Estimation from Burr type XII distribution using progressive first-failure censored data

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Estimation from Burr type XIIdistribution using progressive first-failure censored dataAhmed A. Soliman a b , Ahmed H. Abd Ellah b , Naser A. Abou-Elheggag b & Abdullah A. Modhesh ba Faculty of Science, Islamic University, Madinah, Saudi Arabiab Department of Mathematics, Sohag University, Sohag, 82524,EgyptPublished online: 28 May 2012.

To cite this article: Journal of Statistical Computation and Simulation (2012): Estimation from Burrtype XII distribution using progressive first-failure censored data, Journal of Statistical Computationand Simulation, DOI: 10.1080/00949655.2012.690157

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Journal of Statistical Computation and SimulationiFirst, 2012, 1–21

Estimation from Burr type XII distribution using progressivefirst-failure censored data

Ahmed A. Solimana,b*, Ahmed H. Abd Ellahb, Naser A. Abou-Elheggagb andAbdullah A. Modheshb

aFaculty of Science, Islamic University, Madinah, Saudi Arabia; bDepartment of Mathematics,Sohag University, Sohag 82524, Egypt

(Received 2 August 2011; final version received 29 April 2012)

In this paper, a new life test plan called a progressively first-failure-censoring scheme introduced by Wuand Kus [On estimation based on progressive first-failure-censored sampling, Comput. Statist. Data Anal.53(10) (2009), pp. 3659–3670] is considered. Based on this type of censoring, the maximum likelihood(ML) and Bayes estimates for some survival time parameters namely reliability and hazard functions, aswell as the parameters of the Burr-XII distribution are obtained. The Bayes estimators relative to boththe symmetric and asymmetric loss functions are discussed. We use the conjugate prior for the one-shapeparameter and discrete prior for the other parameter. Exact and approximate confidence intervals with theexact confidence region for the two-shape parameters are derived. A numerical example using the real dataset is provided to illustrate the proposed estimation methods developed here. The ML and the differentBayes estimates are compared via a Monte Carlo simulation study.

Keywords: Burr type XII distribution; progressive first-failure censored sample; Bayesian and non-Bayesian estimations; symmetric and asymmetric loss functions; graphical method; exact confidenceinterval; exact confidence region; Monte Carlo simulation

1. Introduction

Censoring is common in life-distribution work because of time limits and other restrictions ondata. In reliability analysis, the most common censoring schemes are type I and type II censoring.One important characteristic of these two censoring schemes is that they do not allow for unitsto be removed from the test at any point other than the final termination point. However, if anexperimenter desires to remove surviving units at points other than the final termination point ofthe life test, these two traditional censoring schemes will not be of use to the experimenter. Theallowance of removing surviving units from the test before the final termination point is desirable,as in the case of studies of wear, in which the study of the actual aging process requires units to befully disassembled at different stages of the experiment. In addition, when a compromise betweenthe reduced time of experimentation and the observation of at least some extreme lifetimes is

*Corresponding author. Email: [email protected]

ISSN 0094-9655 print/ISSN 1563-5163 online© 2012 Taylor & Francishttp://dx.doi.org/10.1080/00949655.2012.690157http://www.tandfonline.com

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2 A.A. Soliman et al.

sought, such an allowance is also desirable. These reasons lead us into the area of progressivecensoring. The subject of progressive censoring has received considerable attention in the pastfew years, due in part to the availability of high-speed computing resources, which make itboth a feasible topic for simulation studies for researchers, and a feasible method of gatheringlifetime data for practitioners. It has been illustrated by Viveros and Balakrishnan [1] that theinference is feasible, and practical when the sample data are gathered according to a type IIprogressively censored experimental scheme. It is well known that one of the primary goals ofprogressive censoring is to save some live units for other tests, which is particularly useful whenthe units being tested are very expensive. Statistical inferences on the parameters of failure timedistributions under progressive censoring have been studied by several authors such as Viverosand Balakrishnan [1], Balakrishnan and Sandhu [2,3], Balasooriya and Balakrishnan [4], Nget al. [5], Ng [6], Balakrishnan et al. [7] and Soliman [8,9]. A recent account on progressivecensoring schemes can be found in the book by Balakrishnan and Aggarwala [10], or in theexcellent review by Balakrishnan [11].

Johnson [12] described a life test in which the experimenter might decide to group the test unitsinto several sets, each as an assembly of test units, and then run all the test units simultaneouslyuntil occurrence the first failure in each group. Such a censoring scheme is called first-failurecensoring. Jun et al. [13] discussed a sampling plan for a bearing manufacturer. The bearing testengineer decided to save test time by testing 50 bearings in sets of 10 each. The first-failure timesfrom each group were observed. Wu et al. [14] and Wu andYu [15] obtained maximum-likelihoodestimates (MLEs), exact confidence intervals and exact confidence regions for the parameters ofthe Gompertz and Burr type XII distributions based on first-failure-censored sampling, respec-tively, also see [16,17]. Recently, Wu and Kus [18] obtained MLEs, exact confidence intervals andexact confidence regions for the parameters of Weibull distribution under progressive first-failure-censored sampling. Note that a first-failure-censoring scheme is terminated when the first failurein each set is observed. If an experimenter desires to remove some sets of test units before observ-ing the first failures in these sets, this life test plan is called a progressive first-failure-censoringscheme which was recently introduced by Wu and Kus [18] and Soliman et al. [19]. Therefore,the purpose of this paper is to develop Bayes estimation and to construct exact confidence inter-val with the exact confidence region for the parameters of Burr type XII distribution under theprogressive first-failure-censoring plan. The Burr system of distributions includes 12 types ofcumulative distribution functions (cdfs) which yield a variety of density shapes and were listed inBurr [20] and Lewis [21], noted that many standard theoretical distributions, Gompertz, normal,extreme value and uniform are special cases or limiting cases of the Burr system of distributions.The simple closed form of the distribution functions and the inverse of the distribution functionsof Burr’s system play an important role in the selection of a particular family of distributions asa stochastic model in simulation studies.

If X follows a Burr type XII distribution, then the probability density function (pdf) and cdfare given, respectively, by

f (x) = βαxβ−1(1 + xβ)−(α+1), x > 0, (β > 0, α > 0). (1)

F(x) = 1 − (1 + xβ)−α , x > 0, (2)

The corresponding reliability and failure rate functions of this distribution at some t, are given,respectively, by

S(t) = (1 + tβ)−α , t > 0, (3)

h(t) = βαtβ−1(1 + tβ)−1, t > 0, (4)

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Journal of Statistical Computation and Simulation 3

where h(t) has a unique maximum when β > 1. The distribution has a decreasing failure ratefunction when β ≤ 1. Thus, the shape parameter β plays an important role for the distribution. Ithas been applied in areas of quality control, reliability studies, duration and failure time modelling.Other areas of application include the analysis of business failure data, the efficacy of analgesics inclinical trials, and the times to failure of electronic components.Also, it has been used in situationswhere the s-normal distribution is not an appropriate model. Zimmer et al. [22] discussed thestatistical and probabilistic properties of the Burr-XII and its relationship to other distributionsused in reliability analyses. Several authors considered different aspects of this distribution, seefor example Al-Hussaini and Jaheen [23–25] and Soliman [8].

The rest of this paper is organized as follows. In Section 2, we describe the formulation of aprogressive first-failure-censoring scheme as described by Wu and Kus [18]. The point estimationof the parameters of Burr type XII distribution based on the progressive first-failure-censoringscheme are investigated in Section 3. In Section 4, we discuss the approximate and exact intervalestimations with the exact confidence region for the parameters of the Burr type XII distributionunder the progressive first-failure-censored sampling plan. Numerical examples using the realdata set are presented in Section 5 for illustration. In Section 6, we provide a simulation study inorder to give an assessment of the performance of the estimation method. Finally, we concludethe paper in Section 7.

2. A progressive first-failure-censoring scheme

In this section, first-failure censoring is combined with progressive censoring as in Wu andKus [18]. Suppose that n independent groups with k items within each group are put on alife test, R1 groups and the group in which the first failure is observed are randomly removedfrom the test as soon as the first failure (say XR

1:m:n:k) has occurred, R2 groups and the group inwhich the second first failure is observed are randomly removed from the test when the sec-ond failure (say XR

2:m:n:k) has occurred, and finally Rm (m ≤ n) groups and the group in whichthe mth first failure is observed are randomly removed from the test as soon as the mth failure(say XR

m:m:n:k) has occurred. The XR1:m:n:k < XR

2:m:n:k < · · · < XRm:m:n:k are called progressively first-

failure-censored order statistics with the progressive censoring scheme R = (R1, R2, . . . , Rm). Itis clear that n = m + R1 + R2 + · · · + Rm. If the failure times of the n × k items originally in thetest are from a continuous population with distribution function F(x) and pdf f (x), the joint pdffor XR

1:m:n:k , XR2:m:n:k , . . . , XR

m:m:n:k is given by

f1,2,...,m(xR1:m:n:k , xR

2:m:n:k , . . . , xRm:m:n:k) = ckm

m∏j=1

f (xRj:m:n:k)(1 − F(xR

j:m:n:k))k(Rj+1)−1 (5)

0 < xR1:m:n:k < xR

2:m:n:k < · · · < xRm:m:n:k < ∞, (6)

where

c = n(n − R1 − 1)(n − R1 − R2 − 1) · · · (n − R1 − R2 − · · · − Rm−1 − m + 1). (7)

There are four special cases:The first one if R = (0, . . . , 0), Equation (5) reduces to the joint pdf of first-failure-censored

order statistics. The second case if k = 1, Equation (5) becomes the joint pdf of the progressivelytype II censored statistics. The third case if k = 1 and R = (0, . . . , 0), then n = m which corre-sponds to the complete sample. The last one if k = 1 and R = (0, . . . , n − m), then the type IIcensored order statistics are obtained.Also, it can be seen that the progressive first-failure-censored

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order statistics XR1:m:n:k , XR

2:m:n:k , . . . , XRm:m:n:k can be viewed as a progressively type II censored

sample from a population with distribution function 1 − (1 − F(x))k .

3. Maximum-likelihood estimation

Let xRi:m:n:k , i = 1, 2, . . . , m, be the progressively first-failure-censored sample from a Burr type

XII distribution, with a censoring scheme R. For convenience, we will denote the observed valuesof such a progressively first-failure-censored sample by xi, i = 1, 2, . . . , m. From Equation (5),the likelihood function is given by

�(β, α) = ckmβmαmm∏

i=1

xβ−1i (1 + xβ

i )−(αk(Ri+1)+1), (8)

where c is defined in Equation (7).The log-likelihood function may then be written as

L(β, α) = log c + m log k + m log β + m log α + (β − 1)

m∑i=1

log(xi)

−m∑

i=1

(αk(Ri + 1) + 1) log(1 + xβi ). (9)

Differentiating Equation (9) with respect to β, and α, and equating each result to zero, we obtaintwo likelihood equations given by

∂L(β, α)

∂β= m

β+

m∑i=1

log(xi) −m∑

i=1

(αk(Ri + 1) + 1)xβ

i log(xi)

(1 + xβi )

, (10)

and∂L(β, α)

∂α= m

α−

m∑i=1

k(Ri + 1) log(1 + xβi ). (11)

It follows, from Equation (11), that

α = m

(m∑

i=1

k(Ri + 1) log(1 + xβi )

)−1

, (12)

and β the solution of

m

β+

m∑i=1

log(xi) −(

m∑mi=1 k(Ri + 1) log(1 + xβ

i )

)

×m∑

i=1

(Ri+1)xβ

i log(xi)

(1 + xβi )

−m∑

i=1

xβi log(xi)

(1 + xβi )

= 0. (13)

Newton–Raphson iteration is employed to solve Equation (13). The corresponding MLE of thereliability function S(t) and hazard rate function h(t), are given, respectively, by Equations (3)and (4) after replacing β, and α by their MLE’s β, and α. To obtain a starting value for the root

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finding method, we can use the graphical method discussed in Balakrishnan and Kateri [26]. Werewrite Equation (13) in the form

1

β= − 1

m

m∑i=1

log(xi) +(

1∑mi=1 k(Ri + 1) log(1 + xβ

i )

)

×m∑

i=1

(Ri + 1)xβ

i log(xi)

(1 + xβi )

+ 1

m

m∑i=1

xβi log(xi)

(1 + xβi )

= H1(β; x). (14)

We denote the right-hand side of Equation (14) by H1(β; x), and if we can show that, for a givensample xi, i = 1, 2, . . . , m., H1(β; x) is a monotone increasing function of β with a finite andpositive limit as β → ∞. Since 1/β is strictly decreasing with a right limit +∞ at 0, it wouldthen follow that the plots of 1/β and H1(β; x) would intersect exactly once at the value of β, andthe resulting value can be used as a starting value for the Newton–Raphson iterative method.

4. Bayes estimation

Recently, the Bayesian approach has received large attention for analysing failure data and othertime-to-event data, and has been often proposed as a valid alternative to traditional statisticalperspectives. The Bayesian approach to reliability analysis allows prior subjective knowledge onlifetime parameters and technical information on the failure mechanism, as well as experimentaldata, to be incorporated into the inferential procedure. Hence Bayesian methods usually requireless sample data to achieve the same quality of inferences than methods based on sampling theory,which becomes extremely important in the case of expensive testing procedures. In this section,we discuss the Bayesian estimation of the Burr-XII distribution based on progressive first-faliurecensored samples.

4.1. Loss function and prior information

A wide variety of loss functions have been developed in literature to describe various types of lossstructures. The symmetric squared-error (SE) loss is one of the most popular loss functions. It iswidely employed in inference, but its application is motivated by its good mathematical properties,not by its applicability to representing a true loss structure. A loss function should represent theconsequences of different errors. There are situations where over- and under-estimation can leadto different consequences. For example, when we estimate the average reliable working life ofthe components of a spaceship or an aircraft, over-estimation is usually more serious than under-estimation than an underestimation. Being symmetric, the SE loss equally penalizes over- andunder-estimation of the same magnitude. A useful asymmetric loss known as the LINEX lossfunction, was introduced by Prakash and Singh [27], and was widely used in several papers, seefor example, [4,8,9,28,29]. This function rises approximately exponentially on one side of zero,and approximately linearly on the other side. Under the assumption that the minimal loss occursat u = u, the LINEX loss function for u = u(β, α) can be expressed as

L1(�) ∝ ea� − a� − 1, a �= 0, (15)

where � = (u − u), u is an estimate of u.The sign, and magnitude of a represent the direction, and degree of symmetry (a > 0 means

overestimation is more serious than underestimation, and a < 0 means the opposite). For (a) closeto zero, the LINEX loss function is approximately the SE loss, and therefore almost symmetric

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The posterior s-expectation of the LINEX loss function of Equation (15) is

Eu(L(u − u)) ∝ exp(au).Eu[exp(−au)] − a · (u − Eu[u]) − 1. (16)

where Eu is equivalent to the posterior s-expectation with respect to the posterior pdf(u). The Bayesestimator uBL of u under the LINEX loss function is the value u, which minimizes Equation (16)

uBL = −1

alog(Eu[exp(−au)]), (17)

provided that Eu[exp(−au)] exists, and is finite.Another useful asymmetric loss function is the general entropy (GE) loss

L2(u, u) ∝(

u

u

)q

− q log

(u

u

)− 1. (18)

whose minimum occurs at u = u. This loss function is a generalization of the Entropy-loss usedin several papers where q = 1 by Dey et al. [30] and Dey and Liu [31]. When q > 0, a positiveerror (u > u) causes more serious consequences than a negative error. The Bayes estimate uBL ofu under GE loss (18) is

uBG = (Eu[u−q])−q (19)

provided that Eu[u−q] exists, and is finite.

4.2. Prior distribution and posterior analysis

When the parameters β and α, are assumed to be unknown, we assume that the parameter β hasa discrete prior distribution, while α has a conjugate prior distribution.

Suppose that the parameter β is restricted to a finite number of values, say β1, β2, . . . , βN .With prior probability η1, η2, . . . , ηN , respectively, where 0 ≤ ηj ≤ 1 and

∑Nj=1 ηj = 1, thus, the

discrete probability distribution of β is

π(β) = Pr(β = βj) = ηj. (20)

Further, suppose that conditional upon β = βj, j = 1, 2, . . . , N , α has a natural conjugate gammaprior, with density

π(α|β = βj) = abj

j αbj−1 exp(−ajα)

(bj), α; aj, bj > 0. (21)

Combining the likelihood function in Equation (8), and prior density (21), we obtain the posteriordensity of α

π∗(α|β = βj; Tj) = (Tj + aj)(bj+m)αbj+m−1 exp(−α(Tj + aj))

(bj + m), (22)

where

Tj =m∑

i=1

k(Ri + 1) ln(1 + xβj

i ). (23)

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The joint posterior of α and βj by using Equations (8), (20) and (21) is

π∗(α, β = βj, Tj) = βmabj

j vjηjαbj+m−1 exp(−α(Tj + aj))

k2(bj), (24)

and the marginal posterior probability of βj is

Pj = Pr(β = βj|Tj) = βmabj

j vjηj(bj + m)

k2(bj)(Tj + aj)(bj+m)

, (25)

where

k2 =N∑

j=1

βmabj

j vjηj(bj + m)

(bj)(Tj + aj)(bj+m)

, and vj =m∏

i=1

xβj−1i

1 + xβj

i

. (26)

The marginal posterior probability of α is

π∗(α|Tj) =N∑

j=1

π∗(α|β = βj, Tj). (27)

4.3. Symmetric Bayes estimation

SE loss function: Under SE loss function (symmetric), the estimator of a parameters (or a givenfunction of the parameters) is the posterior mean. Thus, Bayes estimators of the parameters,reliability functions, and hazard rate functions are obtained by using the posterior density (22)and (25).

The Bayes estimators βBS and αBS of the parameters β and α are

βBS = Eβ(β|x) =N∑

j=1

βjPj, (28)

and

αBS =∫ ∞

0

N∑j=1

αPjπ∗(α|β = βj, Tj) dα =

N∑j=1

Pj(bj + m)

(Tj + aj). (29)

The Bayes estimators, SBS, and hBS of the reliability function S ≡ S(t), and the hazard rate functionh ≡ h(t), respectively, are

SBS(t) =N∑

j=1

Pj

[1 + ln(1 + tβj )

(Tj + aj)

]−(bj+m)

, (30)

and

hBS =N∑

j=1

Pjβjtβj−1(bj + m)

(1 + tβj )(Tj + aj). (31)

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4.4. Asymmetric Bayes estimation

LINEX loss function: If in Equation (17), u = β, then the Bayes estimate βBL, of parameter β

relative to the LINEX loss function is

βBL = −1

alog

⎡⎣ N∑

j=1

Pj exp(−aβj)

⎤⎦ . (32)

Set u = α in Equation (17) then the Bayes estimate αBL, of parameter α relative to the LINEXloss function is

αBL = −1

alog

⎡⎣ N∑

j=1

Pj

(1 + a

(Tj + aj)

)−(bj+m)

⎤⎦ . (33)

Set u = S(t), in Equation (17), where S(t) is given by Equation (3), then the Bayes estimate SBL is

SBL(t) = −1

alog

⎡⎣ N∑

j=1

∞∑s=1

(−a)s

s! Pj

(1 + s ln(1 + tβj )

(Tj + aj)

)−(bj+m)⎤⎦ · (34)

Also after setting u = h(t), in Equation (17), where h(t) is given by Equation (4), then the Bayesestimate hBL, of the hazard rate function of Equation (4) relative to the LINEX loss function is

hBL = −1

alog

⎡⎣ N∑

j=1

∞∑s=1

Pj

(1 + aβjtβj−1

(1 + tβj )(Tj + aj)

)−(bj+m)⎤⎦ . (35)

GE loss function: Let u = β, in Equation (19), then the Bayes estimate βBG, of parameter β

relative to the GE loss function (19) is

βBG =⎡⎣ N∑

j=1

β−qj Pj

⎤⎦

(−1/q)

· (36)

If in Equation (19), u = α, then the Bayes estimate αBG, of parameter α relative to the GE lossfunction is

αBG =⎡⎣ N∑

j=1

Pj(Tj + aj)

q(bj + m − q)

(bj + m)

⎤⎦

(−1/q)

. (37)

If in Equation (19), u = S(t), where S(t) is given by Equation (3), then the Bayes estimate SBG is

SBG(t) =⎡⎣ N∑

j=1

Pj

(1 − q ln(1 + tβj )

(Tj + aj)

)−(bj+m)⎤⎦

(−1/q)

. (38)

If in Equation (19), u = h(t), where h(t) is given by Equation (4), then the Bayes estimate hBL, is

hBG(t) =⎡⎣ N∑

j=1

Pj

(βjtβj−1

1 + tβj

)−q

× (Tj + aj)q(bj + m − q)

(bj + m)

⎤⎦

(−1/q)

. (39)

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Note that it is not possible always to know the values of the hyper-parameters aj and bj inEquation (21), we use the prior expectation of S(t) thus, from Equations (3) and (21), we obtain

E[S(t)] =(

1 + log(1 + tβj )

aj

)bj

. (40)

Thus, the values of aj and bj for know βj can be obtained numerically from Equation (40).

5. Approximate and exact confidence interval

Early works on interval estimation for censored data are discussed in Lawless [32] and Meekerand Escobar [33]. Interval estimation for the parameters of Burr type XII distribution basedon censored samples were studied by some authors. Soliman [8] obtained the observed Fisherinformation matrix on the basis of progressively type II censored samples in order to construct theapproximate confidence interval. Asgharzadeh and Valiollahi [34] studied the estimation based onprogressively censored data from the Burr type XII model and Wu and Yu [15] provided a pivotalquantity and the exact confidence interval for the parameter β under the first-failure-censoredsampling plan form the Burr type XII distribution. Tes and Xiang [35] discussed and comparedthe performance of seven confidence interval estimators for Weibull distribution under type IIprogressive censored data. Also, Wu [36] constructs an exact confidence intervals and the exactconfidence region for the parameters of the Weibull distribution on the basis of type II progressivecensoring. In this section, approximate and exact confidence intervals for the shape parameters β

and α of the Burr type XII based on progressive first-failure-censored sampling are investigated.

5.1. Approximate interval estimation

The asymptotic variances and covariances of the MLE for parameters β, and α are given byelements of the inverse of the Fisher information matrix

Iij = E

[− ∂2L

∂β ∂α

], i, j = 1, 2. (41)

Unfortunately, the exact mathematical expressions for the above expectations are very difficultto obtain. Therefore, we give the approximate (observed) asymptotic varaince–covariance matrixfor the MLE, which is obtained by dropping the expectation operator E

I−10 (β, α) =

⎡⎢⎢⎣

−∂2L

∂β2− ∂2L

∂β ∂α

− ∂2L

∂α ∂β−∂2L

∂α2

⎤⎥⎥⎦

−1

(β,α)

=[

var(β) cov(β, α)

cov(α, β) var(α)

],

with

∂2L

∂β2= − m

β2−

m∑i=1

(αk(Ri + 1) + 1)xβ

i log2(xi)

(1 + xβi )2

, (42)

∂2L

∂α2= − m

α2(43)

∂2L

∂β ∂α≡ ∂2L

∂α ∂β= −k

m∑i=1

(Ri + 1)xβ

i log(xi)

(1 + xβi )

. (44)

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Approximate confidence intervals for β and α can be found by taking (β, α) to bebivariately normally distributed with mean (β, α) and covariance matrix I−1

0 (β, α), i.e. (β, α) ∼N((β, α), I−1

0 (β, α)).Thus, the (1 − γ )100% confidence intervals for parameters β, and α become

β ± Zγ /2

√var(β) and α ± Zγ /2

√var(α), (45)

where Zγ /2 is the percentile of the standard normal distribution with right-tail probability γ /2.As regards the approximate confidence region for (β, α), the normal approximation for (β, α)

implies that [β − β, α − α]I−10 (β, α)[β − β, α − α]′ is asymptotically distributed as a chi-squared

with two degrees of freedom. Then the (1 − γ )100% approximate confidence region for (β, α)

can be obtained using the following inequality:

A = {(β, α); [β − β, α − α]I−10 (β, α)[β − β, α − α]′ ≤ χ2

(2γ )},

where χ2(2γ ) is the percentile of the chi-squared distribution with a right-tail probability γ and two

degrees of freedom.

5.2. Exact interval estimation

Let XR1:m:n:k < XR

2:m:n:k < · · · < XRm:m:n:k denote a progressively first-failure-censored sample from

a Burr type XII distribution with parameters β and α, and let

URi:m:n:k = kα ln(1 + (XR

i:m:n:k)β), i = 1, 2, . . . , m. (46)

It can be seen that UR1:m:n:k < UR

2:m:n:k < · · · < URm:m:n:k is the progressively censored order

statistics from an exponential distribution with mean 1. Let us consider the followingtransformation:

W1 = nUR1:m:n:k

W2 = (n − R1 − 1)(UR2:m:n:k − UR

1:m:n:k)

W3 = (n − R1 − R2 − 2)(UR3:m:n:k − UR

2:m:n:k)

...

Wm = (n − R1 − · · · − Rm−1 − m + 1)(URm:m:n:k − UR

m−1:m:n:k). (47)

Balakrishnan and Aggarwala [10] proved that the generalized spacing W1, W2, . . . , Wm, as definedin Equation (47), are independent and identically distributed as an exponential distribution withmean 1. Hence,

ζj = 2j∑

i=1

Wi, (48)

has a Chi-squared distribution with (2j) degrees of freedom and

ψj = 2m∑

i=j+1

Wi, (49)

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has a Chi-squared distribution with 2(m − j) degrees of freedom. It is also clear that ζj and ψj

are independently Chi-squared in distribution. In order to obtain the interval estimation of twoparameters, we consider the following pivotal quantities:

ξj = ψj/(2(m − j))

ζj/2j= j

(m − j).2

∑mi=j+1 Wi

2∑j

i=1 Wi

, j = 1, 2, . . . , m − 1

= j

(m − j).

(R1 + R2 + · · · + Rj + j − n)

+ ∑mi=j+1(Ri + 1)(ln(1 + (XR

i:m:n:k)β))/ ln(1 + (XR

j:m:n:k)β)

(n − R1 − R2 − · · · − Rj−1 − j + 1)

+ ∑j−1i=1(Ri + 1)(ln(1 + (XR

i:m:n:k)β))/ ln(1 + (XR

j:m:n:k)β)

(50)

and

η = (ψj + ζj) = 2m∑

i=1

Wi = 2m∑

i=1

(Ri + 1)URi:m:n:k

= 2kα

m∑i=1

(Ri + 1) ln(1 + (XRi:m:n:k)

β). (51)

It is clear that ξj has an F distribution with 2(m − j) and 2j degrees of freedom and η has Chi-squared distribution with 2m degrees of freedom, where j = 1, 2, . . . , m − 1, m > 1. Furthermore,ξj and η are independent.

To obtain the confidence interval for β and the joint confidence region for β and α, we needthe following two lemmas.

Lemma 1 For any positive real numbers b > a > 0, q(δ) = ln(1 + bδ)/ln(1 + aδ) is a strictlyincreasing function of δ, where δ > 0.

Lemma 2 For a given set of observations 0 < xR1:m:n:k < xR

2:m:n:k < · · · < xRm:m:n:k < ∞, the

function ξj is a strictly increasing function of β when β > 0. Furthermore,

(1) For xRm−1:m:n:k ≤ 1, there is a unique solution for the given equation ξj = t, where t > 0.

(2) Let xR0:m:n:k = 0. For xR

l:m:n:k ≤ 1 < xRl+1:m:n:k , there is a unique solution for the given equation

ξj = t,

where

0 < t <j

(m − j).

∑mi=j+1(Ri + 1) ln(xR

i:m:n:k) − (n − R1 − R2 − · · · − Rj − j) ln(xRj:m:n:k)

(n − R1 − R2 − · · · − Rj−1 − j + 1) ln(xRj:m:n:k) + ∑j−1

i=l+1(Ri + 1) ln(xRi:m:n:k)

,

for l = 0, 1, . . . , j − 1 and j = 1, 2, . . . , m − 1.Using the same arguments and notations in Wu et al. [37], we can prove Lemmas 1 and 2.

5.2.1. Exact confidence interval for parameter β

We will provide here the exact confidence interval for parameter β only. However, supposethat xR

i,m,n,k , i = 1, 2, . . . , m, are order statistics of a progressively first-failure-censored sample

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from a sample size n from a two-parameter Burr type XII distribution, with censoring scheme(R1, R2, . . . , Rm). Then for any 0 < γ < 1, a 100(1 − γ )% confidence interval for β as follows:

We know that ξj has an F distribution with 2(m − j) and 2j degrees of freedom, by Lemmas 1and 2 ξj is strictly increasing in β when β > 0, where

(1) For xRm−1:m:n:k ≤ 1, there is a unique solution for the given equation ξj = t, where t > 0.

(2) Let xR0:m:n:k = 0. For xR

l:m:n:k ≤ 1 < xRl+1:m:n:k , there is a unique solution for the equation ξj = t,

when xRl:m:n:k ≤ 1 < xR

l+1:m:n:k and xR0:m:n:k = 0 for l = 0, 1, . . . , j − 1 and j = 1, 2, . . . , m − 1.

Hence, for 0 < γ < 1, from Equation (50), we obtain

F1−(γ /2)(2(m−j),2j) < ξj < F(γ /2)(2(m−j),2j).

Thus, a 100(1 − γ )% confidence interval for β is

(�(XR, F1−(γ /2)(2(m−j),2j)) < β < �(XR, F(γ /2)(2(m−j),2j))),

where XR = (XR1:m:n:k , XR

2:m:n:k , . . . , XRm:m:n:k) and �(XR, t) is the solution of β for the equation

(R1 + R2 + · · · + Rj + j − n) + ∑mi=j+1(Ri + 1)(ln(1 + (XR

i:m:n:k)β))/ ln(1 + (XR

j:m:n:k)β)

(n − R1 − R2 − · · · − Rj−1 − j + 1) + ∑j−1i=1(Ri + 1)(ln(1 + (XR

i:m:n:k)β))/ ln(1 + (XR

j:m:n:k)β)

= t(m − j)

j

5.2.2. Exact confidence region for the parameters β and α

In this subsection, we discuss the exact confidence region for the parameters β and α. By thesame way in the previous subsection, we suppose that xR

i,m,n,k , i = 1, 2, . . . , m, are progressivelyfirst-failure-censored order statistics from a Burr type XII distribution with censoring scheme(R1, R2, . . . , Rm). Then for any 0 < γ < 1, a 100(1 − γ )% confidence region for β and α isdetermined as follows: from Equation (51), It is clear that

η = 2m∑

i=1

(Ri + 1)URi:m:n:k = 2kα

m∑i=1

(Ri + 1) log(1 + (xRi:m:n:k)

β),

has a chi-squared distribution with 2m degrees of freedom, and it is independent of ξj, and for0 < γ < 1, we have

P(F((1+√1−γ )/2)(2(m−j),2j) < ξj < F((1−√

1−γ )/2)(2(m−j),2j)) = √1 − γ ,

and

P(χ2((1+√

1−γ )/2)(2m)< η < χ2

((1−√1−γ )/2)(2m)

) = √1 − γ .

Then, we obtain

P(F((1+√1−γ )/2)(2(m−j),2j) < ξj < F((1−√

1−γ )/2)(2(m−j),2j),

χ2((1+√

1−γ )/2)(2m)< η < χ2

((1−√1−γ )/2)(2m)

) = 1 − γ ,

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this equivalent to

P

(�(XR, F((1+√

1−γ )/2)(2(m−j),2j)) < β < �(XR, F((1−√1−γ )/2)(2(m−j),2j)),

χ2((1+√

1−γ )/2)(2m)

2k∑m

i=1(Ri + 1) log(1 + (xRi:m:n:k)

β)< α <

χ2((1−√

1−γ )/2)(2m)

2k∑m

i=1(Ri + 1) log(1 + (xRi:m:n:k)

β)

)= 1 − γ ,

where �(XR, t) is the solution of Equation (50).It should be noted that the confidence interval for the parameter β does not depend on k. Thus,

the confidence intervals for the parameter β are identical for the progressive type II censored andprogressive first-failure-censored sampling.

6. Illustrative example (real data)

A complete sample from a clinical trial describe a relief time (in hours) for 50 arthritic patientsgiven by Wingo [38] and used recently by Wu et al. [37] is selected. The data are given in Table 1.

To illustrate the use of the proposed method, we have a previous data consistingof 50 arthritic patients from the Burr type XII distribution. Wingo [38] shows thatthe Burr type XII model is acceptable for these data. The data are randomlygrouped into 25 groups with (k = 2) items within each group. Suppose that the pre-determined progressively first-failure censoring scheme is given by (R1 = R2 = 2, R3 =R4 = 0, R5 = · · · = R10 = 1, R11 = · · · = R15 = 0), then a progressively first-failure censoredsample of size 15 out of 25 groups of arthritic patients is obtained as (X1, . . . , X15) =(0.29, 0.35, 0.36, 0.44, 0.46, 0.49, 0.5, 0.52, 0.55, 0.55, 0.57, 0.59, 0.61, 0.70, 0.8). For this exam-ple, 10 groups of patient’s failure times are censored, and 15 first failures are observed. Using thegraphical method introduced by Balakrishnan and Kateri [26], with Equation (14), we obtainedthe estimation of the parameters as β∗ = 5.115 (shown in Figure 1) and α∗ = 7.0651. The MLEsof β and α, using a Newton–Raphson method when solving Equations (10) and (11), with β∗and α∗ as the initial values, are computed. Substituting the MLEs of β and α into Equations (3)and (4), we obtain an MLE of the reliability and hazard rate functions at time t = 0.45, the resultsare listed in Table 2.

Bayes estimates: In this example, there is no prior information about α, and its not possible toknow the values of the hyperparameters aj and bj in Equation (21). So, we use a non-parametricprocedure (S(ti) = (m − i + 0.625)/(m + 0.25)), i = 1, 2, . . . , m, to estimate two values of thereliability function. Using the available data we obtained: S(t1 = 0.36) = 0.87037 and S(t2 =0.61) = 0.376543. These two prior probabilities are substituting into Equation (40), where aj andbj are obtained numerically for each given βj, and ηj, j = 1, 2, . . . , 10 using the Newton–Raphsonmethod. Table 3 summarized the values of aj, bj, Tj, and Pj for each given βj and ηj. The Bayesestimators (BS, BL, BG) for the parameters β, and α, reliability function S(t), and failure ratefunction h(t) are computed using Equations (28)–(39). The results are also given in Table 3.

Table 1. Relief time (in hours) for 50 arthritic patients.

0.70 0.84 0.58 0.50 0.55 0.82 0.59 0.71 0.72 0.610.62 0.49 0.54 0.36 0.36 0.71 0.35 0.64 0.84 0.550.59 0.29 0.75 0.46 0.46 0.60 0.60 0.36 0.52 0.680.80 0.55 0.84 0.34 0.34 0.70 0.49 0.56 0.71 0.610.57 0.73 0.75 0.44 0.44 0.81 0.80 0.87 0.29 0.50

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Figure 1. Plot of 1/β and H1(β; x) functions.

Table 2. The ML estimates and the Bayes estimates of the parameters, (reliability and hazard functions) where(S(t = 0.45) = 0.889) and (h(t = 0.45) = 1.329).

(.)BL (.)MG

a q

(.)ML (.)BS −1 1 2 −1 1 2

β 5.112 5.022 5.0616 4.983 4.947 5.022 5.006 4.999α 7.054 7.052 10.850 5.640 4.832 7.052 6.498 6.226S(t = 0.45) 0.889 0.882 0.882 0.882 0.881 0.882 0.881 0.880h(t = 0.45) 1.330 1.388 1.455 1.329 1.277 1.388 1.298 1.253

Table 3. Prior information and posterior probabilities under k = 2.

C.S j 1 2 3 4 5 6 7 8 9 10

ηj 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1βj 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5aj 0.065 0.052 0.042 0.035 0.029 0.024 0.020 0.017 0.014 0.012bj 1.059 0.948 0.860 0.788 0.728 0.677 0.633 0.595 0.562 0.533Tj 2.717 2.588 2.466 2.350 2.241 2.138 2.040 1.948 1.860 1.778Pj 0.110 0.110 0.109 0.108 0.106 0.102 0.098 0.092 0.086 0.079

For j = 2, we need the percentiles F0.025(28, 2) = 8.49186 and F0.975(28, 2) = 0.3004. to con-struct the 95% confidence interval for β. According to the formula (50), the 95% exact confidenceinterval ofβ is computed as (2.8808, 7.8374)with the confidence length 4.9566.Where the approx-imate confidence interval for β is (3.4175, 6.8057), and the approximate confidence interval for α

is (0.4085, 13.6994). Furthermore, to obtain a 95% joint confidence region for β and α, we need thefollowing percentiles F0.0127(28, 2) = 0.254658, F0.9873(28, 2) = 12.2596, χ2

(0.0127)(30) = 15.3903and χ2

(0.9873)(30) = 49.9138. The 95% joint confidence region for β and α is determined by thefollowing inequalities:

2.6606 < β < 8.3591

15.3903

2k∑m

i=1(Ri + 1) log(1 + (xRi:m:n:k)

β)< α <

49.9138

2k∑m

i=1(Ri + 1) log(1 + (xRi:m:n:k)

β),

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Figure 2. Joint confidence region for β and α for the progressive first failure data in example.

Table 4. The 95% confidence lengths for β and the 95% confidence areafor β and α.

j Length Area j Length Area

1 6.1396 117.0250 8 4.4523 54.20312 4.9566 68.2357 9 4.6025 61.80183 4.1246• 36.8956• 10 4.4888 45.07144 4.6802 68.1047 11 4.5671 40.05785 4.5238 60.7406 12 4.7139 34.50516 4.5880 67.4626 13 4.9272 27.54887 4.4418 55.8035 14 6.5251 47.3212

after the integration of

∫ 8.3591

2.6606

34.5235

2k∑m

i=1(Ri + 1) log(1 + (xRi:m:n:k)

β)dβ,

we can obtain the confidence area for j = 2 by (68.2357). The confidence region for β is displayedin Figure 2. Similarly, the confidence areas for some values of j are listed in Table 4.

For this example, j = 3 marked by (•) gives the smallest confidence length and the smallestconfidence region.

7. Simulation study

In order to compare the parameters, reliability and hazard functions estimators, Monte Carlosimulations were performed utilizing 1000 progressively first-failure-censored samples for eachsimulations. The mean square error (MSE) is used to compare the estimators. The sam-ples were generated by using the algorithm described in Balakrishnan and Sandhu [2] using(β, α) = (1.5, 0.8), (1, 0.5) with different choices of n, m and k. We take into consideration thatthe progressive first-failure-censored order statistics XR

1:m:n:k , XR2:m:n:k , . . . , XR

m:m:n:k is a progres-sively type II censored sample from a population with distribution function 1 − (1 − F(x))k . TheMLE β and α of parameters β and α are then computed from the solution of Equations (10)and (11) using the Newton–Raphson iteration. The corresponding MLEs of the reliability, andhazard rate functions are computed at a given time t, after replacing β and α by their MLEs β

and α in Equations (3) and (4), respectively. Tables 5 and 6 provide the MSE of the estimates ofthe parameters, reliability and hazard functions.

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Table 5. MSE of different estimates of the parameters, S(t) and h(t) with actual value: [β = 1.5, α = 0.8,S(t = 0.3) = 0.885 and h(t = 0.3) = 0.565].

BL BG

a q

k n m Scheme ML BS −1 1 −1 1

1 30 20 (10, 019) β 0.135 0.026 0.028 0.026 0.026 0.026α 0.050 0.036 0.041 0.032 0.036 0.031

S(t) (3.1)a (1.2)a (1.2)a (1.2)a (1.2)a (1.3)a

h(t) 0.036 0.019 0.020 0.017 0.019 0.017

(019, 10) β 0.141 0.028 0.031 0.026 0.028 0.026α 0.052 0.040 0.045 0.032 0.040 0.031

S(t) (3.2)a (1.1)a (1.1)a (1.1)a (1.1)a (1.1)a

h(t) 0.030 0.019 0.020 0.017 0.019 0.017

(06, 5, 06, 5, 06) β 0.138 0.028 0.029 0.026 0.028 0.026α 0.044 0.033 0.038 0.036 0.033 0.035

S(t) (3.1)a (1.2)a (1.2)a (1.2)a (1.2)a (1.2)a

h(t) 0.047 0.017 0.019 0.018 0.017 0.018

5 30 20 (10, 019) β 0.071 0.028 0.031 0.026 0.028 0.026α 0.079 0.048 0.065 0.039 0.048 0.035

S(t) (0.8)a (0.7)a (0.7)a (0.7)a (0.7)a (0.7)a

h(t) 0.026 0.019 0.020 0.017 0.019 0.016

(019, 10) β 0.114 0.028 0.031 0.025 0.028 0.025α 0.246 0.066 0.092 0.054 0.066 0.046

S(t) (0.8)a (0.6)a (0.6)a (0.7)a (0.6)a (0.7)a

h(t) 0.045 0.021 0.024 0.020 0.021 0.017

(06, 5, 06, 5, 06) β 0.083 0.030 0.032 0.029 0.030 0.029α 0.210 0.067 0.086 0.046 0.067 0.041

S(t) (0.9)a (0.7)a (0.7)a (0.7)a (0.7)a (0.7)a

h(t) 0.038 0.022 0.025 0.017 0.022 0.015

1 30 25 (5, 024) β 0.091 0.027 0.029 0.027 0.027 0.027α 0.031 0.025 0.027 0.025 0.025 0.024

S(t) (2.3)a (1.1)a (1.9)a (1.9)a (1.1)a (1.2)a

h(t) 0.023 0.013 0.014 0.014 0.013 0.014

(024, 5) β 0.126 0.025 0.028 0.025 0.025 0.025α 0.039 0.029 0.033 0.025 0.029 0.025

S(t) (3.4)a (1.1)a (1.1)a (1.1)a (1.1)a (1.1)a

h(t) 0.117 0.016 0.017 0.014 0.016 0.015

(08, 3, 08, 2, 07) β 0.153 0.029 0.031 0.028 0.029 0.028α 0.037 0.027 0.029 0.027 0.027 0.026

S(t) (3.2)a (1.1)a (1.0)a (1.2)a (1.1)a (1.2)a

h(t) 0.134 0.015 0.016 0.015 0.015 0.015

5 30 25 (5, 024) β 0.059 0.028 0.031 0.028 0.028 0.028α 0.072 0.033 0.036 0.035 0.033 0.032

S(t) (0.8)a (0.6)a (0.6)a (0.6)a (0.6)a (0.6)a

h(t) 0.020 0.016 0.017 0.014 0.016 0.013

(024, 5) β 0.073 0.028 0.031 0.028 0.028 0.028α 0.102 0.046 0.047 0.044 0.046 0.043

S(t) (0.7)a (0.6)a (0.6)a (0.6)a (0.6)a (0.6)a

h(t) 0.021 0.015 0.016 0.016 0.015 0.015

(08, 3, 08, 2, 07) β 0.062 0.028 0.030 0.027 0.028 0.027α 0.071 0.042 0.051 0.036 0.042 0.033

S(t) (0.7)a (0.6)a (0.6)a (0.7)a (0.6)a (0.7)a

h(t) 0.021 0.016 0.017 0.014 0.016 0.013

aIndicates that the value multiply by 10−3.

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Table 6. MSE of different estimates of the parameters, S(t) and h(t) with an actual value: β = 1, α = 0.5,S(t = 0.5) = 0.816 and h(t = 0.5) = 0.333.

BL BG

a q

k n m Scheme ML BS −2 2 −2 2

1 50 30 (20, 029) β 0.051 0.026 0.029 0.023 0.027 0.024α 0.016 0.015 0.018 0.012 0.017 0.012

S(t) 0.003 0.002 0.002 0.002 0.002 0.002h(t) 0.005 0.005 0.005 0.004 0.005 0.004

(029, 20) β 0.055 0.025 0.029 0.023 0.026 0.023α 0.022 0.024 0.029 0.024 0.028 0.022

S(t) 0.003 0.002 0.002 0.002 0.002 0.002h(t) 0.016 0.005 0.005 0.004 0.005 0.004

(09, 10, 09, 10, 010) β 0.047 0.025 0.027 0.025 0.026 0.023α 0.019 0.019 0.024 0.015 0.022 0.013

S(t) 0.002 0.002 0.002 0.002 0.002 0.002h(t) 0.011 0.004 0.005 0.005 0.005 0.005

5 50 30 (20, 029) β 0.019 0.018 0.019 0.017 0.018 0.017α 0.012 0.013 0.015 0.012 0.014 0.012

S(t) 0.001 0.001 0.001 0.001 0.001 0.001h(t) 0.008 0.007 0.008 0.007 0.008 0.006

(029, 20) β 0.032 0.024 0.028 0.022 0.025 0.022α 0.014 0.012 0.013 0.011 0.013 0.011

S(t) 0.002 0.002 0.002 0.002 0.002 0.002h(t) 0.023 0.019 0.025 0.017 0.025 0.013

(09, 10, 09, 10, 010) β 0.021 0.018 0.020 0.017 0.018 0.017α 0.011 0.013 0.013 0.011 0.013 0.011

S(t) 0.001 0.001 0.001 0.001 0.001 0.001h(t) 0.011 0.011 0.014 0.012 0.014 0.012

1 50 40 (10, 039) β 0.042 0.024 0.029 0.021 0.026 0.022α 0.012 0.011 0.012 0.010 0.012 0.010

S(t) 0.004 0.003 0.003 0.002 0.003 0.002h(t) 0.003 0.003 0.006 0.003 0.006 0.003

(039, 10) β 0.050 0.024 0.028 0.022 0.025 0.022α 0.013 0.014 0.013 0.012 0.012 0.011

S(t) 0.003 0.002 0.002 0.002 0.002 0.002h(t) 0.023 0.003 0.003 0.003 0.003 0.003

(013, 5, 012, 5, 013) β 0.033 0.023 0.026 0.022 0.024 0.023α 0.008 0.009 0.012 0.008 0.011 0.007

S(t) (1.7)a (1.4)a (1.4)a (1.5)a (1.4)a (1.6)a

h(t) 0.003 0.003 0.003 0.003 0.003 0.003

5 50 40 (10, 039) β 0.017 0.016 0.018 0.014 0.016 0.014α 0.009 0.009 0.010 0.008 0.010 0.008

S(t) (0.8)a (0.8)a (0.8)a (0.8)a (0.8)a (0.9)a

h(t) 0.005 0.005 0.006 0.005 0.006 0.004

(039, 10) β 0.018 0.016 0.019 0.015 0.018 0.015α 0.013 0.011 0.012 0.012 0.011 0.011

S(t) (0.8)a (0.8)a (0.8)a (0.9)a (0.8)a (0.9)a

h(t) 0.008 0.007 0.008 0.006 0.008 0.005

(013, 5, 012, 5, 013) β 0.015 0.015 0.016 0.014 0.016 0.014α 0.008 0.008 0.009 0.008 0.009 0.007

S(t) (0.7)a (0.7)a (0.8)a (0.7)a (0.8)a (0.7)a

h(t) 0.005 0.006 0.007 0.005 0.006 0.004

aIndicates that the value multiply by 10−3.

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Table 7. The 95% confidence area for β and α when β = 1.5 and α = 0.8.

j

(k, n, m) Scheme 1 2 3 4 5 6 7 8

(1,30,25) (5, 024) 1.434 1.224 1.150 1.138• 1.154 1.193 1.224 1.263(024, 5) 1.470 1.215 1.159 1.118• 1.136 1.151 1.154 1.216

(08, 3, 08, 2, 07) 1.471 1.237 1.179 1.126• 1.143 1.147 1.170 1.213

9 10 11 12 13 14 15 16(5, 024) 1.328 1.401 1.449 1.508 1.658 1.703 1.732 1.831(024, 5) 1.279 1.304 1.374 1.458 1.501 1.587 1.638 1.770

(08, 3, 08, 2, 07) 1.261 1.300 1.382 1.433 1.493 1.601 1.649 1.693

17 18 19 20 21 22 23 24(5, 024) 1.879 1.962 2.104 2.024 2.274 2.346 2.749 2.990(024, 5) 1.899 2.045 2.289 2.561 2.684 3.295 4.195 7.062

(08, 3, 08, 2, 07) 1.815 1.977 2.227 2.476 2.582 2.611 2.633 2.964

1 2 3 4 5 6 7 8(5, 30, 25) (5, 024) 1.786 1.312 1.172 1.038 1.067 1.057 1.062 1.039

(024, 5) 2.862 1.753 1.488 1.483 1.397 1.367 1.386 1.322(08, 3, 08, 2, 07) 2.014 2.125 1.251 1.127 1.121 1.057 1.068 1.068

9 10 11 12 13 14 15 16(5, 024) 1.042 1.050 1.025• 1.112 1.124 1.153 1.228 1.244(024, 5) 1.296• 1.357 1.405 1.481 1.475 1.488 1.588 2.011

(08, 3, 08, 2, 07) 1.035• 1.121 1.137 1.121 1.132 1.195 1.219 1.317

17 18 19 20 21 22 23 24(5, 024) 1.478 1.450 1.685 1.938 2.141 3.159 3.511 4.121(024, 5) 2.255 2.786 3.183 5.351 6.647 12.599 12.367 16.231

(08, 3, 08, 2, 07) 1.454 1.483 1.540 2.089 2.324 2.514 4.138 5.715

(•)Indicates to the minimum area.

Table 8. Coverage probabilities for the MLEs when β = 1.5 and α = 0.8.

MlE MLE

(k, n, m) Scheme β α (β, α)

(1, 30, 20) (10, 019) 0.957 0.943 0.923(019, 10) 0.939 0.930 0.902

(06, 5, 06, 5, 06) 0.941 0.939 0.895

(5, 30, 20) (10, 019) 0.954 0.958 0.932(019, 10) 0.951 0.934 0.916

(06, 5, 06, 5, 06) 0.948 0.942 0.911

(1, 30, 25) (5, 024) 0.967 0.959 0.932(024, 5) 0.964 0.950 0.912

(08, 3, 08, 2, 07) 0.956 0.955 0.924

(5, 30, 25) (5, 024) 0.955 0.954 0.907(024, 5) 0.941 0.957 0.890

(08, 3, 08, 2, 07) 0.942 0.967 0.911

8. Conclusions

Censoring is a common phenomenon in life-testing, and reliability studies. The subject of pro-gressive censoring has received considerable attention in the past few years, due in part to theavailability of high-speed computing resources, which make it both a feasible topic for simulationstudies for researchers, and a feasible method of gathering lifetime data for practitioners. It hasbeen illustrated by Viveros and Balakrishnan [1] that the inference is feasible, and practical when

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the sample data are gathered according to a type II progressively censored experimental scheme.Combining the concept of first-failure censoring and the concept of progressive censoring, a pro-gressive first-failure censoring scheme has been introduced by Wu and Kus [18]. This censoringscheme has advantages in terms of reducing test cost and test time, in which more items are usedbut only m of n × k items are failures. Based on this new censoring scheme, the present papershows how the things can be routinely managed for the Burr model in a Bayesian and classicalframeworks. We have considered the maximum likelihood (ML), and Bayes estimates for somesurvival time parameters, reliability and hazard functions, as well as the parameters of the BurrXII model using progressively first-failure censored data. The Bayes estimators are discussedunder symmetric and asymmetric loss functions. Also, we have provided pivotal quantities forconstructing exact confidence interval and exact confidence region for the parameters β and α.A simulation study was conducted to examine the performance of the different estimators. Fromthe results, we observe the following:

(i) The results obtained in this paper can be specialized to: (a) first-failure-censored orderstatistics by taking R = (0, . . . , 0). (b) progressively type II censored statistics for k = 1.(c) usually type II censored order statistics for k = 1 and R = (0, . . . , n − m). (d) completesample for k = 1, n = m; and R = (0, . . . , 0).

(ii) The MSE of the Bayes estimations and ML estimations are computed over differentcombination of the censored scheme R as shown in Tables 5 and 6.

(iii) If one uses the Bayesian approach in estimating the parameters, reliability and hazardfunctions, one would expect these estimators to be better (in the sense of MSE’s) than theMLEs. In general, this can be seen in the results for different choices of k, n, m and thecensoring scheme R . Also, to assess the effect of the loss functions, one can see that theMSE of asymmetric Bayes estimates (BL, BG) of the parameters β and α are overestimatesfor (a < 0, q < 0), and when (a > 0, q > 0) the MSE of the parameters are underestimates.Also, the MSE’s of the asymmetric Bayes estimates of reliabiliy function are smaller thanMSE’s of the MLEs. As anticipated, the MSE of the asymmetric Bayes estimates are thesame as the MSE of Bayes estimates relative to SE loss (for a close to 0, and q = −1) .This is one of the useful properties of working with the asymmetric loss functions.

(iv) When the effective sample proportion m/n increases, the MSE of different Bayes estimatorsand MLEs are reduced, also the censoring scheme R = (n − m, . . . , 0) is most efficient forall choices, it usually provides the smallest MSE for all estimators.

(v) The use of discrete distribution for the shape parameter β resulted in a closed-form expres-sion for the posterior distribution, and the equal probabilities in the discrete distributioncased an element of uncertainly, which can be desirable in some cases.

(vi) The results establish that for optimum decision-making, important should be given on thechoice of loss function and not just the choice of prior distribution only.

(vii) From Table 7, for most choices, the censoring scheme R = (n − m, 0, · · · , 0) correspondingto the case of removal in the first stage, it seems to usually provide the smallest confidencearea. The minimum area of the confidence region are marked by (•).

(viii) From Table 8, it is observed that the coverage probabilities of the approximate confidenceintervals for β and α are close to the desired level of 0.95. However, the coverage prob-abilities of the approximate confidence regions for (β, α) are not so close to the desiredlevel.

Acknowledgement

The authors would like to express their thanks to the referees for their useful comments and suggestions on the originalversion of this manuscript.

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Estimation from Burr type XII distribution using progressive first-failure censored data

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