The Marshall-Olkin Flexible Weibull Extension
Distribution
arXiv:1609.08997v1 [] 25 Sep 2016

Abdelfattah Mustafa, B. S. El-Desouky and Shamsan AL-Garash
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Abstract
This paper introduces a new generalization of the flexible Weibull distribution with
three parameters this model called the Marshall-Olkin flexible Weibull extension (MOFWE) distribution which exhibits bathtub-shaped hazard rate. We studied it’s statistical
properties include, quantile function skewness and kurtosis, the mode, rth moments and
moment generating function and order statistics. We used the method of maximum likelihood for estimating the model parameters and the observed Fisher’s information matrix is
derived. We illustrate the usefulness of the proposed model by applications to real data.
Keywords: Weibull distribution; flexible Weibull extension distribution; Marshall-Olkin flexible Weibull;
maximum likelihood estimation.

1

Introduction

The Weibull distribution (WD) introduced by Weibull [20], is a popular distribution for modeling lifetime
data where the hazard rate function is monotone. Recently appeared new classes of distributions were
based on modifications of the Weibull distribution (WD) to provide a good fit to data set with bathtub
hazard failure rate Xie and Lai [18]. Among of these, modified Weibull distribution (MWD), Lai et
al. [8] and Sarhan and Zaindin [14], moreover the beta-Weibull distribution (BWD) has been derived
by Famoye et al. [6], beta modified Weibull distribution (BMWD), Silva et al. [17] and Nadarajah et
al. [12]. Recently, there are many generalization of the WD like a Kumaraswamy Weibull distribution
(KWD), Cordeiro et al. [5], generalized modified Weibull distribution (GMWD), Carrasco et al. [4] and
exponentiated modified Weibull extension distribution (EMWED), Sarhan and Apaloo [15].
The flexible Weibull distribution (FWED), Bebbington et al. [3] has a wide range of applications including life testing experiments, reliability analysis, applied statistics and clinical studies. The origin and
other aspects of this distribution can be found in [3].
A random variable X is said to have the flexible Weibull Extension distribution with parameters α, β > 0
if it’s probability density function (pdf) is given by


n
o
β
β
β
g(x) = α + 2 eαx− x exp −eαx− x , x > 0,
(1.1)
x
while the cumulative distribution function (cdf) is given by

1

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Abdelfattah Mustafa, B. S. El-Desouky and Shamsan AL-Garash

n
o
β
G(x) = 1 − exp −eαx− x ,

x > 0.

(1.2)

The survival function is given by the equation
o
n
β
S(x) = exp −eαx− x ,
the hazard rate function is

x > 0,

(1.3)



β
β
h(x) = α + 2 eαx− x ,
x

and the reversed hazard rate function is


α+

r(x) =

(1.4)

n
o
β
β
eαx− x exp −eαx− x
n
o
.
β
1 − exp −eαx− x

β
x2



(1.5)

In this paper we present a new generalization of the flexible Weibull extension distribution called the
Marshall-Olkin flexible Weibull extension distribution (MO-FWED). By using the Marshall and Olkin’s
[11] method for adding a new parameter to an existing distribution, this new generalized referred to as
the Marshall-Olkin flexible Weibull extension distribution.
Marshall and Olkin [11] proposed a new family of distributions called the Marshall-Olkin extended
(MOE) family by adding a new parameter to the baseline distribution. They defined a new survival
function SM O (x) by introducing the additional shape parameter θ such that θ > 0 and θ̄ = 1 − θ.
Marshall and Olkin called the parameter θ the tilt parameter and they interpreted θ in terms of the
behavior of the hazard rate function of SM O (x). Their ratio is increasing in x for θ ≥ 1 and decreasing
in x for 0 < θ < 1. They consider for any arbitrary continuous distribution called baseline distribution
having cumulative distribution function G(x, ϕ) with the related probability density function pdf g(x, ϕ),
then the cumulative distribution function of the Marshall Olkin (MO) family of distribution is given by
FM O (x) =

G(x, ϕ)
,
1 − θ̄S(x, ϕ)

−∞ < x < ∞,

(1.6)

where θ > 0 and θ̄ = 1 − θ.
The probability density function corresponding to Eq.(1.6) becomes
θg(x, ϕ)
fM O (x) = 
2 ,
1 − θ̄S(x, ϕ)

−∞ < x < ∞.

(1.7)

The survival function, hazard rate function, reversed hazard rate function and cumulative hazard rate
function of the Marshall-Olkin (MO) family of a probability distribution are given by

SM O (x) =
hM O (x) =
rM O (x) =

θS(x, ϕ)
,
1 − θ̄S(x, ϕ)
h(x, ϕ)
,
1 − θ̄S(x, ϕ)
θr(x, ϕ)
,
1 − θ̄S(x, ϕ)

(1.8a)
(1.8b)
(1.8c)


HM O (x) = − log (SM O (x)) = − log

θS(x, ϕ)
1 − θ̄S(x, ϕ)


,

(1.8d)

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Abdelfattah Mustafa, B. S. El-Desouky and Shamsan AL-Garash

respectively, where θ > 0, θ̄ = 1 − θ.
This paper is organized as follows, we define the cumulative distribution, probability density and hazard
functions of the Marshall-Olkin flexible Weibull extension distribution (MO-FWED) in Section 2. In
Sections 3 and 4, we introduced the statistical properties include, quantile function, the mode, skewness
and kurtosis, rth moments and moment generating function. The distribution of the order statistics is
expressed in Section 5. The maximum likelihood estimation of the parameters is determined in Section
6. Real data sets are analyzed in Section 7 and the results are compared with existing distributions. The
conclusions are introduced in Section 8.

2

Marshall-Olkin Flexible Weibull Extension Distribution

In this section, we studied the three parameters Marshall-Olkin flexible Weibull extension distribution.
Substituting from Eqs. (1.2) and (1.3) into Eq. (1.6), the cumulative distribution function of the MarshallOlkin flexible Weibull extension distribution (MO-FWE) is given by
F (x; α, β, θ) =

1 − e−e

β
αx− x
β
αx− x

, x > 0, α, β, θ > 0.

(2.1)

1 − (1 − θ)e−e

Substituting from Eqs. (1.1) and (1.3) in Eq. (1.7), the pdf corresponding to Eq. (2.1) is given by


β
β
αx−
θ α + xβ2 eαx− x e−e x
f (x; α, β, θ) = 
2 , x > 0, α, β, θ > 0.
β
αx− x
−e
1 − (1 − θ)e

(2.2)

The survival function, hazard rate function, reversed-hazard rate function and cumulative hazard rate
function of X ∼ MO-FWED(α, β, θ) are given by
S(x; α, β, θ) =

h(x; α, β, θ) =

θe−e

β
αx− x
αx−

β

,

(2.3a)

β

,

(2.3b)

1 − (1 − θ)e−e x


β
α + xβ2 eαx− x
αx−
θ)e−e x

1 − (1 −


β
β
αx−
α + xβ2 eαx− x e−e x

,
r(x; α, β, θ) = 
β
β
αx−
αx−
1 − (1 − θ)e−e x
1 − e−e x


β
αx− x
−e
θe

H(x; α, β, θ) = − log 
β
αx− x
−e
1 − (1 − θ)e

(2.3c)

(2.3d)

respectively, x > 0 and α, β, θ > 0.
Figures (1–6) display the cdf, pdf, survival function, hazard rate function, reversed hazard rate function
and cumulative hazard rate function of the MO-FWED(α, β, θ) for some parameter values.

Abdelfattah Mustafa, B. S. El-Desouky and Shamsan AL-Garash

Figure 1: The cdf of MO-FWED for different values of parameters.

Figure 2: The pdf of MO-FWED for different values of parameters.

Figure 3: The survival function of MO-FWED for different values of parameters.

4

Abdelfattah Mustafa, B. S. El-Desouky and Shamsan AL-Garash

Figure 4: The hazard rate function of MO-FWED for different values of parameters.

Figure 5: The reversed hazard rate function of MO-FWED for different values of parameters.

Figure 6: The cumulative hazard rate function of MO-FWED for different values of parameters.

5

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Abdelfattah Mustafa, B. S. El-Desouky and Shamsan AL-Garash

3

Statistical Properties

In this section, we study the statistical properties for the MO-FWE distribution, specially quantile and
simulation, median, skewness, kurtosis and moments.

3.1

Quantile and simulation

The quantile xq of the MO-FWE(α, β, θ) random variable is given by
F (xq ; α, β, θ) = q,

0 < q < 1.

(3.1)

Using the cumulative distribution function of the MO-FWE distribution, from (2.1) in Eq. (3.1), we have
αx2q − k(q)xq − β = 0,
where



k(q) = ln − ln

1−q
1 − (1 − θ)q

(3.2)

.

(3.3)

So, the simulation of the MO-FWE random variable is straightforward. Let U be a uniform variate on
the unit interval (0, 1), thus, by means of the inverse transformation method, we consider the random
variable X given by
p
k(u) ± k(u)2 + 4αβ
.
(3.4)
X=
2α
Since the median is 50% quantile then by setting q = 0.5 in Eq. (3.2), the median M of the MO-FWED
can be obtained the median .

3.2

The Mode of MO-FWE

In this subsection, we will derive the mode of the MO-FWED(α, β, θ) by deriving its pdf with respect to
x and equal it to zero thus the mode of the MO-FWED(α, β, θ) can be obtained as a nonnegative solution
of the following nonlinear equation


β

1 − (1 − θ)e

−eαx− x

h


β
2 i
2
β
2
2
−eαx− x
−2βx + αx + β
− αx + β
1 + (1 − θ)e
eαx− x = 0.

(3.5)
From Figure 2, the pdf for MO-FWED has only one peak, It is a unimodal distribution, so the above
equation has only one solution. It is not possible to get an explicit solution of Eq.3.5 in the general case.
Numerical methods should be used such as bisection or fixed-point method to solve it.

3.3

The Skewness and Kurtosis

The analysis of the variability Skewness and Kurtosis on the shape parameters α, β, θ can be investigated
based on quantile measures. The short comings of the classical Kurtosis measure are well-known. The
Bowely’s skewness based on quartiles is given by, Kenney and Keeping [7],
Sk =

q(0.75) − 2q(0.5) + q(0.25)
,
q(0.75) − q(0.25)

(3.6)

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Abdelfattah Mustafa, B. S. El-Desouky and Shamsan AL-Garash

and the Moors Kurtosis based on quantiles, Moors [10],
Ku =

q(0.875) − q(0.625) − q(0.375) + q(0.125)
,
q(0.75) − q(0.25)

(3.7)

where q(.) represents quantile function.

3.4

The Moments

Now in this subsection, we derive the rth moment for MO-FWED. Moments are important in any statistical analysis, especially in applications. It can be used to study the most important features and
characteristics of a distribution (e.g. tendency, dispersion, skewness and kurtosis).
Theorem 3.1. If X has MO-FWED (α, β, θ), then the rth moments of random variable X, is given by
the following


∞ X
∞ X
∞
X
(−1)i+j (k + 1)j+1 β i θ(1 − θ)k Γ(r − i + 1)
0
µr =
+
βΓ(r
−
i
−
1)
.
(3.8)
i!j!(j + 1)r−2i−1 αr−i−1
α(j + 1)2
k=0 j=0 i=0

Proof. The rth moment of the random variable X with probability density function f (x) is given by
Z ∞
0
µr =
xr f (x; α, β, θ)dx.
(3.9)
0

Substituting from Eq. (2.2) into Eq. (3.9) we get




Z ∞
β −2
β
β
0
αx− β
r
−eαx− x
−eαx− x
x
µr =
x θ α+ 2 e
1 − (1 − θ)e
dx.
e
x
0
αx−

β

Since 0 < (1 − θ)e−e x < 1 for x > 0 we can use the binomial series expansion of


β −2
αx− x
−e
1 − (1 − θ)e
yields


∞
β −2
β
X
αx−
−eαx− x
1 − (1 − θ)e
=
(k + 1)(1 − θ)k e−ke x ,
k=0

then we get
Z
∞
X
k
µr =
(k + 1)θ(1 − θ)

∞

0

r



x

0

k=0
β
αx− x

using series expansion of e−(k+1)e



β
αx− x

β

eαx− x e−(k+1)e

dx,

,
β

e

β
α+ 2
x

−(k+1)eαx− x

=

∞
X
(−1)j (k + 1)j

j!

j=0

β

ej(αx− x ) ,

we obtain
0

µr =

Z
∞ X
∞
X
(−1)j (k + 1)j+1 θ(1 − θ)k
k=0 j=0

j!

0

∞


β
xr α + βx−2 e(j+1)αx e−(j+1) x dx,

8

Abdelfattah Mustafa, B. S. El-Desouky and Shamsan AL-Garash

β

using series expansion of e−(j+1) x ,
β

e−(j+1) x =

∞
X
(−1)i (j + 1)i β i

i!

i=0

x−i ,

we obtain
0

µr =
=

Z
∞ X
∞ X
∞
X
(−1)i+j (k + 1)j+1 (j + 1)i β i θ(1 − θ)k
k=0 j=0 i=0
∞ X
∞ X
∞
X
k=0 j=0 i=0

i!j!

∞


xr−i α + βx−2 e(j+1)αx dx,

0

(−1)i+j (k + 1)j+1 (j + 1)i β i θ(1 − θ)k
×
i!j!
Z ∞

Z ∞
r−i (j+1)αx
r−i−2 (j+1)αx
αx e
dx +
βx
e
dx ,
0

0

by using the definition of gamma function ( Zwillinger [19]), in the form,
Z ∞
z
Γ(z) = x
tz−1 ext dt, z, x > 0.
0

Finally, we obtain the rth moment of MO-FWE distribution in the form
0

µr =

∞ X
∞ X
∞
X
(−1)i+j (k + 1)j+1 (j + 1)i β i θ(1 − θ)k

i!j!

k=0 j=0 i=0

×




Γ(r − i + 1)
βΓ(r − i − 1)
+
αr−i (j + 1)r−i+1 αr−i−1 (j + 1)r−i−1

∞
∞
∞
X X X (−1)i+j (k + 1)j+1 β i θ(1 − θ)k  Γ(r − i + 1)
=
+ βΓ(r − i − 1) .
i!j!(j + 1)r−2i−1 αr−i−1
α(j + 1)2
k=0 j=0 i=0

This completes the proof.

4

The Moment Generating Function

The moment generating function (mgf), MX (t), of a random variable X provides the basis of an alternative route to analytic results compared with working directly with the pdf and cdf of X.
Theorem 4.1. The moment generating function (mgf) of MO-FWE distribution is given by


∞
∞ X
∞ X
∞ X
X
(−1)i+j (k + 1)j+1 β i θ(1 − θ)k tr Γ(r − i + 1)
MX (t) =
+ βΓ(r − i − 1) . (4.1)
r!i!j!(j + 1)r−2i−1 αr−i−1
α(j + 1)2
r=0 k=0 j=0 i=0

Proof. The moment generating function of the random variable X with probability density function f (x)
is given by
Z
∞

etx f (x)dx,

MX (t) =
0

(4.2)

9

Abdelfattah Mustafa, B. S. El-Desouky and Shamsan AL-Garash

using series expansion of etx , we obtain
MX (t) =

∞ r Z
X
t
r=0

r!

∞

xr f (x)dx =

0

∞ r
X
t
r=0

r!

0

µr .

(4.3)

Substituting from Eq. (3.8) into Eq. (4.3), we obtain the moment generating function of MO-FWED in
the following form


∞ X
∞ X
∞ X
∞
X
(−1)i+j (k + 1)j+1 β i θ(1 − θ)k tr Γ(r − i + 1)
MX (t) =
+ βΓ(r − i − 1) .
r!i!j!(j + 1)r−2i−1 αr−i−1
α(j + 1)2
r=0 k=0 j=0 i=0

This completes the proof.

5

Order Statistics

In this section, we derive closed form expressions for the probability density function of the rth order
statistic of the MO-FWED. Let X1:n , X2:n , · · · , Xn:n denote the order statistics obtained from a random
sample X1 , X2 , · · · , Xn which taken from a continuous population with cumulative distribution function
F (x; ϕ) and probability density function f (x; ϕ), then the probability density function of Xr:n is given
by
fr:n (x; ϕ) =

1
[F (x; ϕ)]r−1 [1 − F (x; ϕ)]n−r f (x; ϕ),
B(r, n − r + 1)

(5.1)

where f (x; ϕ), F (x; ϕ) are the pdf and cdf of MO-FWED (α, β, θ) given by Eqs. (2.2) and (2.1) respectively, ϕ = (α, β, θ) and B(., .) is the beta function, also we define first order statistics X1:n =
min(X1 , X2 , · · · , Xn ), and the last order statistics as Xn:n = max(X1 , X2 , · · · , Xn ). Since 0 <
F (x; ϕ) < 1 for x > 0, we can use the binomial expansion of [1 − F (x; ϕ)]n−r given as follows.
n−r
X n − r 
n−r
[1 − F (x; ϕ)]
=
(−1)i [F (x; ϕ)]i .
(5.2)
i
i=0

Substituting from Eq. (5.2) into Eq. (5.1), we obtain
fr:n (x; ϕ) =

n−r
X n − r
1
f (x; ϕ)
(−1)i [F (x; ϕ)]i+r−1
B(r, n − r + 1)
i
i=0

=

n−r
X
i=0

(−1)i n!
i!(r − 1)!(n − r − i)!

[F (x, ϕ)]i+r−1 f (x; ϕ).

(5.3)

Substituting from Eqs. (2.1) and (3.2) into Eq. (5.3), we obtain pdf of rth order statistics for MOFWED(α, β, θ).
Relation (5.3), shows that fr:n (x; ϕ) is the weighted average of the Marshall Olkin flexible Weibull
extension MO-FWED withe different shape parameters.

6

Parameters Estimation

In this section, point and interval estimation of the unknown parameters of the MO-FWED are derived
by using the method of maximum likelihood based on a complete sample.

10

Abdelfattah Mustafa, B. S. El-Desouky and Shamsan AL-Garash

6.1

Maximum likelihood estimation

Let x1 , x2 , · · · , xn denote a random sample of complete data from the MO-FWED. The likelihood function is given as
n
Y
L=
f (xi ; α, β, θ),
(6.1)
i=1

substituting from Eq. (2.2) into Eq. (6.1), we have
L=



n
β 
β −2
Y
αxi − x
αxi − x
β
αx − β
i
i
1 − (1 − θ)e−e
.
θ α + 2 e i xi e−e
x
i
i=1

The log-likelihood function is
 X
 X


n
n
n 
β 
X
αxi − x
β
β
αxi − xβ
i
−e
i −2
−
L = n ln(θ) +
αxi −
e
ln 1 − (1 − θ)e
.
ln α + 2 +
x
x
i
i
i=1
i=1
i=1
i=1
(6.2)
The maximum likelihood estimation of the parameters are obtained by differentiating the log-likelihood
function, L, with respect to the parameters α, β and θ and setting the result to zero, we have the following
normal equations.
n
X

∂L
∂α
∂L
∂β

β

n
n
n
αx −
X
X
X
x2i
(1 − θ)xi e i xi
αxi − xβ
i
=
= 0,
+
+2
xi −
xi e
β
αxi − x
β + αx2i
i
i=1
i=1
i=1 1 − θ − ee
i=1
n
n
n
n
αx − β
X
X
X
X
1
1
1 αxi − xβ
(1 − θ)e i xi
i − 2
 = 0,

=
−
+
e
β
αxi − x
x
x
β + αx2i
i
i=1
i=1 i
i=1 i
i=1 xi 1 − θ − ee
n
X

(6.3)

(6.4)
∂L
∂θ

=

n
+2
θ

n
X
i=1

1
1−θ−

ee

β
αxi − x
i

= 0.

(6.5)

The MLEs can be obtained by solving the nonlinear equations previous, (6.3)–(6.5), numerically for α, β
and θ.

6.2

Asymptotic confidence bounds

In this section, we derive the asymptotic confidence intervals when α, β > 0 and θ > 0 as the MLEs
of the unknown parameters α, β > 0 and θ > 0 can not be obtained in closed forms, by using variance
covariance matrix I −1 see Lawless [9], where I −1 is the inverse of the observed information matrix
which defined as follows.

−1 
2

∂2L
∂2L
− ∂∂αL2 − ∂α∂β
− ∂α∂θ
var(α̂) cov(α̂, β̂) cov(α̂, θ̂)
2

∂2L
∂2L 
− ∂∂βL2 − ∂β∂θ
I−1 =  − ∂β∂α
(6.6)
 =  cov(β̂, α̂) var(β̂) cov(β̂, θ̂)  .
∂2L
∂2L
∂2L
cov(θ̂, α̂) cov(θ̂, β̂) var(θ̂)
− ∂θ∂α − ∂θ∂β − ∂θ2

11

Abdelfattah Mustafa, B. S. El-Desouky and Shamsan AL-Garash

where
∂2L
∂α2

= −

n
X

x4i

n
X

n
X

x2i

n
X

i=1

β + αx2i

i=1

∂2L
∂α∂β

= −

∂2L

= 2

∂α∂θ

n
X

x i Ai e

i=1
n
X

∂2L

= −

∂β 2

i=1
n
X

∂2L

2 −
β + αx2i
2 +

x2i e

αxi − xβ

i

+ 2(1 − θ)

i=1

n
X

x2i Bi

(6.7)

i=1

e

αxi − xβ
i

− 2(1 − θ)

n
X

Bi

(6.8)

i=1

i=1

β
αx −
e i xi

(6.9)

1

2 −
β + αx2i

n
n
X
X
1 αxi − xβ
Bi
i + 2(1 − θ)
e
2
2
x
x
i=1 i
i=1 i

(6.10)

β

αxi − x
1
i
Ai ee
= −2
xi
i=1
n 
β −2
X
αx −
n
e i xi
.
= − 2 −2
1−θ−e
θ

∂β∂θ
∂2L
∂θ2

(6.11)

(6.12)

i=1

where
Ai = e

αxi − xβ

i



e

1−θ−e

β
αxi − x
i

−2


and Bi = Ai 1 − θ − e

β
αxi − x
i

e

αxi − xβ

(1 − e

i


) .

We can derive the (1 − δ)100% confidence intervals of the parameters α, β and θ, by using variance
matrix as in the following forms
q
q
p
α̂ ± Z δ var(α̂), β̂ ± Z δ var(β̂), θ̂ ± Z δ var(θ̂),
2

2

2

where Z δ is the upper ( 2δ )-th percentile of the standard normal distribution.
2

7

Application

In this section, we present the analysis of two examples for a real data sets using the MO-FWE (α, β, θ)
model and compare it with the other fitted models like a flexible Weibull extension (FWE), Weibull (W),
linear failure rate (LFR), exponentiated Weibull (EW), generalized linear failure rate (GLFR), exponentiated flexible Weibull (EFW), modified Weibull (MW), reduced additive Weibull (RAW) and Extended
Weibull (EW) distributions using Kolmogorov Smirnov (K-S) statistic, as well as Akaike Information
Criterion (AIC), [2], Akaike Information Citerion with correction (AICC), Bayesian Information Criterion (BIC) and Hannan-Quinn information criterion (HQIC) [16] values.
Example 7.1. 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: Lifetime of 50 devices, see 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

12

Abdelfattah Mustafa, B. S. El-Desouky and Shamsan AL-Garash

Table 2 gives MLEs of parameters of the MO-FWED and K–S Statistics. The values of the log-likelihood
functions, AIC, AICC, BIC and HQIC are presented in Table 3.
Table 2: MLEs and K–S of parameters for Aarset data [1].
Model
MLE of the parameters
K-S
Flexible Weibull
α̂ = 0.0122, β̂ = 0.7002
0.4386
Weibull
α̂ = 0.0223, β̂ = 0.949
0.2397
Linear Failure rate
â = 0.014, b̂ = 2.4 ×10−4
0.1955
Exponentiated Weibull
α̂ = 0.0109, β̂ = 4.69, γ̂ = 0.164
0.1841
Generalized Linear Failure rate â = 0.0038, b̂ = 3.04 ×10−4 , ĉ = 0.533 0.1620
Exponentiated Flexible Weibull α̂ = 0.0147, β̂ = 0.133, θ̂ = 4.22
0.1433
MO-FWE(α, β, θ )
α̂ = 0.017, β̂ = 0.401, θ̂ = 9.043
0.1269

P-value
4.29 ×10−9
0.0052
0.0370
0.0590
0.1293
0.2617
0.3756

Table 3: Log-likelihood, AIC, AICC, BIC and HQIC values of models fitted for Aarset data [1].
Model
L
-2 L
AIC
AICC
BIC
HQIC
FW(α, β)
-250.810 501.620 505.620 505.88 509.448 507.0762
W(α, β )
-241.002 482.004 486.004 486.26 489.828 487.4602
LFR (a, b)
-238.064 476.128 480.128 480.38 483.952 481.5842
EW(α, β, γ )
-235.926 471.852 477.852 478.37 483.588 480.0363
GLFR(a, b, c)
-233.145 466.290 472.290 472.81 478.026 474.4743
EFW(α, β, θ )
-226.989 453.978 459.979 460.65 465.715 462.1623
MO-FWE(α, β, θ ) -223.755 447.510 453.510 454.03 459.246 455.6943
We find that the MO-FWE distribution with three parameters provides a better fit than the previous models flexible Weibull (FW), Weibull (W), linear failure rate (LFR), exponentiated Weibull (EW), generalized linear failure rate (GLFR) and exponentiated flexible Weibull (EFW). It has the largest likelihood,
and the smallest K-S, AIC, AICC, BIC and HQIC values among those considered in this paper.
Substituting the MLE’s of the unknown parameters α, β and θ into (6.6), we get estimation of the variance
covariance matrix as the following


1.523 × 10−6 −1.782 × 10−5 2.177 × 10−3
0.022
−0.061 
I −1 =  −1.782 × 10−5
−3
2.177 × 10
−0.061
8.458
The approximate 95% two sided confidence intervals of the unknown parameters α, β and θ are [0.015, 0.019],
[0.108, 0.694] and [3.343, 14.743], respectively.
To show that the likelihood equation have unique solution, we plot the profiles of the log-likelihood
function of α, β and θ in Figures 7, 8.

Abdelfattah Mustafa, B. S. El-Desouky and Shamsan AL-Garash

13

Figure 7: The profile of the log-likelihood function of α, β for the Aarset data.

Figure 8: The profile of the log-likelihood function of θ for the Aarset data.
The nonparametric estimate of the survival function using the Kaplan-Meier method and its fitted parametric estimations when the distribution is assumed to be MO-FWE, FW, W, LFR, EW, GLFR and EFW
are computed and plotted in Figure 9.

Abdelfattah Mustafa, B. S. El-Desouky and Shamsan AL-Garash

14

Figure 9: The Kaplan-Meier estimate of the survival function for the Aarset(1987) data.
Figures 10 and 11 give the form of the hazard rate and cdf for the MO-FWE, FW, W, LFR, EW, GLFR
and EFW which are used to fit the Aarset(1987) data after replacing the unknown parameters included
in each distribution by their MLE.

Figure 10: The Fitted hazard rate function for the Aarset(1987) data.

Figure 11: The Fitted cumulative distribution function for the Aarset(1987) data.
Example 7.2. The data have been obtained from [13], it is for the time between failures (thousands of
hours) of secondary reactor pumps, Table 4.
Table 4: Time between failures (thousands of hours) of secondary reactor pumps [13]
2.160 0.746 0.402 0.954 0.491 6.560 4.992 0.347
0.150 0.358 0.101 1.359 3.465 1.060 0.614 1.921
4.082 0.199 0.605 0.273 0.070 0.062 5.320

Abdelfattah Mustafa, B. S. El-Desouky and Shamsan AL-Garash

15

Table 5 gives MLEs of parameters of the MO-FWE distribution and K-S Statistics. The values of the
log-likelihood functions, AIC, AICC, BIC and HQIC are in Table 6.
Table 5: MLEs and K–S of parameters for secondary reactor pumps.
Model
α̂
β̂
θ̂
K-S
Flexible Weibull
0.0207 2.5875
–
0.1342
Weibull
0.8077 13.9148
–
0.1173
Modified Weibull
0.1213 0.7924
0.0009 0.1188
Reduced Additive Weibull 0.0070 1.7292
0.0452 0.1619
Extended Weibull
0.4189 1.0212 10.2778 0.1057
MO-FWE
0.2160 0.2350
1.2960 0.0793

Table 6: Log-likelihood, AIC, AICC, BIC and HQIC values of models fitted.
Model
L
-2 L
AIC
AICC
BIC
Flexible Weibull
-83.3424 166.6848 170.6848 171.2848 172.95579
Weibull
-85.4734 170.9468 174.9468 175.5468 177.21779
Modified Weibull
-85.4677 170.9354 176.9354 178.1986 180.34188
Reduced Additive Weibull -86.0728 172.1456 178.1456 179.4088 181.55208
Extended Weibull
-86.6343 173.2686 179.2686 180.5318 182.67508
MO-FWE
-30.2110 60.4220
66.4220
67.6852
69.8285

HQIC
171.2559
175.5179
177.7921
179.0023
180.1253
67.2787

We find that the MO-FWE distribution with the three-number of parameters provides a better fit than
the previous new modified Weibull distributions like a flexible Weibull (FW), Weibull (W), modified
Weibull (MW), reduced additive Weibull (RAW) and extended Weibull (EW) distributions. It has the
largest likelihood, and the smallest K-S, AIC, AICC, BIC and HQIC values among those considered in
this paper.
Substituting the MLE’s of the unknown parameters α, β and θ into (6.6), we get estimation of the variance
covariance matrix as the following


1.996 × 10−3 −7.744 × 10−4 8.987 × 10−3
5.487 × 10−3
−0.022 
I0−1 =  −7.744 × 10−4
−3
8.987 × 10
−0.022
0.326
The approximate 95% two sided confidence intervals of the unknown parameters α, β and θ are [0.128, 0.304],
[0.09, 0.38] and [0.177, 2.415], respectively.
To show that the likelihood equation have unique solution, we plot the profiles of the log-likelihood
function of α, β and θ in Figures 12 and 13.

Abdelfattah Mustafa, B. S. El-Desouky and Shamsan AL-Garash

16

Figure 12: The profile of the log-likelihood function of α and β.

Figure 13: The profile of the log-likelihood function of θ.
The nonparametric estimate of the survival function using the Kaplan-Meier method and its fitted parametric estimations when the distribution is assumed to be MO-FWE, FW, W, MW, RAW and EW are
computed and plotted in Figure 14.

Figure 14: The Kaplan-Meier estimate of the survival function for the data.

Abdelfattah Mustafa, B. S. El-Desouky and Shamsan AL-Garash

17

Figure 15 gives the form of the CDF for the MO-FWE, FW, W, MW, RAW and EW which are used to fit
the data after replacing the unknown parameters included in each distribution by their MLE.

Figure 15: The Fitted cumulative distribution function for the data.

8

Conclusions

A new distribution MO-FWE, it’s generalized of the flexible Weibull extension distribution based on
the Marshall and Olkin’s method, has been proposed and its properties are studied. The idea is to add
parameter to a flexible Weibull extension distribution, so that the hazard function is either increasing or
more importantly, bathtub shaped. Using Marshall and Olkin extended family by adding a new parameter
to the baseline distribution, the distribution has flexibility to model the second peak in a distribution. We
have shown that the Marshall Olkin flexible Weibull extension distribution fits certain well-known data
sets better than existing modifications.

References
[1] Aarset, M. V. (1987). How to identify bathtub hazard rate. IEEE Transactions on Reliability, 36,
106–108.
[2] Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, AC-19, 716–23.
[3] Bebbington, M.S., Lai, C.D. and Zitikis, R. (2007). A flexible Weibull extension. Reliability Engineering & System Safety , 92(6), 719–26.
[4] Carrasco M., Ortega E.M. and Cordeiro G.M. (2008). A generalized modified Weibull distribution
for lifetime modeling. Computational Statistics and Data Analysis, 53(2), 450–62.
[5] Cordeiro, G.M., Ortega, E.M. and Nadarajah, S. (2010). The Kumaraswamy Weibull distribution
with application to failure data. Journal of the Franklin Institute, 347, 1399–429.
[6] Famoye, F., Lee, C. and Olumolade, O. (2005). The beta-Weibull distribution. Journal of Statistical
Theory and Applications, 4(2), 121–36.

Abdelfattah Mustafa, B. S. El-Desouky and Shamsan AL-Garash

18

[7] Kenney, J. and Keeping, E. (1962). Mathematics of Statistics, Volume 1, Princeton.
[8] Lai C.D., Xie, M. and Murthy D.N.P. (2003). A modified Weibull distributions. IEEE Transactions
on Reliability, 52(1), 33–7.
[9] Lawless, J. F. (2003). Statistical Models and Methods for Lifetime Data. John Wiley and Sons, New
York, 20, 1108–1113.
[10] Moors, J.J.A. (1998). A quantile alternative for kurtosis. The Statistician, 37, 25–32.
[11] Marshall, A.N. and Olkin, I. (1997). A new method for adding a parameter to a family of distributions with applications to the exponential and Weibull families, Biometrika 84 (1997), 641-652.
[12] Nadarajah, S., Cordeiro, G.M. and Ortega, E.M.M. (2011). General results for the beta-modified
Weibull distribution. Journal of Statistical Computation and Simulation, 81(10), 1211–32.
[13] Salman Suprawhardana M, Prayoto, Sangadji. (1999). Total time on test plot analysis for mechanical components of the RSG-GAS reactor. Atom Indones, 25(2), 155–61.
[14] Sarhan, A.M. and Zaindin, M. (2009). Modified Weibull distribution. Applied Sciences, 11, 123–
136.
[15] Sarhan, A.M. and Apaloo, J. (2013). Exponentiated modified Weibull extension distribution. Reliability Engineering and System Safety, 112, 137–144.
[16] Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6, 461–4.
[17] Silva, G.O.,Ortega, E.M. and Cordeiro, G.M. (2010). The beta modified Weibull distribution. Lifetime Data Analysis, 16, 409–30.
[18] Xie, M. and Lai, C.D. (1995). Reliability analysis using an additive Weibull model with bathtubshaped failure ratefunction. Reliability Engineering System Safety, 52, 87–93.
[19] Zwillinger, D. (2014). Table of integrals, series and products. Elsevier.
[20] Weibull, W.A. (1951). Statistical distribution function of wide applicability. Journal of Applied
Mechanics, 18, 293–6.

