Post on 12-Jun-2018
International Mathematical Forum, Vol. 11, 2016, no. 19, 943 - 959
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/imf.2016.6793
The Odd Generalized Exponential Modified
Weibull Distribution
Yassmen Y. Abdelall
Department of Mathematical Statistics
Institute of Statistical Studies and Research
Cairo University, Egypt
Copyright © 2016 Yassmen Y. Abdelall. This article is 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
In this paper we propose a new distribution, called the odd generalized
exponential modified Weibull distribution. Some mathematical properties of the
new distribution are studied. The method of maximum likelihood is used for
estimating the model parameters and the observed Fisher's information matrix is
derived. We illustrate the usefulness of the proposed model by application to real
data.
Keywords: Modified Weibull distribution, moments, maximum likelihood
estimation, order statistics
1. Introduction
The modified Weibull (MW) distribution is one of the most important
distributions in lifetime modeling, and some well-known distributions such as the
exponential, Rayleigh, linear failure rate and Weibull distributions are special
cases of it. This distribution was introduced by Lai, Xie, and Murthy (2003) to
which we refer the reader for a detailed discussion as well as applications of the
MW distribution (in particular, the use of the real data set representing failure
times to illustrate the modeling and estimation procedure). Also Sarhan and
Zaindin (2009) introduced the modified Weibull distribution. It can be used to
describe several reliability models. It has three parameters, two scale and one
shape parameters. Recently, Carrasco et al. (2008) extended the MW distribution
by adding another shape parameter and introducing a four parameter generalized
MW (GMW) and log-GMW (LGMW).
944 Yassmen Y. Abdelall
Recently Tahir et al. (2015) proposed a new class of univariate distributions called
the odd generalized exponential (OGE) family and studied each of the OGE-
Weibull (OGE-W) distribution, the OGE-Fréchet (OGE-Fr) distribution and the
OGE-Normal (OGE-N) distribution. This method is flexible because the hazard
rate shapes could be increasing, decreasing, bathtub and upside down bathtub.
In this article we present a new distribution from the odd generalized exponential
distribution and modified Weibull distribution called the Odd Generalized
Exponential-Modified Weibull (OGE-MW) distribution using new family of
univariate distributions proposed by Tahir et al. (2015).
A random variable X is said to have generalized exponential (GE) distribution
with parameters , if the cumulative distribution function (cdf) is given by
.0,0,0,1,; xexF x (1)
The Odd Generalized Exponential family by Tahir et al. (2015) is defined as
follows. Let );( xG is the cdf of any distribution depends on parameter and thus
the survival function is );(1);( xGxG , then the cdf of OGE-family is
defined by replacing x in CDF of GE in Equation (1) by );(
);(
xG
xGto get
.0,0,0,0,1,,; );(
);(
xexF xG
xG
Where , are two additional parameters. This paper is outlined as follows. In
Section 2, we define the cumulative distribution function, density
function,reliability function and hazard function of the Odd Generalized
Exponential-Modified Weibull (OGE-MW) distribution. In Section 3, we
introduce the statistical properties include, the quantile function, the median
andthe moments. Section 4 discusses the distribution of the order statistics for
(OGE-MW) distribution. Moreover, maximum likelihood estimation of the
parameters is determined in Section 5. Finally, an application of (OGE-MW)
using a real data set is presented in Section 6.
2. The OGE-MW Distribution
2.1 OGE-MW specifications
In this section we define new five parameters distribution called Odd Generalized
Exponential-Modified Weibull distribution with parameters ,,,, and
written as OGE-MW(Θ), where the vector Θ is defined by Θ = ( ,,,, ) .
Odd generalized exponential modified Weibull distribution 945
A random variable X is said to have OGE-MW with parameters ,,,, and if
its cumulative distribution function (cdf) given as follows
.0,,,,,0,1;1
xexFxxe
(2)
where ,, are scale parameters and , are shape parameters. Hence, the
corresponding probability density function (pdf) is
.0,,,,,0,1;
111
1
xeeexxfxxxx ee
xx
(3)
2.2 Survival and hazard functions
If a random variable X has cdf in (2), then the corresponding survival function is
given by
.111
xxe
exS
The hazard function of OGE-MW( ) is defined as follow
.
11
1
)(
)(
1
111
1
xx
xxxx
e
eexx
e
eeex
xS
xfxh
Figure 1, 2 and 3 illustrates some of the possible shapes of the pdf, cdf and hazard
function of OGE-MW distribution for some values of the parameters ,,,,
and
946 Yassmen Y. Abdelall
0 2 4 6 8 100
0.2
0.4
0.6
f x 0.25 0.7 0.3 0.2 2( )
f x 0.2 0.2 0.4 0.6 2( )
f x 0.25 0.6 0.2 0.1 2.5( )
f x 0.4 0.3 0.6 0.8 1.5( )
f x 0.3 0.4 0.9 0.5 3( )
x
Figure 1. The pdf’s of various OGE-MW distributions.
0 2 4 6 8 100
0.5
1
F x 0.5 0.4 0.8 0.5 0.9( )
F x 0.4 0.2 0.4 0.2 1( )
F x 0.2 0.6 0.2 0.6 1( )
F x 0.3 0.3 0.6 0.8 1.4( )
F x 0.2 0.7 0.5 0.4 1( )
x
Figure 2.The cdf of various OGE-MW distributions.
1 1.5 2 2.5 3 3.5 40
8
16
24
32
40
h t 1 0.7 0.5 0.4 1.5( )
h t 1.5 0.6 0.4 0.3 1.5( )
h t 1.5 0.8 0.6 0.7 1.5( )
h t 1.5 0.3 0.7 1 1.5( )
h t 2 0.5 0.3 0.8 2( )
t
Figure 3. The hazard function of various OGE-MW distributions.
Odd generalized exponential modified Weibull distribution 947
Note that theOGE-MW distribution is very flexible model that approaches to
different distributions when its parameters are changed. The OGE-MW
distribution contains as special-models with the following well known
distributions. In particular, for 0 we have the odd generalized exponential-
Weibull (OGE-W) distribution as discussed in Tahir et al. (2015). The odd
generalized exponential-exponential(OGE-E) distribution is clearly a special case
for 1,0 and 1 as discussed in Maiti and Pramanik (2015).for 2/
and 2 we have the odd generalized exponential-linear failure rate (OGE-
LFR) distribution as discussed in El-Damcese et al. (2015). When 0 and
2 then the resulting distribution is the odd generalized exponential-Rayleigh
(OGE-R) distribution.
3. Statistical Properties
This section is devoted for studying somestatistical properties for the odd
generalized exponential-modified
Weibull (OGE-MW), specifically quantile, median and the moments.
3.1 Quantile and Median of OGE-MW
The quantile function qx of OGE-MW(Θ) distribution is given by using
qxF q )( (4)
Substituting from (2) into (4), qx is the real solution of the following equation
,10,0
1ln
1ln
1
q
q
xx qq
(5)
The above equation has no closed form solution in qx , so we have to use a
numerical technique such as a Newton- Raphson method to get the quantile. By
putting 5.0q in Equation (5) we can get the median of odd generalized
exponential modified Weibull distribution.
3.2 Moments
Moments are necessary and important in any statistical analysis, especially in
applications. It can be used tostudy the most important features and characteristics
of a distribution (e.g., tendency, dispersion, skewness and kurtosis). In this
subsection, we will derive the rth moments of the OGE-MW(Θ) distribution as
infinite series expansion.
948 Yassmen Y. Abdelall
Theorem1.
The rth moment of a random variable X ~ OGE-MW(Θ), where Θ = ( ,,,,
) is given
LrLrLrLr
i j
j
k L
LjLjkji
r
kj
Lr
kj
Lr
Lj
kji
k
j
i
11
1
!!
111
1
1
0 0 0 0
1'
Proof: The rth moment of a random variable X with pdff(x) is defined by
0
' )( dxxfx r
r (6)
Substituting from (3) into (6), we obtain
.10
111
1'
dxeeexxxxxx ee
xxr
r
(7)
Since 1101
xxe
e for 0x , we obtain
0
11
1
11
1i
eii
e xxxx
ei
e
. (8)
Substituting from (8) into (7), we get
.11
0 0
111'
i
eixxri
r dxeexxi
xx
Using series expansion of
11
xxei
e , we obtain
.1!
11
1
0 0 0
11
'
i j
jxxxxr
jjji
r dxeexxj
i
i
Using binomial expansion of jxxe 1 , we obtain
Odd generalized exponential modified Weibull distribution 949
0 0 0
1'
!
11
1
i j
j
k
jjkji
rj
i
k
j
i
.0
111
dxeexx xkjxkjr
Using series expansion of xkje 1 , we obtain
0 0 0 0
1'
!!
111
1
i j
j
k L
LjLjkji
rLj
kji
k
j
i
.0
11
0
1
dxexdxex xkjLrxkjLr
By using the definition of gamma function in the form, see Zwillinger (2014),
.0,,0
1
xzdttexz zxtz
Finally, we obtain the rth moment of OGE-MW as follows
.
11
1
!!
111
1
1
0 0 0 0
1
'
LrLrLrLr
i j
j
k L
LjLjkji
r
kj
Lr
kj
Lr
Lj
kji
k
j
i
This completes the proof. □
4. Order Statistics
Let nXXX ,...,, 21 be a simple random sample of size n from OGE-MW(Θ)with
cumulative distribution function );( xF and probability density function );( xf
given by (2) and (3) respectively. Let nnnn XXX ::2:1 ..., denote the order
statistics obtained from this sample. The probability density function of nrX : is
given by
,);(1);();()1,(
1;
1
:
rnr
nr xFxFxfrnrB
xf
(9)
950 Yassmen Y. Abdelall
where );( xf and );( xF are the pdf and cdf of OGE-MW(Θ) distribution given
by (2) and (3) respectively and B(., .) is the beta function, also we define first
order statistics nn XXXX ;...;;min 21:1 ,and the last order statistics as
nnn XXXX ;...;;max 21: . Since 1);(0 xF for ,0x we can use the
binomial expansion of rnxF
);(1 given as follows
.);()1();(10
iirn
i
rnxF
i
rnxF
(10)
Substituting from (10) into (9), we obtain
.);()1();()1,(
1;
0
1
:
rn
i
rii
nr xFi
rnxf
rnrBxf (11)
Substituting from (2) and (3) into (11), we obtain
.))(,,,,;()(!)(!)1(!
!)1(,,,,;
0
:
rn
i
i
nr irxfirirnri
nxf
(12)
Relation (12) shows that ,,,,;: xf nr is the weighted average of the odd
generalized exponential-modified Weibull with different shape parameters.
5. Estimation and Inference
Now, we discuss the estimation of the OGE-MW ),,,,( parameters by
using the method of maximum likelihood based on a complete sample.
5.1 Maximum likelihood estimators
Let nXXX ,...,, 21 be a random sample of size n from OGE-MW(Θ), where Θ = (
,,,, ) , then the likelihood function l of this sample is defined as
.),,,,;(1
i
n
i
xfl
(13)
Substituting from (3) into (13), we get
.11
111
1
n
i
eexx
i
ixixixix
ii eeexl
The log-likelihood function, L, becomes:
Odd generalized exponential modified Weibull distribution 951
.1ln1
1lnlnln
1
1
111
1
n
i
e
n
i
xxn
i
ii
n
i
i
ixix
ii
e
exxxnnL
(14)
The maximum likelihood estimates of the parameters are obtained by
Differentiating the log-likelihood function L, with respect to the parameters
,,,, and setting the result to zero.
,0,,,,
1,,,ln11,,,
11
n
i i
in
i
ix
xx
nL
(15)
,0
,,,,
,,,1,,,
1
1111
1
n
i i
iin
i
ii
n
i i
n
i
ix
xxxx
xx
L
(16)
,0
,,,,
,,,1,,,
1111
1
1
n
i i
iin
i
ii
n
i i
in
i
ix
xxxx
x
xx
L
(17)
,0
,,,,
,,,ln1
,,,lnln
ln
1
111
11
1
n
i i
iii
n
i
iii
n
i i
iiin
i
ii
x
xxx
xxxx
xxxxx
L
(18)
and
,01ln1
1
n
i
e ixix
enL
(19)
Where the nonlinear functions ,,,ix and ,,,,ix are given by
,,,,
ii xx
i ex
1,,,,1
ixix
e
i ex .
952 Yassmen Y. Abdelall
From equation (19), we obtain the maximum likelihood estimate of in a closed
form as follow
.
1ln1
1
n
i
e ixix
e
n
(20)
Substituting from (20) into (15), (16), (17) and (18), we get the MLEs of
,,,, by solving the following system of non-linear equations
,0
,,,,
1,,,
ln11,,,11
n
ii
in
i
i
x
x
xn
,0
,,,,
,,,
1,,,1
111 11
n
ii
iin
i
ii
n
ii
n
i
i
x
xx
xx
x
x
,0
,,,,
,,,
1,,,111 1
1
1
n
ii
iin
i
ii
n
ii
in
i
i
x
xx
xx
x
xx
,0
,,,,
,,,ln
1
,,,lnln
ln
1
11 1
11
1
n
ii
iii
n
i
iii
n
ii
iiin
i
ii
x
xxx
xxx
x
xxxxx
where ,,,,
ii xx
i ex and
1,,,,
1
ixixe
i ex . These
equations cannot be solved analytically and statistical software can be used to
solve the equations numerically. We can use iterative techniques such as Newton
Raphson type algorithm to obtain the estimate
.
Odd generalized exponential modified Weibull distribution 953
5.2 Asymptotic confidence bounds
In this subsection, we derive the asymptotic confidence intervals of the unknown
parameters ,,,, and . As the sample size n , then
),,,,(
approaches a multivariate normal vector with
zero means and covariance matrix ,1
0
I where 1
0
I is the inverse of the
observedinformation matrix which defined as follows
1
2
2
2
2
2
2
2
2
2
2
2
2
22
2
222
222
2
22
2222
2
2
1
0
L
L
L
L
L
L
L
L
L
L
LLLLL
LLLLL
LLLLL
I
Var
Var
Var
Var
Var
,cov
,cov,cov
,cov
,cov
,cov
,cov
,cov
,cov,cov,cov,cov
,cov,cov,cov,cov
,cov,cov,cov,cov
(21)
The second partial derivatives included in 1
0
I are given as follows
,22
2
nL
,
,,,,
,,,
1
2
n
ii
ii
x
xxL
954 Yassmen Y. Abdelall
,
,,,,
,,,ln
1
2
n
ii
iii
x
xxxL
,
,,,,
,,,
1
2
n
ii
ii
x
xxL
,
,,,,
1,,,
1
2
n
ii
i
x
xL
,
,,,,
1,,,
11
2
1,,,2
22
2
n
i
i
x
i
x
exnL
i
1,,,1
2
n
i
ii xxL
,
,,,,
1,,,
1
,,,,
,,,
1
1,,,
n
ii
x
i
i
ii
x
ex
x
xxi
1,,,1
2
n
i
ii xxL
,
,,,,
1,,,
1
,,,,
,,,
1
1,,,
n
i
i
x
i
i
ii
x
ex
x
xxi
Odd generalized exponential modified Weibull distribution 955
1,,,ln1
2
n
i
iii xxxL
,
,,,,
1,,,
1
,,,,
,,,ln
1
1,,,
n
i
i
x
i
i
iii
x
ex
x
xxxi
1,,,1
2
1
2
1
1
2
2
n
i
ii
n
i i
i xxx
xL
,
,,,,
,,,
1
,,,,
,,,
1
1,,,2
n
i
i
x
i
i
ii
x
ex
x
xxi
1,,,1
1
121
12
n
i
ii
n
ii
i xxx
xL
,
,,,,
,,,
1
,,,,
,,,
1
1,,,1
n
i
i
x
i
i
ii
x
ex
x
xxi
1,,,ln1
lnln1
ln1
1
111
1
1
12
n
i
iiii
n
i
ii
n
i i
ii
i
i
i
xxxx
xxx
xxx
x
xL
,
,,,,
,,,
1
,,,,
,,,ln
1
1,,,
n
i
i
x
ii
i
i
iii
x
exx
x
x
xxxi
956 Yassmen Y. Abdelall
1,,,1
1
2
1212
2
n
i
ii
n
ii
xxx
L
,
,,,,
,,,
,,,,
,,,
1
1,,,
n
i
i
x
ii
i
i
ii
x
exx
x
x
xxi
,,,lnln11
1
121
12
i
n
i
ii
n
i
i
i
i xxxxx
xL
,
,,,,
,,,
1
,,,,
,,,ln
11
1,,,1
n
i
i
x
i
i
iii
x
ex
x
xxxi
1,,,ln1
lnln1
lnln2
1
2
1
2
11
2
2
1
1
2
2
n
i
iiii
n
i
ii
n
i i
i
ii
i
i
xxxx
xxx
xxx
x
xL
,
,,,,
,,,
1
,,,,
,,,ln
1
1,,,2
n
i
i
x
ii
i
i
iii
x
exx
x
x
xxxi
where ,,,,
ii xx
i ex and
1,,,,
1
ixixe
i ex .
The asymptotic 1100 % confidence intervals of ,,,, and are
,2
Varz ,2
Varz ,2
Varz ,2
Varz and
Odd generalized exponential modified Weibull distribution 957
Varz2
respectively, where 2
z is the upper
2
th percentile of the
standard normal distribution.
6. Data Analysis
In this section, we perform an application to real data to illustrate that the OGE-
MW can be a good lifetime model, comparing with many known distributions
such as the Exponential (E), Generalized Exponential (GE), Linear Failure Rate
(LFR), and Weibull (W). Consider the data have been obtained from Aarset [1],
and widely reported in many literatures. It represents the lifetimes of 50 devices,
and also, possess a bathtub-shaped failure rate property, Table 1.
Table 1: The data from Aarset [1].
0.1 0.2 1 1 1 1 1 2 3 6 7 11
12 18 18 18 18 18 21 32 36 40 45 46
47 50 55 60 63 63 67 67 67 67 72 75
79 82 82 83 84 84 84 85 85 85 85 85
86 86
The maximum likelihood estimates (MLEs) of the unknown parameters for the
five models is given in Table 2.
Table 2.MLEs of the parameters
MLE of the parameter(s) The Model
022.0
)(E
901.0,021.0
),( GE
4104.2,014.0
),( LFR
949.0,022.0
),( W
946.0,02.0,012.0,448.0,793.0
),,,,( OGEMW
The values of log-likelihood functions (-L), Akaike Information Criteria (AIC),
Bayesian Information Criteria (BIC), and the Consistent Akaike Information
Criteria (CAIC) are given in Table 3 for the five models in order to verify which
distribution fits better to these data.
958 Yassmen Y. Abdelall
Table 3.The -L, AIC, BIC, and CAIC for devices data.
The Model
Measures
L AIC BIC CAIC
E 241.090 484.179 486.091 484.263
GE 240.386 484.771 488.595 485.026
LFR 238.064 480.128 483.952 480.383
W 241.002 486.004 489.828 486.259
OGE-MW 233.393 476.786 486.346 478.150
Based on Table 2and 3, it is shown that OGE-MW ),,,,( model provide
better fit to the data rather than other distributions which we compared with
because it has the smallest value of AIC, BIC, CAIC.
Substituting the MLEs of the unknown parameters ),,,,( into (21), we get
estimation of the variance covariance matrix as the following:
6
5
5
4
8
6
5
4
5
4
86856
54543
5463
1
0
10976.4
10968.5
10968.5
10044.1
10766.3
10914.5
1014.2
10568.2
10851.7
10047.9
10766.310914.510446.210713.210623.7
1014.210568.210713.21052.210239.3
10851.710047.910623.710239.3014.0
I
The approximate 95% two sided confidence intervals of the unknown parameters
,,,, and are [0.714,1.178], [0,0.043], [0,0.02], [0.428,0.468], and
[0.789,0.797], respectively.
7. Conclusions
In this paper, we have introduced a new five-parameter model called odd
generalized exponential modified Weibull (OGE-MW) distribution and studied its
different properties. Some statistical properties of this distribution have been
derived and discussed. We provide the pdf, the cdf, and the hazard rate function
for the new model also we provide an explicit expression for the moments. The
distributions of the order statistics are discussed. Both point and asymptotic
confidence interval estimates of the parameters are derived using maximum
likelihood method and we obtained the observed fisher information matrix. We
use application on set of real data to compare the OGE-MW with other known
distributions such as Exponential (E), Generalized Exponential (GE), Linear
Failure Rate (LFR), and Weibull (W). Applications on set of real data showed that
the OGE-MW is the best distributionfor fitting these data sets compared with
other distributions considered in this article.
Odd generalized exponential modified Weibull distribution 959
References
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Reliability, 36 (1987), no. 1, 106-108.
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Received: July 22, 2016; Published: October 3, 2016