International Journal of Advanced Statistics and Probability, 2 (2) (2014) 108-113
©Science Publishing Corporation
www.sciencepubco.com/index.php/IJASP
doi: 10.14419/ijasp.v2i2.3423
Research Paper
Analysis of Generalized Exponential Distribution
Under Adaptive Type-II Progressive Hybrid
Censored Competing Risks Data
S. K. Ashour
1
, M. M. A. Nassar *
2
1
Department of Mathematical Statistics, Institute of Statistical Studies & research. Cairo University, Egypt
2
Department of Statistics, Faculty of Commerce, Zagazig University, Egypt
*Corresponding author E-mail: mezo10011@gmail.com
Copyright © 2014 S.K. Ashour, M.M.A.Nassar. This is an open access article distributed under the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
This paper presents estimates of the parameters based on adaptive type-II progressive hybrid censoring scheme (AT-II
PHCS) in the presence of the competing risks model. We consider the competing risks have generalized exponential
distributions (GED). The maximum likelihood method is used to derive point and asymptotic confidence intervals for
the unknown parameters. The relative risks due to each cause of failure are investigated. A real data set is used to
illustrate the theoretical results and to test the hypothesis that the causes of failure follow the generalized exponential
distributions against the exponential distribution (ED).
Keywords: Competing Risks; Adaptive Type-II Progressive Hybrid Censoring; Generalized Exponential Distribution; Maximum Likelihood
Estimation.
1. Introduction
In the context of life testing experiments, hybrid censoring scheme was introduced at first by Epstein [3]. Since the
introduction by Epstein [3], extensive works and different types of hybrid censoring scheme has been appeared.
Recently, Kundu and Joarder [8] and Childs et al [2] both investigated the type-II progressive hybrid censoring scheme
(T-II PHCS), where the life testing experiment with progressive censoring scheme
12
( , ,..., )
m
R R R
is terminated at time
::
min ,
mmn
T X T
, where
(0, )T
is the prefixed time and
::mmn
X
denotes the m-th failure time when n items are
place on a life test experiment. Briefly, If
::mmn
XT
the experiment terminate at time
::mmn
X
and m failures occurs;
otherwise, the experiment stops at time T and only J failures occur before time T, where
: : 1: :J m n J m n
X T X
and
0 Jm
.
The drawback of the T-II PHCS is that the effective number of failures is random and it can be a very small number
(even equal to zero), so that usual statistical inference procedures will not be applicable or they will have low
efficiency. For this reason, Ng et al [14] suggested an adaptive type-II progressive hybrid censoring scheme in which
the effective number of failures m is fixed in advance and the progressive censoring scheme
12
, ,...,
m
R R R
is provided,
but the values of some of the
i
R
may be change accordingly during the experiment. Suppose the experimenter
provides a time T, which is an ideal total test, but the experimental time is allowed to run over time T. If the m-th
progressively censored observed failures occurs before time T (i.e.
::mmn
XT
), the experiment stops at this time
::mmn
X
, and we will have a usual type-II progressive censoring scheme with the prefixed progressive censoring scheme
12
, ,...,
m
R R R
. Otherwise, once the experimental time passes time T but the number of observed failures has not reached
m, then we adapt the number of items progressively removed from the experiment upon failure by setting
1 2 1
, ,..., 0
J J m
R R R
and
1
J
mi
i
R n m R
, where
: : 1: :J m n J m n
X T X
, and
::J m n
X
is the J-th failure time occur
International Journal of Advanced Statistics and Probability
109
before time T and
1Jm
. Thus the effectively applied scheme is
1
1
,..., ,0,...,0,
J
Ji
i
R R n m R
. This formulate on
leads to terminate the experiment as soon as possible if the (J+1)-th failure time is greater than T, and the total test time
will not be too far away from the time T. If
0T
, the scheme will lead us to the case of conventional type-II censoring
scheme, and if
T
, we will have a usual progressive type-II censoring scheme.
We should mention here that many authors studied the statistical properties of some life time models under AT-II PHCS
in the presence of one and two causes of failures. Ng et al [14] developed inferential methods for the case when the
lifetime distribution is exponential. They observed that the MLE always exists in this case. Lin et al [5], considered the
adaptive progressive censoring scheme when the lifetime distribution is Weibull, and discussed the corresponding
inferential issues. They have also discussed confidence intervals for the model parameters through the use of the
asymptotic distribution of the MLEs as well as by the bootstrap method. Hemmati and Khorram [11], studied the
competing risk model based on exponential distributions under the adaptive type-II progressively censoring scheme.
They obtained the maximum likelihood and the Bayes estimators of the exponential distribution parameter, and two
sides Bayesian probability intervals of the parameter are also obtained. Hemmati and Khorram [10], obtained the
maximum likelihood estimators of the parameters from a two-parameter log-normal distribution based on the adaptive
Type-II progressive hybrid censoring scheme, they compared the results with corresponding estimators of the type-II
progressive hybrid censoring scheme. Mahmoud et al [15], obtained the maximum likelihood estimators for the
unknown parameters of Pareto distribution based on the adaptive type-II progressive censoring scheme, point estimation
and confidence intervals based on maximum likelihood also proposed.
The main aim of this paper is analyzing the AT-II PHCS under the competing risk model when lifetimes have
independent GED. We derive the maximum likelihood estimates (MLE) and we obtain the approximate two sided
confidence intervals of these different parameters. We use the likelihood ratio test to test the ED against the GED. We
consider a real data set and see how the different models work in the practical situation.
The rest of this paper is organized as follows: In section (2), we introduce the model and the notation used throughout
this paper. In section (3), we discuss the maximum likelihood estimation; confidence intervals are presented in section
(4). In section (5), Goodness of fit test for testing a competing risks model where causes follow a GED against ED. In
section (6), a real data set is used to illustrate the theoretical results.
2. Model description and notation
In reliability analysis, the failure of items may be attributable to more than one cause at the same time. These "causes"
are competing for the failure of the experimental unit. Consider a life time experiment with
nN
identical units,
where its lifetimes are described by independent and identically distributed (i.i.d) random variables
12
, ,...,
n
X X X
.
Without loss of generality; assume that there are only two causes of failure. We have
12
min ,
i i i
X X X
for
1,...,in
,
where
, 1,2
ki
Xk
, denotes the latent failure time of the i-th unit under the k-th cause of failure. We assume that the
latent failure times
1i
X
and
2i
X
are independent, and the pairs
12
,
ii
XX
are i.i.d. Assume that the failure times
follows the GED introduced by Gupta and Kundu [16] as generalization of the exponential distribution with the
probability density function
()
k
fx
as
1
..
( ) . . 1 , 0, . 0
k
kk
xx
k k k k k
f x e e x
(1)
where
k
is the scale parameter and
k
is the shape parameters. The cumulative distribution function
()
k
Fx
and
failure hazard function
()
k
hx
have the form
.
( ) 1
k
k
x
k
F x e
(2)
and
1
1
. . .
( ) . . . 1 1 1
kk
k k k
x x x
k k k
h x e e e
(3)
Under AT-II PHCS and in presence of competing risks data we have the following observation:
1: : 1 1 : : 1: : 1 1: : 1 : :
( , , ),..,( , , ),( , ,0),..,( , ,0),( , , )
m n J m n J J J m n J m m n m m m n m m
X c R X c R X c X c X c R
where
::
max :
J m n
J J X T
,
1
J
mi
i
R n m R
and
1,2
i
c
. Here,
1,2
i
c
means the unit
i
has failed at time
::i m n
X
due to the first and the second cause of failures, respectively. Let
12
1, 1 1, 2
1 , 2
00
ii
ii
cc
I c I c
else else
thus the random variables
11
1
1
m
i
i
m I c
and
22
1
2
m
i
i
m I c
describe the number of failures due to the first
and the second cause of failures, respectively and
12
m m m
. Now we can write the likelihood function of the
observed data as follows
110
International Journal of Advanced Statistics and Probability
12
1 2 2 1 1 2 1 2
11
( ) ( ) ( ) ( ) ( ) ( )
i i i m
mJ
I c I c R R
i i i i i i m m
ii
L C f x F x f x F x F x F x F x F x
(4)
where
::i i m n
xx
for simplicity of notation,
( ) 1 ( )
kk
F x F x
and C is a constant doesn’t depend on the parameters.
3. Maximum likelihood estimation
From (1), (2) and (4), the likelihood function ignoring the normalized constant can be written as follows
12
12
1 2 2 1 1 2 1 2
11
1 1 2 2 1 1 2 2 2 1 1 2 1 2
1 1 1
. . 1 1 1 1 1 1
i
m
R
mm
J
R
mm
i i i i i i i i m m
i i i
L z u u z u u u u u u
(5)
where
.
( ) 1
ki
x
ki ki k
u u e
,
.
()
ki
x
ki ki k
z z e
and
1,2k
and the log-likelihood function is
1
21
2
1 2 1 2 1 2
1 1 1 2 2 2 1 1 2 1
1
2 2 1 2 1 2 1 2
11
ln (ln ln ) (ln ln ) ( 1)ln ln 1 .
( 1)ln ln 1 . .ln 1 1 .ln 1 1 (6)
m
i i i
i
m
D
i i i i i i m m m
ii
L m m u u x
u u x R u u R u u
The first order derivations of (6) with respect to
k
and
k
,
1,2k
are given, respectively, by
3
1 1 1 1
ln
( 1) . . ,
k
kk
m
mm
J
k
i k ki ki i ki m km
i i i i
kk
Lm
x v s R s R s
and
3
1 1 1
1 1 1
ln
ln( ) (1 ) .ln( ) .(1 ) .ln( ) .(1 ) .ln( )
k
k
k k k
m
m
J
k
ki ki ki i ki ki m km km
i i i
kk
m
L
u u u R u u R u u
. (7)
where
( ) . /
ki ki k i ki ki
v v x z u
,
1
( , ) . . . /( 1)
kk
ki ki k k k i ki ki ki
s s x z u u
. Equating the first derivations in (7) to zero, one
can obtain the MLE of the unknown parameters
1 2 1
,,
and
2
. As it seems, the system of non-linear equations (7)
has no closed form solution in
1 2 1
,,
and
2
. So a numerical method technique is required for computing the MLE of
the parameters
1 2 1
,,
and
2
.
The asymptotic variance-covariance matrix for
1 2 1
,,
and
2
can be obtained by inverting the information matrix
with the elements that are negative of the expected values of the second order derivatives of logarithms of the likelihood
functions. Cohen (1965) concluded that the approximate variance covariance matrix may be obtained by replacing
expected values by their MLEs. Now the approximate sample information matrix will be
1 2 1 2
22
2
1 1 1
22
2
2 2 2
22
2
1 1 1
22
2
ˆˆ
2 2 2
ˆˆ
, , ,
ln ln
00
ln ln
00
ˆ
()
ln ln
00
ln ln
00
LL
LL
LL
LL
I
The elements of the
44
matrix,
( ), , 1,2,..,4
ij
ijI
can be obtained as follows
3
2
1
1 1 1
1
22
11
11
1 1 1
1
ln
( 1) ( 1) 1 ( 1)
( 1) 1 ( 1) ( 1) 1 ( 1)
k
k
kk
k k k k
m
m
k i ki ki i ki k ki ki k ki ki ki
ii
kk
J
i i ki k ki ki k ki ki ki m m km k km km k km km km
i
Lm
x v u x s z u u z u
R x s z u u z u R x s z u u z u
1
,
3
2
2 2 2 2 2 2
22
11
ln
(1 ) ln( ) (1 ) ln( ) (1 ) ln( ) ,
k
k k k k k k
m
J
k
ki ki ki i ki ki ki m ki km km
ii
kk
Lm
u u u R u u u R u u u
and
3
2
1 1 1
1 1 1
1 1 1
ln
1 ln( ) 1 1 ln( ) 1 1 ln( ) 1 . (8)
k
k
k k k
m
m
J
ki k ki k ki ki i k ki k ki ki m k km k km km
i i i
kk
L
v s u u R s u u R s u u
where
1
( , ) .( 1). . . . 1 ,
ki ki k k k k i ki ki k i
w w x z u x
1,2k
. Using the independence of the latent failure times
International Journal of Advanced Statistics and Probability
111
12
,
ii
XX
,
1,...,in
, we can obtain the relative risk rate due to a particular cause (say, cause 1) as follows
12
1 1 2
1 1 2 1 2
0
1
. . .
11
0
( ) . ( ).
1 . . . 1 . 1 .
ii
x x x
P X X f x F x dx
e e e dx
use the binomial expansion of
2
2
.
1
x
e
, we have
1
1 1 2
1
. . . .
1 1 1
0
0
2
1 . 1 . 1 .
i
x x i x
i
j
e e e dx
use the transformation
1
.x
ye
, we have
2
1 1 1
0
1
2
.
1 1 . 1,
i
i
j
i
B
(9)
Once
1
is computed, we determine
2
using the relation
21
1
. As the integral in the right side of (9) has no
analytical solution, we have to use a numerical technique to solve the integral. According to the invariance property of
the MLE, the MLE of the relative risk rates
1
, can be obtained by replacing the MLE of
1 2 1
,,
and
2
in (9). Based
on the above results, When
12
1
, the MLE's of
1
and
2
and the relative risk rates
1
and
2
, corresponds to
the results of the exponential distribution obtained by Hemmati and Khorram [11], when the cause of failure is known.
4. Asymptotic confidence intervals
In this section we derive the confidence intervals of the vector of the unknown parameters
1 2 1 2
, , ,
. Based on
the asymptotic distribution of the MLE of the parameters, it is known that
1
4
ˆ
0, ( )NI
where
()I
is the Fisher information matrix. The elements of
44
matrix
()
ij
I
can be approximated by
ˆ
()
ij
I
, where
2
ˆ
ln
ˆ
( ) , , 1,2,..,4
ij
ij
L
I i j
and
2
ln /
ij
L
is the second derivations obtained in (8). The
100(1 )
approximate confidence intervals of the
vector of the unknown parameters
1 2 1 2
, , ,
can be obtained as follows
/2
ˆˆ
. var( ), 1,...,4.
jj
zj
where
ˆ
var( )
j
is the elements on the main diagonal of
1
ˆ
()I
and
/2
z
is the upper
( /2)-th
percentile of a standard
normal distribution.
5. Goodness of fit
We now discuss the problem of testing goodness of fit of a competing risks model when the causes of failures follow
the GED against the ED to illustrate whether the GED can better fit a real data set rather than the ED studied by
Hemmati and Khorram [11]. Because the ED can be derived as a special case of the GED, the likelihood ratio test will
be used to test the adequacy of generalized exponential distributions competing risks. The null and alternative
hypotheses are
0 1 2
:1H
, the causes of death follow ED,
0 1 2
:1H
, the causes of death follow GED.
The test statistic is the ratio of the likelihood of
0
H
and the likelihood of
1
H
, and given by the expression
0
( )/ ( )LL
where
is a vector of parameters, and
0
is a subset of
. Under the null hypothesis the log-likelihood ratio test
statistic is
0
2ln 2 ( ) ( )
L
where
()
and
0
()
are the log-likelihood functions under
1
H
and
0
H
, respectively. Asymptotically, the test
statistic is distributed as a chi-squared distribution with
degree of freedom. Now, we can write the log-likelihood
ratio test statistics as follows
2
L GED ED
112
International Journal of Advanced Statistics and Probability
where
ED
and
GED
are the log-likelihood functions under
0
H
and
1
H
, respectively, after replacing the unknown
parameters with their MLE.
For comparison purposes between the candidate models, we can use two model criterion selection, the Akaike
information criterion (AIC) (Akaike [13]) and Bayes information criterion (BIC) (Schwarz [12]) defined as
AIC 2 2 and BIC 2 .ln( )p p n
where p is the number of parameters in the model, and is the maximized value of the likelihood function for the
model. As a model selection criterion, the researcher should choose the model that minimizes AIC and BIC.
6. Numerical results
In this section, we analyze one data set which was originally analyzed by Hoel [6] and later by Kundu et al [7], Pareek
et al [4], Cramer and Schmiedt [9], Hemmati and Khorram [11] and Ashour and Nassar [17]. The data was obtained
from a laboratory experiment in which male mice received a radiation dose of 300 roentgens at 35 days to 42 days (5-6
weeks) of age. The cause of death for each mouse was determined by reticulum cell sarcoma as cause 1 and other
causes of death as cause 2, there were
77n
observations remain in the analysis. Using the censoring scheme
25m
and
1 2 25
... 2R R R
, the progressive type-II censored sample from the original data is given by
(40, 2), (42, 2), (62, 2), (163, 2), (179, 2), (206, 2), (222, 2), (228, 2), (252, 2), (259,2), (318, 1), (385, 2), (407, 2), (420,
2), (462, 2), (517, 2), (517, 2), (524, 2), (525, 1),( 536, 1),( 558, 1), (605, 1), (612, 1), (620,2), (621, 1). All of the
computations were performed using MATHCAD program version 2007.
The first component denotes the life time and the second component indicate the cause of failure.
Example 1: Considering
550T
, then
20J
,
1
7m
and
2
18m
. From the above data, the MLEs of the unknown
parameters , the corresponding approximate 95% two sided confidence intervals distributions, the log-likelihood
values ( ), AIC and BIC shown given in table (1).
Table 1: The MLE and Approximate 95% Two Sided Confidence Intervals of the Parameters in Each Model (ED and GED).
Model
Estimates
Statistics
1
2
1
2
AIC
BIC
ED
0.000239
(0.00006, 0.00042)
0.000615
(0.00033, 0.0009)
--
--
-215.9
435.9
440.6
GED
0.0047
(0.0021, 0.007)
0.0011
(0.0003, 0.002)
25.885
(0, 62.47)
1.527
(0.728, 2.326)
-204.4
416.8
426.3
and the relative risk due to cause one for ED and GED are 0.28 and 0.5577, respectively. The value of the likelihood
ratio test statistic is
24.844
L
, and the corresponding p-value is 0.00005.
Example 2: Now we use the same data, but use
610T
instead of
550T
, while m and
i
R
's are same as before, then
22J
,
1
7m
and
2
18m
. the MLEs of the unknown parameters, the corresponding approximate 95% two sided
confidence intervals distributions, the log-likelihood values, AIC and BIC shown given in table (2).
Table 2: The MLE and Approximate 95% Two Sided Confidence Intervals of the Parameters in Each Model (ED and GED).
Model
Estimates
Statistics
1
2
1
2
AIC
BIC
ED
0.000241
(0.00006, 0.000419)
0.000619
(0.00033,0.000904)
--
--
-215.8
435.7
440.4
GED
0.0049
(0.0022, 0.008)
0.0011
(0.0003, 0.002)
28.09
(0, 68.109)
1.5433
(0.735, 2.352)
-203.8
415.8
425.1
and the relative risk due to cause one for ED and GED are 0.28 and 0.5597, respectively. The value of the likelihood
ratio test statistic is
25.63
L
, and the p-value is 0.00003.
The analysis of the previous real data set demonstrates the importance and usefulness of adaptive type-II progressive
hybrid censoring scheme and inferential procedures based on them. From example 1 and 2, it is observed that T plays a
major role in the estimation and for the construction of the corresponding confidence intervals, because when T
increases some additional information is gathered. We also conclude that based on the values of
L
, the p-value, AIC
and BIC, the GED fits the data better than ED. We have observed that the assumptions that the generalized exponential
distributions may be used to analyze this set of real data better that the exponential distribution.
