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Modiﬁed Weibull model: A Bayes study using MCMC approach based on

progressive censoring data

Ahmed A. Soliman

a,b,

n

, Ahmed H. Abd-Ellah

b

, Naser A. Abou-Elheggag

b

, Essam A. Ahmed

b

a

Faculty of Science, Islamic University, Madinah, Saudi Arabia

b

Mathematics Department, Sohag University, Sohag 82524, Egypt

article info

Article history:

Received 8 October 2010

Received in revised form

10 December 2011

Accepted 16 December 2011

Available online 31 December 2011

Keywords:

Modiﬁed Weibull distribution

Progressive type-II censoring

Balanced loss

Maximum likelihood estimation

Bayesian estimation

Gibbs and Metropolis–Hasting samplers

Hybrid MCMC approach

Bootstrap

abstract

In this paper, we investigate the problem of point and interval estimations for the modiﬁed Weibull

distribution (MWD) using progressively type-II censored sample. The maximum likelihood (ML), Bayes,

and parametric bootstrap methods are used for estimating the unknown parameters as well as some

lifetime parameters (reliability and hazard functions). Also, we propose to apply Markov chain Monte

Carlo (MCMC) technique to carry out a Bayesian estimation procedure. Bayes estimates and the credible

intervals are obtained under the assumptions of informative and noninformative priors. The results of

Bayes method are obtained under both the balanced squared error loss (bSEL) and balanced linear-

exponential (bLINEX) loss. We show that these loss functions are more general, which include the MLE

and both symmetric and asymmetric Bayes estimates as special cases. Finally, Two real data sets have

been analyzed for illustrative purposes.

&2011 Elsevier Ltd. All rights reserved.

1. Introduction

Models with bathtub-shaped failure rate function are useful in

reliability analysis and particularly in reliability related decision

making and cost analysis. There are a number of papers dealing

with models for bathtub-shaped failure rate. For example, Xie and

Lai [27] and Xie et al. [29]. A modiﬁed Weibull distribution

(MWD) was recently proposed by Xie et al. [28] as a general-

ization of the two-parameter Weibull distribution. This distribu-

tion has both the two-parameter Weibull and the type I extreme

value distributions as special cases. As discussed in [28], this

lifetime distribution is able to model data with bathtub-shaped

hazard rate, which is an important feature for engineering

reliability analysis.

The probability density function (pdf), cumulative distribution

function (cdf), reliability function SðtÞ, and hazard function HðtÞof

the MWD are given, respectively, by

fðx;

a

,

b

,

l

Þ¼

a

ð

b

þ

l

xÞxb

1

expð

l

x

a

xbel

x

Þ,

a

40,

l

40,

b

Z0,

ð1Þ

Fðx;

a

,

b

,

l

Þ¼1expð

a

xbel

x

Þ,ð2Þ

SðtÞ¼expð

a

tbel

t

Þð3Þ

and

HðtÞ¼

a

ð

b

þ

l

tÞtb

1

expð

l

tÞ,ð4Þ

where

a

may be looked upon as the scale parameter and both

b

and

l

are responsible to determine the shape parameters of the

distribution. From Eq. (1), it should be noted that when

l

¼0,

MWD reduces to the two-parameter Weibull distribution, when

b

¼0, its reduces to a type I extreme-value distribution and is also

known as a log-gamma distribution or log-Weibull distribution.

Also, when

l

¼0 and

b

¼2, MWD reduces to the one-parameter

Rayleigh distribution.

It can be seen that the shape of the hazard function H(t) in (4)

depends only on

b

, in which its monotonically increasing when

b

Z1, and when 0o

b

o1 its a bathtub shape, see [28]. Lai et al.

[15] plotted the hazard function and the mean residual life

function of the MWD, and calculated the change point of

the hazard function. The relationships between the parameters

of the MWD and both the change points of the hazard function,

and mean residual function are discussed in details in [30]. Lai

et al. [15] discussed a graphical approach to estimate the MWD

parameters for complete and censored samples. Recently, Jiang

et al. [14] prove that the MLEs of the parameters of the MWD

given a progressively type-II censored samples are exist and are

unique.

There are many situations in life-testing and reliability studies

in which the experimenter may be unable to obtain complete

information on failure times of all experimental items. There are

Contents lists available at SciVerse ScienceDirect

journal homepage: www.elsevier.com/locate/ress

Reliability Engineering and System Safety

0951-8320/$ - see front matter &2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ress.2011.12.013

n

Corresponding author. Tel.: þ2 173597859; fax: þ2 934601159.

E-mail address: soliman1957@live.com (A.A. Soliman).

Reliability Engineering and System Safety 100 (2012) 48–57

Author's personal copy

also situations wherein the removal of items prior to failure is

pre-planned in order to reduce the cost and time associated with

testing. The most common censoring schemes are type-I and

type-II censoring, but the conventional type-I and type-II censor-

ing schemes do not have the ﬂexibility of allowing removal of

items at points other than the terminal point of the experiment.

For this reason, we consider a more general censoring scheme

called progressive type-II right censoring. It can be described as

follows. Suppose that nindependent items are put on a life test

with continuous identically distributed failure times X

1

,X

2

,...,X

n

.

Suppose further that a censoring scheme ðR

1

,R

2

,...,R

m

Þis pre-

viously ﬁxed such that immediately following the ﬁrst failure

X

1

,R

1

surviving items are removed from the experiment at

random, and immediately following the second failure X

2

,R

2

surviving items are removed from the experiment at random. This

process continues until, at the time of the mth observed failure

X

m

, the remaining R

m

surviving items are removed from the test.

The mordered observed failure times denoted by X

ðR

1

,...,R

m

Þ

1:m:n

,

X

ðR

1

,...,R

m

Þ

2:m:n

...,X

ðR

1

,...,R

m

Þ

m:m:n

are called progressively type-II right cen-

sored order statistics of size mfrom a sample of size nwith

progressive censoring scheme ðR

1

,R

2

,...,R

m

Þ. It is clear that

n¼mþP

m

i¼1

R

i

.

The special case when R

1

¼R

2

¼¼R

m1

¼0 so that

R

m

¼nmis the case of conventional type-II right censored

sampling. Also when R

1

¼R

2

¼¼R

m

¼0, so that m¼n, the

progressively type-II right censoring scheme reduces to the case

of no censoring (ordinary order statistics). Many authors have

discussed inference under progressive type-II censored using

different lifetime distributions, see for example, [4,21,16]. A

recent account on progressive censoring schemes can be obtained

the excellent review article by Balakrishnan [2].

In Bayesian approach, we need to integrate over the posterior

distribution and the problem is that the integrals are usually

impossible to evaluate analytically. Markov chain Monte Carlo

(MCMC) technique is a Monte Carlo integration method which

draws samples from the target posterior distribution. MCMC meth-

odology provided a convenient and efﬁcient way to sample from

complex, high-dimensional statistical distributions. Recently, appli-

cation of the MCMC method to the estimation of parameters or

some other vital properties about statistical models is very common.

Green et al. [8] using the MCMC method for estimating the three -

parameters Weibull distribution, and they showed that the MCMC

method is better than the ML method, when given a proper prior

distribution of the parameters. As a generalization of the two-

parameter Weibull model, Gupta et al. [10] gave a complete

Bayesian analysis of the Weibull extension model using MCMC

simulation and complete sample. Ng [19] obtained MLE and con-

ﬁdence intervals for the parameters of the MWD based on a

progressively type-II censored sample, and showed that MLE

behaves statistically better than the least-square regression estima-

tors. Jiang et al. [13] discussed the MCMC method for estimating the

parameters of MWD based on complete sample and independent

generalized uniform prior distributions for the three parameters.

Upadhyay and Gupta [24] discussed some Bayes analysis of MWD

using MCMC technique, complete sample and independent vague

priors for the unknown parameters. Nadarajah [18] is another

work on the MWD but the author was conﬁned to obtaining the

moments of the distribution. The inferences to the MWD are meager

because of the fact that the model has been proposed recently and

the form of its pdf is a bit complicated. Also, it is to be noted that

most of the cited literatures are conﬁned to classical developments

and the Bayesian results are rarely seen. We are not aware of any

discussions concerning Bayes inference of the MWD based on

progressive censoring samples. The importance of the Bayesian

methods is well known in the context of engineering reliability

studies. Among several advantages, the most important is the fact

that the Bayesian methods are equally well applicable for small

sample sizes and censored sample data problems.

In general, there are several potential issues about the meth-

ods of estimation. Among them the Bayesian method. In the case

of progressive censoring, analytical Bayesian estimation using a

full likelihood is difﬁcult because of the complexity of the like-

lihood equation and the numeric integrations could be very time

consuming. Some discussions for reducing the number of numeric

integrations required in Bayesian approach can be found in Touw

[23]. Recently, Guan et al. [9] proposed some analytical Bayesian

estimation methods using Laplace approximation and inverse

ﬁrst-order computations. The Bayesian Markov chain Monte Carlo

(MCMC) method is an alternative to the analytical Bayesian

method and is especially appropriate for progressive censored

data. Therefore, it would be convenient to develop a general

sensitivity method that can be applied to estimate local para-

metric sensitivities in Bayesian models solved by MCMC techni-

ques. The MCMC simulations have been extensively used in

reliability studies, see for example [7,20,26].

Building on previous work, in this paper, we consider the

estimation of the MWD based on progressively type-II censoring.

Bayes and maximum likelihood estimators are considered. Esti-

mation of some lifetime parameters such as reliability and hazard

functions are also considered. In Bayesian framework, we pro-

poses to use the MCMC techniques for the Bayes analysis of the

modiﬁed Weibull model. We have developed a hybrid strategy

combining the Metropolis algorithm within the Gibbs sampler for

obtaining the samples from the posterior distributions. The Bayes

estimators have been obtained under balanced loss function and

both the squared error loss function and linear-exponential

(LINEX) loss function as special cases of balanced loss function.

Also, based on this type of censoring, we use the parametric

bootstrap method to construct conﬁdence intervals (CIs) for the

parameters of the model. Numerical examples using two real data

sets are presented to illustrate all the methods of estimation

developed here.

The rest of the paper is organized as follows. In the next

section, the ML estimators of the unknown parameters, reliability

and hazard functions are presented. The corresponding bootstrap

conﬁdence intervals for the parameters, reliability and hazard

functions are given in Section 3.Section 4 begins with a short

description of the priors, posteriors, Gibbs sampling and Metro-

polis–Hastings (MH) algorithms. In the same section the proposed

hybrid algorithm with the resulting Bayes estimators are dis-

cussed. Two real data sets have been analyzed in Section 5. Finally

conclusions appear in Section 6.

2. Maximum likelihood estimation

Maximum likelihood estimation (MLE) is one of the most

common parameter estimation methods for statistical models. If

the failure times of the nitems originally on the test are from a

continuous population with cdf FðxÞand pdf fðxÞ. The joint

probability density function of a progressively type-II censored

sample xðx

ðR

1

,...,R

m

Þ

1:m:n

,x

ðR

1

,...,R

m

Þ

2m:n

,...,x

ðR

1

,...,R

m

Þ

m:m:n

Þof size mfrom a

sample of size nwith progressive censoring scheme ðR

1

,R

2

,

...,R

m

Þis given by

f

X

1

,X

2

,...,X

m

ðx

1

,x

2

,...,x

m

Þ¼AY

m

i¼1

fðx

i

Þ½1Fðx

i

Þ

R

i

,ð5Þ

where x

i

is used instead of x

ðR

1

,...,R

m

Þ

i:m:n

,R

i

Z0ði¼1;2,...,mÞand

A¼nðn1R

1

Þðn2R

1

R

2

Þ nX

m1

i¼1

ðR

i

þ1Þ

!

:ð6Þ

A.A. Soliman et al. / Reliability Engineering and System Safety 100 (2012) 48–57 49

Author's personal copy

Substituting (1) and (2) in (5), the likelihood function can be

written as

Lðx;

a

,

b

,

l

Þ¼A

a

m

Y

m

i¼1

ð

b

þ

l

x

i

Þxb

1

i

!

exp

l

mx

a

X

m

i¼1

ðR

i

þ1Þxb

i

el

x

i

"#

,

ð7Þ

where Ais deﬁned in (6), the log-likelihood function may then be

written as

‘¼logðA

a

m

Þþ X

m

i¼1

logð

b

þ

l

x

i

Þþð

b

1ÞX

m

i¼1

logðx

i

Þ

þm

l

x

a

X

m

i¼1

ðR

i

þ1Þxb

i

el

x

i

,ð8Þ

The MLEs of the parameters

a

,

b

and

l

are obtained by solving the

following likelihood equations simultaneously:

~

a

ML

¼m

P

m

i¼1

ðR

i

þ1Þx

~

b

i

e

~

l

x

i

:ð9Þ

X

m

i¼1

1

ð~

b

ML

þ~

l

ML

x

i

ÞþX

m

i¼1

logðx

i

Þ ~

a

ML

X

m

i¼1

ðR

i

þ1Þx

~

b

ML

i

e

~

l

ML

x

i

logðx

i

Þ¼0

ð10Þ

and

X

m

i¼1

x

i

ð~

b

ML

þ~

l

ML

x

i

Þþmx~

a

ML

X

m

i¼1

ðR

i

þ1Þx

~

b

ML

þ1

i

e

~

l

ML

x

i

¼0:ð11Þ

Since Eqs. (10) and (11) cannot be solved analytically for

b

and

l

,

some numerical methods such as Newton’s method must be

employed. The least-square regression method discussed in Ng

[19] can be used for obtaining the initial values in the iteration

procedure. By invariance property of the ML estimators, the ML

estimators of reliability function and hazard rate function can be

obtained by substituting the MLE’s of the parameters

a

,

b

and

l

in

(3) and (4), respectively.

3. Bootstrap conﬁdence intervals

In this section, we use the parametric bootstrap percentile

method suggested by Efron [5] to construct conﬁdence intervals

for the parameters as well as the reliability and hazard functions.

The following steps are followed to obtain a progressive type-II

right censoring bootstrap sample from MWD based on a real

progressive type-II censored data set.

Step 1 From the original data xx

ðR

1

,...,R

m

Þ

1:m:n

,x

ðR

1

,...,R

m

Þ

2:m:n

,...,x

ðR

1

,...,R

m

Þ

m:m:n

compute the ML estimates of the parameters: ~

a

,~

b

and ~

l

by solving the nonlinear equations (9)–(11).

Step 2 Using the values of ~

a

,~

b

and ~

l

in Eqs. (3) and (4) with some

mission time t, we obtain the ML estimates ~

SðtÞand ~

HðtÞof

the reliability and hazard functions .

Step 3 Use ~

a

,

~

b

and ~

l

to generate a bootstrap sample x

n

with the

same values of R

i

,m;ði¼1;2,...,mÞusing algorithm

presented in [3].

Step 4 As in step 1, based on x

n

compute the bootstrap sample

estimates of

a

,

b

,

l

,S(t) and HðtÞ, say ~

a

n

,~

b

n

,~

l

n

,~

S

n

ðtÞand

~

H

n

ðtÞ.

Step 5 Repeat steps 3 and 4 Ntimes representing Nbootstrap

MLE’s of ð

a

,

b

,

l

,SðtÞ,HðtÞÞ based on Nbootstrap samples.

Step 6 Arrange all ~

a

n

0

s,

~

b

n

0

s,

~

l

n

0

s,

~

S

n

ðtÞ

0

sand ~

H

n

ðtÞ

0

sin an ascending

order to obtain the bootstrap sample ð

j

1

l

,

j

½2

l

,...,

j

½N

l

Þ,

l¼1;2,3;4,5, where

j

1

~

a

n

,

j

2

~

b

n

,

j

3

~

l

n

,

j

4

~

S

n

ðtÞ

and

j

5

~

H

n

ðtÞ.Let GðzÞ¼Pð

j

l

rzÞbe the cumulative dis-

tribution function of

j

1

. Deﬁne

j

lboot

¼G

1

ðzÞfor given z.

The approximate bootstrap 100ð1

g

Þ%conﬁdence interval

of

j

l

is given by ½

j

lboot

ð

g

=2Þ,

j

lboot

ðð1

g

Þ=2Þ.

4. Bayesian estimation

Now we will propose the Bayesian estimator of parameters

(

a

,

b

,

l

Þas well as the reliability SðtÞand hazard H(t) functions of

the MWD. Let us consider independent priors for the parameters

a

,

b

and

l

as

p

1

ð

a

Þp

ag

1

1

expð

Z

1

a

Þ,

a

40,

g

1

40,

Z

1

40,ð12Þ

p

2

ð

b

Þp

b

g

2

1

expð

Z

2

b

Þ,

b

40,

g

2

40,

Z

2

40,ð13Þ

p

3

ð

l

Þ¼ 1

M,0o

l

oM,ð14Þ

where

g

1

,

Z

1

,

g

2

,

Z

2

, and Mare chosen to reﬂect prior knowledge

about

a

,

b

and

l

. Note that when

g

1

¼

g

2

¼

Z

1

¼

Z

2

¼0, they are

the non-informative priors of

a

and

b

, respectively.

From Eqs. (7) and (12)–(14), the joint posterior density func-

tion of

a

,

b

and

l

is thus

p

nð

a

,

b

,

l

9xÞ¼

p1

ð

a

Þ

p2

ð

b

Þ

p3

ð

l

ÞLðx;

a

,

b

,

l

Þ

R

a

R

b

R

l

p1

ð

a

Þ

p2

ð

b

Þ

p3

ð

l

ÞLðx;

a

,

b

,

l

Þd

a

d

b

d

l

p

b

g

2

1aðmþ

g

1

1Þ

exp

l

mx

Z2

b

aZ

1

þX

m

i¼1

ðR

i

þ1Þx

b

i

e

l

x

i

!"#

Y

m

i¼1

ð

b

þ

l

x

i

Þx

b

1

i

()

:

ð15Þ

In Bayesian statistics the posterior distribution

p

n

ð

a

,

b

,

l

9xÞ

contains all relevant information on the unknown parameters

given the observed data x. All statistical inference can be deduced

from the posterior distribution. The problem is that the integrals

in (15) are usually impossible to evaluate analytically. And the

numerical methods may fail. The MCMC method provides an

alternative method for parameter estimation. It is more ﬂexible

when compared with the traditional methods. Moreover, prob-

ability intervals are available. The probability intervals provide us

a reasonable interval estimate about the unknown parameter. In

the following subsections, we propose using the MCMC technique

to obtain Bayes estimates of the unknown parameters and to

construct the corresponding credible intervals.

4.1. Markov chain Monte Carlo estimation

Markov chain Monte Carlo (MCMC) methods use computer

simulation of Markov chains in the parameter space. The Markov

chains are deﬁned in such a way that the posterior distribution

in the given statistical inference problem is the asymptotic

distribution. This allows to use ergodic averages to approximate

the desired posterior expectations. The MCMC method works

successfully in Bayesian computing. In addition, the simulation

algorithm can be easily extensible to models with a large number

of parameters or high complexity. Several standard approaches

to deﬁne such Markov chains exist, including Gibbs sampling,

MH and reversible jump. Using these algorithms it is possible

to implement posterior simulation in essentially any problem

which allow pointwise evaluation of the prior distribution and

likelihood function. Gibbs sampler is a special case of a Monte

Carlo Markov chain algorithm. It generates a sequence of

samples from the full conditional probability distributions of

two or more random variables. Gibbs sampling requires decom-

posing the joint posterior distribution into full conditional

A.A. Soliman et al. / Reliability Engineering and System Safety 100 (2012) 48 –5750

Author's personal copy

distributions for each parameter and then sampling from them.

The sampler can be efﬁcient when the full conditional distribu-

tions are easy to sample from. The MH algorithm is a very general

MCMC method ﬁrst developed by Metropolis et al. [17] and later

extended by Hastings [11]. It can be used to obtain random

samples from any arbitrarily complicated target distribution of

any dimension that is known up to a normalizing constant. In fact,

Gibbs sampler is just a special case of the MH algorithm. Details of

the MCMC method can be found in [6,20].

4.2. Algorithm of the hybrid MCMC method for modiﬁed

Weibull model

In this section we derived the full conditional posterior

distributions for all model parameters and the censored data.

Then we will show how the hybrid MCMC sampling method

works for our case.

Assuming independent priors as in (12)–(14), the full condi-

tional distribution for any parameter can be obtained, to within a

constant, by factoring out from the likelihood function Lðx;

a

,

b

,

l

Þ

any terms containing the relevant parameter and multiplying by

its prior. From (15), the marginal posterior density of

a

is

proportional to

p

n

1

ð

a

9

b

,

l

,xÞp

a

ðmþ

g

1

1Þ

exp

aZ

1

þX

m

i¼1

xb

i

el

x

i

ðR

i

þ1Þ

!"#

:ð16Þ

Similarly, the full posterior conditional distributions for

b

and

l

are, respectively

p

n

2

ð

b

9

a

,

l

,xÞp

b

g

2

1

Y

m

i¼1

ð

b

þ

l

x

i

Þx

b

1

i

!

exp

Z

2

b

a

X

m

i¼1

x

b

i

e

l

x

i

ðR

i

þ1Þ

"#

ð17Þ

and

p

n

3

ð

l

9

a

,

b

,xÞpY

m

i¼1

ð

b

þ

l

x

i

Þ

!

exp

l

mx

a

X

m

i¼1

xb

i

el

x

i

ðR

i

þ1Þ

"#

:ð18Þ

It can be seen that Eq. (16) is a gamma density with shape

parameter ðmþ

g

1

Þand scale parameter ð

Z

1

þP

m

i¼1

xb

i

el

x

i

ðR

i

þ1ÞÞ

and, therefore, samples of

a

can be easily generated using any

gamma generating routine. However, in our case, both conditional

posterior distributions of

b

and

l

cannot be reduced analytically

to well known distributions and therefore it is not possible to

sample directly by standard methods, but the plots of them show

that they are similar to normal distribution. So, as suggested by

Tierney [22], a common way to solve this problem is to use the

hybrid algorithm by combined a Metropolis sampling with the

Gibbs sampling scheme using normal proposal distribution. Thus,

our resulting algorithm for simulating from (17) and (18) may be

termed as a hybrid algorithm that combines Metropolis steps

within the Gibbs sampler. To sample from (17) we generated a

proposal value from a normal distribution Nð

b

ðj1Þ

,KbVbÞ, and to

sample from (18) we generated a proposal value from a normal

distribution Nð

l

ðj1Þ

,KlVlÞ, where

b

ðj1Þ

and

l

ðj1Þ

are, respectively,

the current values of

b

and

l

,Vband Vlare variances–covar-

iances matrix, and where Kband Klare scaling factors. The hybrid

MH and Gibbs sampler works as follows:

Algorithm of the hybrid MCMC method:

(1) Start with initial guess ð

a

ð0Þ

,

b

ð0Þ

,

l

ð0Þ

Þ.

(2) Set j¼1.

(3) Generate

a

ðjÞ

from Gamma(mþ

g

1

,

Z

1

þP

m

i¼1

xb

ðj1Þ

i

el

ðj1Þ

x

i

ðR

i

þ1Þ).

(4) Using Metropolis–Hastings, generate

b

ðjÞ

1

from

p

n

2

(

b

ðj1Þ

9

a

ðjÞ

,

l

ðj1Þ

,x) with normal proposal distribution, Nð

b

ðj1Þ

,KbVbÞ.

(5) Using Metropolis–Hastings, generate

l

ðjÞ

1

from

p

n

3

ð

l

ðj1Þ

9

a

ðjÞ

,

b

ðj1Þ

,xÞwith normal proposal distribution, Nð

l

ðj1Þ

,KlVlÞ.

(6) Compute S(t) and H(t) from (3) and (4).

(7) Set j¼jþ1.

(8) Repeat steps 3–7, N

1

times.

It well known that rapid convergence is facilitated by choosing

appropriate starting values. The starting values for the parameters

a

,

b

and

l

cane be obtained using a least squares regression

method, see [19].

4.3. Estimation based on balanced loss functions

Under ‘‘balanced’’ loss functions of the form, see [12]

L

r

,w,

d

0

ð

y

,

d

Þ¼

or

ð

d

,

d

0

Þþð1

o

Þ

r

ð

y

,

d

Þ,ð19Þ

where

r

is an arbitrary loss function, while

d

0

is a chosen a prior

‘‘target’’ estimator of

y

, obtained for instance using the criterion of

maximum likelihood, least-squares, or unbiasedness. Loss L

r

,w,

d

0

,

which depends on the observed value of

d

0

ðXÞ, reﬂects a desire of

closeness of

d

to both: (i) the target estimator

d

0

and (ii) the

unknown parameter

y

; with the relative importance of these

criteria governed by the choice of

o

A½0;1Þ. In (19), the choice

r

ð

y

,

d

Þ¼ð

d

y

Þ

2

leads to balanced squared error loss function in

the form

L

w,

d

0

ð

y

,

d

Þ¼

o

ð

d

d

0

Þ

2

þð1

o

Þð

d

y

Þ

2

,ð20Þ

and the Bayes estimator of the unknown parameter

y

under

balanced squared error loss (bSEL) is given by

d

o

ðxÞ¼

od

0

ðxÞþð1

o

ÞEð

y

9xÞ:ð21Þ

The balanced LINEX loss function (bLINEX) with shape parameter

c(ca0), is obtained with the choice of

r

ð

y

,

d

Þ¼e

cð

d

y

Þ

c

ð

d

y

Þ1;ca0 (Zellner [31]). Hence the Bayes estimation of the

unknown parameter

y

under balanced LINEX loss function is

given by

d

o

ðxÞ¼1

clog½

o

e

c

d

0

ðxÞ

þð1

o

ÞE

x

ðe

c

y9xÞ:ð22Þ

It is clear that the balanced loss functions are more general, which

include the MLE and both symmetric and asymmetric Bayes

estimates as special cases. For examples, from (21), with

o

¼1,

the Bayes estimate under balanced squared error loss function

reduces to ML estimate, and for

o

¼0,it reduces to the Bayes

estimate relative to squared error loss function (symmetric). Also,

the Bayes estimate under balanced LINEX loss function in (22)

reduces to ML estimate when

o

¼1, and for

o

¼0,it reduces to

the case of LINEX loss function (asymmetric).

If

y

¼ð

a

,

b

,

l

,Sðt,

a

,

b

,

l

Þ,Hðt,

a

,

b

,

l

ÞÞ and suppose that we judge

convergence to have been reached after Miterations of an MCMC

algorithm have been performed.

We discard the observations

y

ð1Þ

,

y

ð2Þ

,...,

y

ðMÞ

and work with

the observations {

y

ðjÞ

,MojrN

1

}, which are viewed as being an

independent sample from the stationary distribution of the

Markov chain which is typically the posterior distribution. Now

the approximate posterior mean under balanced squared error

loss become

E½

y

¼

od

0

ðxÞþ ð1

o

Þ

N

1

MX

N

1

j¼Mþ1

y

ðjÞ

,ð23Þ

A.A. Soliman et al. / Reliability Engineering and System Safety 100 (2012) 48–57 51

Author's personal copy

and the approximate posterior mean under balanced LINEX error

loss is then given by

E½

y

¼1

clog

o

e

c

d

0

ðxÞ

þð1

o

Þ

N

1

MX

N

1

j¼Mþ1

e

c

y

ðjÞ

0

@1

A:ð24Þ

5. Application to real life data

In this section, we present two examples using two different

real data sets to illustrate the computations of the methods

proposed in this article.

Example 1. For illustrative purposes, we considered the real data

set of 50 observed failure times that was initially reported in Aarset

[1] and later by a number of authors including Xie et al. [28],Ng

[19], Jiang et al [13] and Upadhyay and Gupta [24]. Xie et al. [28]

indicated that MWD ﬁts this data set better than the exponential

distribution and another extension of the Weibull distribution.

Based on progressively type-II censored sample generated from

thesamesetofdata,withm¼35, n¼50 and censoring scheme

R

4

¼R

11

¼R

18

¼R

25

¼R

32

¼3, R

i

¼0,ia4;11,18;25,32. Ng [19]

obtained the MLE’s of the parameters as: ð~

a

ML

,~

b

ML

,~

l

ML

Þ¼

ð0:0714,0:398,0:01702Þ. In our study, we consider the same set of

data with the same censoring scheme as given in Table 1.

The computations for different methods of estimations are as

follows:

(I) Bootstrap method: Using the algorithm of the percentile

bootstrap method, described in section (3), we present the mean

of 1000 bootstrap samples of the parameters, the result becomes

(~

a

,~

b

,~

l

,~

S(t¼10), ~

Hðt¼10ÞÞ ¼ ð0:070,0:4095 ,0:0178,0:8097,

0:0124Þ. The 90% and 95% bootstrap conﬁdence interval (BCIs) are

displayed in Table 2.

(II) MCMC method: Under assumption that these data are from

MWD, we run the Gibbs sampler with in MH algorithm to

generate a Markov chain with 30,000 observations. Discarding

the ﬁrst 5000 values as ‘burn-in’ and taking every tenth variate as

iid observations. We used the non-informative gamma priors for

a

and

b

, that is, when the hyperparameters are 0. We call it prior 0:

g

1

¼

g

2

¼

Z

1

¼

Z

2

¼0. For computing Bayes estimators, other than

prior 0, we used the method of moment to estimate the values of

hyperparameters

g

1

,

Z

1

,

g

2

and

Z

2

. We have different informative

prior (prior 1), where ð

g

1

,

Z

1

,

g

2

,

Z

2

Þ¼ð1:33,0:036,1:33,0:036Þ. The

hyperparameters Mfor

l

was seen that does not have a con-

siderable effect on the results, so several values were assigned to

it. Vital statistics about the parameters are then computed from

the 2500 generated sample values. The marginal posterior density

estimates of the parameters, reliability and hazard functions and

their histograms based on samples of size 2500 are shown in

Fig. 1 using the Gaussian kernel. It is evident from the estimates

that all the marginal distributions are almost symmetrical. We

can see from the empirical posterior distribution of

a

in Fig. 1,itis

quite skewed. A traceplot is a plot of the iteration number against

the value of the draw of the parameter at each iteration. Fig. 2

display 25,000 chain values for the three parameters

a

,

b

and

l

as

well as reliability S(t) and hazard H(t) functions, with their

sample mean and 90% credible intervals. Interval estimates of

the unknown concentrations are easily obtained from the per-

centiles of the posterior distributions. For example, A 100ð1

g

Þ%

probability interval for the unknown parameter may be estimated

by taking the ð100ð

g

=2ÞÞ th and ð100ð1ð

g

=2ÞÞÞ th percentiles of

the sample as the lower ðLLÞand upper ðULÞbounds, respectively,

of the interval estimation. Table 3 lists the 90% and 95% prob-

ability intervals for the parameters, reliability and hazard

functions.

Table 4 provides the corresponding values of parameter

estimates and some of the other important posterior character-

istics based on sample of size 2500. The result of Bayes estimates

relative to both balanced squared error loss function (bSEL) and

balanced LINEX loss function (bLINEX) with different values of the

shape parameter ðcÞof LINEX loss function and various values of

o

for the parameters

a

,

b

,

l

, as well as the reliability function

Sðt¼15Þand hazard function H(t¼15) are displayed in Table 5.

Example 2. ThedataaretakenfromWang[25], and they represent

the failure times of 18 electronic devices put on a life test experiment.

The data that was used by Upadhyay and Gupta [24].TheMLEsof

parameters, reliability and hazard functions based on the complete

sample are obtained to be ( ~

a

,

~

b

,

~

l

,

~

Sðt¼100Þ,

~

Hðt¼100ÞÞ ¼ ð0:0114,

0:7017,0:0035,0:6660,0:0043Þ, respectively. In our example, let us

consider the following progressively type-II censored sample of size

m¼8 generated randomly from the n¼18 observations. The obser-

vations and the tree-stage removal pattern applied are reported

in Table 6. Based on this progressively type-II censored sample, the

MLEs of

a

,

b

,

l

,SðtÞand HðtÞ,arecomputedas:(

~

a

,

~

b

,

~

l

,

~

Sðt¼100Þ,

~

Hðt¼100ÞÞ ¼ ð0:0145,0:5736,0:0036,0:7468,0:0027Þ.

(I) Bootstrap method: Using the algorithm described in Section

3 of the percentile bootstrap method, we present the mean of 1000

Table 1

Progressively censored sample generated from data in Aarset [1].

i1 2 3456789101112131415161718

x

i:35:50

0.10.211111236711181818182132

R

i

0 0 0300000030000003

i19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

x

i:35:50

36 45 47 50 55 60 63 63 67 67 75 79 82 84 84 85 86

R

i

0 0 000030000003000

Table 2

Two-sided 90% and 95% BCIs of

a

,

b

,

l

,SðtÞand HðtÞfor Example 1.

Parameters 90% CI 95% CI

LL UL Length LL UL Length

a

0.0380 0.1216 0.0836 0.0355 0.1346 0.0991

b

0.3062 0.5292 0.2230 0.2933 0.5790 0.2857

l

0.0107 0.0255 0.0148 0.0093 0.0282 0.0189

Sðt¼10Þ0.6515 0.8522 0.2007 0.6265 0.8739 0.2474

Hðt¼10Þ0.0076 0.018 0.0104 0.0064 0.0194 0.0130

A.A. Soliman et al. / Reliability Engineering and System Safety 100 (2012) 48 –5752

Author's personal copy

bootstrap samples of the parameters, the result becomes ( ~

a

,~

b

,~

l

,

~

S(t¼100), ~

Hðt¼100ÞÞ ¼ ð0:0120,0:7120,0:0020,0:7770,0:0022). The

90% and 95% bootstrap conﬁdence interval (BCIs) are displayed in

Table 7.

(II) MCMC method: We run the Gibbs sampler with in MH

algorithm to generate a Markov chain with 30,000 observations.

Discarding the ﬁrst 5000 values as ‘burn-in’ and taking every

tenth variate as iid observations. We used the non-informative

gamma priors for

a

and

b

, that is, when the hyperparameters are

0. We call it prior 0:

g

1

¼

g

2

¼

Z

1

¼

Z

2

¼0. For computing Bayes

estimators, other than prior 0, we also used different informative

prior including prior 2, ð

g

1

,

Z

1

,

g

2

,

Z

2

Þ¼ð1:2,0:01,1:2,0:01Þ.Table 8

lists the 90% and 95% probability intervals for the parameters,

reliability and hazard functions. The MCMC results of the poster-

ior mean, median, mode, standard deviation (SD) and skewness of

the parameters, reliability and hazard functions are displayed in

Table 9. Also, the result of the Bayes estimates relative to both

ðbSEL Þand (bLINEX) loss functions are reported in Table 10.

It is clear from the Table 5,when

o

goes to one all results of Bayes

estimates under both balanced square loss function (bSEL) and

balanced LINEX loss function (bLINEX) of the parameters

a

,

b

and

l

, using prior 0 are equal to corresponding MLEs, given by Ng [19].

Fig. 1. Histogram and kernel density estimates of

a

,

b

,

g

,S(t) and H(t) from Example 1.

A.A. Soliman et al. / Reliability Engineering and System Safety 100 (2012) 48–57 53

Author's personal copy

6. Concluding remarks

Modiﬁed Weibull model is able to model lifetime data with

monotonic and bathtub-shaped failure rates. Lifetimes of modern

mechanic and electronic units are two examples for such data.

The inferences to the Modiﬁed Weibull model are meager because

of the fact that the model has been proposed recently and also

because of the complicated form of its pdf. Also, it is to be noted

that most of the cited literatures are conﬁned to classical devel-

opments and the Bayesian results are rarely seen. In this paper we

have considered the Bayesian inference of the modiﬁed Weibull

lifetime parameters when the data are progressively censored. It

well known that when all parameters are unknown, the Bayes

estimates cannot be obtained in explicit form. We used the MCMC

and parametric bootstrap techniques to compute the approximate

Bayes estimates and the corresponding credible intervals. We

compared the Bayes estimates based on noninformative priors

(prior 0) with the corresponding MLEs and found that their

behaviors were very similar, as expected. However, the results

are improved by adopting proper prior (prior 1 and prior 2)

distributions. The results obtained using MCMC methods are not

only reliable, but also complete with all aspects of parameter

inference: exact marginal distributions, means, mod, conﬁdence

interval, etc. It have been shown that a Bayesian analysis of the

MWD is achievable using MCMC methods, but that the applica-

tion is not straightforward and requires a hybrid algorithm. Two

Fig. 2. MCMC output of

a

,

b

,

g

,S(t) and H(t). Dashed lines (y) represent the posterior means and soled lines (—) represent lower, and upper bounds 90% probability

interval from Example 1.

A.A. Soliman et al. / Reliability Engineering and System Safety 100 (2012) 48 –5754

Author's personal copy

real examples are presented to illustrate how the MCMC and

parametric bootstrap methods are work based on progressive

censored data. The results obtained in this paper can be

specialized to the estimation problem of MWD based on complete

samples and usually type-II censoring samples.

As a reference, the MLEs of the parameters for complete sample

ðn¼50Þobtained by Ng [19] are: ð~

a

ML

,

~

b

ML

,

~

l

ML

Þ¼ð0:0624,0:355,

0:02332Þ. In light of this result we conclude that the proposed Bayes

estimates using prior 0 and prior 1, for some c40 are better than the

corresponding MLEs, see Tables 4 and 5. When the shape parameter

of the LINEX loss function is small (c¼0:001), the estimates will be

close to the case of quadratic loss.

Table 4

MCMC results for some posterior characteristics for Example 1.

Parameters Priors Mean Median Mode SD Skewness

a

Prior 0 0.0728 0.0676 0.0573 0.0303 1.0451

Prior 1 0.0705 0.0669 0.0598 0.0246 0.7969

b

Prior 0 0.3980 0.3931 0.3832 0.1210 0.4017

Prior 1 0.3906 0.3803 0.3597 0.1092 0.4286

l

Prior 0 0.0177 0.0177 0.0176 0.0054 0.1215

Prior 1 0.0182 0.0180 0.0175 0.0053 0.1552

Sðt¼15ÞPrior 0 0.7686 0.7730 0.7818 0.0536 0.4209

Prior 1 0.7736 0.7765 0.7822 0.0485 0.2892

Hðt¼15ÞPrior 0 0.0115 0.0113 0.0107 0.0028 0.6307

Prior 1 0.0113 0.0110 0.0105 0.0026 0.5292

Table 5

MLE and Bayes MCMC estimates under balanced square loss function (bSEL) and balanced LINEX loss function (bLINEX) for Example 1.

Parameters MLEs

o

Priors bSEL bLINEX

c¼9c¼5c¼0:001 c¼3c¼7

a

0.0715 0 Prior 0 0.0728 0.0773 0.0752 0.0728 0.0714 0.0698

Prior 1 0.0705 0.0734 0.0721 0.0705 0.0696 0.0684

0.3 Prior 0 0.0724 0.0756 0.0741 0.0724 0.0714 0.0703

Prior 1 0.0708 0.0728 0.0719 0.0708 0.0702 0.0693

0.9 Prior 0 0.0716 0.0721 0.0718 0.0716 0.0715 0.0713

Prior 1 0.0714 0.0717 0.0715 0.0714 0.0713 0.0712

b

0.3985 0 Prior 0 0.3980 0.4739 0.4377 0.3980 0.3770 0.3520

Prior 1 0.3906 0.4520 0.4228 0.3906 0.3735 0.3530

0.3 Prior 0 0.3981 0.4561 0.4267 0.3981 0.3833 0.3644

Prior 1 0.3929 0.4385 0.4158 0.3929 0.3808 0.3652

0.9 Prior 0 0.3984 0.4088 0.4028 0.3984 0.3963 0.3931

Prior 1 0.3977 0.4051 0.4010 0.3977 0.3959 0.3932

l

0.0170 0 Prior 0 0.0173 0.0173 0.0173 0.0173 0.0173 0.0172

Prior 1 0.0182 0.0183 0.0183 0.0182 0.0182 0.0181

0.3 Prior 0 0.0175 0.0176 0.0175 0.0175 0.0175 0.0174

Prior 1 0.0178 0.0179 0.0179 0.0178 0.0178 0.0178

0.9 Prior 0 0.0171 0.0171 0.0171 0.0171 0.0171 0.0171

Prior 1 0.0171 0.0171 0.0171 0.0171 0.0171 0.0171

Sðt¼15Þ0.7623 0 Prior 0 0.7686 0.7807 0.7755 0.7686 0.7642 0.7580

Prior 1 0.7736 0.7837 0.7793 0.7736 0.7700 0.7651

0.3 Prior 0 0.7667 0.7755 0.7717 0.7667 0.7636 0.7593

Prior 1 0.7702 0.7777 0.7744 0.7702 0.7677 0.7643

0.9 Prior 0 0.7629 0.7643 0.7637 0.7629 0.7625 0.7619

Prior 1 0.7634 0.7646 0.7641 0.7634 0.7631 0.7626

Hðt¼15Þ0.0118 0 Prior 0 0.0115 0.0116 0.0116 0.0115 0.0115 0.0115

Prior 1 0.0113 0.0113 0.0113 0.0113 0.0113 0.0113

Prior 0 0.0116 0.0116 0.0116 0.0116 0.0116 0.0116

0.3 Prior 1 0.0115 0.0115 0.0115 0.0115 0.0115 0.0114

Prior 0 0.0118 0.0118 0.0118 0.0118 0.0118 0.0118

0.9 Prior 1 0.0118 0.0118 0.0118 0.0118 0.0118 0.0118

Table 6

Progressively censored sample based on data from Example 2.

i12345678

x

i::8:18

5 11 31 98 122 195 224 330

R

i

2030 0 1 0 4

Table 7

Two-sided 90% and 95% BCIs of

a

,

b

,

l

,SðtÞand HðtÞfor Example 2.

Parameters 90% CI 95% CI

LL UL Length LL UL Length

a

0.0006 0.0311 0.0305 0.0004 0.037 0.0366

b

0.4292 1.1967 0.7675 0.3831 1.2960 0.9129

l

0.0008 0.0056 0.0064 0.0012 0.0076 0.0089

Sðt¼10Þ0.6152 0.9212 0.3060 0.5881 0.9375 0.3494

Hðt¼10Þ0.0009 0.0038 0.0029 0.0008 0.0044 0.0036

Table 3

90% and 95% MCMC conﬁdence intervals of

a

,

b

,

l

,SðtÞand H(t) for Example 1.

Parameters Priors 90% CI 95% CI

LL UL Length LL UL Length

a

Prior 0 0.0325 0.1301 0.0976 0.0281 0.1447 0.1166

Prior 1 0.0359 0.1148 0.0789 0.0313 0.1263 0.0949

b

Prior 0 0.2109 0.6152 0.4043 0.1798 0.662 0.4821

Prior 1 0.2250 0.5838 0.3588 0.1994 0.6243 0.4249

l

Prior 0 0.0088 0.0269 0.0181 0.0073 0.0286 0.0213

Prior 1 0.0099 0.0272 0.0173 0.0084 0.0287 0.0203

Sðt¼15ÞPrior 0 0.6726 0.8506 0.1780 0.6539 0.8639 0.2101

Prior 1 0.6901 0.8494 0.1592 0.6715 0.8615 0.1899

Hðt¼15ÞPrior 0 0.0076 0.0165 0.0089 0.0070 0.0176 0.0106

Prior 1 0.0075 0.0159 0.0085 0.0070 0.0170 0.0100

A.A. Soliman et al. / Reliability Engineering and System Safety 100 (2012) 48–57 55

Author's personal copy

Table 8

90% and 95% MCMC conﬁdence intervals of

a

,

b

,

l

,SðtÞand H(t) for Example 2.

Parameters 90% CI 95% CI

LL UL Length LL UL Length

a

Prior 0 0.0057 0.0199 0.0142 0.0049 0.022 0.0170

Prior 2 0.0069 0.0213 0.0144 0.0061 0.0235 0.0173

b

Prior 0 0.6239 0.6910 0.0671 0.6220 0.6934 0.0714

Prior 2 0.6510 0.6890 0.0379 0.6489 0.6926 0.0437

l

Prior 0 0.0016 0.0021 0.0005 0.0015 0.0021 0.0006

Prior 2 0.0014 0.0026 0.0012 0.0014 0.0027 0.0013

Sðt¼100ÞPrior 0 0.6143 0.8636 0.2493 0.5848 0.8796 0.2949

Prior 2 0.5725 0.8267 0.2541 0.5449 0.8456 0.3007

HðtÞPrior 0 0.0012 0.0041 0.0029 0.0011 0.0045 0.0035

Prior 2 0.0017 0.0048 0.0031 0.0015 0.0052 0.0038

Table 9

MCMC results for some posterior characteristics for Example 2.

Parameters Priors Mean Median Mode SD Skewness

a

Prior 0 0.0119 0.0111 0.0095 0.0045 0.9101

Prior 1 0.0133 0.0126 0.0114 0.0045 0.8487

b

Prior 0 0.6595 0.6628 0.6693 0.0207 0.2963

Prior 1 0.6694 0.6690 0.6681 0.0110 0.2268

l

Prior 0 0.0019 0.0019 0.0020 0.0002 0.7885

Prior 1 0.0021 0.0020 0.0018 0.0004 0.0010

Sðt¼100ÞPrior 0 0.7477 0.7548 0.7689 0.0768 0.5169

Prior 1 0.7067 0.7129 0.7251 0.0771 0.3580

Hðt¼100ÞPrior 0 0.0025 0.0024 0.0021 0.0009 0.8642

Prior 1 0.0031 0.0030 0.0027 0.0010 0.6363

Table 10

MLE and Bayes MCMC estimates under balanced square loss function (bSEL) and balanced LINEX loss function (bLINEX).

Parameters MLEs

o

Priors bSEL bLINEX

c¼9c¼5c¼0:001 c¼3c¼7

a

0.0120 0 Prior 0 0.0119 0.0119 0.0119 0.0119 0.0118 0.0118

Prior 2 0.0133 0.0133 0.0133 0.0133 0.0132 0.0132

0.3 Prior 0 0.0119 0.0120 0.0119 0.0119 0.0119 0.0119

Prior 2 0.0129 0.0129 0.0129 0.0129 0.0129 0.0128

0.9 Prior 0 0.0120 0.0120 0.0120 0.0120 0.0120 0.0120

Prior 2 0.0121 0.0121 0.0121 0.0121 0.0121 0.0121

b

0.6771 0 Prior 0 0.6595 0.6614 0.6606 0.6595 0.6589 0.6580

Prior 2 0.6694 0.6700 0.6697 0.6694 0.6692 0.6690

0.3 Prior 0 0.6648 0.6664 0.6657 0.6648 0.6642 0.6635

Prior 2 0.6717 0.6722 0.6720 0.6717 0.6716 0.6714

0.9 Prior 0 0.6753 0.6711 0.6755 0.6753 0.6752 0.6751

Prior 2 0.6763 0.6764 0.6764 0.6763 0.6763 0.6763

l

0.0014 0 Prior 0 0.0019 0.0019 0.0019 0.0019 0.0019 0.0019

Prior 2 0.0021 0.0021 0.0021 0.0021 0.0021 0.0021

0.3 Prior 0 0.0017 0.0017 0.0017 0.0017 0.0017 0.0017

Prior 2 0.0019 0.0019 0.0019 0.0019 0.0019 0.0019

0.9 Prior 0 0.0015 0.0015 0.0015 0.0015 0.0015 0.6751

Prior 2 0.0015 0.0015 0.0015 0.0015 0.0015 0.0015

Sðt¼100Þ0.7313 0 Prior 0 0.7477 0.7714 0.7615 0.7477 0.7385 0.7251

Prior 2 0.7067 0.7313 0.7209 0.7067 0.6976 0.6847

0.3 Prior 0 0.7428 0.7608 0.7529 0.7428 0.7363 0.7269

Prior 2 0.7141 0.7313 0.7241 0.7141 0.7073 0.6971

0.9 Prior 0 0.7329 0.7360 0.7345 0.7329 0.7320 0.7306

Prior 2 0.7288 0.7313 0.7303 0.7288 0.7278 0.7259

Hðt¼100Þ0.0026 0 Prior 0 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025

Prior 2 0.0031 0.0031 0.0031 0.0031 0.0031 0.0031

A.A. Soliman et al. / Reliability Engineering and System Safety 100 (2012) 48 –5756

Author's personal copy

Acknowledgments

The authors would like to thank the two anonymous referees

for their constructive comments on an early version of this paper.

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Table 10 (continued )

Parameters MLEs

o

Priors bSEL bLINEX

c¼9c¼5c¼0:001 c¼3c¼7

0.3 Prior 0 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025

Prior 2 0.0029 0.0029 0.0029 0.0029 0.0029 0.0029

0.9 Prior 0 0.0026 0.0026 0.0026 0.0026 0.0026 0.0026

Prior 2 0.0026 0.0026 0.0026 0.0026 0.0026 0.0026

A.A. Soliman et al. / Reliability Engineering and System Safety 100 (2012) 48–57 57