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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