.01410.01990.03820.0465(50,30,15)10.06690.02650.02990.00060.00210.01750.020220.06420.02320.02610.00330.00240.01400.0162(50,40,20)10.05150.01670.01840.00650.00640.00880.009920.05270.01870.02050.00390.00360.01100.0122(50,20,10)1(10,20)0.09920.04130.04920.00770.01180.02930.035720.10600.04640.05560.01270.01790.03440.0419(50,30,15)10.03190.00300.00250.00050.00190.01110.011120.04800.01130.01300.00300.00220.00300.0041(50,40,20)10.04640.01500.01660.00510.00610.00790.008920.04740.01680.01850.00350.00320.00990.0110(50,20,10)1(15,30)0.20710.22430.24170.24840.26770.23030.248020.16740.18840.20400.21380.23140.19500.2111(50,30,15)10.02870.00270.00230.02440.02540.01000.010020.04700.01020.01170.01190.01190.00270.0037(50,40,20)10.05200.02330.02540.00440.00540.01680.018420.05100.02250.02450.00360.00460.01600.0176Table 2
The values of MSE and EB of ML and Bayesian estimates forθ based on the different Type-II unified UHCSs.
n,m,kSch.T1,T2θ^MLθ^Bθ^BSθ^BEθ^BLIPNIPIPNIPIPNIPMSE(50,20,10)1(5,10)0.08200.06040.06730.06710.06740.05440.059820.09110.06710.06740.06720.06750.06040.0664(50,30,15)10.07710.05690.06340.05700.05720.05120.056320.08570.06320.06350.06330.06360.05690.0626(50,40,20)10.07580.06020.06040.06030.06050.05420.059620.08420.06010.06030.06020.06040.05410.0595(50,20,10)1(10,20)0.07380.05440.06060.06040.06070.04890.053820.08200.06040.06070.06050.06080.05440.0598(50,30,15)10.06940.05120.06310.05130.05150.04610.050720.07710.06300.06310.06310.06330.05670.0623(50,40,20)10.06820.06000.06010.06010.06020.05400.059420.07580.06000.06010.06000.06020.05400.0594(50,20,10)1(15,30)0.06640.04890.05450.05440.05460.04400.048420.07380.05440.05460.05440.05470.04890.0538(50,30,15)10.06250.04610.05680.04610.04640.04150.045620.06940.05670.05680.05680.05700.05100.0561(50,40,20)10.06140.05400.05410.06000.05420.04860.053520.06820.05990.06000.05400.06010.04860.0593EB(50,20,10)1(5,10)0.05550.00050.00100.00060.00210.00040.000520.06170.00060.00090.00050.00200.00050.0006(50,30,15)10.05190.00340.00480.00450.00590.00310.003420.05770.00370.00520.00480.00630.00340.0037(50,40,20)10.05260.00310.00450.00410.00560.00280.003020.05840.00290.00430.00390.00530.00260.0028(50,20,10)1(10,20)0.05000.00050.00090.00050.00190.00040.000520.05550.00050.00080.00050.00180.00050.0005(50,30,15)10.04670.00310.00430.00410.00530.00280.003120.05190.00360.00490.00460.00600.00320.0035(50,40,20)10.04730.00200.00330.00300.00430.00180.002020.05260.00190.00320.00290.00420.00170.0019(50,20,10)1(15,30)0.04500.00040.00080.00050.00170.00030.000420.05000.00050.00070.00040.00160.00040.0005(50,30,15)10.04210.00280.00390.00360.00480.00250.002820.04670.00320.00440.00410.00540.00290.0032(50,40,20)10.04260.00190.00310.00290.00410.00170.001920.04730.00170.00310.00260.00380.00150.0017Table 3
The ACL of95% and 99% confidence intervals and corresponding CP for λ^ML and λ^B at the different priors and Type-II unified PHCSs.
n,m,kSch.95%99%λ^MLλ^Bλ^MLλ^BIPNIPIPNIPT1,T2ACLCPACLCPACLCPACLCPACLCPACLCP(50,20,10)1(5,10)0.9880.9850.9400.9790.9680.9961.3910.9901.2400.9791.2571.00020.9880.9770.9400.9710.9680.9931.3910.9861.2400.9771.2530.989(50,30,15)10.8000.9810.7610.9730.7860.9811.1340.9911.0030.9881.0890.98320.6820.9850.6500.9810.6540.9970.9240.9860.8200.9820.8690.975(50,40,20)10.5321.0000.5060.9790.5130.9780.7580.9950.6721.0000.7160.98920.4100.9860.3840.9770.5091.0000.7521.0000.6671.0000.7111.000(50,20,10)1(10,20)0.6770.9910.6430.9880.6750.9950.9540.9760.8490.9800.8750.98520.7110.9860.6750.9820.7070.9891.0010.9770.8911.0000.9170.978(50,30,15)10.7890.9950.7501.0000.7750.9831.1090.9840.9890.9741.0040.98920.8001.0000.7611.0000.7860.9751.1250.9861.0030.9821.0180.984(50,40,20)10.6010.9760.5720.9800.5830.9890.8130.9840.7260.9790.7270.99020.7930.9770.7540.9970.7770.9721.1141.0000.9930.9741.0061.000(50,20,10)1(15,30)0.5410.9840.5140.9740.5440.9850.7640.9830.6780.9740.7100.97120.5630.9860.5340.9820.5640.9780.7940.9910.7050.9870.7370.994(50,30,15)10.6810.9840.6480.9790.6730.9890.9381.0000.8540.9880.7031.00020.7021.0000.6670.9740.6920.9840.9690.9780.8790.9860.7510.995(50,40,20)10.7930.9830.7540.9740.7770.9901.1140.9920.9930.9921.0030.98420.7930.9910.7540.9870.7771.0001.1170.9750.9931.0001.0320.986Table 4
The ACL of95% and 99% confidence intervals and corresponding CP for θ^ML and θ^B at the different priors and Type-II unified PHCSs.
n,m,kSch.95%99%θ^MLθ^Bθ^BIPNIPθ^MLIPNIPT1,T2ACLCPACLCPACLCPACLCPACLCPACLCP(50,20,10)1(5,10)0.2690.9630.2490.9430.2630.9450.4050.9920.3540.9900.3750.99020.3700.9630.3510.9440.3540.9450.5120.9910.4560.9900.4660.989(50,30,15)12.4470.9692.3790.9502.2820.9502.6990.9832.4780.9832.3850.98022.7570.9532.7490.9452.5030.9233.0190.9652.8450.9762.5950.951(50,40,20)14.1880.9734.0730.9553.9040.9534.5360.9824.1700.9824.0040.97824.2760.9784.1510.9573.9940.9604.6290.9844.2460.9844.0930.980(50,20,10)1(10,20)0.3260.9860.2980.9650.3220.9680.5050.9950.4370.9930.4730.99320.3090.9820.2830.9620.3050.9640.4800.9950.4160.9940.4490.993(50,30,15)10.2940.9640.2760.9420.2850.9480.4280.9880.3780.9860.3920.98720.4370.9620.4140.9420.4180.9450.5770.9880.5150.9860.5250.987(50,40,20)14.5740.9844.4690.9634.2440.9664.9440.9934.5650.9914.3440.99124.5640.9874.5230.9664.1690.9694.9330.9964.6190.9944.2690.994(50,20,10)1(15,30)0.3950.9830.3610.9650.3920.9630.6090.9950.5260.9940.5720.99320.3780.9850.3460.9670.3750.9650.5840.9950.5040.9930.5480.994(50,30,15)10.2800.9680.2610.9480.2730.9500.4230.9900.3720.9880.3900.98820.2870.9670.2680.9490.2790.9480.4270.9900.3770.9890.3930.988(50,40,20)12.7740.9762.9400.9562.3430.9573.0410.9913.0370.9892.4430.99022.4020.9732.6440.9531.9310.9552.6480.9912.7410.9892.0310.990


## 8. Numerical Example

In this section, we use the real data set to show the performance of the inferential results established for the Pareto distribution based on the Type-II unified PHSC, in addition to comparing ML and Bayesian estimates through Monte Carlo simulations. This real data set contains the failure times (in hours) of one plane’s ac system from a pair of real data sets collected by Bain and Engelhardt [30]. Moreover, Guo and Gui [31] demonstrated that these data sets closely matched the inverse Pareto distribution. For further proceeding, before using these data, we ran Kolmogorov-Smirnov (KS) goodness of fit tests to see if they followed the Pareto distribution or not. For these data sets, the KS test statistics with their related p-values are more than 0.05, so we can assume that these data sets follow Pareto distribution at a 0.05% level of significance. This real data are ordered in Table 5.We will use these data to generate the Type-II unified PHCS, supposem=9, k=6, Ri=2 for i=3,6,9, and Ri=0 otherwise with different values of T1and T2with T2=2T1. Table 6 shows different Type-II unified PHCSs.Table 5
The real data.
1.22.12.62.72.92.94.85.75.97.07.415.332.638.650.2Table 6
The different Type-II unified HPCS withm,k=9,6 and different choices of T1 and T2.
Scheme1t1,t2=2,4T∗=Xk:m:nd∗=6Y¯=1.2,2.1,2.6,2.7,2.9,4.8R~=0,0,2,0,0,7R~t1,R~t2=0,0Scheme2t1,t2=3,6T∗=t2d∗=7Y¯=1.2,2.1,2.6,2.7,2.9,4.8,5.7R~=0,0,2,0,0,2,0R~t1,R~t2=0,4Scheme3t1,t2=6,12T∗=Xm:m:nd∗=9Y¯=1.2,2.1,2.6,2.7,2.9,4.8,5.7,7.0,7.4R~=0,0,2,0,0,2,0,0,2R~t1,R~t2=0,0Scheme4t1,t2=10,20T∗=t1d∗=9Y¯=1.2,2.1,2.6,2.7,2.9,4.8,5.7,7.0,7.4R~=0,0,2,0,0,2,0,0,0R~t1,R~t2=2,0After generating the Type-II unified PHC data with the different unified PHCS, we ran KS goodness of fit tests for all Type-II unified PHC data to see if they followed the Pareto distribution or not. For all these Type-II unified PHC data sets, the KS test statistics with their relatedp-values are more than 0.05, so we can assume that these data and all generated Type-II unified PHC data sets from it follow Pareto distribution at a 0.05% level of significance.Based on the Type-II unified PHCS and two different choicesIP and NIP priors, the ML and Bayesian estimates for the unknown parameters λ and θ are presented in Tables 7 and 8. Moreover, the 95% and 99% asymptotic confidence intervals and the credible intervals are presented in Tables 9 and 10. Finally, Tables 11 and 12 present the point predictor with 95% and 99% Bayesian prediction bounds of Ys:ℓ:N from the future progressively censored sample of size ℓ=10 from a sample of size N=20 with progressive censoring scheme S=0,2,0,2,0,2,0,2,0,2 for four different choices of censoring schemes.Table 7
The ML and Bayesian estimates ofλ based on the different Type-II unified PHCSs from real data.
Sch.λ^MLλ^Bλ^BSλ^BEλ^BLIPNIPIPNIPIPNIP10.38310.35040.31920.31080.27180.34580.314220.43200.39640.37800.36200.33780.39190.373030.51400.46410.45690.42800.41430.45860.450540.48980.44930.44080.41770.40430.44460.4355Table 8
The95% and 99% confidence intervals estimates of λ based on the different Type-II unified PHCSs from real data.
Sch.95%99%λ^MLλ^Bλ^BIPNIPλ^MLIPNIPLBUBLBUBLBUBLBUBLBUBUB10.24940.75970.13620.66390.10370.65390.14740.86180.09730.79950.06880.804120.34530.91820.17740.70200.15200.70520.23081.03280.13390.82990.11000.845630.34530.91820.21860.80050.19720.82370.23081.03280.16830.93950.14680.978540.31130.78020.22090.75750.20160.77210.21750.87400.17280.88340.15340.9100Table 9
The ML and Bayesian estimates ofθ based on the different Type-II unified PHCSs from real data.
Sch.θ^MLθ^Bθ^BSθ^BEθ^BLIPNIPIPNIPIPNIP11.20001.00930.96230.90920.86741.00090.949921.20001.00930.96230.90920.86741.00090.949931.20001.05511.03170.95010.92921.05011.025141.20001.05221.02840.94750.92621.04711.0217Table 10
The95% and 99% confidence intervals estimates of θ based on the different Type-II unified PHCSs from real data.
Sch.95%99%θ^MLθ^Bθ^MLθ^BIPNIPIPNIPLBUBLBUBLBUBLBUBLBUBLBUB10.49851.31430.52971.19500.38401.19370.31261.31890.31261.19900.16761.198720.49851.31430.52971.19500.38401.19370.31261.31890.31261.19900.16761.198730.63461.31580.68421.19620.60561.19560.49611.31920.49611.19920.40091.199140.62971.31560.67921.19610.60221.19540.49441.31910.49441.19920.40301.1991Table 11
Bayesian point predictor with95% and 99% ET prediction intervals for Yq:R~j for q=1,⋯,R~j, with j=1,⋯,d∗,R~t1, and R~t2.
Sch.jq95%99%X^q:Rj∗IPNIPX^q:Rj∗IPNIPLBUBLBUBLBUBLBUB1315.221.8838.5611.6959.4177.8303.30512.3542.97513.59027.614.27110.9483.84312.04310.2175.69214.7415.12316.215618.334.99311.6714.49412.83810.9396.41515.4645.77317.010212.198.85115.5297.96617.08214.79810.27319.3229.24621.255318.0914.75021.42713.27523.57020.69616.17225.22114.55527.743422.2418.90325.58117.01328.13924.84920.32529.37418.29232.311526.7923.44930.12721.10433.13929.39624.87133.92022.38437.312633.6330.29536.97227.26540.66936.24131.71640.76528.54544.842762.5159.16965.84753.25272.43165.11660.59169.64054.53276.6042317.394.04610.7243.64211.7969.9935.46814.5174.92115.969210.907.55714.2346.80115.65813.5038.97918.0288.08119.8306112.589.24215.9208.31817.51215.18910.66419.7139.59821.684218.8815.54622.22413.99224.44621.49316.96826.01715.27128.619t2111.788.44415.1227.60016.63414.3919.86618.9158.88020.807217.1613.82220.49912.44022.54919.76815.24424.29313.71926.722323.0719.72626.40417.75429.04425.67321.14830.19719.03333.217430.6827.33734.01424.60337.41633.28328.75937.80825.88341.5893319.045.70612.3845.13613.62211.6537.12816.1776.41517.795214.4311.09317.7709.98319.54717.03912.51421.56311.26323.7206110.447.10513.7836.39515.16113.0528.52717.5767.67519.334215.4112.07018.74710.86320.62218.01613.49222.54112.14224.7959116.6613.32520.00211.99322.00319.27114.74723.79613.27226.175226.1522.81529.49220.53332.44128.76124.23633.28621.81336.61443123.1319.79126.46817.81129.11525.73721.21230.26119.09133.287238.1434.79841.47531.31845.62340.74436.22045.26932.59849.7966117.7914.45321.13113.00823.24420.40015.87524.92414.28827.417226.7123.36930.04621.03233.05129.31524.79033.83922.31137.223t1131.1427.80034.47725.02037.92533.74629.22238.27126.29942.098236.9933.64840.32530.28344.35839.59435.07044.11931.56348.530Table 12
Bayesian point predictor with95% and 99% ET prediction intervals for Ys:10 for s=1,⋯,10.
Sch.s95%99%Y^s:10IPNIPY^s:10IPNIPLBUBLBUBLBUBLBUB111.2250.6292.0260.3893.0371.3430.4842.1850.2313.69421.5730.7403.0880.4795.5412.5320.6003.6650.3108.14632.4380.8735.0820.59111.3507.1030.7446.8320.41721.23844.6131.0188.5380.72023.78019.7100.90613.1820.54857.920512.1881.19316.6700.88162.35859.5511.10530.8520.720214.386632.6531.38234.2631.062174.841151.7491.32077.5510.921878.9497111.0721.63396.6641.290789.276431.1751.595290.9791.1816816.2268322.7521.956311.6341.5364.3E+031.0E+031.9401.3E+031.4676.9E+0491.4E+032.5472.9E+031.8564.4E+033.3E+032.5702.2E+041.8396.0E+06104.8E+033.5825.6E+042.5045.7E+048.6E+033.6839.3E+052.4976.2E+06211.2060.7001.8600.4842.5451.1870.5981.9080.3632.73221.4310.7972.6120.5664.0361.4540.6982.7820.4414.65331.7930.9133.8850.6676.9601.9950.8194.3450.5398.78242.3731.0405.8530.78012.1373.1630.9546.9140.65316.89753.7901.1949.9240.92125.0746.9661.12012.5970.79939.71767.2341.36217.4791.07954.22117.2871.30224.0360.96699.035721.8061.58639.6211.279168.94459.0181.53861.0601.184378.186869.0391.87399.5081.502599.893179.0771.833174.9021.4301.7E+039427.9452.388591.6691.8037.7E+03881.0132.3581.3E+031.7663.3E+04102.0E+033.2696.2E+032.1358.0E+033.3E+033.2571.9E+042.1393.5E+04311.2000.7641.7330.5652.2381.1800.6861.7430.4672.30321.3790.8512.2950.6423.2581.3670.7752.3480.5423.47131.6410.9533.1840.7345.0671.6560.8803.3340.6335.64942.0001.0634.4580.8357.9392.0940.9954.7930.7359.30552.6601.1956.8770.95914.2693.0821.1337.6600.86317.86063.8711.33810.9371.09626.5675.3200.26112.6760.26135.78177.9781.52621.4161.26866.59414.1770.38826.2610.32899.524820.7911.76245.5751.457185.14641.9220.26259.6760.262312.0949135.9392.175198.0621.7101.5E+03250.6322.118291.3178.9373.1E+03107.5E+022.8571.4E+031.9921.5E+031.2E+032.7832.4E+031.9703.2E+03411.2000.7581.7440.5622.2501.1790.6811.7530.4672.30621.3830.8452.3140.6373.2661.3690.7692.3610.5403.44931.6500.9473.2130.7285.0581.6540.8743.3470.6295.55042.0101.0594.5000.8307.8782.0630.9894.7970.7309.01652.6511.1956.9430.95414.0512.8961.1317.6340.85716.99963.7621.34311.0341.09325.9194.6211.28812.5641.00033.35673.7621.34311.0341.09325.91911.2111.49125.8521.18690.282818.0761.79145.9461.468176.05232.4991.74258.2401.398273.9419121.9892.232200.6301.7401.4E+03212.1322.180281.9211.6942.6E+03107.2E+022.9691.4E+032.0511.4E+031.1E+032.9062.2E+032.0352.7E+03SinceR1=0, then y1=1.2 is not removed from censored data in all Type-II unified PHCS, and since θ^ML=y1 so that θ^ML=1.2 in all four Type-II unified PHCS.


## 9. Conclusions and Discussion

From Tables1 and 2, we observe that the MSEs of the Bayesian estimates based on the LINEX, GE, and SE loss functions are smaller than those of the ML estimates. Furthermore, the MSEs and EBs of all estimates decrease with increasing m and k when T1 and T2 are fixed. Also, the MSEs and EBs of all estimates decrease with increasing T1 and T2 when m and k are fixed. Moreover, a comparison of the results for the informative priors with the corresponding ones for non-informative priors reveals that the former produces more precise results.From the results in Tables10 and 11, we notice that the point predictor of mean is between the upper and lower bounds of the prediction intervals. Additionally, as we would expect, a comparison of the results for the informative prior with the corresponding ones for non-informative prior reveals that the former produces more precise results, because the interval length in the informative prior case is short than in non-informative case. Moreover, the 95% prediction intervals seem to be more precise than the 99% prediction intervals, Finally when we use the same value of T1 and T2 but increasing k and m. , the Bayesian prediction bounds become tighter as expected since the duration of the life-testing experiment is longer in this case.



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*Source: 2073067-2022-04-29.xml*