# Estimation for Parameters of Life of the Marshall-Olkin Generalized-Exponential Distribution Using Progressive Type-II Censored Data

Authors: Ahmed Elshahhat; Abdisalam Hassan Muse; Omer Mohamed Egeh; Berihan R. Elemary
Journal: Complexity
 (2022)

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## Abstract

A new three-parameter extension of the generalized-exponential distribution, which has various hazard rates that can be increasing, decreasing, bathtub, or inverted tub, known as the Marshall-Olkin generalized-exponential (MOGE) distribution has been considered. So, this article addresses the problem of estimating the unknown parameters and survival characteristics of the three-parameter MOGE lifetime distribution when the sample is obtained from progressive type-II censoring via maximum likelihood and Bayesian approaches. Making use of the s-normality of classical estimators, two types of approximate confidence intervals are constructed via the observed Fisher information matrix. Using gamma conjugate priors, the Bayes estimators against the squared-error and linear-exponential loss functions are derived. As expected, the Bayes estimates are not explicitly expressed, thus the Markov chain Monte Carlo techniques are implemented to approximate the Bayes point estimates and to construct the associated highest posterior density credible intervals. The performance of proposed estimators is evaluated via some numerical comparisons and some specific recommendations are also made. We further discuss the issue of determining the optimum progressive censoring plan among different competing censoring plans using three optimality criteria. Finally, two real-life datasets are analyzed to demonstrate how the proposed methods can be used in real-life scenarios.


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## Body

## 1. Introduction

Several extensions of the exponential distribution such as Weibull, gamma, generalized-exponential, and Nadarajah-Haghighi distributions among others have been proposed in literature. A general technique that allows to adding a new shape parameter to expand a family of distributions is originally proposed by Marshall and Olkin [1]. Applying the Marshall-Olkin technique to generalized-exponential distribution, Ristić and Kundu [2] introduced the three-parameter Marshall-Olkin generalized-exponential (MOGE) distribution. However, suppose that the lifetime random variable X of an individual testing item follows a three-parameter MOGEα,β,θ. Hence, its cumulative distribution function (CDF) F⋅, probability density function (PDF) f⋅, reliability function (RF) R⋅, and hazard function (HF) h⋅, at distinct time t, are given, respectively, by(1)Fx;α,β,θ=1−e−θxαβ+β¯1−e−θxα,x>0,(2)fx;α,β,θ=αβθe−θx1−e−θxα−1β+β¯1−e−θxα2,x>0,(3)Rt;α,β,θ=β1−1−e−θtαβ+β¯1−e−θtα,t>0,and(4)ht;α,β,θ=αθe−θt1−e−θtα−1β+β¯1−e−θtα1−1−e−θtα,t>0,where β¯=1−β, α>0, and β>0 are the shape parameters and θ>0 is the scale parameter. Ristić and Kundu [2] derived several properties of the MOGE distribution and estimated its parameters using the likelihood method.Using some specified values on the range ofα, β, and θ, different shapes of the density and failure rate functions of the MOGE distribution are displayed in Figure 1. It shows that the density shapes are decreasing or unimodal while the HF shapes can be increasing, decreasing, bathtub, or inverted tub. Since the bathtub HF shape is quite beneficial in several lifetime data, therefore the MOGE distribution is justly flexible and can be used to provide a good description of different types of censored data, for details, it can be seen in Ristić and Kundu [2]. From (1), three lifetime submodels can be obtained as special cases from the MOGE distribution, namely,(i)
Marshall–Olkin exponential distribution, by Marshall and Olkin [1]; if setting α=1.(ii)
Generalized-exponential distribution, by Gupta and Kundu [3]; if setting θ=1.(iii)
Exponential distribution, discussed by Johnson and Kotz [4]; if setting α=θ=1.Figure 1
Plots of the density and hazard rate functions of the MOGE distribution.Statistical life distributions under censoring plans have received great interest owing to their wide application in numerous fields such as engineering, social sciences, marketing, and medicine. One of the most popular censoring mechanisms, which is of great importance in reliability studies, is known as type-II progressive censoring scheme (PCS-T2). This censoring scheme can be described as follows: suppose thatn identical and independent units are put on a life-testing experiment at time zero and the progressive censoring scheme R=R1,R2,…,Rm is prefixed, where m<n is the number of observed failures that is decided by the experimenter in advance. At the time of the first failure occur (say x1:m:n), R1 of survival items are randomly removed from the remaining live n−1 units. Again, at the time of the next failure occur (say x2:m:n), R2 of survival items are randomly removed from the remaining live n−R1−2 units and continue the test. This procedure continues until all remaining live Rm units are removed from the experiment at the time of the mth failure observed. It is clear that n−m=∑i=1mRi. For more details, one may refer to an excellent monograph presented by Balakrishnan and Cramer [5].However, the likelihood function of the PCS-T2 samplexi:m:n for i=1,2,…,m, is defined as (5)LΘ|x=C∏i=1mfxi:m:n;Θ1−Fxi:m:n;ΘRi,where C=nn−R1−1…n−∑i=1m−1Ri+1 and Θ is parameter vector.In literature, several authors including Amin [6], Kim et al. [7], Huang and Wu [8], Dey et al. [9], Panahi [10], Muhammed and Almetwally [11], Elshahhat and Rastogi [12], Maiti and Kayal [13], Dey et al. [14], Maiti et al. [15], and Elshahhat and Abu El Azm [16] studied a wide variety of inferential problems for different lifetime distributions under type-II progressively censored samples.In the context of censoring mechanisms, to the best of our knowledge, we have not encountered any work related to the estimation of parameters and/or survival characteristics of the MOGE distribution under incomplete (censored) sampling, which is of considerable interest and practical significance in many practical situations. So, in this study, we shall exclusively focus on both classical and Bayesian approaches of estimation to derive the both point and interval estimators of the MOGE parametersα, β, and θ, as well as its time parameters Rt and ht under PCS-T2. Our objectives in this study are as follows: first, deriving the maximum likelihood estimators (MLEs) for the unknown MOGE parameters or any function of them. Correspondingly, using observed Fisher information matrix, two-sided asymptotic confidence intervals are also established. Second, obtain the Bayes estimators of the unknown quantities α, β, θ, Rt, and ht using independent gamma priors based on the two popular symmetric and asymmetric loss functions. Markov chain Monte Carlo (MCMC) techniques are also used to carry out the Bayes estimates and associated credible interval estimates from the complex posterior functions. Third, determine the optimum progressive censoring plan from a class of all possible removal patterns that contain a magnificent amount of information regarding the unknown model parameter under consideration. Lastly, the performance of the proposed methods is compared through an extensive simulation study in terms of their simulated root mean squared-error, mean relative absolute bias, average confidence lengths, and coverage percentages. To discuss how the proposed methods can be applied in a real phenomenon, two real-life data sets are analyzed. We also suggest the use of “maxLik” and “coda” statistical packages in R programming software to evaluate the theoretical findings. Finally, some specific recommendations are made.The rest of the paper is arranged as follows: classical and Bayes estimations of unknown parameters and the reliability characteristics are presented in Sections2 and 3, respectively. Optimum progressive censoring plans are discussed in Section 4. Monte Carlo simulation is performed in Section 5. Two applications using real data sets are presented in Section 6. Finally, we conclude the paper in Section 7.


## 2. Likelihood Inference

In this section, the point and interval estimators of the unknown parameters and the reliability characteristics of the MOGE distribution based on PCS-T2 are derived.


### 2.1. Point Estimators

Supposex=x1:m:n,R1,x2:m:n,R2,…,xm:m:n,Rm is a PCS-T2 data obtained from MOGEα,β,θ. From (1) and (2), with ignoring any additive constant, the likelihood function (5) becomes (6)Lα,β,θ|x∝βnαθme−θ∑i−1mxi∏i=1mψiα−1θ1−ψiαθRiβ+β¯ψiαθ−Ri+2,where xi is used instead of xi:m:n and. ψiθ=1−exp−θxi,i=1,2,…,m.From (6), the log-likelihood function, ℓ⋅=logL⋅, can be written as(7)ℓα,β,θ|x∝nlogβ+mlogαθ−θmx¯+α−1∑i=1mlogψiθ+∑i=1mRilog1−ψiαθ−∑i=1mRi+2logβ+β¯ψiαθ.Setting the first-partial derivatives of (7) with respect to each unknown parameter to zero, the respective MLEs of α, β, and θ are the solutions to the following normal equations as(8)0=mα^+∑i=1mlogψ^iθ^−∑i=1mRilogψ^iθ^ψ^iα^θ^1−ψ^iα^θ^−1−β^¯∑i=1mRi+2logψ^iθ^ψ^iα^θ^β^+β^¯ψ^iα^θ^−1,(9)0=nβ^−∑i=1mRi+21−ψ^iα^θ^β^+β^¯ψ^iα^θ^−1,And,(10)0=mθ^−mx¯+α^−1∑i=1mψ^′iθ^ψ^i−1θ^−α^∑i=1mRiψ^′iθ^ψ^iθ^α^−11−ψ^iα^θ^−1−α^β^¯∑i=1mRi+2ψ^′iθ^ψ^iθ^α^−1β^+β^¯ψ^iα^θ^−1,where ψ^′iθ^=xi1−ψ^iθ^ for i=1,…,m.Obviously, from (8) and (10), the MLEs α^, β^, and θ^ of the parameters α, β, and θ, respectively, cannot be obtained analytically but can be evaluated using any iterative approximation technique. Once α^, β^, and θ^ calculated, using the invariance property of MLEs, the corresponding MLEs R^t and h^t of Rt and ht at any given time t>0 can be easily derived by replacing α, β, and θ with α^, β^, and θ^ in (3) and (4), respectively.


### 2.2. Asymptotic Intervals

To construct the1001−τ% two-sided ACIs for the unknown parameters α, β, and θ, or any life function such as Rt and ht (say φ), the asymptotic variances or covariances (by inverting the Fisher information matrix I⋅) of the MLEs α^, β^, and θ^ must be obtained as (11)I−1φ^=ℒ11ℒ12ℒ22ℒ31ℒ32ℒ33φ=φ^−1=σ^11σ^21σ^22σ^31σ^32σ^33.Under some mild regularity conditions of the MLEs, using the concept of large sample theory, we have the asymptotic normality approximation (NA) ofφ^ is approximately multivariate normal with mean vector φ and variance-covariance matrix I−1φ, i.e., φ^∼N3φ,I−1φ^, where φ=α,β,θ⊤ is the parametric vector of α, β, and θ. By differentiating (7) partially with respect to α, β, and θ, locally at their MLEs α^, β^, and θ^, the Fisher’s elements ℒij for i,j=1,2,3, are obtained and provided in Appendix A.Similarly, based on NA of the MLEsR^t and h^t, it is clear that R^∼NRt,σ^Rt2 and h^t∼Nht,σ^ht2. Thus, ACIs for Rt and ht can be constructed using the corresponding normality. To obtain ACIs of Rt and ht, we need to compute their variances. Following Greene [17], the delta method is used to find σ^R2 and σ^h2 of Rt and ht, respectively, as:(12)σ^R2=∇R,I−1φ,∇R⊤|φ=φ^,σ^h2=∇h,I−1φ,∇h⊤|φ=φ^,where,(13)∇R=∂∂αRt,∂∂βRt,∂∂θRt,∇h=∂∂αht,∂∂βht,∂∂θht.Then, based on NA, the1001−τ% two-sided ACIs for the unknown parameter α, β, θ, Rt or ht (say κ) is given by:(14)κ^±zτ/2σ^κ^2,where zτ/2 is the upper τ/2th quantile of the standard normal distribution.Though, the main problem of asymptotic NA confidence interval is that it may give a negative value in lower bound for a lifetime parameter. To handling this drawback, Meeker and Escobar [18] developed the normal approximation for the log-transformed (NL) of MLE. They also showed that ACI has better CP based on NL compared to NA. However, based on NL, the 1001−τ% two-sided ACI of κ is given by (15)κ^exp∓zτ/2σ^κ^2κ^.


## 2.1. Point Estimators

Supposex=x1:m:n,R1,x2:m:n,R2,…,xm:m:n,Rm is a PCS-T2 data obtained from MOGEα,β,θ. From (1) and (2), with ignoring any additive constant, the likelihood function (5) becomes (6)Lα,β,θ|x∝βnαθme−θ∑i−1mxi∏i=1mψiα−1θ1−ψiαθRiβ+β¯ψiαθ−Ri+2,where xi is used instead of xi:m:n and. ψiθ=1−exp−θxi,i=1,2,…,m.From (6), the log-likelihood function, ℓ⋅=logL⋅, can be written as(7)ℓα,β,θ|x∝nlogβ+mlogαθ−θmx¯+α−1∑i=1mlogψiθ+∑i=1mRilog1−ψiαθ−∑i=1mRi+2logβ+β¯ψiαθ.Setting the first-partial derivatives of (7) with respect to each unknown parameter to zero, the respective MLEs of α, β, and θ are the solutions to the following normal equations as(8)0=mα^+∑i=1mlogψ^iθ^−∑i=1mRilogψ^iθ^ψ^iα^θ^1−ψ^iα^θ^−1−β^¯∑i=1mRi+2logψ^iθ^ψ^iα^θ^β^+β^¯ψ^iα^θ^−1,(9)0=nβ^−∑i=1mRi+21−ψ^iα^θ^β^+β^¯ψ^iα^θ^−1,And,(10)0=mθ^−mx¯+α^−1∑i=1mψ^′iθ^ψ^i−1θ^−α^∑i=1mRiψ^′iθ^ψ^iθ^α^−11−ψ^iα^θ^−1−α^β^¯∑i=1mRi+2ψ^′iθ^ψ^iθ^α^−1β^+β^¯ψ^iα^θ^−1,where ψ^′iθ^=xi1−ψ^iθ^ for i=1,…,m.Obviously, from (8) and (10), the MLEs α^, β^, and θ^ of the parameters α, β, and θ, respectively, cannot be obtained analytically but can be evaluated using any iterative approximation technique. Once α^, β^, and θ^ calculated, using the invariance property of MLEs, the corresponding MLEs R^t and h^t of Rt and ht at any given time t>0 can be easily derived by replacing α, β, and θ with α^, β^, and θ^ in (3) and (4), respectively.


## 2.2. Asymptotic Intervals

To construct the1001−τ% two-sided ACIs for the unknown parameters α, β, and θ, or any life function such as Rt and ht (say φ), the asymptotic variances or covariances (by inverting the Fisher information matrix I⋅) of the MLEs α^, β^, and θ^ must be obtained as (11)I−1φ^=ℒ11ℒ12ℒ22ℒ31ℒ32ℒ33φ=φ^−1=σ^11σ^21σ^22σ^31σ^32σ^33.Under some mild regularity conditions of the MLEs, using the concept of large sample theory, we have the asymptotic normality approximation (NA) ofφ^ is approximately multivariate normal with mean vector φ and variance-covariance matrix I−1φ, i.e., φ^∼N3φ,I−1φ^, where φ=α,β,θ⊤ is the parametric vector of α, β, and θ. By differentiating (7) partially with respect to α, β, and θ, locally at their MLEs α^, β^, and θ^, the Fisher’s elements ℒij for i,j=1,2,3, are obtained and provided in Appendix A.Similarly, based on NA of the MLEsR^t and h^t, it is clear that R^∼NRt,σ^Rt2 and h^t∼Nht,σ^ht2. Thus, ACIs for Rt and ht can be constructed using the corresponding normality. To obtain ACIs of Rt and ht, we need to compute their variances. Following Greene [17], the delta method is used to find σ^R2 and σ^h2 of Rt and ht, respectively, as:(12)σ^R2=∇R,I−1φ,∇R⊤|φ=φ^,σ^h2=∇h,I−1φ,∇h⊤|φ=φ^,where,(13)∇R=∂∂αRt,∂∂βRt,∂∂θRt,∇h=∂∂αht,∂∂βht,∂∂θht.Then, based on NA, the1001−τ% two-sided ACIs for the unknown parameter α, β, θ, Rt or ht (say κ) is given by:(14)κ^±zτ/2σ^κ^2,where zτ/2 is the upper τ/2th quantile of the standard normal distribution.Though, the main problem of asymptotic NA confidence interval is that it may give a negative value in lower bound for a lifetime parameter. To handling this drawback, Meeker and Escobar [18] developed the normal approximation for the log-transformed (NL) of MLE. They also showed that ACI has better CP based on NL compared to NA. However, based on NL, the 1001−τ% two-sided ACI of κ is given by (15)κ^exp∓zτ/2σ^κ^2κ^.


## 3. Bayesian Inference

In this section, we discuss the Bayes inference of the model parametersα, β, and θ, as well as, the reliability parameters Rt and ht using progressively type-II censoring.


### 3.1. Prior Information and Loss Functions

Since the gammaG⋅ density provides various shapes based on parameter values and is flexible in nature, so the utilizing of independent gamma priors are relatively simple which may yield to results with more explicit posterior density expressions Muse et al. [19]; for more details, it can be seen in [14, 20, 21]. Thus, the gamma conjugate priors Gαa1,b1, Gβa2,b2, and Gθa3,b3 are used to adapt the support of MOGE parameters α, β, and θ, respectively. However, the joint prior PDF of α, β, and θ becomes (16)πα,β,θ∝αa1−1βa2−1θa3−1e−b1α+b2β+b3θ,α,β,θ>0,where the hyperparameters ai,bi,i=1,2,3, are chosen to reflect prior knowledge about the unknown parameters α, β, and θ, and they assumed to be known as non-negative.The choice of (symmetric and/or symmetric) loss function is an important issue in Bayesian analysis. So, to develop the objective estimates, two well-known loss functions namely, squared-error loss (SEL) and linear-exponential loss (LL), are considered. However, any other loss function can be easily incorporated. The SEL function is defined as(17)ℑSϕ,ϕ˜=ϕ˜−ϕ2,where ϕ˜ being an estimate of ϕ. Under (13), the Bayes estimate ϕ˜ is the posterior mean of ϕ.On the other hand, the LL function,ℑL⋅, is defined as (18)ℑLϕ,ϕ˜=expυϕ˜−ϕ−υϕ˜−ϕ−1,υ≠0.The sign and magnitude ofυ represent the direction and degree of symmetry, respectively (if υ>0, overestimation is more serious than underestimation and υ<0 means the opposite). Using (14), the Bayes estimator ϕ˜L of ϕ is given by:(19)ϕ˜L=−υ−1lnEϕe−υϕ|x,υ≠0,provided the above exception exists, and is finite. When υ close to 0, the LL function is approximately the SEL function and it can become almost symmetric, as can be seen in Pandey and Rai [22].


### 3.2. Posterior Analysis

Using (6) and (12), the joint posterior PDF of α, β, and θ can be written as (20)πα,β,θ|x=K−1αa1+m−1βa2+n−1θa3+m−1e−αb1+βb2+θb3+mx¯×∏i=1mψiα−1θ1−ψiαθRiβ+β¯ψiαθ−Ri+2,where K is the normalizing constant of (15).Due to nonlinear expression in (6), the closed-form of the marginal PDF for each unknown parameter is not possible. Thus, we purpose to apply some simulation algorithms such as the MCMC methods to approximate the Bayesian estimates and to find associated credible intervals. So the MCMC methods have been widely used in Bayesian computational analysis. At first, from (15), we derive the conditional posterior distributions of α, β, and θ, respectively, as (21)πα∗α|β,θ,x∝αa1+m−1e−αb1∗θ∏i=1m1−ψiαθRiβ+β¯ψiαθ−Ri+2,(22)πβ∗β|α,θ,x∝βa2+n−1e−βb2∏i=1mβ+β¯ψiαθ−Ri+2,And,(23)πθ∗θ|α,β,x∝θa3+m−1e−θb3+mx¯∏i=1mψiα−1θ1−ψiαθRiβ+β¯ψiαθ−Ri+2,where b1∗θ=b1−∑i=1mlogψiθ.It can be seen, from (16)–(18), that the conditional distributions of α, β, and θ cannot be reduced to any standard distribution so that the exact inferences from their marginal densities cannot be easily obtained. Thus, to generate samples from posterior distribution whenever the posterior PDF cannot be reduced to any familiar distribution, the Metropolis–Hastings algorithm can be easily used for this purpose (see Gelman et al. [23] for detail).


### 3.3. M-H Algorithm

The Metropolis-Hastings (M-H) algorithm is a general member of the MCMC family of simulation techniques. In this subsection, the M-H algorithm with normal proposal distributions is conducted to generate samples ofα, β, and θ from (16), (17), and (18), respectively. Precisely, Figure 2 indicates that the distribution of MOGE parameters behaves similar to the normal distribution. Next, to obtain the Bayes estimates and the associated credible intervals, do the following generation steps:Step 1: Putg=1.Step 2: Set initial valuesα0,β0,θ0=α^,β^,θ^.Step 3: Generateα∗, β∗, and θ∗ from (16)–(18) with normal distributions Nα^,σ^α2, Nβ^,σ^β2, and Nθ^,σ^θ2, respectively, as follows:(a)
CalculateQα=πα∗α∗|βg−1,θg−1,x/πα∗αg−1|βg−1,θg−1,x, Qβ=πβ∗β∗|αg,θg−1,x/πβ∗βg−1|αg,θg−1,x and Qθ=πθ∗θ∗|αg,βg,x/πθ∗θg−1|αg,βg,x.(b)
Generate sample variatesu1, u2, and u3 from the uniform U0,1 distribution.(c)
Ifu1⩽min1,Qα, set αg=α∗, else set αg=αg−1.(d)
Ifu2⩽min1,Qβ, set βg=β∗, else set βg=βg−1.(e)
Ifu3⩽min1,Qθ, set θg=θ∗, else set θg=θg−1.Step 4: ObtainRgt and hgt, for a specified time t>0 by replacing α, β, and θ with their αg, βg, and θg, respectively.Step 5: Putg=g+1.Step 6: Redo Steps 3–5G times to obtain the MCMC simulated variates of α, β, θ, Rt, or ht (say κ) as κg=αg,βg,θg,Rgt,hgt for g=1,2,…,G.Step 7: Use outputs in Step 6, compute the Bayesian estimates ofκ under SEL and LL functions given in (13) and (14), respectively, as:(24)κ˜S=∑g=G0+1GκgG−G0,And,(25)κ˜L=−1υlog∑g=G0+1Gexp−υκgG−G0,υ≠0,where G0 is the burn-in period.Figure 2
Plots of the conditional distributions ofα, β, and θ.


### 3.4. Credible Intervals

To construct the Bayes credible interval (BCI) and highest posterior density (HPD) credible interval for the parameter of interest, the MCMC simulated variates of the target parameter are used. Under HPD credible interval, for every point inside the interval, the study distribution must be greater than that for every point outside the interval. Therefore, it has the shortest length among all possible credible intervals. According to the procedure proposed by Chen and Shao [24], the HPD credible interval can be constructed. Now, to establish the BCIs (or HPD credible intervals) of the unknown parameters α, β, and θ or the survival characteristics Rt and ht, for short say κ, one can be easily perform the following process:Step 1: Order the simulated MCMC variates ofκg for g=G0+1,…,G, as κG0+1,κG0+2,…,κG.Step 2: Set the level of significanceτ.Step 3: Obtain the1001−τ BCI of κ as κG−G0τ/2,κG−G01−τ/2.Step 4: Obtain the1001−τ% HPD credible interval κ is given by,(26)κg∗,κg∗+1−τG−G0,where g∗=G0+1,…,G is chosen, such that,(27)κg∗+1−τG−G0−κg∗=min1⩽g⩽τG−G0κg+1−τG−G0−κg.Here, x denotes the largest integer less than or equal to x.


## 3.1. Prior Information and Loss Functions

Since the gammaG⋅ density provides various shapes based on parameter values and is flexible in nature, so the utilizing of independent gamma priors are relatively simple which may yield to results with more explicit posterior density expressions Muse et al. [19]; for more details, it can be seen in [14, 20, 21]. Thus, the gamma conjugate priors Gαa1,b1, Gβa2,b2, and Gθa3,b3 are used to adapt the support of MOGE parameters α, β, and θ, respectively. However, the joint prior PDF of α, β, and θ becomes (16)πα,β,θ∝αa1−1βa2−1θa3−1e−b1α+b2β+b3θ,α,β,θ>0,where the hyperparameters ai,bi,i=1,2,3, are chosen to reflect prior knowledge about the unknown parameters α, β, and θ, and they assumed to be known as non-negative.The choice of (symmetric and/or symmetric) loss function is an important issue in Bayesian analysis. So, to develop the objective estimates, two well-known loss functions namely, squared-error loss (SEL) and linear-exponential loss (LL), are considered. However, any other loss function can be easily incorporated. The SEL function is defined as(17)ℑSϕ,ϕ˜=ϕ˜−ϕ2,where ϕ˜ being an estimate of ϕ. Under (13), the Bayes estimate ϕ˜ is the posterior mean of ϕ.On the other hand, the LL function,ℑL⋅, is defined as (18)ℑLϕ,ϕ˜=expυϕ˜−ϕ−υϕ˜−ϕ−1,υ≠0.The sign and magnitude ofυ represent the direction and degree of symmetry, respectively (if υ>0, overestimation is more serious than underestimation and υ<0 means the opposite). Using (14), the Bayes estimator ϕ˜L of ϕ is given by:(19)ϕ˜L=−υ−1lnEϕe−υϕ|x,υ≠0,provided the above exception exists, and is finite. When υ close to 0, the LL function is approximately the SEL function and it can become almost symmetric, as can be seen in Pandey and Rai [22].


## 3.2. Posterior Analysis

Using (6) and (12), the joint posterior PDF of α, β, and θ can be written as (20)πα,β,θ|x=K−1αa1+m−1βa2+n−1θa3+m−1e−αb1+βb2+θb3+mx¯×∏i=1mψiα−1θ1−ψiαθRiβ+β¯ψiαθ−Ri+2,where K is the normalizing constant of (15).Due to nonlinear expression in (6), the closed-form of the marginal PDF for each unknown parameter is not possible. Thus, we purpose to apply some simulation algorithms such as the MCMC methods to approximate the Bayesian estimates and to find associated credible intervals. So the MCMC methods have been widely used in Bayesian computational analysis. At first, from (15), we derive the conditional posterior distributions of α, β, and θ, respectively, as (21)πα∗α|β,θ,x∝αa1+m−1e−αb1∗θ∏i=1m1−ψiαθRiβ+β¯ψiαθ−Ri+2,(22)πβ∗β|α,θ,x∝βa2+n−1e−βb2∏i=1mβ+β¯ψiαθ−Ri+2,And,(23)πθ∗θ|α,β,x∝θa3+m−1e−θb3+mx¯∏i=1mψiα−1θ1−ψiαθRiβ+β¯ψiαθ−Ri+2,where b1∗θ=b1−∑i=1mlogψiθ.It can be seen, from (16)–(18), that the conditional distributions of α, β, and θ cannot be reduced to any standard distribution so that the exact inferences from their marginal densities cannot be easily obtained. Thus, to generate samples from posterior distribution whenever the posterior PDF cannot be reduced to any familiar distribution, the Metropolis–Hastings algorithm can be easily used for this purpose (see Gelman et al. [23] for detail).


## 3.3. M-H Algorithm

The Metropolis-Hastings (M-H) algorithm is a general member of the MCMC family of simulation techniques. In this subsection, the M-H algorithm with normal proposal distributions is conducted to generate samples ofα, β, and θ from (16), (17), and (18), respectively. Precisely, Figure 2 indicates that the distribution of MOGE parameters behaves similar to the normal distribution. Next, to obtain the Bayes estimates and the associated credible intervals, do the following generation steps:Step 1: Putg=1.Step 2: Set initial valuesα0,β0,θ0=α^,β^,θ^.Step 3: Generateα∗, β∗, and θ∗ from (16)–(18) with normal distributions Nα^,σ^α2, Nβ^,σ^β2, and Nθ^,σ^θ2, respectively, as follows:(a)
CalculateQα=πα∗α∗|βg−1,θg−1,x/πα∗αg−1|βg−1,θg−1,x, Qβ=πβ∗β∗|αg,θg−1,x/πβ∗βg−1|αg,θg−1,x and Qθ=πθ∗θ∗|αg,βg,x/πθ∗θg−1|αg,βg,x.(b)
Generate sample variatesu1, u2, and u3 from the uniform U0,1 distribution.(c)
Ifu1⩽min1,Qα, set αg=α∗, else set αg=αg−1.(d)
Ifu2⩽min1,Qβ, set βg=β∗, else set βg=βg−1.(e)
Ifu3⩽min1,Qθ, set θg=θ∗, else set θg=θg−1.Step 4: ObtainRgt and hgt, for a specified time t>0 by replacing α, β, and θ with their αg, βg, and θg, respectively.Step 5: Putg=g+1.Step 6: Redo Steps 3–5G times to obtain the MCMC simulated variates of α, β, θ, Rt, or ht (say κ) as κg=αg,βg,θg,Rgt,hgt for g=1,2,…,G.Step 7: Use outputs in Step 6, compute the Bayesian estimates ofκ under SEL and LL functions given in (13) and (14), respectively, as:(24)κ˜S=∑g=G0+1GκgG−G0,And,(25)κ˜L=−1υlog∑g=G0+1Gexp−υκgG−G0,υ≠0,where G0 is the burn-in period.Figure 2
Plots of the conditional distributions ofα, β, and θ.


## 3.4. Credible Intervals

To construct the Bayes credible interval (BCI) and highest posterior density (HPD) credible interval for the parameter of interest, the MCMC simulated variates of the target parameter are used. Under HPD credible interval, for every point inside the interval, the study distribution must be greater than that for every point outside the interval. Therefore, it has the shortest length among all possible credible intervals. According to the procedure proposed by Chen and Shao [24], the HPD credible interval can be constructed. Now, to establish the BCIs (or HPD credible intervals) of the unknown parameters α, β, and θ or the survival characteristics Rt and ht, for short say κ, one can be easily perform the following process:Step 1: Order the simulated MCMC variates ofκg for g=G0+1,…,G, as κG0+1,κG0+2,…,κG.Step 2: Set the level of significanceτ.Step 3: Obtain the1001−τ BCI of κ as κG−G0τ/2,κG−G01−τ/2.Step 4: Obtain the1001−τ% HPD credible interval κ is given by,(26)κg∗,κg∗+1−τG−G0,where g∗=G0+1,…,G is chosen, such that,(27)κg∗+1−τG−G0−κg∗=min1⩽g⩽τG−G0κg+1−τG−G0−κg.Here, x denotes the largest integer less than or equal to x.


## 4. Optimum Progressive Censoring Plans

In reliability field, the problem to determine the “optimal” censoring scheme from a set of available censoring schemes is main issue. So, to great the information included in the unknown parameters under consideration, Balakrishnan and Aggarwala [25] first discussed the optimal censoring plan via several setups. Following Ng et al. [26], the values of n (total test units) and m (effective sample) must be fixed in advance, then one can be easily determine the optimal censoring design (R1,R2,…,Rm where ∑i=1mRi=n−m) under type-II progressive censoring by considering the following optimum criteria:(i)
Criterion-A:Minimize⟶traceI−1⋅.(ii)
Criterion-B:Minimize⟶detI−1⋅.(iii)
Criterion-C:Maximize⟶traceI⋅.Regarding to criteria A and B, our goal is to minimize the trace and determinant ofI−1⋅ while regarding to the criterion C our goal is to maximize the main diagonal elements of I⋅, with respect to MLE φ^ of φ. The optimized PCS-T2 plan that provides more information corresponds to the smallest value of criteria (A and B) with the highest value of C-criterion.


## 5. Monte Carlo Simulation

To show the performance of the proposed estimators of the unknown parametersα, β, and θ as well as the survival characteristics Rt and ht, a Monte Carlo simulation study is carried out. Using various combinations of n (total test units), m (effective sample size), and R (progressive censoring), a large 1,000 PCS-T2 samples are generated from the MOGE distribution when the true value of parameters α,β,θ is taken as 0.8,0.5,0.2. Thus, for time t=0.1, the actual value of the reliability and hazard functions is used as 0.91683 and 0.68856, respectively. In this study, some specified values of n, m, and R are also taken in account such as n=30 (small), 60 (moderate), and 90 (large). As soon as the number of failed subjects achieves (or exceeds) m, where the failure proportion (FP), m/n100%, is taken as 30, 60, and 90%, the experiment is terminated.Moreover, for givenn and m, different censoring schemes R to remove survival items during the test are also considered as:(28)CS−1:R1=n−m,Ri=0fori≠1,CS−2:Rm/2=n−m,Ri=0fori≠m2,ifmis even,Rm+1/2=n−m,Ri=0fori≠m+12,ifmis odd,CS−3:Rm=n−m,Ri=0fori≠m.To examine the effects of the prior parametersai and bi for i=1,2,3 on the Bayesian analysis, two informative sets of the hyperparameters are used; called Prior-1 (say P1): a1,a2,a3=1.6,1,0.4 and bi=2,i=1,2,3, and Prior-II (say P2): a1,a2,a3=4,2.5,1 and bi=5,i=1,2,3. These values are specified in such a way that the expected value of prior became the expected value of the corresponding parameter. If one setting ai,bi=0,i=1,2,3, the posterior distribution (20) is reduced in proportion to the corresponding likelihood function (6). So, if one does not have prior information on the unknown parameter, we recommend to use the frequentist estimates instead of the Bayes estimates because the latter are more computationally expensive.To develop the Bayesian calculations, following the M–H algorithm proposed in Section3.3, we generate G=12,000 MCMC samples and then the first G0=2,000 iterations have been discarded as burn-in. Hence, using 10,000 MCMC samples, the average Bayes point estimates and associated BCI/HPD credible intervals of the unknown parameters α, β, θ, Rt, and ht are computed. Here, the Bayesian estimates relative to SEL and LL υ=−3,−0.03,+3 are developed.For each considered setting, the average point estimates (APEs) using the proposed frequentist and Bayes approaches ofα, β, θ, Rt, or ht (say η) are computed by(29)APEη^q=1ℬ∑j=1ℬη^τj,q=1,2,3,4,5,where η^ is the desired estimate of the parametric function η, η^qj denotes the obtained estimate at the j−th sample of the unknown parameter ηq, ℬ is number of generated sequence data, η1=α, η2=β, η3=θ, η4=Rt, and η5=ht.Comparison between different point estimates ofα, β, θ, Rt, or ht (say η) is made based on their root mean squared-error (RMSE) and mean relative absolute bias (MRAB) values using the following formulae, respectively, as(30)RMSEη^q=1ℬ∑j=1ℬη^qj−ηq2,q=1,2,3,4,5,And,(31)MRABη^q=1ℬ∑j=1ℬη^qj−ηqηq,q=1,2,3,4,5.Further, the performances of different type of the1−τ% two-sided asymptotic/credible intervals estimates are compared using their average confidence lengths (ACLs) and the average coverage percentages (CPs) as:(32)ACLηq=1ℬ∑j=1ℬUη^qj−ℒη^qj,q=1,2,3,4,5,And,(33)CPηq=1ℬ∑j=1ℬ1ℒη^qj;Uη^qjηq,q=1,2,3,4,5,where ℒ⋅ and U⋅ denote the lower and upper bounds, respectively, of 1−τ% asymptotic (or credible) interval of the unknown parameter ηq.All numerical computations were performed usingR 4.0.4 software with three useful packages, namely: (i) “coda” package proposed by Plummer et al. [27]; (ii) “maxLik” package proposed by Henningsen and Toomet [28]; (iii) “GoFKernel” package by Pavia [29]. Recently, the same packages are also recommended by Elshahhat and Nassar [30].The simulation results ofα, β, θ, Rt, and ht are presented in Tables 1–10 and are also displayed with heatmaps in Figures 3–7. In each heatmap plot, the “x-lab” display the proposed estimation point (or interval) methods while the “y-lab” represents the different choices of n, FP%, and censoring schemes. For specification, using P1 for example, we have used the notation “SEL-P1” for the Bayes (SEL) estimates; “LL1-P1” for the Bayes (LL) estimates when υ=−3; “LL2-P1” for the Bayes (LL) estimates when υ=−0.03; “LL3-P1” for the Bayes (LL) estimates when υ=+3; “BCI-P1” and “HPD-P1” denote to the BCI and HPD credible intervals, respectively. Also, “ACI-NA” and “ACI-NL” denote to ACI based on NA and NL methods, respectively. All simulation results are presented in Appendix B. Now, we can make the following observations:(i)
In general, all estimates of the unknown parametersα, β, and θ or the time parameters Rt and ht are very good in term of minimum RMSEs and MRABs.(ii)
Asn (or m) increases, the proposed estimate become more better. This pattern is also observed when ∑i=1mRi decreases.(iii)
Due to gamma prior information, the Bayesian estimates perform better than the competing frequentist estimates.(iv)
Bayes estimates using LL function of all unknown parameters are or may be overestimates (or underestimates) whenυ<0 (or υ>0).(v)
RMSEs and MRABs of the Bayes (LL) estimates are smaller than the Bayes (SEL) estimates, in most cases, whenυ close to a positive value.(vi)
On the basis of smallest ACLs and highest CPs, it is noted that the ACI-NA approach is the best compared to those obtained based on the ACI-NL approach, similarly, the interval estimates based on the HPD method are the best compared to those obtained based on the BCI method.(vii)
Comparing CSs 1 and 3, it is observed that RMSEs, MRABs, and ACLs ofα, β, and θ are greater based on CS-3 than CS-1, thus the associated CPs of the same unknown parameters are greater based on CS-1 than CS-3.(viii)
It is also a predictable fact because of the expected duration of the test items using CS-1 (wheren−m subjects at the time of 1st failure observed are removed) is greater than the CS-3 (where n−m subjects are withdrawn at the time of mth failure observed).(ix)
Estimates ofRt and ht have the lowest RMSEs, MRABs, and ACLs as well as the highest CPs based on CS-3 than CS-1.(x)
Since the corresponding variance of P2 is less than the variance of P1, one can be seen that the Bayes (point/interval) estimates performed satisfactory based on P2 than P1 as expected.(xi)
In summary, the Bayesian inference via M-H algorithm of the unknown life parametersα, β, θ, Rt, and ht of the Marshall-Olkin generalized-exponential lifetime model is recommended.Table 1
The APEs (first-line), RMSEs (second-line), and MRABs (third-line) ofα.
tFPSchemeMLESELLLPrior⟶IIIIIIυ⟶−3−0.03+3−3−0.03+33030%10.84650.72420.79550.73050.72430.71000.80220.79570.77590.30010.15080.08980.10810.11900.13610.03060.03070.05570.27900.14840.05120.13470.14820.16810.03440.03570.039420.90380.82810.80560.83600.82820.80500.81240.80570.78920.36620.11130.09490.06320.07190.09060.03500.04360.06590.31440.09240.07290.07850.08880.10770.04270.05240.072030.93730.74760.82410.75470.74770.72680.83120.82420.80260.52910.11480.09430.05990.06950.09190.03210.03060.05360.39340.08610.05790.07440.08580.10850.03910.03600.056360%10.86690.76510.82010.77640.76530.73960.82710.82020.80230.27990.11050.08790.06980.07660.09560.00780.01860.05030.25730.09480.03120.08690.09470.11250.00870.01570.053120.87420.80390.79800.81360.80400.77710.80500.79810.78270.27550.10290.08470.03710.03510.06330.01620.02050.04820.25680.06400.04940.04500.04110.06210.01900.02310.040430.84470.79630.73120.80340.79640.78370.74050.73140.71320.27010.10450.08190.04620.05590.09070.01990.02780.04520.25730.07000.05350.05670.06530.09140.01990.03270.046690%10.86410.79060.68130.79610.79070.77670.69220.68140.66550.24820.10940.07710.02450.03770.07690.00770.01760.04420.23820.07970.02710.02950.04340.07550.00650.01290.030120.85800.72880.75800.73720.72900.71380.76590.75810.74240.25660.09940.08480.01650.02310.05850.01530.01930.04630.23830.06020.04850.01760.02240.04840.01770.02080.038030.87140.77850.81760.78410.77860.76560.82610.81770.80010.25760.09260.07960.02890.02900.05210.01980.02480.04260.24120.06580.04340.03320.02680.04300.02070.02380.03466030%10.89040.84700.80630.85430.84720.82800.81410.80640.78240.20910.10160.09560.05630.05320.06090.01600.02030.06020.20030.08420.04360.06780.06070.07180.01860.02280.049420.88190.75750.76430.76370.75760.74060.77040.76440.74910.22310.09620.08910.04930.05610.07600.03020.03830.06310.20980.06930.05420.06080.06710.08300.03700.04450.063630.87700.77920.84810.78470.77930.76610.85510.84820.82910.25920.10150.08430.06540.06170.06220.02060.03030.06020.22620.09560.03410.08050.07280.07370.02350.03060.050760%10.86900.81540.79350.82190.81550.79580.79940.79360.77610.19400.08900.08350.02380.02550.05280.00900.02000.05320.18360.04660.02480.02770.03020.05190.00880.01580.037420.85920.74620.81430.75130.74630.73360.82000.81440.79970.19190.09330.08770.04190.04550.07400.02990.02980.05130.18290.06720.04750.05110.05300.07430.03550.03520.054330.84610.77550.85810.78120.77560.75940.86440.85820.84220.18380.10130.08030.05690.05390.06150.01910.02530.05090.17730.08640.03400.06890.06180.07270.02220.02590.047590%10.86210.79340.79120.79860.79350.78100.79590.79120.77960.18010.07430.07150.00980.01860.04180.00820.01710.03950.17210.03170.02300.00900.01440.02910.00590.01160.026020.85680.82210.83390.82840.82220.80320.84080.83410.81320.18480.09140.08010.03690.03970.05600.02190.02280.04120.17670.05320.04430.04540.04600.06330.02530.02700.042130.84360.81020.77700.81640.81030.79200.78240.77710.76320.18080.08580.07820.02010.02920.05500.01660.02260.04920.17260.03940.03100.02200.02860.04600.01920.02650.04249030%10.89780.78930.76820.79450.78940.77530.77370.76830.75350.19150.08400.07830.02720.03530.06130.01740.02590.05120.18050.03970.02590.03290.03960.05810.01930.02520.041620.87050.77420.79400.77940.77430.76070.79940.79410.77910.18890.08250.08090.02330.03170.05760.01360.01970.04750.18160.03570.03020.02800.03480.05310.01590.02160.040930.87510.80200.80110.80760.80210.78660.80670.80120.78560.19100.08300.07480.01810.02760.05510.01170.01930.04700.17720.03260.02470.01990.02650.04470.01380.01970.038560%10.87620.79180.78050.79780.79190.77460.78570.78060.76670.16860.08350.07700.02500.03320.05880.00940.02030.04960.15980.03700.02530.03010.03670.05480.00740.01450.041020.85160.77210.80360.77760.77220.75750.80930.80370.78760.15900.07970.07840.02210.03050.05550.00860.01860.04730.15340.03220.02010.02580.03210.04920.00820.01470.033230.85070.77870.78840.78410.77880.76420.79320.78850.77640.15050.08040.07290.01350.02090.05010.01020.01890.04290.14930.02710.02070.01570.02170.04140.01200.01730.029690%10.84890.77970.77050.78460.77980.76720.77590.77060.75620.15040.08250.07570.01600.02510.05360.00890.01950.04690.14600.02260.01730.01790.02430.03600.00680.01320.030920.84540.79810.79200.80350.79820.78350.79700.79210.77910.14130.07800.07450.01050.01890.04590.00830.01790.04310.13730.02310.01930.01200.01800.03590.00600.01200.028130.84710.80100.80070.80710.80110.78450.80580.80080.78750.14740.08000.08000.01270.01950.04680.01010.01680.04100.14320.02910.02810.01490.02070.03930.00850.01440.0334Table 2
The APEs (first-line), RMSEs (second-line), and MRABs (third-line) ofβ.
nFPSchemeMLESELLLPrior⟶IIIIIIυ⟶−3−0.03+3−3−0.03+33030%11.99850.56290.54270.56910.56300.55710.54710.54280.53901.97260.08940.07490.06920.06300.05710.04800.04320.04022.47220.13270.10670.13830.12600.11420.09410.08550.078821.84320.58340.55140.58700.58340.58040.55860.55150.54491.51050.09570.08590.08740.08360.08050.07410.06800.06432.14970.16680.13550.17390.16680.16080.14540.13500.128236.05280.53040.40640.53360.53040.52790.41450.40650.40091.65220.11950.11610.08990.09040.09610.08350.09540.10012.94690.20790.20700.17100.17770.19140.15390.18700.198360%11.03930.54790.49520.54900.54800.54700.50550.49530.48871.79480.06720.05460.04930.04820.04720.03810.03920.03931.55250.09730.09610.09800.09590.09400.07290.07620.078020.91050.55650.56750.55830.55650.55480.57270.56750.56411.45390.08520.06600.05920.05660.05490.05860.05220.04581.31150.13500.11290.11710.11290.10970.11670.10300.089831.69280.41100.56780.42430.41110.40430.57150.56780.56412.11200.08410.06120.07150.06790.06420.03530.03160.02862.15060.14200.08030.14290.13560.12820.06710.06090.055790%10.82500.51880.46190.52140.51880.51690.46350.46190.46061.10760.05010.04290.02380.02010.01770.02820.01680.01551.11410.08730.04900.04270.03760.03390.04080.03050.028920.80930.48510.50930.48690.48510.48370.51170.50930.50710.92110.04090.03630.01580.01630.01710.01540.01250.01001.07250.06230.04840.02950.03020.03260.02330.01860.014230.79590.48100.51450.48660.48110.47700.51800.51450.51220.89400.05360.04610.02610.02540.02700.02290.01700.01361.10520.06960.04130.04860.04600.04780.03590.02900.02456030%10.82890.46090.52710.46960.46090.45630.52920.52710.52541.27920.07750.06480.04840.04580.04570.03020.03120.03471.18950.11020.09730.09550.09100.08740.05840.06060.068820.82210.51960.55240.52260.51970.51770.55340.55240.55151.00560.06740.05820.05360.05250.05220.02460.02060.01811.14910.11500.10690.10480.10490.10310.04510.03930.035332.39640.49630.43630.49960.49640.49430.44510.43630.43191.26050.09180.05050.02750.02790.02890.06320.06610.06911.78700.15500.07250.05080.05120.05510.11590.12730.136160%10.64040.54080.46960.54400.54080.53870.47560.46970.46560.62400.05850.05820.04520.04350.04400.02800.02770.02580.73390.09100.08170.08800.08170.08760.05190.05430.050920.65070.45150.51020.45520.45150.44900.51450.51020.50720.64290.05030.04850.04750.05040.05160.02030.01360.00930.76730.05640.08970.04770.09700.10200.02900.02050.014530.87050.47460.52830.47800.47470.47250.53250.52830.52571.23450.05520.05000.02120.02070.02170.03550.02940.02931.50160.07260.05950.04040.03850.04210.06510.05660.056290%10.57370.51000.54550.51390.51000.50780.54780.54550.54380.45760.04580.04470.04380.04140.03900.01970.01290.00910.60490.06390.03170.08200.07940.07730.02770.02010.015520.57920.48800.50300.49200.48810.48570.50720.50300.50040.45280.04780.04480.01820.01560.01570.01790.01070.00680.61100.04400.04620.03420.02940.02910.02080.01400.010530.61530.48140.47410.48550.48140.47900.47800.47420.47190.53900.05230.04210.01430.01000.00850.02840.02840.02610.71380.05960.03480.02350.01840.01590.05310.05250.05149030%10.69180.49630.49650.49880.49630.49470.49890.49660.49501.00700.04000.03700.01840.01380.01100.01430.00970.00700.87840.02990.03050.02710.02210.01900.01860.01500.013220.66030.51380.49040.51630.51380.51220.49360.49040.48850.71140.04420.04350.02830.02450.02230.02220.01630.01310.84850.04690.02850.05180.04670.04350.03430.02760.024531.46940.49880.50480.50260.49880.49670.50820.50490.50291.20010.04480.04250.01810.01540.01490.01760.01120.00751.01830.04250.02500.03340.02880.02720.01900.01360.010660%10.53220.51100.50590.51360.51110.50950.50830.50590.50440.40710.03870.03650.01590.01070.00830.01200.00820.00670.53940.02770.01980.02250.01810.01580.01660.01250.009920.55730.51930.51380.52120.51940.51810.51710.51380.51180.39030.04400.03910.02340.02050.01870.01710.01430.01300.54670.03990.03880.04240.03870.03620.03170.02700.023730.62490.49000.50050.49350.49010.48790.50330.50050.49880.80330.04420.04030.01750.01040.00760.01550.00960.00611.07690.02510.02040.02470.01780.01410.01870.01260.008890%10.54750.50750.50080.51030.50750.50570.50290.50080.49940.34300.03770.03450.01330.00980.00750.01100.00800.00590.48280.02280.01710.02060.01500.01140.01650.01180.008720.53320.49690.52340.50040.49690.49480.52590.52340.52170.32580.04260.03780.01980.01560.01380.01650.01100.00880.47290.02960.02810.03250.02760.02520.02640.01990.016630.54890.50360.50490.50730.50360.50150.50790.50490.50310.41350.04380.03900.01680.01010.00640.01370.00880.00600.56690.02030.01980.01980.01260.00850.01620.01020.0067Table 3
The APEs (first-line), RMSEs (second-line), and MRABs (third-line) ofθ.
nFPSchemeMLESELLLPrior⟶IIIIIIυ⟶−3−0.03+3−3−0.03+33030%10.37170.15970.12970.16470.15970.15620.13650.12970.12500.39980.09430.06710.03980.04210.04450.06760.07190.07581.16660.40280.25520.18300.20150.21880.31740.35130.375020.50220.09240.22620.10260.09250.08680.23410.22620.22120.63590.12990.07120.10310.10930.11390.04620.03840.03391.78610.60280.21240.48700.53760.56580.20900.18170.164531.12300.13310.19820.13780.13320.12980.20000.19820.19681.80960.08540.06540.06550.06860.07130.03540.03620.03811.68110.37900.23780.31080.33420.35090.16510.16770.184960%10.27170.16720.22350.17300.16730.16360.22560.22350.22180.18250.06530.04300.03460.03560.03770.02730.02460.02250.63360.23010.14410.16050.16410.18180.12810.11770.109220.29580.16430.23630.17190.16430.15910.24180.23630.23290.24290.07530.06560.03760.03960.04300.04270.03110.02450.76440.27760.20080.17380.17980.20450.17050.13120.105830.34770.19670.16650.20170.19670.19220.17240.16660.16300.42420.06830.04010.00740.00740.00990.01040.00890.00821.44170.31040.10880.03130.03160.04400.04270.03860.036390%10.24430.21570.19690.21910.21570.21330.19860.19700.19570.13920.04670.03160.02340.01800.01460.00900.00740.00670.48360.12470.07880.09550.07850.06670.03940.03440.031820.25460.21670.21850.21960.21680.21480.22150.21850.21650.15390.04330.04500.02180.01780.01520.02540.02080.01790.53520.12020.11010.09780.08380.07380.10730.09260.082730.25060.26210.19670.26330.26210.26090.19950.19670.19470.16980.05640.03350.06350.06220.06100.00950.00670.00640.63280.21470.10310.31670.31050.30470.03770.03070.02976030%10.26330.18890.14080.19030.18890.18780.14690.14080.13720.19080.08270.05740.05860.06140.06390.02430.01490.01320.61700.35390.10950.26830.29600.31420.09650.06850.061420.32010.14520.21570.15260.14520.14060.22070.21580.21290.29670.08420.05380.05410.05720.06060.02920.02040.01570.88390.33420.11290.24740.27380.29720.10370.07890.064530.67000.21260.19770.21470.21260.21120.19940.19770.19651.10810.06260.05430.04080.04310.04520.02180.01330.00831.28870.25530.14970.19210.21030.22370.07710.05080.033660%10.23050.19450.19900.20110.19460.19120.20390.19900.19580.11060.05510.04770.03980.03470.03190.01930.01270.01010.40230.16750.10500.18530.16760.15660.06980.05380.046820.24090.21210.22670.21350.21210.21090.22980.22670.22450.14420.06400.04280.04670.04840.05000.01820.01670.01680.48700.28080.11130.21990.23660.24720.08550.07760.076030.26760.15790.21010.16160.15790.15530.21540.21020.20670.27630.05090.03330.02190.02070.02130.00930.00700.00601.04590.14640.07340.10290.09500.09670.03900.03220.028790%10.21210.20980.23350.21400.20980.20730.23710.23350.23130.08460.05260.03150.01970.01280.01230.01280.01270.00870.31630.14790.09710.06980.05930.05790.06010.04920.036620.21480.15270.18750.15600.15270.15060.19040.18750.18560.09090.05020.03190.03240.02810.02520.01480.01280.01130.33450.14780.09310.14920.13360.12250.06760.06040.054630.21570.18360.19730.18770.18370.18100.19920.19730.19600.11150.03680.03140.01830.01470.01240.00890.00680.00580.42290.07210.06910.07370.06320.05610.03750.03110.02749030%10.24350.21010.19280.21260.21010.20850.19570.19280.19110.13840.04240.03980.01750.01290.01010.01650.01240.01150.47710.08960.05430.07630.06400.05840.04430.05880.053820.26160.21030.19860.21340.21040.20850.20090.19860.19720.19910.04510.04130.02430.01900.01600.01530.01180.01080.63240.10750.08400.10250.08630.07640.06850.05600.050630.47310.20100.20660.20350.20100.19950.20900.20660.20520.78700.04940.04220.03670.03330.03130.01940.01400.01070.99130.16310.08240.17550.16300.15490.06150.04650.037160%10.21070.19250.20570.19570.19260.19060.20860.20580.20400.08370.04100.03820.01690.01370.01270.01620.01100.00770.31320.08530.05250.06280.05040.04260.04320.02910.020420.22030.18920.19330.19290.18920.18710.19640.19330.19150.09730.04200.03570.01980.01460.01440.01250.00900.00710.35220.09480.05490.08130.06900.06660.04860.03880.033030.20940.19400.23260.19620.19400.19260.23510.23260.23100.20800.04420.03660.02630.02180.01930.01260.01040.00960.83950.10300.07370.11670.10240.09360.05770.04920.045190%10.20930.20570.20600.20880.20570.20390.20860.20600.20440.06720.04060.03810.01610.01120.00810.01540.01020.00660.25840.05050.04930.04480.03220.02440.04010.02890.019620.20870.20210.21720.20440.20220.20070.22050.21730.21530.07180.04100.03540.01740.01400.01060.01160.00730.00470.26940.06070.05100.06720.05180.04250.03310.02260.016230.20530.20930.22050.21230.20930.20740.22330.22050.21870.09250.03690.03550.01560.01110.00820.01290.00840.00580.35630.04530.04340.04540.03370.02630.04290.03120.0241Table 4
The APEs (first-line), RMSEs (second-line), and MRABs (third-line) ofRt.
nFPSchemeMLESELLLPrior⟶IIIIIIυ⟶−3−0.03+3−3−0.03+33030%10.92120.90940.93650.91510.90950.89420.94300.93660.91770.14540.09840.07970.05640.06250.07610.00950.02030.06960.13940.06730.02360.06120.06720.07880.00990.01660.044420.91910.95650.91120.96360.95660.93490.91640.91120.89910.14610.10130.07420.04800.04560.07170.02090.02800.04730.13770.06710.03250.05100.04610.06120.02180.02690.039030.92400.92060.90160.92850.92070.89290.90700.90170.88690.14250.10060.08540.03240.03670.07360.01240.02240.05130.13650.03800.03400.03460.03800.05240.01070.01660.032660%10.92370.91460.90960.92260.91480.88820.91350.90960.90070.13850.09420.06580.02800.02840.05100.00900.01790.04770.13420.04500.01760.02860.02930.04610.00600.01080.027120.92000.92840.90970.93630.92850.90040.91380.90970.90070.13640.10020.06560.03620.04250.06150.01130.01960.04190.13150.04780.02980.03900.04460.05660.00970.01430.026430.92460.89790.90740.90010.89790.89420.91410.90750.88980.13710.07930.06510.01790.02440.04660.01050.01840.05020.13280.03190.02360.01830.02310.04490.01080.01400.029790%10.92420.90720.85510.91130.90730.89780.86070.85520.84460.13460.08890.06430.00930.01680.03530.00740.01660.03370.13030.03140.01710.00650.01040.02070.00520.01060.020220.92160.87580.89210.88110.87590.86490.89680.89210.88100.13480.08190.05420.02200.02620.05900.01060.01780.03480.13050.04010.02460.02210.02680.05550.00790.01170.021030.92400.88200.91950.88510.88200.87570.92350.91960.91080.13540.06360.04840.01610.02130.02750.00690.01470.02940.13160.02280.02130.01630.02060.02470.00450.01080.02286030%10.92680.92280.93270.92660.92290.91430.94030.93280.90730.13210.09850.07780.02550.02780.06560.01290.01990.04820.12850.04360.02370.02580.02880.05370.01320.01600.028920.92060.92000.90310.92610.92010.90240.90820.90310.88910.12910.08440.07770.03410.03960.05450.01580.02160.04860.12480.04090.02950.03650.04080.05090.01600.01990.034430.92560.90110.91670.90470.90110.89330.92050.91680.90790.12950.07650.06580.01800.02100.04350.01090.01830.03840.12580.02990.02060.01820.01940.03060.01110.01600.026860%10.92480.92230.90420.92810.92240.90560.90950.90430.89030.12850.08220.06290.01470.02220.04870.01230.01590.02910.12540.02850.02320.01500.01960.03700.00800.01370.024620.92160.87930.91020.88340.87940.87010.91380.91030.90220.12680.07740.07700.02090.02230.05160.01180.01960.04020.12350.03560.01770.02110.02310.03780.00940.01490.030230.92190.91440.92960.91960.91450.90050.93310.92970.92220.12710.07620.06220.01400.02030.04300.00920.01690.03090.12380.02220.01750.01330.01710.02840.00930.01350.025890%10.92400.90960.90710.91290.90960.90240.91040.90710.90000.12700.05860.05840.00970.01610.03010.00830.01480.02880.12420.01250.01140.00700.01060.01840.00470.00820.016020.92260.93070.92440.93580.93070.91700.92970.92450.91010.12610.07450.06130.01370.02040.04260.00840.01540.03090.12290.02720.01620.01410.01930.03600.00560.00930.017930.92070.91650.90130.92170.91660.90270.90520.90140.89220.12630.06330.06060.01390.01870.03530.00850.01480.02670.12300.01820.01700.01270.01690.02570.00850.01300.02249030%10.92950.90500.90320.90910.90510.89530.90790.90330.89110.12750.07690.06770.01420.02240.04550.01070.01850.03850.12470.01780.01330.01300.01780.03020.00850.01380.023820.92070.90280.90900.90700.90280.89260.91320.90910.89910.12280.07200.06960.01270.02060.04050.00880.01690.03650.12010.01940.01540.01080.01530.02640.00540.00990.023330.92460.91200.91080.91620.91210.90210.91510.91090.90060.12430.07120.06760.01590.02170.04060.01060.01900.03510.12170.01840.01960.01580.01940.02420.00780.01240.027360%10.92570.91250.90500.91780.91260.89840.90920.90510.89500.12390.07320.06660.01200.02100.04470.01070.01840.03740.12100.01790.01300.00970.01480.02920.00830.01280.023520.92080.90940.91680.91410.90950.89740.92130.91690.90580.12260.06840.06700.00980.01710.04030.00830.01620.03610.12010.01440.01310.01000.01430.02610.00520.01020.020830.92180.90540.89900.90970.90540.89490.90240.89910.89180.12140.06890.06140.00980.01850.03950.00930.01630.03420.11900.01440.01410.00710.01190.02400.00670.01100.022190%10.92320.90550.90040.90920.90550.89690.90490.90050.88910.12300.07170.06350.01090.01890.04400.00910.01790.03470.12030.01530.01260.00830.01280.02810.00840.01230.021720.92120.91040.90840.91430.91050.90130.91210.90840.90000.12140.06450.06250.00860.01590.03360.00810.01570.03240.11880.01210.01170.00560.00980.01980.00520.00920.018430.92210.90940.90740.91400.90940.89800.91100.90740.89940.12130.06670.06090.00850.01660.03660.00830.01600.03200.11870.01400.01160.00700.01130.02220.00640.01030.0190Table 5
The APEs (first-line), RMSEs (second-line), and MRABs (third-line) ofht.
nFPSchemeMLESELLLPrior⟶IIIIIυ⟶−3−0.03+3−3−0.03+33030%10.67860.64900.47301.00810.65030.60530.87720.47460.41350.41890.39830.30370.35060.22970.27580.25500.09780.12350.46430.42150.19690.38220.31120.39950.27730.13510.179220.71980.31300.70490.79860.31490.25351.06130.70640.63470.43000.42580.32640.41340.38630.35590.36090.14000.08740.42250.39550.25920.41860.32670.31820.35590.19180.124330.70280.56040.79020.97260.56190.51211.03980.79110.74880.40350.35190.27610.32280.24990.22260.23780.11380.09160.39730.29680.20130.33620.25960.25630.22160.14900.132960%10.63730.61890.73111.07220.62060.56520.92400.73180.70190.28170.24890.22730.29670.22060.25240.20530.06960.02350.33260.26140.13570.25400.26300.26440.24210.06550.028120.67910.53510.71581.00680.53680.48750.97940.71680.66780.27230.27600.21960.21660.23100.20260.20840.09090.06820.30640.26250.23740.24330.23630.29210.22900.09600.086330.62950.82160.66250.91910.82210.79051.08460.66420.59710.25690.24160.22930.25190.15470.17960.23070.07120.08250.30640.26010.18670.27920.20700.22310.21020.09820.118890%10.63540.73111.00630.98780.73200.69681.22381.00740.93950.24350.21810.20730.21600.07350.08350.20310.06850.01080.28300.24000.12800.23460.10070.12090.23160.06310.011920.64780.91530.81951.17310.91640.84941.09310.82060.77420.23920.22360.20480.20420.17670.16250.18680.07840.04120.27830.24210.19510.20370.22100.15940.18760.07430.050430.63290.93570.64201.06170.93620.91100.87710.64290.60680.25200.21010.18110.23090.13680.10210.22380.06740.06050.29060.25890.17220.24190.19400.14810.20470.08990.08746030%10.65720.63340.50080.81370.63410.60850.95920.50250.45030.27700.22160.19360.24280.20620.23920.22690.09650.08130.31120.28770.18980.24080.27600.24610.23340.10310.116220.70690.58810.73860.97640.58960.54091.09290.73980.70370.25090.23350.19530.22910.16000.16690.20320.10930.12010.26860.25600.11080.21250.17550.21440.18710.14840.172430.66640.77340.68810.97060.77410.74820.87780.68880.66400.22400.21680.21070.21250.11480.13140.18380.07970.07780.24950.17890.15400.20960.15460.19010.20330.10710.112360%10.64590.61040.74901.00760.61180.57271.06450.75020.70220.20240.19420.13160.18600.11000.11750.18460.07530.03180.23900.22360.15660.19530.15090.16820.16220.08950.033820.66940.91380.72681.09090.91450.88490.94770.72760.69620.18120.16450.14030.17610.12950.16560.15580.08550.01920.20990.17190.12480.18440.15220.20520.17630.07440.022030.66360.64730.58870.94320.64830.61120.84120.58960.55770.19030.17280.16860.18140.09760.07580.17540.07120.05360.21940.16730.11290.19950.12430.10410.17230.09820.077090%10.64740.71800.74010.94050.71880.69310.96830.74090.71370.17900.14050.12030.18000.07520.02720.15480.05970.00990.21220.09680.06910.16620.07600.03650.15970.04450.008220.65560.54210.60300.85060.54320.50950.90750.60410.56980.16680.16080.13930.15370.12880.14960.15050.06750.01460.19160.15620.12200.15890.13280.16000.14420.05670.013630.66400.65650.76570.96800.65760.61690.96710.76640.74160.17190.16500.13410.16670.09180.06000.16160.05200.02790.19750.12600.09660.19570.11300.08660.15220.06100.03679030%10.63730.74680.73890.98900.74760.72051.01400.73980.71060.21880.18620.16470.20790.09630.05130.16620.07910.04610.25410.12260.08640.19920.11350.07330.17270.08760.061820.69820.75290.71461.01050.75380.72620.94980.71540.68880.18490.15490.13860.15430.08760.06400.14800.06220.02680.20730.14160.10100.16750.09750.09020.14810.06410.037330.67280.69610.70500.93580.69690.66980.94880.70590.67850.17620.15990.14230.14810.12360.08800.13200.07240.02360.20170.16640.08490.16520.16680.12740.14010.06500.030460%10.65040.67710.73890.98360.67820.64600.99260.73980.71210.16480.16050.15030.15670.07910.03290.15550.07350.02570.19100.11220.08480.15240.08570.04640.14920.07740.034220.67590.69280.65490.99330.69380.66290.92780.65590.62640.15090.14480.12580.13600.07150.03950.12910.05850.01180.17210.09480.07100.14940.09470.05470.14020.04990.020530.67220.73140.80270.96860.73220.70501.00550.80340.77630.14620.14190.11240.13590.07340.02810.12490.06960.02280.17060.12770.08410.14400.07680.03930.12680.06340.029690%10.65260.73770.76580.97470.73860.71191.02710.76670.73900.14590.14220.12590.15260.07640.02510.13940.06960.02490.16860.08890.07830.14410.07440.03390.14160.07260.032020.67020.70820.72490.93230.70890.68330.96520.72580.69800.13910.12320.11420.13050.06390.01210.12020.05620.01050.16160.08800.06650.14710.05400.01440.13880.04870.013730.66260.71480.74060.98110.71570.68220.96490.74140.71560.13800.12590.11090.12110.06110.01900.12010.05830.01690.16370.08820.08330.14070.06260.02390.10910.05870.0226Table 6
The ACLs/CPs of asymptotic/credible interval estimates ofα.
nFPSchemeACI-NABCIACI-NLHPDPrior⟶IIIIIIACLCPACLCPACLCP3030%11.51360.9630.31980.9880.25070.9992.01530.9350.27050.9590.19420.97221.62410.9630.30850.9880.27100.9992.04660.9330.24960.9570.23160.97031.86420.9300.29660.9540.28040.9672.07520.9310.25690.9550.23930.96860%11.25200.9660.29670.9910.22570.9991.41320.9420.26440.9660.17060.97921.22880.9700.28350.9950.26680.9991.36720.9470.24010.9720.22020.98431.15030.9480.27760.9730.23440.9851.65740.9350.23200.9590.17900.97290%11.11560.9820.23890.9970.20690.9991.22040.9580.19920.9830.15950.99621.10420.9780.26300.9940.24200.9991.20400.9490.20740.9740.19590.98631.12520.9660.23470.9910.21470.9991.26210.9410.17990.9650.16970.9786030%11.05780.9710.26070.9970.24950.9991.13750.9390.20030.9640.19680.97721.04230.9650.23270.9910.20830.9991.11560.9430.18410.9680.15770.98231.33120.9530.24420.9790.20280.9921.60500.9300.19170.9550.15970.96860%10.87040.9790.20090.9950.18970.9990.91270.9470.14520.9730.13490.98620.83330.9710.22490.9970.19620.9990.87130.9490.16870.9730.14600.98530.94570.9560.22410.9820.19080.9951.03450.9380.17700.9630.13860.97690%10.76580.9900.19030.9970.17300.9990.79450.9610.14570.9870.12940.99920.75140.9870.20590.9960.19250.9990.77890.9490.16000.9740.14040.98830.77660.9690.19270.9950.18480.9990.80870.9420.14190.9670.13340.9809030%10.87490.9770.18650.9970.17300.9990.91560.9450.13380.9730.12430.98820.82020.9680.18670.9970.18010.9990.85480.9450.13760.9730.12830.98831.03930.9580.20300.9870.17990.9991.14900.9380.15170.9660.12920.98160%10.70250.9810.18180.9990.16740.9990.72320.9490.13220.9770.11770.99220.66790.9850.17830.9990.17020.9990.68660.9500.12810.9790.11930.99330.79370.9740.18690.9990.16580.9990.80150.9400.13580.9680.11830.98390%10.62170.9960.18170.9990.15750.9990.63690.9690.13180.9980.11040.99920.60810.9980.17310.9980.16640.9990.62240.9610.12470.9900.11880.99930.63640.9810.17300.9990.16450.9990.65310.9470.12300.9750.11790.990Table 7
The ACLs/CPs of asymptotic/credible interval estimates ofβ.
nFPSchemeACI-NABCIACI-NLHPDPrior⟶IIIIIIACLCPACLCPACLCP3030%12.10290.9350.24700.9590.18850.9702.89780.9460.21560.9650.17370.97622.51560.9330.22010.9680.18470.9792.96650.9470.20800.9700.16930.98132.67280.9300.26280.9410.16670.9513.28310.9310.22380.9450.16120.95560%12.06460.9470.19410.9680.11590.9752.84500.9590.18830.9680.09980.97922.14570.9490.18110.9690.12100.9802.90670.9530.15410.9750.10880.98632.25940.9350.22880.9490.15980.9593.11350.9350.20300.9670.14360.97890%12.04770.9610.12890.9680.10220.9792.42180.9670.10990.9700.09250.98122.03450.9590.14010.9750.10550.9862.78360.9680.12660.9820.08990.99332.07790.9420.19860.9560.13690.9672.80340.9520.17460.9770.11340.9886030%11.22600.9420.20370.9660.14020.9751.33980.9600.16960.9660.11550.98121.54920.9430.15700.9680.15010.9841.48730.9760.13510.9730.12470.98631.62450.9360.18870.9460.13530.9571.82390.9460.15310.9650.11280.96160%11.19000.9490.19130.9690.12960.9841.14090.9680.16520.9750.11290.98421.29700.9590.14350.9710.12330.9851.25370.9810.11770.9830.09680.99231.15200.9410.14080.9550.12590.9651.38750.9650.11520.9710.10050.98390%10.89320.9640.12730.9780.11360.9841.03090.9730.10070.9790.09480.98620.97010.9700.12250.9760.09470.9921.13680.9880.10100.9850.08090.99931.01420.9470.13390.9660.12020.9721.21750.9780.10740.9800.09270.9949030%10.66420.9450.10610.9790.09050.9850.71520.9790.08330.9770.06760.98520.75680.9450.10330.9750.09000.9870.94290.9790.07990.9780.06710.99230.83740.9400.11880.9770.10220.9651.10040.9480.09380.9800.07530.98460%10.52770.9580.10210.9800.08740.9880.62720.9810.08110.9790.06430.99520.59500.9670.10310.9750.08170.9900.68200.9820.07750.9850.06120.99830.80450.9470.10460.9850.09790.9740.85270.9790.07710.9840.07370.99090%10.32160.9690.08820.9830.08000.9980.44310.9860.06580.9800.05890.99920.48850.9810.10190.9880.09960.9960.60320.9890.07700.9870.07160.99830.59970.9580.10370.9860.09730.9850.62630.9800.07600.9860.07240.998Table 8
The ACLs/CPs of asymptotic/credible interval estimates ofθ.
nFPSchemeACI-NABCIACI-NLHPDPrior⟶IIIIIIACLCPACLCPACLCP3030%11.05550.9560.20680.9670.18620.9731.67420.9470.18220.9600.15890.97021.57850.9510.23010.9640.22960.9721.92510.9430.20420.9560.20040.96931.84260.9500.20610.9620.18210.9732.12180.9470.19500.9600.15560.97260%10.62230.9590.17480.9720.13690.9780.91520.9520.15040.9690.11030.97820.74890.9590.22420.9660.17690.9751.20990.9500.18940.9630.14970.97331.41600.9550.18050.9690.13490.9781.87410.9510.15770.9640.11560.97390%10.47880.9650.14910.9780.10360.9850.61360.9560.12240.9710.08710.97420.51260.9670.13780.9720.12910.9780.65980.9510.11680.9640.10650.97631.20410.9570.11610.9720.10290.9841.51740.9620.09790.9740.09510.9756030%10.62540.9600.17500.9690.15450.9741.52860.9480.14550.9650.12100.97720.85380.9580.19750.9710.14600.9781.49000.9620.16680.9740.11710.97831.60510.9590.16800.9630.15610.9751.61170.9580.14090.9650.13380.97660%10.40830.9730.16790.9760.12540.9840.48280.9600.14210.9730.10880.98120.47310.9600.13050.9740.12340.9800.59530.9650.11000.9780.09980.98331.18660.9580.13920.9720.09400.9801.46100.9600.11430.9690.07590.97790%10.31810.9790.12050.9850.09160.9860.35630.9620.09560.9750.07710.98120.33240.9680.11930.9800.09920.9820.37830.9680.09650.9820.08550.98430.45270.9620.09710.9750.09020.9871.02510.9640.07730.9750.07260.9829030%10.49020.9710.11260.9840.10410.9860.96750.9610.08770.9740.07990.98320.62960.9620.10870.9750.10660.9790.92740.9670.08390.9800.07840.98931.21450.9650.11530.9780.10170.9821.44970.9670.09120.9800.07930.98460%10.32220.9760.09790.9860.09590.9870.36270.9670.07220.9800.07070.98420.37070.9760.09540.9890.09450.9910.42930.9720.07130.9850.07090.98630.91490.9680.10030.9810.08630.9900.98670.9690.07560.9820.06430.99290%10.25810.9800.09360.9890.08850.9900.27760.9680.07040.9830.06560.98620.26860.9790.09270.9900.09200.9920.29110.9750.06940.9880.06950.99530.36400.9780.08800.9880.08250.9910.52620.9700.06500.9830.05960.990Table 9
The ACLs/CPs of asymptotic/credible interval estimates ofRt.
nFPSchemeACI-NABCIACI-NLHPDPrior⟶IIIIIIACLCPACLCPACLCP3030%10.24520.9470.21380.9610.19300.9700.18230.9480.19360.9620.17310.97120.22630.9520.19810.9660.16010.9750.16580.9500.16050.9640.15360.97330.23550.9410.22970.9550.21010.9640.21340.9460.17700.9600.16990.96960%10.19680.9540.16370.9680.15550.9770.16390.9500.15930.9640.14490.97320.22150.9440.18650.9580.15220.9670.16210.9470.15240.9610.14270.97030.18480.9490.18040.9630.13860.9720.18570.9480.13490.9620.10330.97190%10.15710.9550.15040.9690.11120.9790.15060.9520.11500.9660.11070.97520.20600.9420.17610.9560.14940.9650.16080.9450.14960.9590.13650.96830.15710.9560.16090.9700.12910.9800.14670.9530.11630.9670.10120.9766030%10.22250.9680.17970.9810.14120.9890.15990.9610.14130.9730.12670.98220.19520.9620.16930.9750.11440.9830.14190.9680.12000.9810.11450.98930.17640.9650.14250.9780.14170.9860.12720.9680.10280.9810.10090.98960%10.18680.9690.14230.9820.12100.9900.13560.9650.12110.9780.10230.98620.16940.9650.16030.9780.10920.9860.12030.9700.11560.9830.10780.99130.17380.9670.14500.9800.13160.9880.12460.9710.11040.9840.09960.99290%10.12710.9720.13040.9850.10960.9930.10970.9740.09110.9870.08960.99520.16640.9690.14410.9820.10770.9900.11820.9730.10920.9860.10480.99430.14080.9680.14650.9810.12830.9890.11300.9760.11510.9890.09610.9979030%10.16410.9640.13910.9770.11650.9880.11660.9600.10320.9720.09640.98420.15420.9700.14760.9830.09330.9940.10680.9680.10180.9810.09340.99230.09690.9680.13980.9810.11290.9920.11530.9740.09920.9870.09790.99860%10.15020.9720.13890.9850.10020.9960.11440.9760.10030.9890.09560.99820.14110.9740.13980.9870.09040.9980.09800.9730.09610.9860.09040.99730.09320.9770.13970.9900.11030.9980.09330.9750.10070.9880.09400.99990%10.14590.9650.12560.9780.09110.9890.09860.9610.08960.9730.08830.98520.13440.9750.12440.9880.08990.9990.09400.9760.09220.9890.08990.99830.08950.9780.13760.9910.10910.9980.08950.9760.10910.9890.08880.999Table 10
The ACLs/CPs of asymptotic/credible interval estimates ofht.
nFPSchemeACI-NABCIACI-NLHPDPrior⟶IIIIIIACLCPACLCPACLCP3030%11.50130.9490.99650.9620.98590.9731.84750.9580.80690.9660.80420.97421.32250.9461.02540.9611.00640.9701.52600.9510.81970.9620.84680.97231.18620.9601.01910.9650.91900.9721.32470.9480.83630.9600.73160.97160%11.07500.9600.88590.9640.83990.9741.22290.9620.77550.9680.66270.97520.98210.9660.94810.9670.85180.9731.07850.9610.80870.9640.71090.97530.97530.9500.75170.9620.66600.9730.99630.9580.61330.9650.58980.97290%10.91780.9550.72190.9660.63990.9771.01020.9650.57380.9740.49760.98220.89500.9680.93270.9700.82430.9820.97780.9620.72210.9750.67180.98430.89530.9540.73280.9680.56620.9800.98280.9620.56700.9710.45780.9836030%11.05100.9590.91130.9740.76100.9861.17160.9610.71260.9720.56930.98320.88890.9590.85810.9720.71330.9830.94950.9590.67820.9740.53310.98530.80950.9640.73320.9750.67130.9820.86540.9570.50110.9690.52540.98060%10.77870.9660.83580.9750.56220.9850.83010.9640.67150.9730.43590.98420.67700.9600.67300.9740.66900.9850.70760.9600.52480.9750.51020.98630.70820.9660.69310.9760.56250.9860.75270.9600.56890.9700.42430.98190%10.66060.9690.57230.9760.53620.9860.69170.9650.42530.9740.39390.98520.64050.9670.65260.9780.64840.9890.66800.9610.50930.9730.49600.98530.64880.9680.65900.9750.51910.9870.67650.9670.55280.9780.38160.9849030%10.84860.9660.63360.9770.57390.9860.91530.9640.46510.9730.41200.98420.69940.9660.60880.9770.57800.9880.72930.9710.43370.9820.41820.99030.65690.9690.56690.9790.54240.9880.68570.9680.43240.9760.40620.99060%10.64340.9670.54520.9780.54340.9890.67130.9710.39470.9780.38890.99020.56410.9690.55250.9820.54020.9900.56850.9700.41250.9810.38210.99130.60810.9720.55080.9830.53790.9920.65710.9690.40150.9790.39040.99090%10.53990.9680.53390.9820.53000.9930.55620.9730.37890.9840.37280.99520.54480.9700.52720.9820.51530.9920.54140.9720.39590.9830.36810.99330.52990.9770.54750.9840.52060.9940.54470.9710.39640.9800.37380.989Figure 3
Heatmaps for the simulated results. (a) RMSE, (b) RMAB, (c) ACL, and (d) CP ofα.
(a)(b)(c)(d)Figure 4
Heatmaps for the simulated results. (a) RMSE, (b) RMAB, (c) ACL, and (d) CP ofβ.
(a)(b)(c)(d)Figure 5
Heatmaps for the simulated results. (a) RMSE, (b) RMAB, (c) ACL, and (d) CP ofθ.
(a)(b)(c)(d)Figure 6
Heatmaps for the simulated results. (a) RMSE, (b) RMAB, (c) ACL, and (d) CP ofRt.
(a)(b)(c)(d)Figure 7
Heatmaps for the simulated results. (a) RMSE, (b) RMAB, (c) ACL, and (d) CP ofht.
(a)(b)(c)(d)


## 6. Real-Life Applications

To show the potentiality and flexibility of the proposed methodologies to real phenomenon, in this section we shall provide two real data sets obtained from different fields. First data set (denoted by Data-I) represents the failure times ofn=18 electronic devices, as can be seen in Wang [31]. Recently, this data set has also been discussed by Elshahhat and Abu El Azm [16]. Other data set (denoted by data-II), reported by Hinkley [32], consists of the amount of precipitation (inches) having n=30 successive values in March at Minneapolis/St Paul. The lifetime points (in order) of both data sets I and II are presented in Table 11.Table 11
Failure times of data sets I and II.
DataTimesI5112131467598122145165196224245293321330350420II0.320.470.520.590.770.810.810.900.961.181.201.201.311.351.431.511.621.741.871.891.952.052.102.202.482.813.003.093.374.75Using Table11, plots of the empirical/estimated scaled TTT-transform of the MOGE distribution are provided in Figure 8. It shows that the scaled TTT-transform is (a) convex and then concave for data-I while is concave for data-II. It also suggests that increasing and bathtub shaped failure rates are suitable for the fitting MOGE lifetime model.Figure 8
Empirical and estimated scaled TTT-transform plot of the MOGE distribution. (a) Data-I. (b) Data-II.
(a)(b)Before discussing our proposed estimators, the MOGE distribution is compared with seven popular Marshall-Olkin extended distributions as competitors, namely: Marshall-Olkin extended Weibull (MOEW), Marshall-Olkin logistic exponential (MOLE), Marshall-Olkin alpha power exponential (MOAPE), Marshall-Olkin Nadarajah-Haghighi (MONH), Marshall-Olkin Gompertz (MOG), Marshall-Olkin Lomax (MOL), and Marshall-Olkin exponential (MOE) distributions. The corresponding probability densities of the competing models (forx>0 and α,β,θ>0) are tabulated in Table 12.Table 12
Some useful Marshall-Olkin life models.
ModelfxAuthor(s)MOEWαβθxα−1exp−θxα/1−β¯exp−θxα2Cordeiro and Lemonte [33]MOLEαβθexpθxexpθx−1−α−1/1+βexpθx−1−α2Mansoor et al. [34]MOAPEθβlogαexp−θxα1−exp−θx/α−1β+β¯1/α−1α1−exp−θx−12Nassar et al. [35]MONHθβα1+θxα−1exp1−1+θxα/1−β¯exp1−1+θxα2Lemonte et al. [36]MOGαβexpθxexp−α/θexpθx−1/1−β¯exp−α/θexpθx−12Eghwerido et al. [37]MOLαβθ1+θxα−1/1+θxα−β¯2Ghitany et al. [38]MOEθβexp−θx/β+β¯1−exp−θx2Marshall and Olkin [1]To judge the performance of the considered models, several goodness-of-fit criteria, namely, estimated negative log-likelihood (ENL), Akaike information (AI), consistent Akaike information (CAI), Bayesian information (BI), Hannan-Quinn information (HQI), Anderson-Darling (AD), Cramer von Mises (CvM) and Kolmogorov-Smirnov (KS) distance with itsP-value are used. For each model, we estimate the parameters with their standard errors (SEs) by utilizing the method of maximum likelihood, see Table 13. Generally, the lowest values of these statistics (except the highest P-value) represent the best fit to the given data. The calculated statistics based on data sets I and II are reported in Table 14. It indicates that the MOGE model provides the best fit than other competing models for the given data sets. Plots of the estimated densities and reliability functions of all fitted models are displayed in Figure 9. It shows that the MOGE distribution for both data sets I and II captures the general pattern of the histograms and also supports the numerical findings established here. Also, quantile-quantile plots of the fitted models are also investigated in Figure 10. It yields that the MOGE model fits the given data sets more closely and supports the numerical findings.Table 13
MLEs with their (SEs) of the competing distributions.
ModelMLE (SE)αβθData-IMOGE0.6490 (0.5526)4.5237 (6.4116)0.0090 (0.0030)MOEW0.6927 (0.1490)0.2761 (0.2874)0.0114 (0.0157)MOLE0.8497 (0.2631)1.9676 (1.5960)0.0086 (0.0048)MOAPE1.0462 (2.6066)2.3850 (3.5866)0.0086 (0.0029)MONH1.0053 (0.5547)2.6755 (2.7688)0.0088 (0.0106)MOG0.0139 (0.0148)4.8900 (7.7044)0.0014 (0.0045)MOL2.4924 (1.0329)2.6491 (2.4099)0.0055 (0.0035)MOE—9.8556 (3.5453)0.0132 (0.0051)Data-IIMOGE3.3527 (1.2959)1.2211 (1.5027)1.2271 (0.4267)MOEW2.4651 (0.5673)0.1748 (0.2675)0.0624 (0.0951)MOLE1.9930 (1.0694)0.9614 (4.5808)0.4598 (0.7649)MOAPE17.022 (34.498)2.4018 (2.9264)1.4241 (0.3697)MONH0.7013 (0.4966)24.895 (59.086)4.5347 (11.584)MOG1.9963 (0.9514)13.623 (12.192)0.1311 (0.2301)MOL12.687 (10.343)12.129 (8.4767)0.1508 (0.1575)MOE—9.8857 (6.2279)1.5511 (0.3205)Table 14
Summary fit for the competing models.
ModelENLAICAIBIHQIADCvMKS (P-value)Data-IMOGE109.9511225.9023227.6165228.5734226.27060.26020.03490.0994 (0.9863)MOEW115.9579237.9373239.6516240.6084238.30560.65530.10590.2019 (0.4019)MOLE110.0574226.1147227.8290228.7858226.48300.29560.04190.0857 (0.9977)MOAPE110.1670226.3339228.0482229.0051226.70220.29660.04170.1122 (0.9580)MONH110.1754226.3507228.0650229.0218226.71900.28800.04010.1199 (0.9309)MOG110.9341227.8682229.5825230.5393228.23650.33990.04810.1288 (0.8898)MOL111.6748229.3495231.0638232.0206229.71780.45790.06730.1349 (0.8565)MOE111.7332227.4664228.2664229.2471227.71190.23950.03100.1993 (0.4172)Data-IIMOGE38.081482.162883.085886.366483.50750.10300.01410.0601 (0.9999)MOEW38.143582.287183.210286.490783.63180.11070.01490.0602 (0.9999)MOLE38.369082.738183.661186.941784.08280.11630.01570.0585 (0.9999)MOAPE38.934883.869684.792788.073285.21440.17440.02280.0538 (0.9999)MONH39.090884.181785.104788.385285.52640.18600.02460.0549 (0.999)MOG39.186984.373885.296888.577385.71850.17400.02300.0661 (0.9994)MOL39.239484.478885.401988.682485.82360.10390.01400.0603 (0.9999)MOE39.377182.754283.198685.556683.65070.24010.03280.0646 (0.9996)Figure 9
Plots of histogram and fitted PDFs (left-panel) fitted/empirical RFs (right-panel) of MOGE and competing models under data sets I and II. (a) Data-I. (b) Data-II.
(a)(b)Figure 10
The Q-Q plots of MOGE and competing models under data sets I and II. (a) Data-I. (b) Data-II.
(a)(b)Now, using some specified choices ofm and R, different PCS-T2 samples are generated from data sets I and II, see Table 15. For brevity, the PCS 2,0,0,0,2 is abbreviated by 2,03,2. Using the M-H algorithm described in Section 3.3, we generate 30,000 MCMC samples and discard the first 5,000 samples as burn-in. Under SEL and LL (with υ=−2,−0.02,+2), the Bayesian MCMC estimates and associated credible intervals of α, β, θRt, and ht using noninformative priors are developed. Also, MLEs and two-sided 95% asymptotic (NA and NL) interval estimates of the unknown parameters are computed. The estimates of RF and HF are calculated when the mission time t is taken as 100 and 1 for data sets I and II, respectively. The point estimates (with their SEs) and interval estimates with their confidence lengths (CLs) and reported in Tables 16 and 17, respectively. It is noted that the proposed point and interval estimates, obtained based on data sets I and II, of α, β, θRt, and ht are very similar to each other, as expected.Table 15
Three generated PCS-T2 samples from data sets I and II.
DatanmSchemeCensored dataI18 (9)R1=9,085, 196, 224, 245, 293, 321, 330, 350, 420R2=04,9,045, 11, 21, 31, 46, 321, 330, 350, 420R3=08,95, 11, 21, 31, 46, 75, 98, 122, 145II30 (15)R1=15,0140.77, 1.31, 0.32, 0.59, 0.81, 2.81, 1.87, 1.18, 1.35, 4.75, 2.48, 0.96, 1.89, 0.90, 2.05R2=07,15,070.77, 1.74, 0.81, 1.20, 1.95, 1.20, 0.47, 1.43, 1.35, 4.75, 2.48, 0.96, 1.89, 0.90, 2.05R3=014,150.77, 1.74, 0.81, 1.20, 1.95, 1.20, 0.47, 1.43, 3.37, 2.20, 3.00, 3.09, 1.51, 2.10, 0.52Table 16
The frequentist and bayes estimates (with their SEs) under data sets I and II.
SchemeParameterMLESELLLυ⟶−2−0.022Data-IR1α0.2442 (0.3133)0.2441 (1.10×10−5)0.2442 (2.38×10−8)0.2442 (3.33×10−8)0.2442 (9.16×10−8)β302.82 (2.9810)302.82 (1.09×10−5)302.82 (1.14×10−7)302.82 (5.71×10−8)302.82 (1.19×10−9)θ0.0156 (0.0042)0.0157 (9.77×10−6)0.0157 (5.05×10−7)0.0157 (4.90×10−7)0.0157 (4.75×10−7)R1000.9471 (0.0451)0.9462 (5.84×10−5)0.9463 (6.21×10−6)0.9462 (6.75×10−6)0.9461 (7.29×10−6)h1000.0010 (0.0006)0.0009 (1.54×10−5)0.0010 (2.24×10−7)0.0010 (2.23×10−7)0.0010 (2.23×10−7)R2α0.6520 (0.4854)0.6520 (1.90×10−5)0.6520 (3.68×10−8)0.6519 (1.94×10−8)0.6520 (7.68×10−8)β2.7522 (4.3540)2.7521 (1.89×10−5)2.7521 (4.54×10−8)2.7522 (1.02×10−7)2.7521 (1.59×10−7)θ0.0057 (0.0031)0.0057 (5.44×10−6)0.0057 (2.06×10−7)0.0057 (2.02×10−7)0.0057 (1.97×10−7)R1000.6659 (0.1229)0.6646 (2.45×10−4)0.6661 (5.21×10−6)0.6646 (4.17×10−6)0.6661 (1.37×10−5)h1000.0039 (0.0016)0.0039 (4.16×10−6)0.0039 (2.52×10−7)0.0039 (2.49×10−7)0.0039 (2.47×10−7)R3α0.9315(0.5133)0.9315 (6.31×10−6)0.9315 (1.10×10−7)0.9315 (1.16×10−7)0.9315 (1.23×10−7)β0.5257 (2.5912)0.5257 (6.29×10−6)0.5257 (1.36×10−7)0.5257 (1.42×10−7)0.5257 (1.48×10−7)θ0.0026 (0.0098)0.0027 (2.81×10−6)0.0027 (8.15×10−7)0.0027 (8.13×10−7)0.0027 (8.12×10−7)R1000.6057 (0.1072)0.5974 (2.75×10−4)0.5993 (5.39×10−5)0.5974 (6.59×10−5)0.5955 (7.79×10−5)h1000.0043(0.0020)0.0044 (3.69×10−6)0.0044 (9.90×10−7)0.0044 (9.88×10−7)0.0044 (9.86×10−7)Data-IIR1α4.2551 (1.4664)4.2535 (5.09×10−4)4.2600 (3.12×10−5)4.2536 (9.42×10−6)4.2471 (5.09×10−5)β0.3270 (0.8587)0.3455(4.39×10−4)0.3504 (1.48×10−4)0.3456 (1.17×10−4)0.3407 (8.69×10−5)θ0.8510 (0.8031)0.8586 (4.28×10−4)0.8632 (7.69×10−5)0.8586 (4.82×10−5)0.8540 (1.90×10−5)R10.7602 (0.0747)0.7600 (3.06×10−4)0.7623 (1.39×10−5)0.7600 (6.12×10−7)0.7576 (1.58×10−5)h10.7139 (0.2696)0.7119 (8.57×10−4)0.7310 (1.07×10−4)0.7121 (1.18×10−5)0.6942 (1.25×10−4)R2α4.9887 (2.1453)4.9941 (1.25×10−3)5.0334 (2.83×10−4)4.9945 (3.63×10−5)4.9548 (2.14×10−4)β0.9138 (2.3699)0.9500 (1.15×10−3)0.9835 (4.41×10−4)0.9503 (2.31×10−4)0.9173 (2.20×10−5)θ1.0688 (0.9528)1.0867 (7.30×10−4)1.1002 (1.98×10−4)1.0868 (1.14×10−4)1.0735 (2.98×10−5)R10.8674 (0.0514)0.8619 (2.42×10−4)0.8633 (2.57×10−5)0.8619 (3.47×10−5)0.8604 (4.42×10−5)h10.4216 (0.1402)0.4359 (6.97×10−4)0.4485 (1.70×10−4)0.4360 (9.14×10−5)0.4242 (1.64×10−5)R3α4.3478 (1.9903)4.3514 (1.92×10−3)4.4437 (6.07×10−4)4.3523 (2.83×10−5)4.2601 (5.55×10−4)β1.6147 (2.4821)1.6355 (1.82×10−3)1.7195 (6.63×10−4)1.6363 (1.37×10−4)1.5536 (3.86×10−4)θ1.4146 (0.5795)1.4307 (1.04×10−3)1.4583 (2.77×10−4)1.4310 (1.04×10−4)1.4042 (6.55×10−5)R10.7918 (0.0788)0.7835 (3.61×10−4)0.7867 (3.23×10−5)0.7835 (5.24×10−5)0.7802 (7.35×10−5)h10.5857 (0.2394)0.6097 (9.78×10−4)0.6348 (3.11×10−4)0.6099 (1.53×10−4)0.5869 (7.61×10−6)Table 17
The 95% asymptotic/credible interval estimates [with their CLs] under data sets I and II.
SchemeParameterACI-NAACI-NLBCIHPDData-IR1α(0.0000, 0.8595) [0.8582](0.0198, 3.0176) [2.9978](0.2382, 0.2501) [0.0118](0.2382, 0.2501) [0.0118]β(296.14, 307.82) [11.680](296.19, 307.88) [11.691](302.81, 302.83) [0.0117](302.81, 302.83) [0.0117]θ(0.0073, 0.0239) [0.0166](0.0092, 0.0265) [0.0173](0.0127, 0.0188) [0.0061](0.0126, 0.0187) [0.0060]R100(0.8588, 0.9999) [0.1411](0.8628, 0.9999) [0.1371](0.9260, 0.9620) [0.0360](0.9280, 0.9636) [0.0356]h100(0.0000, 0.0022) [0.0022](0.0003, 0.0036) [0.0033](0.0006, 0.0015) [0.0009](0.0006, 0.0015) [0.0009]R2α(0.0000, 1.6033) [1.6033](0.1515, 2.8051) [2.6536](0.6460, 0.6578) [0.0118](0.6461, 0.6579) [0.0118]β(0.0000, 11.286) [11.286](0.1239, 61.137) [61.137](2.7463, 2.7581) [0.0118](2.7465, 2.7583) [0.0118]θ(0.0000, 0.0118) [0.0118](0.0019, 0.0166) [0.0147](0.0041, 0.0075) [0.0034](0.0041, 0.0075) [0.0034]R100(0.4251, 0.9067) [0.4816](0.4638, 0.9560) [0.4922](0.5889, 0.7407) [0.1518](0.5900, 0.7417) [0.1516]h100(0.0007, 0.0070) [0.0063](0.0017, 0.0088) [0.0071](0.0027, 0.0053) [0.0026](0.0026, 0.0052) [0.0026]R3α(0.0000, 1.9375) [1.9375](0.3163, 2.7428) [2.4264](0.9296, 0.9335) [0.0039](0.9295, 0.9334) [0.0039]β(0.0000, 5.6043) [5.6043](0.0001, 8244.3) [8244.3](0.5237, 0.5276) [0.0039](0.5237, 0.5276) [0.0039]θ(0.0000, 0.0219) [0.0219](0.0001, 4.0602) [4.0601](0.0019, 0.0036) [0.0017](0.0019, 0.0036) [0.0017]R100(0.3956, 0.8159) [0.4203](0.4282, 0.8569) [0.4287](0.5170, 0.6866) [0.1696](0.5164, 0.6854) [0.1690]h100(0.0004, 0.0082) [0.0078](0.0017, 0.0107) [0.0090](0.0033, 0.0056) [0.0023](0.0033, 0.0056) [0.0023]Data-IIR1α(1.3811, 7.1292) [5.7481](2.1656, 8.3608) [6.1952](4.0952, 4.4105) [0.3153](4.0955, 4.4107) [0.3152]β(0.0000, 2.0101) [2.0101](0.0019, 56.228) [56.226](0.2136, 0.4867) [0.2731](0.2147, 0.4873) [0.2726]θ(0.0000, 2.4250) [2.4250](0.1338, 5.4104) [5.2765](0.7269, 0.9931) [0.2662](0.7294, 0.9952) [0.2659]R1(0.6137, 0.9066) [0.2929](0.6270, 0.9216) [0.1371](0.6563, 0.8470) [0.1906](0.6664, 0.8539) [0.1875]h1(0.1854, 1.2424) [1.0570](0.3405, 1.4967) [1.1562](0.4684, 1.0015) [0.5331](0.4530, 0.9793) [0.5263]R2α(0.7840, 9.1933) [8.4093](2.1475, 11.588) [9.4409](4.6080, 5.3856) [0.7776](4.6064, 5.3816) [0.7752]β(0.0000, 5.5587) [5.5587](0.0057, 147.40) [147.39](0.6005, 1.3118) [0.7113](0.5975, 1.3079) [0.7103]θ(0.0000, 2.9362) [2.9362](0.1862, 6.1331) [5.9469](0.8663, 1.3248) [0.4585](0.8644, 1.3219) [0.4574]R1(0.7667, 0.9680) [0.2013](0.7723, 0.9741) [0.2018](0.7776, 0.9269) [0.1494](0.7851, 0.9325) [0.1474]h1(0.1468, 0.6964) [0.5496](0.2197, 0.8090) [0.5894](0.2458, 0.6759) [0.4301](0.2370, 0.6642) [0.4272]R3α(0.4469, 8.2488) [7.8018](1.7726, 10.664) [8.8915](3.7588, 4.9459) [1.1871](3.7527, 4.9383) [1.1856]β(0.0000, 6.4795) [6.4795](0.0794, 32.851) [32.772](1.0759, 2.2085) [1.1326](1.0858, 2.2128) [1.1270]θ(0.2788, 2.5505) [2.2717](0.6338, 3.1576) [2.5238](1.1212, 1.7705) [0.6493](1.1070, 1.7542) [0.6472]R1(0.6373, 0.9463) [0.3090](0.6515, 0.9624) [0.3109](0.6612, 0.8832) [0.2220](0.6740, 0.8934) [0.2194]h1(0.1165, 1.0548) [0.9383](0.2629, 1.3048) [1.0419](0.3420, 0.9462) [0.6042](0.3258, 0.9194) [0.5936]To examine the convergence of M-H algorithm, trace plot based on 25,000 MCMC simulated variates of each unknown parameter (forR1 as an example) using data sets I and II, is plotted in Figure 11. For each trace plot, the sample mean and two bounds of 95% HPD credible intervals are mentioned by solid (—) and dashed (- - -) lines, respectively. It indicates that the M-H algorithm converges very well. Using the Gaussian kernel density, the corresponding histogram plot of each unknown parameter (for R1 as an example) is also plotted and represented in Figure 12. In each histogram, the sample mean is depicted as vertical dash-dotted line (:). It represents that the simulated posteriors of α, β, and θ are fairly symmetric while the simulated posteriors of Rt and ht are close to negative and positive skewed, respectively.Figure 11
Trace plots for MCMC draws of the unknown parameters under data sets I and II. (a)R1 of data-I. (b) R1 of data-II.
(a)(b)Figure 12
Histogram plots of the unknown parameters under data sets I and II. (a)R1 from data-I. (b) R1 from data-II.
(a)(b)Using the optimum criteria given in Section4, using the observed Fisher information (11) obtained from both data sets I and II, the concept of optimal censoring is illustrated, see Table 18. It shows that the scheme R3 is the optimal censoring (for data set I) while R1 is the optimal censoring (for data set II) than other competing scheme plans. However, the parameter and optimal estimates examined here support the same findings reported in the simulation study.Table 18
Optimum censoring scheme under data sets I and II.
DataSchemeCriterionABCIR18.98242.822×10−6306091.5R219.1934.203×10−6330194.6R36.97767.775×10−7943744.4IIR13.53261.988×10−2138.9751R211.1273.890×10−176.48990R310.4586.801×10−131.96376


## 7. Conclusion Remarks

This article investigates the problem of estimating the unknown parameters and reliability characteristics of the MOGE distribution when the data are collected under type-II progressive censoring. Besides the likelihood inference, the Bayes MCMC estimates of the unknown parameters, reliability, and hazard functions are derived. As expected the joint posterior density cannot be obtained explicitly. Therefore, using gamma informative priors under different loss functions, the Bayes point and interval estimates of the unknown parametric functions are approximated utilizing the Metropolis–Hasting algorithm. From the simulation outputs, the Bayes MCMC estimation method for any unknown function of the MOGE parameters is recommended. To show the practical utility of the proposed methods in real-life phenomenon and to determine the optimum censoring plan, two data sets from engineering and physics are analyzed for this purpose. Additionally, based on the given data sets, we showed that the proposed model provides an adequate fit with respect to its competing Marshall–Olkin extended models. We hope that the results and methodologies discussed in this paper will be beneficial to data analyst and reliability practitioners. [39, 40].



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