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Estimating the Pareto parameters underprogressive censoring data for constant-partially accelerated life testsTahani A. Abushala & Ahmed A. Solimanb

a Department of Mathematics, Umm Al-Qura University, Makkah,Saudi Arabiab Department of Mathematics, Islamic University, Madinah, SaudiArabiaPublished online: 14 Nov 2013.

To cite this article: Tahani A. Abushal & Ahmed A. Soliman (2013): Estimating the Paretoparameters under progressive censoring data for constant-partially accelerated life tests, Journal ofStatistical Computation and Simulation, DOI: 10.1080/00949655.2013.853768

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Journal of Statistical Computation and Simulation, 2013http://dx.doi.org/10.1080/00949655.2013.853768

Estimating the Pareto parameters under progressive censoringdata for constant-partially accelerated life tests

Tahani A. Abushala and Ahmed A. Solimanb∗

aDepartment of Mathematics, Umm Al-Qura University, Makkah, Saudi Arabia; bDepartment ofMathematics, Islamic University, Madinah, Saudi Arabia

(Received 31 January 2013; accepted 7 October 2013)

Accelerated life testing is widely used in product life testing experiments since it provides significantreduction in time and cost of testing. In this paper, assuming that the lifetime of items under use conditionfollow the two-parameter Pareto distribution of the second kind, partially accelerated life tests based onprogressively Type-II censored samples are considered. The likelihood equations of the model parametersand the acceleration factor are reduced to a single nonlinear equation to be solved numerically to obtainthe maximum-likelihood estimates (MLEs). Based on normal approximation to the asymptotic distributionof MLEs, the approximate confidence intervals (ACIs) for the parameters are derived. Two bootstrap CIsare also proposed. The classical Bayes estimates cannot be obtained in explicit form, so we propose toapply Markov chain Monte Carlo method to tackle this problem, which allows us to construct the credibleinterval of the involved parameters. Analysis of a simulated data set has also been presented for illustrativepurposes. Finally, a Monte Carlo simulation study is carried out to investigate the precision of the Bayesestimates with MLEs and to compare the performance of different corresponding CIs considered.

Keywords: Pareto distribution; constant-partially accelerated life tests; progressive Type-II censor-ing; maximum-likelihood estimation; asymptotic confidence intervals; bootstrap; Bayesian estimation;confidence intervals

1. Introduction

Due to the continual improvement in the manufacturing design, it is more difficult to obtaininformation about the lifetime of products or materials with high reliability at the time of testingunder normal conditions. This makes the lifetime testing under normal conditions very costlyand take a long time. For this reason, accelerated life tests (ALTs) are preferred to be used inmanufacturing industries to obtain enough failure data, in a short period of time, necessary tomake inferences regarding its relationship with external stress variables. In ALTs, the test itemsare tested only at accelerated conditions, viz. higher than usual levels of stress, to induce earlyfailures. Data collected at such accelerated conditions are then extrapolated through a physicallyappropriate statistical model to estimate the lifetime distribution at normal use conditions.

According to Nelson,[1] there are mainly three ALT methods. The first method is called theconstant-stress ALT, the stress is kept at a constant level throughout the life of test products.[2–5]The second one is referred to as progressive-stress ALT, the stress applied to a test product is

∗Corresponding author. Email: a_a_sol@hotmail.com

© 2013 Taylor & Francis

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2 T.A. Abushal and A.A. Soliman

continuously increasing in time.[6–8] The third is the step-stress ALT, in which the test conditionchanges at a given time or upon the occurrence of a specified number of failures, this type hasbeen studied by several authors. Miller and Nelson [9] obtained the optimal simple step-stressALT plans for the case where test products have exponentially distributed lives and are observedcontinuously until all test products fail; Bai and Chung [10] extended their results to the case ofcensoring. The optimal step-stress test under progressive Type-I censoring, assuming exponentiallifetime distribution, was considered by Gouno et al.[11] For more recent researches on ALTs,see among others [12–17].

When the acceleration factor cannot be assumed as a known value, the partially acceleratedlife test (PALT) will be a good choice to perform the life test. In PALTs, items are tested at bothaccelerated and use conditions. Also, there are three major stress types of PALTs: constant-stress,step-stress and progressive-stress. When a test involves two levels of stress with the first one beingthe normal level and at a specific time point the stress will be change, it is referred to as a step-stressPALT. Several authors have dealt with this type of ALT, including Abdel-Hamid and Al-Hussaini[8] and Bhattacharyya and Soejoeti.[18] The constant-stress PALT (which is the main topic of thispaper) runs each item at either use or accelerated condition only. Optimum constant-stress PALTplans for the Pareto distribution under Type-I censoring have been discussed by Ismail et al.[16]

The probability density function (PDF), cumulative distribution function (CDF), reliabilityfunction (SF), and hazard rate function (HRF) of the two-parameter Pareto distribution (Pareto(α, β)) are given, respectively, by

f1(t) = αβα(β + t)−(α+1), t > 0; α > 0, β > 0, (1)

F1(t) = 1 − βα(β + t)−α , (2)

S1(t) = βα(β + t)−α , (3)

and

H1(t) = α(β + t)−1. (4)

Equation (1) is a special form of Pearson Type-VI distribution. The Pareto distribution of thesecond type (also known as the Lomax distribution) has been widely used in economic studiesand to analyse business failure data. The Pareto distribution has been studied by several authors.Balkema and de Haan [19] showed that the CDF (2) arises as a limit distribution of residuallifetime at great age.According toArnold,[20] the Pareto distribution is well adapted for modellingreliability problems, since many of its properties are interpretable in that context and could bean alternative to the well-known distributions used in reliability. This distribution was used formodelling size spectra data in aquatic ecology by Vidondo et al.[21] Childs et al. [22] consideredorder statistics from non-identical right-truncated Lomax distributions and provided applicationsfor this situation. Al-Awadhi and Ghitany [23] used the Pareto distribution as a mixing distributionfor the Poisson parameter and obtained the discrete Poisson–Pareto distribution. Howlader andHossain [24] investigated the Bayesian estimation of the Pareto survival function. More recently,Abd-Elfattah et al. [25] discussed some Bayesian inferences based on censored samples from thePareto distribution. Hassan and Al-Ghamdi [26] determined the optimal times of changing stresslevel for simple stress plans under a cumulative exposure model using the Pareto distribution.

In life testing and reliability studies, the experimenter may not always obtain complete infor-mation on failure times for all experimental units. Data obtained from such experiments are calledcensored data. Saving the total time on test and the cost associated with it are some of the majorreasons for censoring. A censoring scheme (CS), which can balance between total time spent forthe experiment, number of units used in the experiment and the efficiency of statistical inferencebased on the results of the experiment, is desirable. The most common CSs are Type-I (time)censoring, and Type-II (item) censoring. The conventional Type-I and Type-II CSs do not have

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the flexibility of allowing removal of units at points other than the terminal point of the exper-iment. Because of that, a more general CS called progressive Type-II right censoring has beenused in this article. Censored data are of progressively Type-II right type when they are censoredby the removal of a prespecified number of survivors whenever an individual fails; this contin-ues until a fixed number of failures has occurred, at which stage the remainder of the survivingindividuals are also removed or censored. This scheme includes ordinary Type-II censoring andcomplete scheme as special cases. A general account of theoretical developments and applicationsconcerning progressive censoring is given in the book by Balakrishnan and Aggarwala [27] andBalakrishnan.[28]

Next, we provide a brief description of the model, along with some basic assumptions. Section 3deals with the derivation of the maximum-likelihood estimators (MLEs) of the involved param-eters as well as the corresponding approximate confidence intervals (ACIs). The two parametricbootstrap confidence intervals (CIs) for the parameters are also discussed in this section. Section 4deals with the Bayesian approach that uses the well-known Markov chain Monte Carlo (MCMC)methods. A simulation example to illustrate the approach is given in the same section. MonteCarlo simulation results are presented in Section 5. Finally, Section 6 provides some concludingremarks.

2. Model description and basic assumptions

2.1. Model description

In constant-stress PALTs, n1 items are randomly chosen among n test items which are allocatedto use condition and n2 = n − n1 remaining items are subjected to an accelerated condition.Progressive Type-II censoring is applied as follows.At the time of the first failure T Rj

j1;mj ,nj, Rj1 items

are randomly withdrawn from the remaining nj − 1 surviving items. At the second failure T Rjj2;mj ,nj

,Rj2 items from the remaining nj − 2 − Rj1 items are randomly withdrawn. The test continues

until the mjth failure T Rjjmj ;mj ,nj

at which time, all remaining Rjmj = nj − mj − ∑mj−1k=1 Rjk items are

withdrawn for j = 1, 2. In our study, Rji are fixed prior and mj < nj. If the failure times of the nj

items originally in the test are from a continuous population with distribution function Fj(x) andPDF fj(x), the joint PDF for T Rj

j1;mj ,nj< T Rj

j2;mj ,nj< · · · < T Rj

jmj ;mj ,njand j = 1, 2 is given by

L(α, β, λ|t) =2∏

j=1

Cj

{ mj∏i=1

fj(tji;mj ,nj )(Sj(tji;mj ,nj ))Rji

}, (5)

where

Cj = nj(nj − 1 − Rj1)(nj − 2 − Rj1 − Rj2) · · · (nj − mj −mj−1∑k=1

Rjk).

It is clear from Equation (5) that the constant PALTs’ progressively Type-II censored schemeincludes ordinary Type-II right censoring and complete data as special cases.

2.2. Assumptions

We assume that the lifetime of an items tested at use condition follows a Pareto(α, β) distributionwith PDF, CDF, SF and HRF given in Equations (1)–(4). The hazard rate of an item tested ataccelerated condition is given by h2(t) = λh1(t), where λ is an acceleration factor satisfying

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4 T.A. Abushal and A.A. Soliman

λ > 1. Therefore, the HRF, SF, CDF and PDF under accelerated condition are given, for t > 0,(α, β) > 0, λ > 1, respectively, by

H2(t) = λα(β + t)−1, (6)

S2(t) = exp

(−

∫ t

0h2(z)

)dz = βλα(β + t)−λα , (7)

F2(t) = 1 − βλα(β + t)−λα , (8)

and

f2(t) = αλβλα(β + t)−(λα+1). (9)

3. Maximum-likelihood estimation

Let, for j = 1, 2, T Rjj1;mj ,nj

< T Rjj2;mj ,nj

< · · · < T Rjjmj ;mj ,nj

denote two progressively Type-II censoredsamples from two populations whose CDFs and PDFs are as given in Equations (1), (2) and (8), (9),with Rj = (Rj1, Rj2, . . . , Rj1). The log-likelihood function �(α, β, λ|t) = log L(α, β, λ|t) withoutnormalized constant is then given by

�(α, β, λ|t) = (m1 + m2) log α + m2 log λ + α(n1 + λn2) log β −m1∑i=1

(α(R1i + 1) + 1)

× log(β + t1i) −m2∑i=1

(αλ(R2i + 1) + 1) log(β + t2i). (10)

Calculating the first partial derivatives of Equation (10) with respect to α, β, and λ and equatingeach to zero, we get the likelihood equations as

∂�(α, β, λ|t)∂α

= m1 + m2

α+ (n1 + λn2) log β −

m1∑i=1

(R1i + 1) log(β + t1i)

− λ

m2∑i=1

(R2i + 1) log(β + t2i) = 0, (11)

∂�(α, β, λ|t)∂β

= α(n1 + λn2)

β−

m1∑i=1

α(R1i + 1) + 1

β + t1i−

m2∑i=1

λα(R2i + 1) + 1

β + t2i= 0, (12)

and

∂�(α, β, λ|t)∂λ

= m2

λ+ αn2 log β −

m2∑i=1

α(R2i + 1) log(β + t2i) = 0. (13)

From Equations (11)–(13) we obtain the MLEs of λ, α, and β as

α(β) = m1

D1, (14)

λ(α, β) = m2

αD2, (15)

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Journal of Statistical Computation and Simulation 5

and

m1

D1

m1∑i=1

(R1i + 1)

[t1i

β(β + t1i)

]+ m2

D2

m2∑i=1

(R2i + 1)

[t2i

β(β + t2i)

]−

m1∑i=1

1

β + t1i−

m2∑i=1

1

β + t2i= 0,

(16)

where

Ds =ms∑i=1

(Rsi + 1) log

(1 + tsi

β

), s = 1, 2. (17)

The likelihood equations (11)–(13) are reduced to a single nonlinear equation (16), which couldbe solved numerically with respect to β using any iteration procedure such as Newton–Raphson,to get the MLE of β. Thus, once the values of β is determined, estimates of α and λ are easilyobtained by using Equations (14) and (15), respectively.

3.1. Approximate CIs

As indicated by Vander Wiel and Meeker,[29] the most common method to set confidence boundsfor the parameters is to use the asymptotic normal distribution of the MLEs. In relation to theasymptotic variance–covariance matrix of the maximum-likelihood (ML) estimators of the param-eters, it can be approximated by numerically inverting the Fisher information matrix F, where itis composed of the negative second derivatives of the natural logarithm of the likelihood functionevaluated at (α, β, λ), the MLEs of the parameters.

From the log-likelihood function in Equation (10), we have

∂2�(α, β, λ|t)∂α2

= −m1 + m2

α2, (18)

∂2�(α, β, λ|t)∂β2

= −α(n1 + λn2)

β2+

m1∑i=1

α(R1i + 1) + 1

(β + t1i)2+

m2∑i=1

λα(R2i + 1) + 1

(β + t2i)2, (19)

∂2�(α, β, λ|t)∂λ2

= −m2

λ2, (20)

∂2�(α, β, λ|t)∂α∂β

= ∂2�(α, β, λ|t)∂β∂α

= n1 + λn2

β−

m1∑i=1

(R1i + 1)

β + t1i− λ

m2∑i=1

(R2i + 1)

β + t2i, (21)

∂2�(α, β, λ|t)∂α∂λ

= ∂2�(α, β, λ|t)∂λ∂α

= n2 log β − λ

m2∑i=1

(R2i + 1) log(β + t2i), (22)

and

∂2�(α, β, λ|t)∂β∂λ

= ∂2�(α, β, λ|t)∂λ∂β

= αn2

β− α

m2∑i=1

(R2i + 1)

β(β + t2i). (23)

Therefore, the asymptotic variance–covariance matrix can be written as follows:

F−1 =

⎡⎢⎢⎢⎢⎢⎢⎣−∂2�(α, β, λ|t)

∂α2− ∂2�(α, β, λ|t)

∂α∂β− ∂2�(α, β, λ|t)

∂α∂λ

−∂2�(α, β, λ|t)∂β∂α

− ∂2�(α, β, λ|t)∂β2

− ∂2�(α, β, λ|t)∂β∂λ

−∂2�(α, β, λ|t)∂λ∂α

− ∂2�(α, β, λ|t)∂λ∂β

− ∂2�(α, β, λ|t)∂λ2

⎤⎥⎥⎥⎥⎥⎥⎦

−1

↓(α,β,λ)

.

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6 T.A. Abushal and A.A. Soliman

Thus, the 100(1 − γ )% ACls for α, β, and λ are obtained as

α ∓ zγ /2√

v11, β ∓ zγ /2√

v22, and λ ∓ zγ /2√

v33, (24)

respectively, where v11, v22, and v33 are the elements on the main diagonal of the variance–covariance matrix F−1 and zγ /2 is the percentile of the standard normal distribution with right-tailprobability γ /2.

The problem with applying normal approximation of the MLE is that when the sample size issmall, the normal approximation may be poor. However, a different transformation of the MLEcan be used to correct the inadequate performance of the normal approximation. Meeker andEscobar [30] suggested the use of the normal approximation for the log-transformed MLE. Letωi, i = 1, 2, 3, with (ω1 = α, ω2 = β, ω3 = λ). A two-sided 100(1 − γ )% normal approximationCIs for ωi, are given by

(ωi. exp

(− zγ /2

√var(ωi)

ωi

), ωi. exp

(z γ

2

√var(ωi)

ωi

)). (25)

3.2. Bootstrap CIs

The bootstrap is a resampling method for statistical inference. It is commonly used to estimateCIs, but it can also be used to estimate bias and variance of an estimator or calibrate hypothesistests. For more survey of the non-parametric and parametric bootstrap methods, see [31,32] anda more recently reviewed article by Kreiss and Paparoditis.[33] It is observed by Kundu et al.[34] that the non-parametric bootstrap method does not work well. In this section, we use theparametric bootstrap method to construct CIs for the unknown parameters α, β, and λ. We presenttwo parametric bootstrap methods, percentile bootstrap CI (PBCI) suggested by Efron [35] andbootstrap-t CI (BTCI) suggested by Hall.[36] The following steps are followed to obtain bootstrapsamples for both methods:

(1) Based on the original progressively Type-II sample, (tj1;mj ,nj < tj2;mj ,nj < · · · < tjmj ;mj ,nj ), j =1, 2, obtain α, β, and λ.

(2) Based on the values of nj and mj (1 < mj < nj) with the same CS in step 1, Rji (i =1, 2, . . . , mj), j = 1, 2, generate two independent progressive samples of sizes m1 and m2 fromPareto distribution, t∗ = (t∗j1;mj ,nj

< t∗j2;mj ,nj< · · · < t∗jmj ;mj ,nj

), by using the algorithm describedin [37].

(3) As in step 1 based on t∗ compute the bootstrap sample estimates of α, β, and λ say α∗, β∗,and λ∗.

(4) Repeat the above steps 2 and 3 N times representing N different bootstrap samples. The valueof N has been taken to be 1000.

(5) Arrange all α∗, β∗, and λ∗ in an ascending order to obtain the bootstrap sample (ϕ∗[1]k ,

ϕ∗[2]k , . . . , ϕ∗[N]

k ), k = 1, 2, 3, where (ϕ∗1 = α∗, ϕ∗

2 = β∗, ϕ∗3 = λ∗).

Percentile bootstrap CI:Let G(z) = P(ϕ∗

k � z) be CDF of ϕ∗k . Define ϕ∗

kboot = G−1(z) for given z. The approximatebootstrap 100(1 − γ )% CI of ϕ∗

k is given by

[ϕ∗

kboot

(γ

2

), ϕ∗

kboot

(1 − γ

2

)]. (26)

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Journal of Statistical Computation and Simulation 7

Bootstrap-t CI:First, find the order statistics δ

∗[1]k < δ

∗[2]k < · · · < δ

∗[N]k , where

δ∗[j]k = ϕ

∗[j]k − ϕk√

var(ϕ

∗[j]k

) , j = 1, 2, . . . , N , k = 1, 2, 3, (27)

where ϕ1 = α, ϕ2 = β, and ϕ3 = λ.Let H(z) = P(δ∗

k < z) be the CDF of δ∗k . For a given z, define

ϕkboot−t = ϕk +√

Var(ϕk)H−1(z). (28)

The approximate 100(1 − γ )% CI of ϕk is given by(ϕkboot−t

(γ

2

), ϕkboot−t

(1 − γ

2

)). (29)

4. Bayes estimation

Let us consider independent vague priors for the parameters α, β, and λ, as follows:

π(α) ∝ α−1, α > 0,

π(β) ∝ β−1, β > 0,

and π(λ) ∝ λ−1, λ > 1.

(30)

Therefore, the joint prior of the three parameters can be expressed by

π(α, β, λ) ∝ (α, β, λ)−1, α > 0, β > 0, λ > 1. (31)

It is to be noted that our objective is to consider vague priors so that the priors do not have anysignificant roles in the analyses that follow. However, if one uses the prior beliefs different fromEquation (30) and resorts to sample-based approaches for analysing the posterior, one may use theconcept of sampling-importance-resampling without working afresh with the new prior-likelihoodsetup.[38]

The joint posterior density function of α, β, and λ given the data, denoted by π∗(α, β, λ|t), canbe written as

π∗(α, β, λ|t) = L(α, β, λ|t) × π(α, β, λ)∫ ∞0

∫ ∞0

∫ ∞0 L(α, β, λ|t) × π(α, β, λ) dα dβ dλ

; (32)

therefore, the Bayes estimate of any function of the parameters, say ϕ(α, β, λ), under squarederror loss function is

ϕ(α, β, λ) = Eα,β,λ|t(ϕ(α, β, λ))

=∫ ∞

0

∫ ∞0

∫ ∞0 ϕ(α, β, λ)L(α, β, λ|t) × π(α, β, λ) dα dβ dλ∫ ∞

0

∫ ∞0

∫ ∞0 L(α, β, λ|t) × π(α, β, λ) dα dβ dλ

. (33)

Generally, the ratio of two integrals given by Equation (33) cannot be obtained in a closed form.In this case, we use the MCMC technique to generate samples from the posterior distributionsand then compute the Bayes estimators of the individual parameters.

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8 T.A. Abushal and A.A. Soliman

4.1. MCMC approach

A wide variety of MCMC schemes are available, and it can be difficult to choose among them. Animportant sub-class of MCMC methods are Gibbs sampling and more general Metropolis within-Gibbs samplers. The advantage of using the MCMC method over the MLE method is that wecan always obtain a reasonable interval estimate of the parameters by constructing the probabilityintervals based on the empirical posterior distribution. This is often unavailable in ML estimation.Indeed, the MCMC samples may be used to completely summarize the posterior uncertainty aboutthe parameters α, β, and λ, through a kernel estimate of the posterior distribution. This is alsotrue of any function of the parameters.

The joint posterior density function of α, β, and λ can be written as

π∗(α, β, λ|t) ∝ αm1+m2−1λm2−1βα(n1+λn2)−1 exp

(−

m1∑i=1

(α(R1i + 1) + 1) log(β + t1i)

)

× exp

(−

m2∑i=1

(αλ(R2i + 1) + 1) log(β + t2i)

). (34)

The conditional posterior densities of α, β, and λ are as follows:

π∗1 (α|β, λ, t) ∼ Gamma

⎛⎝m1 + m2,2∑

j=1

mj∑i=1

δj(Rji + 1) log(β + tji) − (n1 + λn2) log β

⎞⎠ , (35)

where

δj ={

1, if j = 1,

λ, if j = 2,(36)

π∗2 (λ|α, β, t) ∼ Gamma

(m2,

m2∑i=1

α(R2i + 1) log(β + t2i) − αn2 log β

), (37)

and

π∗3 (β|α, λ, t) ∝ βα(n1+λn2)−1 exp

(−

m1∑i=1

(α(R1i + 1) + 1) log(β + t1i)

)

× exp

(−

m2∑i=1

(αλ(R2i + 1) + 1) log(β + t2i)

). (38)

It can be easily seen that both Equations (35) and (37) are gamma distributed and, therefore,samples of α and λ can be easily generated using any of the gamma generating routines. Theposterior of β in Equation (38) is not known, but the plot of it shows that it is similar to normaldistribution. Therefore, to generate from this distribution, we use the Metropolis–Hastings method(Metropolis et al. [39] with normal proposal distribution). For details regarding the implementationof Metropolis–Hastings algorithm, the readers may refer to [40].

To run the Gibbs sampler algorithm we started with the MLEs. We then drew samples fromvarious full conditionals, in turn, using the most recent values of all other conditioning variablesunless some systematic pattern of convergence was achieved.

The algorithm of Gibbs sampling is as follows:

Step 1: Start with an (β(0) = β and λ(0) = λ) and set I = 1.

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Step 2: Generate α(I) from Gamma distribution (π∗1 (α|β(I−1), λ(I−1), t)).

Step 3: Generate λ(I) from Gamma distribution π∗2 (λ|α(I), β(I−1), t).

Step 4: Using Metropolis–Hastings method, generate β(I) from Equation (38) with theN(β(I−1), σ) proposal distribution, where σ is from a variance–covariance matrix.

Step 5: Compute α(I), β(I), and λ(I).Step 6: Set I = I + 1.Step 7: Repeat steps 2 − 5 N times.Step 8: We obtain the Bayes MCMC point estimate of ϕl (ϕ1 = α, ϕ2 = β, and ϕ3 = λ) as

E(ϕl|data) = 1

N − M

N∑i=M+1

ϕ(i)l , (39)

where M is the burn-in period (that is, a number of iterations before the stationary distribution isachieved) and the posterior variance of ϕ becomes

V(ϕl|data) = 1

N − M

N∑i=M+1

(ϕ

(i)l − E(ϕl|data)

)2. (40)

Step 9: To compute the credible intervals (CRIs) of ϕl, we usually take the quantiles of thesample as the endpoints of the interval. Order ϕ

(M+1)

l , ϕ(M+2)

l , . . . , ϕ(N)

l as ϕl(1), ϕl(2), . . . , ϕl(N−M).Then, the 100(1 − 2γ )% symmetric CRI is

(ϕl(γ (N−M)), ϕl((1−γ )(N−M))

). (41)

4.2. Illustrative example

In this subsection, for illustrative purposes, we present a simulation example to check the esti-mation procedures. In this example, by using the algorithm described in [37], we generate twosamples from Pareto (α, β) distribution with parameters (α, β, λ) = (0.3, 3, 2), using progressiveCSs : (m1 = 20, m2 = 30, n1 = n2 = 50, R1 = (5, 0, 0, 5, 0, 0, 3, 0, 0, 0, 5, 2, 2, 2, 1, 1, 1, 1, 1,1) and R2 = (3, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 3, 0, 1, 0, 1, 0, 1, 0, 0, 2, 1). Thefollowing two progressively censored data sets were observed:Set 1: 0.046, 0.063, 0.136, o.189, 0.453, 1.086, 1.151, 1.458, 1.622, 1.738, 1.738, 2.799, 3.403,3.414, 11.492, 20.517, 48.172, 50.883, 86.442, 104.561.Set 2: 0.066, 0.106, 0.186, 0.432, 0.773, 1.010, 1.088, 1.104, 1.133, 1.541, 1.906, 1.977, 2.065,2.620, 3.048, 3.159, 3.279, 3.568, 5.532, 6.632, 7.395, 10.545, 10.563, 13.662, 14.245, 15.322,15.687, 17.584, 18.908, 19.304.

Figure 1 shows the PDFs under normal and accelerated conditions. Figure 2 plots the profile log-likelihood function of β. It is a unimodal function. We use the quasi-Newton–Raphson algorithmto compute the MLE from Equation (16) with the initial guess of β as 2.5; the MLEs of α andλ can be easily obtained from Equations (14) and (15). The MLEs and bootstrap point estimatesof the parameters as well as ACI, PBCIs, and BTCIs are obtained. In the MCMC approach, werun the chain for 11, 000 times and discard the first 1000 values as ‘burn-in’. The Bayes pointestimates and 90% CRIs for parameters α, β, and λ are computed. The results are presented inTable 1. Figures 3–8 show simulation number of α, β, and λ generated by MCMC samples andthe corresponding histograms. We observed that the BTCIs and CRIs are narrower than the PBCIsand always include the population parameter values.

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10 T.A. Abushal and A.A. Soliman

Figure 1. f1(t) with dashed line and f2(t) with black line.

Figure 2. Profile log-likelihood function of β.

Table 1. Different point estimates and 90% CIs for (α, β, λ) = (0.3, 3, 2).

90% bootstrap CIs 90% CRI

Pa.s (·)ML (·)Boot (·)MCMC 90% ACI PB BT MCMC

α 0.2980 0.3639 0.3109 (0.1527, 0.4433) (0.204, 0.646) (0.151, 0.412) (0.1827, 0.4789)β 2.3556 3.23207 2.8100 (0.2822, 4.429) (1.165, 6.833) (0, 3.759) (1.0508, 5.7552)λ 1.9655 2.10264 2.1036 (0.9894, 2.9416) (1.175, 3.417) (0.586, 2.775) (1.216, 3.3592)

5. Simulation studies

In this section, we present some results based on Monte Carlo simulations to compare the perfor-mance of the different methods. All computations were performed using (MATHEMATICA ver.8.0). To generate progressively Type-II censored Pareto samples, we used the algorithm proposedby Balakrishnan and Aggarwala.[27] The performance of the resulting estimators of α, β, and λ

has been considered in terms of their average estimates (AVG), absolute relative bias (ARB), and

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Figure 3. Simulation number of α generated by the MCMC method.

Figure 4. Simulation number of β generated by the MCMC method.

Figure 5. Simulation number of λ generated by the MCMC method.

mean square error (MSE), which they computed, for k = 1, 2, 3 and (ϕ1 = α, ϕ2 = β, ϕ3 = λ), as

AVG = 1

M

M∑i=1

ϕ(i)k , ARB = |ϕk − ϕk|

ϕk, MSE = 1

M

M∑i=1

(ϕ(i)k − ϕk)

2.

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Figure 6. Histogram of α generated by the MCMC method.

Figure 7. Histogram of β generated by the MCMC method.

Figure 8. Histogram of λ generated by the MCMC method.

Also, we compare CIs obtained by using asymptotic distributions of the MLEs, two bootstrap CIs,and MCMC CRIs. The comparison of them are made in terms of the average CI lengths/credibleinterval lengths (ACL) and coverage percentages (CP). For each simulated sample, we computed

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90% CIs and checked whether the true value lies within the interval and recorded the length of theCI. This procedure was repeated 1000 times. The estimated coverage probability was computedas the number of CIs that covered the true values divided by 1000, while the estimated expected

Table 2. AVG,ARB, and MSE of ML and Bayes MCMC estimates for the parameters with α = 0.5, β = 5, and λ = 1.5.

MLE MCMC

(n, m) CS α β λ α β λ

(30, 15) I 0.574 5.687 1.611 0.536 5.283 1.5710.147 (0.318) 0.137 (3.510) 0.074 (0.674) 0.058 (0.157) 0.108 (1.515) 0.054 (0.341)

II 0.567 6.202 1.662 0.517 5.403 1.5510.133 (0.214) 0.240 (3.290) 0.108 (0.608) 0.044 (0.099) 0.102 (1.291) 0.038 (0.201)

III 0.592 6.116 1.619 0.532 5.512 1.5690.185 (0.284) 0.223 (3.309) 0.079 (0.655) 0.055 (0.147) 0.105 (1.442) 0.044 (0.321)

IV 0.595 6.225 1.619 0.535 5.555 1.5490.190 (0.275) 0.245 (3.316) 0.080 (0.652) 0.057 (0.157) 0.108 (1.455) 0.045 (0.333)

(30, 25) I 0.572 6.376 1.595 0.533 5.400 1.5350.144 (0.196) 0.275 (3.406) 0.064 (0.513) 0.045 (0.100) 0.101 (1.323) 0.041 (0.221)

II 0.544 5.760 1.575 0.514 5.322 1.5410.088 (0.163) 0.152 (2.834) 0.050 (0.456) 0.023 (0.072) 0.091 (1.001) 0.022 (0.099)

III 0.558 6.089 1.602 0.544 5.312 1.5000.117 (0.179) 0.218 (3.047) 0.068 (0.511) 0.032 (0.105) 0.095 (1.032) 0.029 (0.221)

IV 0.566 6.026 1.564 0.522 5.425 1.5330.133 (0.185) 0.205 (3.008) 0.043 (0.472) 0.038 (0.107) 0.099 (1.155) 0.031 (0.233)

(50, 25) I 0.575 6.022 1.566 0.545 5.332 1.5510.150 (0.241) 0.204 (3.390) 0.044 (0.535) 0.049 (0.102) 0.105 (1.343) 0.040 (0.220)

II 0.559 6.170 1.595 0.522 5.411 1.5370.117 (0.170) 0.234 (2.975) 0.063 (0.463) 0.025 (0.077) 0.098 (1.011) 0.024 (0.100)

III 0.560 5.805 1.601 0.534 5.411 1.5230.119 (0.200) 0.161 (3.222) 0.069 (0.519) 0.037 (0.110) 0.099 (1.033) 0.028 (0.220)

IV 0.565 6.100 1.511 0.533 5.435 1.5300.118 (0.210) 0.224 (3.220) 0.074 (0.515) 0.039 (0.109) 0.100 (1.137) 0.029 (0.230)

(50, 40) I 0.575 6.022 1.566 0.523 5.055 1.5500.150 (0.145) 0.204 (2.493) 0.044 (0.426) 0.021 (0.077) 0.088 (0.770) 0.014 (0.099)

II 0.559 6.170 1.595 0.511 5.301 1.5110.117 (0.111) 0.234 (2.462) 0.063 (0.386) 0.011 (0.041) 0.058 (0.772) 0.013 (0.091)

III 0.560 5.805 1.601 0.524 5.011 1.5240.119 (0.140) 0.161 (2.481) 0.068 (0.391) 0.018 (0.055) 0.060 (0.933) 0.017 (0.111)

IV 0.565 6.100 1.511 0.521 5.332 1.5090.118 (0.138) 0.224 (2.472) 0.074 (0.403) 0.019 (0.058) 0.065 (0.942) 0.019 (0.111)

(100, 50) I 0.536 5.699 1.547 0.524 5.151 1.5310.071 (0.111) 0.140 (2.070) 0.031 (0.346) 0.019 (0.065) 0.081 (0.888) 0.017 (0.111)

II 0.523 5.529 1.322 0.533 5.351 1.5660.051 (0.097) 0.100 (1.071) 0.038 (0.309) 0.009 (0.035) 0.041 (0.670) 0.010 (0.082)

III 0.534 5.592 1.555 0.501 5.055 1.6240.067 (0.108) 0.118 (1.939) 0.036 (0.314) 0.016 (0.047) 0.050 (0.833) 0.017 (0.100)

IV 0.553 5.930 1.553 0.531 5.344 1.5220.105 (0.102) 0.186 (1.996) 0.035 (0.328) 0.017 (0.060) 0.068 (0.855) 0.017 (0.101)

(100, 75) I 0.522 5.561 1.536 0.511 5.022 1.5210.045 (0.271) 0.112 (1.816) 0.015 (0.055) 0.012 (0.044) 0.051 (0.688) 0.012 (0.088)

II 0.535 5.733 1.544 0.512 5.051 1.5630.070 (0.070) 0.147 (1.091) 0.029 (0.241) 0.004 (0.020) 0.025 (0.471) 0.07 (0.064)

III 0.545 5.831 1.574 0.530 5.155 1.5210.079 (0.080) 0.157 (1.492) 0.042 (0.262) 0.010 (0.040) 0.030 (0.531) 0.009 (0.097)

IV 0.521 5.493 1.545 0.532 5.343 1.5110.041 (0.082) 0.099 (1.529) 0.030 (0.269) 0.011 (0.045) 0.034 (0.655) 0.011 (0.100)

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14 T.A. Abushal and A.A. Soliman

width of the CI was computed as the sum of the lengths for all intervals divided by 1000. In ourstudy, we consider the following CS:

Scheme I: Rm = n − m, Ri = 0 for i �= m.Scheme II: R1 = n − m, Ri = 0 for i �= 1.

Table 3. Comparisons of ACL and CP of 90% CIs for the parameters with (α, β, λ) = (0.5, 5, 1.5).

MLE Boot-P Boot-t MCMC

(n, m) CS α β λ α β λ α β λ α β λ

(30, 15) I 1.568 18.915 2.113 1.030 12.901 2.070 1.003 12.339 2.009 0.990 8.330 1.021(0.90) (0.89) (0.89) (0.92) (0.92) (0.89) (0.88) (0.89) (0.87) (0.89) (0.89) (0.88)

II 0.676 12.023 2.091 0.657 10.820 2.011 0.602 10.003 2.000 0.402 6.003 1.001(0.95) (0.92) (0.91) (0.89) (0.95) (0.89) (0.91) (0.88) (0.89) (0.91) (0.89) (0.88)

III 0.830 12.475 2.520 0.820 11.613 2.222 0.729 10.817 2.008 0.529 7.811 1.149(0.95) (0.90) (0.88) (0.89) (0.95) (0.89) (0.90) (0.87) (0.86) (0.91) (0.87) (0.87)

IV 1.005 15.79 2.448 0.893 12.255 2.245 0.889 11.208 2.111 0.555 6.208 1.145(0.93) (0.90) (0.90) (0.93) (0.99) (0.91) (0.86) (0.88) (0.86) (0.89) (0.88) (0.92)

(30, 25) I 0.611 11.823 1.558 0.591 9.251 1.471 0.537 9.056 1.402 0.407 8.001 1.202(0.94) (0.91) (0.90) (0.89) (0.95) (0.88) (0.91) (0.86) (0.89) (0.90) (0.88) (0.88)

II 0.484 9.324 1.508 0.482 8.463 1.447 0.445 8.391 1.409 0.300 5.392 0.991(0.93) (0.90) (0.90) (0.87) (0.92) (0.89) (0.92) (0.88) (0.91) (0.92) (0.89) (0.90)

III 0.523 9.699 1.535 0.517 8.590 1.463 0.476 8.012 1.401 0.401 7.015 1.001(0.92) (0.93) (0.90) (0.86) (0.90) (0.89) (0.91) (0.88) (0.87) (0.91) (0.89) (0.88)

IV 0.533 9.632 1.499 0.513 8.615 1.426 0.487 8.014 1.429 0.412 6.999 1.029(0.94) (0.92) (0.89) (0.86) (0.91) (0.90) (0.92) (0.88) (0.89) (0.91) (0.89) (0.87)

(50, 25) I 1.135 14.419 1.558 0.973 10.147 0.902 1.125 10.335 1.065 0.998 7.333 1.000(0.92) (0.88) (0.89) (0.95) (0.99) (0.89) (0.876) (0.85) (0.89) (0.88) (0.89) (0.89)

II 0.502 9.837 1.531 0.501 8.630 1.552 0.461 8.185 1.497 0.360 5.105 0.984(0.95) (0.93) (0.89) (0.88) (0.92) (0.87) (0.93) (0.88) (0.84) (0.91) (0.88) (0.88)

III 0.578 8.884 1.529 0.610 8.280 1.552 0.517 8.143 1.509 0.401 6.149 1.009(0.92) (0.91) (0.91) (0.88) (0.92) (0.89) (0.90) (0.87) (0.88) (0.92) (0.89) (0.88)

IV 0.588 8.877 1.499 0.620 8.277 1.544 0.521 8.147 1.500 0.520 6.142 1.003(0.93) (0.91) (0.93) (0.89) (0.90) (0.89) (0.92) (0.89) (0.91) (0.92) (0.92) (0.93)

(50,40) I 0.603 9.395 1.283 0.629 7.346 1.263 0.377 6.936 1.218 0.309 3.930 0.780(0.93) (0.93) (0.91) (0.88) (0.89) (0.90) (0.92) (0.91) (0.90) (0.91) (0.91) (0.92)

II 0.355 5.395 9.950 0.325 5.375 9.650 0.320 5.345 1.590 0.220 3.310 0.602(0.91) (0.91) (0.90) (0.89) (0.89) (0.91) (0.92) (0.89) (0.88) (0.89) (0.88) (0.91)

III 0.403 7.395 1.183 0.409 7.346 1.173 0.377 6.936 1.518 0.321 3.936 0.701(0.92) (0.93) (0.92) (0.88) (0.87) (0.90) (0.93) (0.92) (0.89) (0.93) (0.91) (0.88)

IV 0.401 7.372 1.183 0.431 7.362 1.267 0.375 6.902 1.223 0.375 3.902 0.733(0.92) (0.94) (0.89) (0.88) (0.89) (0.88) (0.92) (0.89) (0.88) (0.91) (0.90) (0.89)

(100, 50) I 0.329 6.343 1.040 0.317 6.330 1.106 0.310 6.128 1.074 0.301 3.128 0.775(0.92) (0.93) (0.89) (0.88) (0.88) (0.89) (0.92) (0.91) (0.88) (0.90) (0.90) (0.89)

II 0.229 4.343 0.0950 0.207 4.200 0.9111 0.198 4.000 0.899 0.190 3.001 0.590(0.91) (0.92) (0.90) (0.89) (0.88) (0.89) (0.92) (0.91) (0.90) (0.91) (0.91) (0.88)

III 0.374 5.879 1.040 0.301 5.288 1.050 0.300 5.130 1.001 0.201 3.109 0.720(0.92) (0.93) (0.92) (0.87) (0.88) (0.90) (0.91) (0.92) (0.90) (0.88) (0.91) (0.91)

IV 0.417 7.589 1.050 0.406 7.456 1.107 0.392 6.983 1.020 0.222 3.009 0.739(0.95) (0.94) (0.90) (0.88) (0.89) (0.90) (0.93) (0.92) (0.88) (0.93) (0.93) (0.89)

(100, 75) I 0.258 5.035 0.840 0.247 5.111 0.830 0.240 4.965 0.820 0.122 2.961 0.510(0.90) (0.89) (0.89) (0.88) (0.89) (0.89) (0.90) (0.90) (0.91) (0.90) (0.92) (0.90)

II 0.239 4.258 0.857 0.231 4.514 0.904 0.214 3.884 0.816 0.100 1.884 0.416(0.90) (0.93) (0.89) (0.89) (0.88) (0.91) (0.92) (0.91) (0.90) (0.90) (0.89) (0.92)

III 0.278 4.921 0.845 0.301 4.344 0.886 0.251 4.037 0.829 0.115 2.740 0.499(0.90) (0.91) (0.91) (0.88) (0.89) (0.89) (0.90) (0.89) (0.91) (0.91) (0.88) (0.92)

IV 0.268 4.901 0.846 0.303 5.248 0.888 0.278 4.797 0.869 0.120 2.750 0.511(0.90) (0.91) (0.91) (0.88) (0.89) (0.89) (0.90) (0.89) (0.91) (0.92) (0.89) (0.90)

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Scheme III: R(m+1)/2 = n − m, Ri = 0 for i �= (m + 1)/2, if m odd; and Rm/2 = n − m, Ri = 0for i �= m

2 , if m even.scheme IV: R((2m−n)/2)+1 = · · · = Rn/2 = 1, other Ri = 0.For population parameters and sample sizes, we consider the following two cases:

Table 4. AVG, ARB, and MSE of ML and Bayes estimates for the parameters with α = 0.2, β = 3, and λ = 2.

MLE MCMC

(n, m1, m2) CS α β λ α β λ

(20, 10, 20) I 0.261 4.543 2.246 0.242 3.511 2.3110.303 (0.158) 0.514 (3.504) 0.123 (1.168) 0.300 (0.089) 0.401 (2.003) 0.099 (0.889)

II 0.233 3.921 2.130 0.230 3.320 2.1110.163 (0.099) 0.307 (2.664) 0.065 (0.937) 0.150 (0.077) 0.211 (1.009) 0.053 (0.377)

III 0.254 4.389 2.266 0.255 3.319 2.2150.268 (0.135) 0.463 (3.025) 0.133 (1.106) 0.202 (0.099) 0.301 (1.895) 0.076 (0.786)

IV 0.244 4.446 2.281 0.232 3.441 2.1180.220 (0.136) 0.482 (3.210) 0.141 (1.100) 0.203 (0.100) 0.278 (1.789) 0.074 (0.780)

(20, 15, 30) I 0.226 3.976 2.174 0.231 3.335 2.1140.128 (0.082) 0.325 (2.509) 0.087 (0.809) 0.115 (0.055) 0.125 (1.588) 0.073 (0.708)

II 0.221 3.800 2.119 0.204 3.333 2.1120.103 (0.074) 0.267 (2.264) 0.059 (0.731) 0.085 (0.065) 0.163 (1.000) 0.049 (0.331)

III 0.227 3.944 2.125 0.204 3.144 2.0220.137 (0.081) 0.315 (2.394) 0.062 (0.781) 0.102 (0.060) 0.115 (1.335) 0.061 (0.501)

IV 0.224 4.122 2.210 0.221 3.533 2.1440.121 (0.075) 0.374 (2.433) 0.105 (0.795) 0.101 (0.058) 0.108 (1.311) 0.060 (0.544)

(40, 20, 40) I 0.243 4.337 2.165 0.241 3.377 2.1280.215 (0.097) 0.446 (2.849) 0.083 (0.742) 0.155 (0.075) 0.128 (1.681) 0.092 (0.738)

II 0.214 3.442 2.013 0.214 3.355 2.1320.108 (0.064) 0.151 (1.549) 0.022 (0.500) 0.092 (0.072) 0.176 (1.055) 0.060 (0.341)

III 0.224 3.542 2.063 0.234 3.158 2.1170.118 (0.074) 0.181 (1.649) 0.032 (0.640) 0.112 (0.083) 0.115 (1.344) 0.072 (0.531)

IV 0.222 3.687 2.069 0.225 3.534 2.3340.112 (0.071) 0.229 (2.033) 0.035 (0.605) 0.109 (0.065) 0.117 (1.322) 0.065 (0.554)

(40, 30, 60) I 0.218 3.624 2.053 0.210 3.221 2.0500.088 (0.054) 0.208 (1.603) 0.026 (0.520) 0.075 (0.041) 0.128 (1.303) 0.020 (0.420)

II 0.209 3.406 2.031 0.202 3.422 2.2210.056 (0.047) 0.142 (1.312) 0.015 (0.502) 0.041 (0.038) 0.112 (0.999) 0.013 (0.222)

III 0.215 3.456 2.037 0.204 3.321 2.1250.076 (0.049) 0.152 (1.470) 0.019 (0.509) 0.056 (0.055) 0.100 (1.070) 0.014 (0.500)

IV 0.225 3.556 2.039 0.200 3.188 2.0330.078 (0.051) 0.154 (1.471) 0.018 (0.511) 0.067 (0.041) 0.101 (1.371) 0.020 (0.412)

(60, 30, 60) I 0.221 3.513 2.053 0.221 3.513 2.0530.049 (0.054) 0.134 (1.603) 0.027 (0.520) 0.049 (0.054) 0.134 (1.603) 0.027 (0.320)

II 0.212 3.313 2.043 0.212 3.313 2.0430.042 (0.047) 0.104 (1.312) 0.021 (0.502) 0.042 (0.047) 0.098 (0.754) 0.010 (0.102)

III 0.213 3.413 2.051 0.241 3.321 2.1110.047 (0.049) 0.124 (1.470) 0.024 (0.509) 0.045 (0.049) 0.121 (1.111) 0.021 (0.150)

IV 0.214 3.399 2.052 0.224 3.185 2.1440.048 (0.051) 0.125 (1.471) 0.025 (0.511) 0.046 (0.051) 0.122 (1.144) 0.024 (0.160)

(60, 50, 90) I 0.211 3.370 2.050 0.221 3.321 2.0080.042 (0.042) 0.099 (1.111) 0.022 (0.476) 0.031 (0.033) 0.085 (1.003) 0.027 (0.220)

II 0.207 3.270 2.033 0.210 3.513 2.1410.038 (0.032) 0.090 (1.073) 0.017 (0.376) 0.030 (0.037) 0.086 (0.754) 0.008 (0.100)

III 0.210 3.290 2.041 0.242 3.121 2.1550.040 (0.038) 0.095 (1.099) 0.020 (0.399) 0.039 (0.039) 0.089 (1.0011) 0.018 (0.120)

IV 0.209 3.280 2.040 0.221 3.155 2.3210.039 (0.037) 0.093 (1.095) 0.019 (0.397) 0.033 (0.032) 0.089 (1.101) 0.015 (0.150)

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16 T.A. Abushal and A.A. Soliman

Table 5. Comparisons of ACL and CP of 90% CIs for the parameters with (α, β, λ) = (0.2, 3, 1.5).

MLE Boot-P Boot-t MCMC

(n, m1, m2) CS α β λ α β λ α β λ α β λ

(20, 10, 20) I 0.457 13.731 3.380 0.456 9.906 3.312 0.405 10.297 3.307 0.308 6.290 2.018(0.94) (0.89) (0.88) (0.88) (0.96) (0.88) (0.87) (0.87) (0.86) (0.89) (0.88) (0.91)

II 0.270 8.140 2.808 0.302 8.258 2.919 0.254 8.014 2.201 0.201 5.012 2.009(0.92) (0.91) (0.87) (0.88) (0.91) (0.88) (0.91) (0.87) (0.86) (0.92) (0.89) (0.88)

III 0.325 8.612 3.054 0.315 8.589 3.037 0.298 8.201 3.007 0.275 5.299 3.015(0.93) (0.93) (0.85) (0.86) (0.87) (0.88) (0.91) (0.88) (0.87) (0.90) (0.89) (0.92)

IV 0.325 9.917 3.165 0.300 9.046 3.037 0.292 9.037 3.010 0.282 5.237 3.009(0.93) (0.92) (0.89) (0.88) (0.87) (0.88) (0.91) (0.87) (0.88) (0.92) (0.89) (0.88)

(20, 15, 30) I 0.222 7.405 2.520 0.270 7.910 2.717 0.207 7.100 2.489 0.187 6.009 2.004(0.92) (0.92) (0.89) (0.88) (0.88) (0.88) (0.92) (0.88) (0.86) (0.91) (0.87) (0.89)

II 0.203 6.550 2.272 0.201 6.087 2.503 0.197 6.005 2.211 0.108 5.003 1.548(0.91) (0.89) (0.87) (0.87) (0.88) (0.89) (0.92) (0.86) (0.87) (0.90) (0.88) (0.91)

III 0.217 6.626 2.301 0.221 7.354 2.553 0.208 6.454 2.005 0.158 5.954 1.905(0.92) (0.93) (0.89) (0.85) (0.87) (0.90) (0.92) (0.90) (0.85) (0.91) (0.92) (0.89)

IV 0.213 6.969 2.406 0.215 7.482 2.652 0.204 6.750 2.114 0.155 5.850 1.813(0.94) (0.93) (0.91) (0.86) (0.85) (0.90) (0.92) (0.89) (0.87) (0.91) (0.89) (0.88)

(40, 20, 40) I 0.265 8.584 2.191 0.2567 8.253 2.202 0.232 8.183 2.161 0.165 6.111 1.862(0.95) (0.94) (0.90) (0.88) (0.85) (0.88) (0.93) (0.89) (0.87) (0.92) (0.90) (0.88)

II 0.142 4.569 1.900 0.129 4.628 1.990 0.116 4.372 1.861 0.100 4.079 1.231(0.92) (0.92) (0.89) (0.87) (0.88) (0.89) (0.90) (0.89) (0.88) (0.91) (0.89) (0.93)

III 0.192 4.769 1.918 0.229 5.628 2.099 0.186 4.672 2.061 0.133 5.474 2.009(0.92) (0.93) (0.88) (0.87) (0.88) (0.89) (0.91) (0.89) (0.86) (0.88) (0.89) (0.88)

IV 0.196 5.527 1.956 0.203 5.571 2.005 0.185 5.322 1.903 0.137 5.321 1.993(0.92) (0.92) (0.90) (0.87) (0.86) (0.90) (0.92) (0.89) (0.87) (0.91) (0.89) (0.88)

(40, 30, 60) I 0.148 4.623 1.584 0.147 4.702 1.520 0.144 4.538 1.451 0.098 4.008 1.059(0.92) (0.93) (0.90) (0.87) (0.87) (0.89) (0.92) (0.91) (0.87) (0.91) (0.88) (0.89)

II 0.138 4.223 1.384 0.137 4.202 1.320 0.124 4.018 1.251 0.075 3.119 1.001(0.92) (0.92) (0.91) (0.89) (0.88) (0.89) (0.92) (0.91) (0.88) (0.91) (0.92) (0.90)

III 0.143 4.293 1.546 0.149 4.077 1.547 0.141 4.070 1.518 0.087 3.970 1.222(0.91) (0.92) (0.87) (0.86) (0.89) (0.88) (0.89) (0.89) (0.88) (0.87) (0.89) (0.89)

IV 0.146 4.395 1.548 0.149 4.347 1.547 0.141 4.072 1.518 0.088 3.888 1.233(0.91) (0.92) (0.87) (0.85) (0.89) (0.88) (0.89) (0.89) (0.86) (0.91) (0.88) (0.88)

(60, 30, 60) I 0.149 4.630 1.591 0.145 4.712 1.521 0.145 4.539 1.452 0.118 4.118 1.150(0.91) (0.92) (0.91) (0.88) (0.87) (0.88) (0.91) (0.91) (0.88) (0.93) (0.89) (0.89)

II 0.134 4.032 1.537 0.137 4.061 1.517 0.133 4.017 1.411 0.078 3.129 1.018(0.93) (0.89) (0.88) (0.89) (0.90) (0.88) (0.93) (0.87) (0.88) (0.92) (0.91) (0.88)

III 0.144 4.332 1.537 0.147 4.569 1.517 0.143 4.027 1.511 0.088 3.889 1.234(0.92) (0.88) (0.87) (0.88) (0.89) (0.87) (0.91) (0.87) (0.86) (0.87) (0.91) (0.89)

IV 0.147 4.385 1.549 0.148 4.348 1.551 0.142 4.071 1.519 0.088 3.888 1.233(0.92) (0.91) (0.88) (0.87) (0.89) (0.89) (0.89) (0.90) (0.87) (0.92) (0.88) (0.88)

(60, 50, 90) I 0.133 3.990 1.508 0.130 3.888 1.408 0.1250 3.718 1.210 0.119 3.229 1.109(0.93) (0.88) (0.90) (0.89) (0.88) (0.91) (0.91) (0.88) (0.87) (0.91) (0.92) (0.88)

II 0.103 3.229 1.208 0.102 3.268 1.158 0.100 3.218 1.195 0.055 2.515 1.009(0.92) (0.89) (0.91) (0.89) (0.87) (0.90) (0.91) (0.87) (0.88) (0.91) (0.88) (0.92)

III 0.123 3.529 1.338 0.112 3.468 1.258 0.110 3.319 1.200 0.098 3.049 1.055(0.91) (0.88) (0.90) (0.89) (0.88) (0.92) (0.91) (0.88) (0.88) (0.91) (0.93) (0.89)

IV 0.125 3.527 1.335 0.122 3.477 1.261 0.112 3.321 1.211 0.089 3.111 1.115(0.92) (0.87) (0.92) (0.88) (0.88) (0.90) (0.91) (0.89) (0.87) (0.92) (0.87) (0.88)

(i) The population parameter values (α = 0.5, β = 5, λ = 1.5), with sample sizes (n1 = n2 = n)

and observed failure times (m1 = m2 = m). The results are shown in Tables 2 and 3.(ii) The population parameter values (α = 0.2, β = 3, λ = 2), the sample sizes (n2 = 2n1 = 2n),

and the results are shown in Tables 4 and 5.

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Journal of Statistical Computation and Simulation 17

6. Concluding remarks

The progressive censoring is of great importance in planning duration experiments in reliabilitystudies. It has been shown by Viveros and Balakrishnan [41] that the inference is possible andpractical when the sample data are gathered according to a progressively Type-II censored scheme.Based on progressively Type-II censored samples, this article is related to full Bayes and non-Bayes procedures for the analysis of the constant-partially accelerated life testing using the Paretofailure model. The classical Bayes estimates cannot be obtained in explicit form. One can clearlysee the scope of MCMC-based Bayesian solutions which make every inferential developmentroutinely available. A simulation study was conducted to examine and compare the performanceof the proposed methods for different sample sizes, different CSs, different acceleration factors,and different parameter values. From the results, we observe the following:

(1) In general, for increasing effective sample size m/n, the MSEs and ARBs of the consideredparameters decrease.

(2) For fixed values of the sample and failure time sizes, the scheme II in which the censoringoccurs after the first observed failure gives more accurate results through the MSEs and RABsthan the other schemes and this coincides with Theorem [2.2] by Burkschat et al.[42]

(3) Results in the CSs III and IV are close to each other.(4) The MCMC CRIs give more accurate results than the approximate CIs and bootstrap CIs

since the lengths of the former are less than the lengths of latter, for different sample sizes,observed failures and schemes.

(5) For fixed sample sizes and observed failures, the second scheme II, in which censoring occursafter the first observed failures, gives lower lengths for the three methods of the CIs otherthan the other three schemes.

(6) For small sample sizes, the MCMC CRIs and the bootstrap-t CIs are better than the percentilebootstrap CIs in the sense of having smaller widths.

Acknowledgements

The authors would like to express their thanks to the referees for their useful comments in revising the paper.

Funding

This research was supported by the Institute of Scientific Research and Revival of Islamic Heritage, Umm Al-QuraUniversity, under the project (No. 43205014).

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