/compile/output.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 8, No. 1, 2015, 1-14

ISSN 1307-5543 – www.ejpam.com

Exponentiated Transmuted Modified Weibull Distribution

Manisha Pal1,∗, Montip Tiensuwan2

1 Department of Statistics, University of Calcutta, India
2 Department of Mathematics, Faculty of Science, Mahidol University, Thailand

Abstract. The paper introduces the exponentiated transmuted modified Weibull distribution, which

contains a number of distributions as special cases. The properties of the distribution are discussed and

explicit expressions for the quantiles, mean deviations and the reliability are derived. The distribution

and moments of order statistics are also studied. Estimation of the model parameters by the methods

of least squares and maximum likelihood are discussed. Finally, the usefulness of the distribution for

modeling data is illustrated using real data.

2010 Mathematics Subject Classifications: 90B25; 62N05

Key Words and Phrases: Reliability function, moment generating function, quantiles, mean deviation,

order statistics, least square estimation, maximum likelihood estimation.

1. Introduction

Modelling and analysis of lifetime data have become very crucial in different areas of re-

search, like engineering, medicine, reliability, etc. In this regard, it is observed that the Weibull

distribution is extensively used as it is found to provide reasonable fit in many practical situ-

ations. Attempts at generalization of the distribution have led to the exponentiated Weibull

distribution, the modified Weibull distribution and the exponentiated modified Weibull distri-

bution. An interesting idea of generalizing a distribution, which is known in the literature as

transmutation has been used to develop further distributions. A random variable T is said to

have a transmuted distribution if its distribution function is given by

Z(t) = (1+λ)G(t)−λ[G(t)]2, (1)

where G(t) denotes the base distribution, and λ ∈ [-1, 1] denotes the transmuting parameter.

Aryall and Tsokos [2] introduced the transmuted Weibull distribution, Ebraheim [4] studied

the exponentiated transmuted Weibull distribution, Pal and Tiensuwan [7] developed the beta

transmuted Weibull distribution, Khan and King [6] investigated the transmuted modified

∗Corresponding author.

Email addresses: manishapal2@gmail.com (M. Pal), montip.tie@mahidol.ac.th (M. Tiensuwan)

http://www.ejpam.com 1 c© 2015 EJPAM All rights reserved.



M. Pal and M. Tiensuwan / Eur. J. Pure Appl. Math, 8 (2015), 1-14 2

Weibull distribution, and Ashour and Eltehiwy [3] proposed the transmuted exponentiated

modified Weibull distribution.

In this paper, we introduce and study several mathematical properties of a new reliabil-

ity model referred to as the exponentiated transmuted modified Weibull distribution. The

modified Weibull distribution, introduced by Sarhan and Zaindin [8], has the cumulative dis-

tribution function (c.d.f.)

FMW (t) = 1− ex p(−αt − γtβ ), t ≥ 0,α,β ,γ > 0. (2)

The c.d.f. of the transmuted modified Weibull distribution [6] is given by

FT MW (t) = [1− ex p(−αt − γtβ )][1+λex p(−αt − γtβ )], t ≥ 0,α,β ,γ > 0. (3)

The above distribution has three shape parameters α, β and γ, and λ denotes the transmut-

ing parameter. The exponentiated transmuted modified Weibull distribution generalizes this

distribution by introducing another shape parameter.

The paper is organized as follows. In Section 2 we introduce the distribution. In Sections 3,

we obtain the quantile function of the distribution. The moment generating function and the

moments are derived in Sections 4. Mean deviation is discussed in Section 5. Order statistics

and their moments are studied in Sections 6. In Section 7, the stress-strength reliability is

obtained. Estimation of parameters by the least square method and the maximum likelihood

method are discussed in Sections 8 and 9. In Section 10, a simulation study is carried out to

compare the two methods of estimation. The usefulness of the distribution for modeling real

life data is illustrated in Section 11. Finally, in Section 12, we make some concluding remarks

on our study.

2. Exponentiated Transmuted Modified Weibull Distribution

The five parameter exponentiated transmuted modified Weibull (ETMW) distribution is

given by the c.d.f.

F(t) = [1− ex p(−αt − γtβ )1+λex p(−αt − γtβ )]δ, t ≥ 0,α,β ,γ > 0,λ ∈ [−1,1], (4)

where α, β , γ, δ are all shape parameters, and λ is the transmuting parameter.

The density function of the distribution is obtained as

f (t) =δ[{1− ex p(−αt − γtβ )}{1+λex p(−αt − γtβ )}]δ−1(α+ βγtβ−1)ex p(−αt − γtβ )

× [1−λ+ 2λex p(−αt − γtβ )], t ≥ 0.

(5)

The hazard rate and the hazard function of the distribution are as follows:

r(t) =δ[{1− ex p(−αt − γtβ )}1+λex p(−αt − γtβ )]δ−1(αt + βγtβ−1)ex p(−αt − γtβ )

× [1−λ+ 2λex p(−αt − γtβ )][1− {ex p(−αt − γtβ )}(δ){1+λex p(−αt − γtβ )}δ]( − 1), t ≥ 0,

H(t) =− ln[1−λ+ 2λex p(−αt − γtβ )][1− {−ex p(−αt − γtβ )}δ{1+λex p(−αt − γtβ )}δ], t ≥ 0.



M. Pal and M. Tiensuwan / Eur. J. Pure Appl. Math, 8 (2015), 1-14 3

Plots of the p.d.f. and the hazard rate are given in Figure 1. Figure 1a exhibits the diverse

shapes of the exponentiated transmuted modified Weibull density for different choices of the

parameters. Figure 1b shows that for almost all the parameter combinations considered the

distribution exhibits monotonic hazard rate.

(a) Probability density function. (b) Hazard rate function.

Figure 1: Exponentiated Transmuted Modified Weibull Distribution when α= 1.

By proper selection of the model parameters we can get a number of distributions as special

cases as shown below:

Parameters Distribution

δ = 1 Transmuted modified Weibull

δ = 1,α= 0 Transmuted Weibull

δ = 1,β = 1 Transmuted exponential distribution

α= 0 Exponentiated transmuted Weibull

β = 1 Exponentiated transmuted exponential

λ = 0 Exponentiated modified Weibull

λ = 0,α = 0 Exponentiated Weibull

λ = 0,β = 1 Exponentiated exponential

δ = 1,λ = 0 Modified Weibull

δ = 1,λ = 0,β = 2 Linear failure rate distribution

δ = 1,λ = 0,α = 0 Weibull

δ = 1,λ = 0,α= 0,β = 2 Rayleigh

δ = 1,λ = 0,β = 1 Exponential

3. Quantile Function and Simulation

For any random variable X with cumulative distribution function F(·), for a given q,

0≤ q ≤ 1, the quantile function returns the threshold value x such that F(x) = q. This function

is useful in statistical applications and Monte Carlo simulation. For statistical applications, one

needs to know the key percentage points of a given distribution, like the median and the 25%

and 75% quartiles. The quantile function is also helpful in examining the fit of a given data

set to a theoretical distribution. This is done by using the quantile-quantile (Q-Q) plot.



M. Pal and M. Tiensuwan / Eur. J. Pure Appl. Math, 8 (2015), 1-14 4

The quantile function corresponding to the ETMW distribution (4) is given by

tq = F−1(q) = G−1(q1/δ), (6)

where G(x) = [1 − ex p(−αt − γtβ )][1 + λex p(−αt − γtβ )] is the cumulative distribution

function of the transmuted modified Weibull distribution investigated by Khan and King [6].

Thus, tq is the (q1/δ)-th quantile of a transmuted modified Weibull distribution, and, from

Khan and King [6], tq is the real solution to the equation

γtβq +αtq + ln(z∗) = 0, (7)

where

z∗ = 1− (1+λ)−
p

(1+λ)2 − 4λq1/δ

2λ
. (8)

For β = 2, tq is given by

tq =
−α+pα2 − 4γ ln(z∗)

2γ
, (9)

where z∗ is given by (8).

Thus, for β = 2, the median of the distribution has the form

t0.5 =
−α+
q

α2 − 4γ ln[{
p

(1+λ)2 − 22−1/δλ− (1−λ)}/2λ]
2γ

.

In order to simulate from the ETMW distribution, we have to solve for tq from (7) for a random

proportion q. However, for β = 2, simulation is straight forward from (9).

4. Moment Generating Function

We can express the moment generating function M(t∗) of the ETMW distribution in terms

of the moment generating function of the modified Weibull distribution as follows:

M(t∗) =
∫ ∞

0

ex p(t∗ t) f (t)d t

=δ

∞
∑

i=0

∞
∑

j=0

A(i, j;δ,λ)[
1−λ

(i + j + 1)
MMW (t

∗; (i + j + 1)α, (i + j + 1)γ,β)

+
2λ

(i + j + 2)
MMW (t

∗; (i + j + 2)α, (i + j + 2)γ,β)],

where MMW (t
∗;α,γ,β) denotes the moment generating function of a modified Weibull distri-

bution with c.d.f. (2), and

A(i, j;δ,λ) = (−1)i
{Γ(δ− 1)}2

Γ(i)Γ( j)Γ(δ− i − 1)Γ(δ− j − 1)
λ j .



M. Pal and M. Tiensuwan / Eur. J. Pure Appl. Math, 8 (2015), 1-14 5

From Sarhan and Zaindin [8], we have

MMW (t
∗;α,γ,β) =

∞
∑

k=0

(−γ)k
k!

�

αΓ(kβ + 1)

(α− t∗)kβ+1
+
γβΓ(k+ 1)β

(α− t∗)(k+1)β

�

, for α,γ > 0,α > t∗

=

∞
∑

k=0

t∗kΓ( k
β+1)

γ
k
β

for α= 0,γ > 0

=
α

α− t∗
, for γ= 0,α > t∗

Thus,

M(t∗) =δ
∞
∑

i=0

∞
∑

j=0

A(i, j;δ,λ)[
1−λ

(i + j + 1)

∞
∑

k=0

(−(i + j + 1)γ)k

k!
{ (i + j + 1)αΓ(kβ + 1)

((i + j + 1)α− t∗)kβ+1

=+
(i + j + 1)γβΓ(k+ 1)β

((i + j + 1)α− t∗)(k+1)β
+

2λ

(i + j + 2)

∞
∑

k=0

(−(i + j + 2)γ)k

k!
×

{ (i + j + 2)αΓ(kβ + 1)

((i + j + 2)α− t∗)kβ+1
+
(i + j + 2)γβΓ(k+ 1)β

((i + j + 2)α− t∗)(k+1)β
}], for α,γ > 0,α > t∗,

=δ

∞
∑

i=0

∞
∑

j=0

A(i, j;δ,λ)[
1−λ

(i + j + 1)

∞
∑

k=0

t∗kΓ( k
β+1)

{(i + j + 1)γ} k
β

+
2λ

(i + j + 2)

∞
∑

k=0

t∗kΓ( k
β+1)

{(i + j + 2)γ} k
β

, for alpha = 0,γ > 0,

=δ

∞
∑

i=0

∞
∑

j=0

A(i, j;δ,λ)[
1−λ

(i + j + 1)

(i + j + 1)α

{(i + j + 1)α− t∗}

+
2λ

(i + j + 2)

(i + j + 2)α

{(i + j + 2)α− t∗}], for γ= 0,α > t∗ (10)

The moments can be independently derived as follows:

µr =E{X r}

=δ

∞
∑

i=0

∞
∑

j=0

A(i, j;δ,λ)[
1−λ

i + j + 1
µMW

r ((i + j + 1)α, (i + j + 1)γ,β)

+
2λ

i + j + 2
µMW

k
((i + j + 2)α, (i + j + 2)γ,β),

where µMW
r (α,γ,β) denotes the r-th moment of the modified Weibull distribution (2).

From Sarhan and Zaindin [8] we have

µMW
r (α,γ,β) =













∞
∑

k=0

(−β)k
k!

�

Γ(kγ+r+1)

αkγ+r + βγ
Γ(r+kγ+γ)

αkγ+γ+r

�

, for α,β > 0

Γ( r
γ+1)

β
r
γ

, for α= 0,β > 0

Γ(r+1)
αr , for α > 0,β = 0.

(11)



M. Pal and M. Tiensuwan / Eur. J. Pure Appl. Math, 8 (2015), 1-14 6

Hence, we get

µr =δ

∞
∑

i=0

∞
∑

j=0

A(i, j;δ,λ)[
1−λ

i + j + 1

∞
∑

k=0

(−β)k
k!
{ Γ(k(i + j + 1)γ+ r + 1)

{(i + j + 1)α}k(i+ j+1)γ+r

(i + j + 1)βγ
Γ(r + {k(i + j + 1) + 1}γ)
{(i + j + 1)α}{k(i+ j+1)+1}γ+r

}+ 2λ

i + j + 2

∞
∑

k=0

(−β)k
k!

× Γ(k(i + j + 2)γ+ r + 2)

{(i + j + 2)α}k(i+ j+2)γ+r
+ (i + j + 2)βγ

Γ(r + {k(i + j + 2) + 1}γ)
{(i + j + 2)α}{k(i+ j+2)+1}γ+r

}, for α,β > 0

=δ

∞
∑

i=0

∞
∑

j=0

A(i, j;δ,λ)[
1−λ

i + j + 1
(

r

(i + j + 1)γ
+ 1)/β

r
(i+ j+1)γ

+
2λ

i + j + 2
(

r

(i + j + 2)γ
+ 1)/β

r
(i+ j+2)γ , for α= 0,β > 0

=δ

∞
∑

i=0

∞
∑

j=0

A(i, j;δ,λ)[
1−λ

i + j + 1

Γ(r + 1)

{(i + j + 1)α}r +
2λ

i + j + 2

Γ(r + 1)

{(i + j + 2)α}r ], for α > 0,β = 0

(12)

5. Mean Deviation

The amount of scatter in a population is evidently measured to some extent by the totality of

deviations from the mean and the median. If X has a ETMW distribution, then we can derive

the mean deviations about the mean µ= E(X ) and about the median M as

η1 =

∫ ∞

0

| x −µ | f (x)d x ,η2 =

∫ ∞

0

| x −M | f (x)d x .

The mean of the distribution is obtained from (12) by putting r = 1, and the median is obtained

by solving the equation

γMβ +αM = − ln(ν0),

where ν0 is given by

ν0 =
−(1−λ) +
p

(1−λ)2 + 4λ(1− (0.5)1/δ)

2λ
, if λ 6= 0

= 1− (0.5)1/δ, if λ = 0.



M. Pal and M. Tiensuwan / Eur. J. Pure Appl. Math, 8 (2015), 1-14 7

6. Order Statistics

Let T(1) < T(2) < . . .< T(n) be the ordered observations in a random sample of size n drawn

from the exponentiated transmuted modified Weibull distribution with cdf F(t), given by (4)

and density f (t), given by (5).

The pdf of T (r), 1≤ r ≤ n, is given by

f(r)(t) =
n!

(r − 1)!(n− r)!
[F(t)]r−1[1− F(t)]n−r f (t)

=
n!

(r − 1)!(n− r)!
δ[{1− ex p(−αt − γtβ )}{1+λex p(−αt − γtβ}]δr−1

× [1− {1− ex p(−αt − γtβ )}δ{1+λex p(−αt − γtβ )}δ]n−r(αt + βγtβ−1)ex p(−αt − γtβ )

× [1−λ+ 2λex p(−αt − γtβ )], t ≥ 0,α,β ,δ > 0,λ ∈ [−1,1].

Hence the pdf of the smallest and the largest order statistics are as follows:

f(1)(t) =nδ[{1− ex p(−αt − γtβ )}{1+λex p(−αt − γtβ )}]δ−1

× [1− {1− ex p(−αt − γtβ )}δ{1+λex p(−αt − γtβ )}δ]n−1(αt + βγtβ−1)ex p(−αt − γtβ )

× [1−λ+ 2λex p(−αt − γtβ )],

f(n)(t) =nδ[{1− ex p(−αt − γtβ )}{1+λex p(−αt − γtβ )}]δn−1(αt + βγtβ−1)

× [1−λ+ 2λex p(−αt − γtβ],

t ≥ 0,α,β ,δ > 0,λ ∈ [−1,1].

The density of the (r+1)-th order statistic can be expressed as a function of the density of

the r-th order statistic from the following relation:

f(r+1)(t) =
n− r

r

�{1− (1− ex p(−αt − γtβ ))δ(1+λex p(−αt − γtβ ))δ}]−1 − 1
�

f(r)(t).

The moments of the order statistics can be easily written in terms of the moments of the

modefied Weibull distribution by proceeding as follows:

We can write f(r)(t) as

f(r)(t) =
n!

(r − 1)!(n− r)!
δ

∞
∑

i=0

(−1)i
�

n− r

i

�

[{1− ex p(−αt − γtβ )}{1+λex p(−αt − γtβ )}]δ(i+r)−1

× (α+ γβ tβ−1)ex p(−αt − γtβ )[1−λ+ 2λex p(−αt − γtβ )]

=
n!

(r − 1)!(n− r)!

∞
∑

i=0

(−1)i
�

n− r

i

�

g(t;α,γ,β ,λ,δ(i + r)), t ≥ 0,α,β ,γ,δ > 0,λ ∈ [−1,1],

where g(t;α,γ,β ,λ,δ(i + r)) denote the density function of an exponentiated transmuted

modified Weibull distribution with shape paameters (α,β ,γ), transmuting parameter λ and

exponentiatinf parameter δ(i + r).



M. Pal and M. Tiensuwan / Eur. J. Pure Appl. Math, 8 (2015), 1-14 8

Hence, using (12) we have

E
�

T r
(s)

�

=
n!

(s− 1)!(n− s)!

∞
∑

u=0

(u+ s)δ

× [
∞
∑

i=0

∞
∑

j=0

A(i, j; (u+ s)δ,λ)[
1−λ

i + j + 1

∞
∑

k=0

(−β)k
k!
{ Γ(k(i + j + 1)γ+ r + 1)

{(i + j + 1)α}k(i+ j+1)γ+r

+ (i + j + 1)βγ
Γ(r + {k(i + j + 1) + 1}γ)
{(i + j + 1)α}{k(i+ j+1)+1}γ+r

}+ 2λ

i + j + 2

∞
∑

k=0

(−β)k
k!

× { Γ(k(i + j + 2)γ+ r + 2)

{(i + j + 2)α}k(i+ j+2)γ+r
+ (i + j + 2)βγ

Γ(r + {k(i + j + 2) + 1}γ)
{(i + j + 2)α}{k(i+ j+2)+1}γ+r

}],
for α,β > 0

=
n!

(s− 1)!(n− s)!

∞
∑

u=0

(u+ s)δ[

∞
∑

i=0

∞
∑

j=0

A(i, j; (u+ s)δ,λ)

× { 1−λ
i + j + 1

(
r

(i + j + 1)γ
+ 1)/β

r
(i+ j+1)γ +

2λ

i + j + 2
(

r

(i + j + 2)γ
+ 1)/β

r
(i+ j+2)γ }],

for α= 0,β > 0

=
n!

(s− 1)!(n− s)!

∞
∑

u=0

(u+ s)δ[

∞
∑

i=0

∞
∑

j=0

A(i, j; (u+ s)δ,λ){ 1−λ
i + j + 1

Γ(r + 1)

{(i + j + 1)α}r

+
2λ

i + j + 2

Γ(r + 1)

{(i + j + 2)α}r }]tex t, f orα > 0,β = 0.

In addition, we can calculate the L-moments [5], which are summary statistics for probabil-

ity distributions and data samples but have several advantages over ordinary moments. For

example, they apply for any distribution having a finite mean and no higher-order moments

need be finite. The rth L-moment is computed from the linear combinations of the ordered

data values as given below:

ρr =

∞
∑

u=0

(−1)r−u−1

�

r − 1

u

��

r + u− 1

u

�

θu,

where θu = E[T F(T )u].

Thus, ρ1 = θ0, ρ2 = 2θ1 − θ0, ρ3 = 6θ2 − 6θ1 + θ0, and ρ4 = 20θ3 − 30θ2 + 12θ1 − θ0. In

general, θk = (k+ 1)−1E(Tk+1:k+1), which can be computed from (6) by substituting

n= s = k+ 1 and r = 1.



M. Pal and M. Tiensuwan / Eur. J. Pure Appl. Math, 8 (2015), 1-14 9

7. Reliability

A stress-strength model describes the life of a component having a random strength X1 and

subjected to a random stress X2. The component functions satisfactorily for X1 > X2 and fails

when X1 < X2. The probability R = Pr(X1 > X2) defines the component reliability. Stress-

strength models have many applications especially in engineering concepts such as structures,

deterioration of rocket motors, static fatigue of ceramic components, fatigue failure of aircraft

structures and the aging of concrete pressure vessels.

Consider X1 and X2 to be independently distributed, with X1 ∼ ET MW (α1,γ1,β ,λ1,δ1)

and X2 ∼ ET MW (α2,γ2,β ,λ2,δ2). The c.d.f. F1 of X1 and the pdf f2 of X2 are obtained from

(4) and (5), respectively. Then,

R=Pr(X1 > X2) =

∫ ∞

0

f2(y)[1− F1(y)]d y

=1−
∞
∑

k,l=0

w
(1)

k,l
A(k, l),

where

w
(1)

k,l
= (−1)k
�

δ1

k

��

δ1

l

�

λl
1, w

(2)

k,l
= (−1)k
�

δ2 − 1

k

��

δ2 − 1

l

�

λl
2,

and

A(k, l) =

∫ ∞

0

f2(y)ex p(−(k+ l)α1 y − (k+ l)γ2 yβ )d y

=

∞
∑

i, j=0

w
(2)

i, j

∫ ∞

0

(α2 + γ2β yβ−1)ex p(−{(k+ l)α1 + (i + j + 1)α2}y

− {(k+ l)γ1 + (i + j + 1)γ2}yβ )(1−λ2 + 2λ2ex p(−α2 y − γ2 yβ ))d y

=

∞
∑

i, j=0

w
(2)

i, j
[α2{(1−λ2)µ

MW
1 ((k+ l)α1 + (i + j + 1)α2, (k+ l)γ1 + (i + j + 1)γ2,β)

+ 2λ2µ
MW
1 ((k+ l)α1 + (i + j + 2)α2, (k+ l)γ1 + (i + j + 2)γ2,β)}

+ γ2{(1−λ2)µ
MW
β ((k+ l)α1 + (i + j + 1)α2, (k+ l)γ1 + (i + j + 1)γ2,β)

+ 2λ2µ
MW
β ((k+ l)α1 + (i + j + 2)α2, (k+ l)γ1 + (i + j + 2)γ2,β)}],

with µMW
r (α,γ,β) given by (11).

In particular, if α1 = α2 and γ1 = γ2, we have

A(k, l) =

∞
∑

i, j=0

w
(2)

i, j
[

(1−λ2)

(i + j + k+ l + 1)
+

2λ2

(i + j + k+ l + 2)
].



M. Pal and M. Tiensuwan / Eur. J. Pure Appl. Math, 8 (2015), 1-14 10

8. Least Squares Estimation

Let T(1) < T(2) < . . . < T(n) denote the ordered observations in a random sample of size n

drawn from the ETMW(α,γ,β ,λ,δ) distribution with distribution function F(·), given by (4).

Then,

E[F(T(i))] =
i

n+ 1
, i = 1,2, . . . , n.

The least square estimators are obtained by minimizing

D(θ ) =

n
∑

i=1

�

F(T(i))−
i

n+ 1

�2

=

n
∑

i=1

�

(1− ex p(−αT(i) − γT
β

(i)
))δ(1+λex p(−αT(i) − γT

β

(i)
))δ − i

n+ 1

�2

.

Writing Vi = ex p(−αT(i)−γT
β

(i)
), the normal equations to be satisfied by the estimators are as

follows:

n
∑

i=1

�

(1− Vi)
δ(1+λVi)

δ − i

n+ 1

�

(1− Vi)
δ−1(1+λVi)

δ−1T(i)Vi(1−λ+ 2λVi) = 0

n
∑

i=1

�

(1− Vi)
δ(1+λVi)

δ − i

n+ 1

�

(1− Vi)
δ−1(1+λVi)

δ−1T
β

(i)
Vi(1−λ+ 2λVi) = 0

n
∑

i=1

�

(1− Vi)
δ(1+λVi)

δ − i

n+ 1

�

(1− Vi)
δ−1(1+λVi)

δ−1T
β

(i)
ln(T(i))Vi(1−λ+ 2λVi) = 0

n
∑

i=1

�

(1− Vi)
δ(1+λVi)

δ − i

n+ 1

�

(1− Vi)
δ(1+λVi)

δ−1Vi = 0

n
∑

i=1

�

(1− Vi)
δ(1+λVi)

δ − i

n+ 1

�

(1− Vi)
δ(1+λVi)

δ[ln(1− Vi) + ln(1+λVi)] = 0.

9. Maximum Likelihood Method of Estimation

Consider a random sample (T1, T2, . . . , Tn) of size n taken from the distribution ETMW(α,γ,β ,λ,δ)

with density function (5).

For given Ti = t i , i = 1,2, . . . , n, the log-likelihood function for θ = (α,γ,β ,λ,δ) is

l(θ ) =n lnδ+ (δ− 1){
n
∑

i=1

ln(1− ex p(−αt i − γt
β

i
)) +

n
∑

i=1

ln(1+λex p(−αt i − γt
β

i
))}

+

n
∑

i=1

ln(α+ γβ t
β−1

i
) +

n
∑

i=1

ln(1−λ+ 2λex p(−αt i − γt
β

i
))−

n
∑

i=1

(αt i + γt
β

i
).



M. Pal and M. Tiensuwan / Eur. J. Pure Appl. Math, 8 (2015), 1-14 11

Writing νi = ex p(−αt i − γt
β

i
)), i = 1,2, . . . , n, the log-likelihood equations are obtained as

follows:

n
∑

i=1

t i =− 2λ

n
∑

i=1

t iνi

(1−λ+ 2λνi)
+ (1−δ)

n
∑

i=1

t iνi(1−λ+ 2λνi)

(1− νi)(1+λνi)
+

n
∑

i=1

1

(α+ γβ t
β−1

i
)

(13)

n
∑

i=1

t
β

i
=− 2λ

n
∑

i=1

t
β

i
νi

(1−λ+ 2λνi)
+ (1−δ)

n
∑

i=1

t
β

i
νi(1−λ+ 2λνi)

(1− νi)(1+λνi)
+ β

n
∑

i=1

t
β−1

i

(α+ γβ t
β−1

i
)

(14)

n
∑

i=1

t
β

i
ln t i =− 2λ

n
∑

i=1

(t
β

i
ln t i)νi

(1−λ+ 2λνi)
+ (1−δ)

n
∑

i=1

(t
β

i
ln t i)νi(1−λ+ 2λνi)

(1− νi)(1+λνi)
(15)

+

n
∑

i=1

t
β−1

i
(1+ β ln t i)

(α+ γβ t
β−1

i
)

(16)

(δ− 1)

n
∑

i=1

νi

1+λνi

=

n
∑

i=1

1− 2νi

1−λ+ 2λνi

(17)

n
∑

i=1

ln(1− νi) +

n
∑

i=1

ln(1+λex pνi) = −
n

δ
. (18)

Solving the non-linear system of equations (13)-(18) we obtain the maximum likelihood esti-

mate θ̂ = (α̂, γ̂, β̂ , λ̂, δ̂) of θ .

Under certain regularity conditions,
p

n(θ̂ − θ ) d−→ Normal(0, I−1(θ )) (here
d−→ stands for

convergence in distribution), where I(θ ) denotes the information matrix given by

I(θ ) = E

�

∂ 2l(θ )

∂ θ∂ θ ′

�

. (19)

This information matrix I(θ ) may be approximated by the observed information matrix

I(θ̂ ) = E

�

∂ 2l(θ )

∂ θ∂ θ ′

�

θ=θ̂

. (20)

Hence, using the approximation
p

n(θ̂ −θ )∼ Normal(0, I−1(θ̂ )), one can carry out tests and

find confidence regions for functions of some or all of the parameters in θ .

10. Simulation Study

A simulation study is carried out to investigate the performance of the least square (LS)

estimators and the ML estimators. We take sample sizes to be n = 15,25,50,100,250,500

and generate observations from a ETMW distribution with parameters α = 1, γ = 1, β = 2.5,

λ = 0.5, δ = 0.75. For each sample we compute the estimates of the parameters using the



M. Pal and M. Tiensuwan / Eur. J. Pure Appl. Math, 8 (2015), 1-14 12

two methods. The process is repeated N = 1000 times, and the average mean squared error

is computed as follows:

AMSE(T ) =
1

N

N
∑

i=1

(θ̂ i(T )− θ )′(θ̂ i(T )− θ ), (21)

where T denotes the method used to estimate θ = (α,γ,β ,λ,δ), i denotes the sample repeti-

tion number, i = 1,2, . . . , N , and θ̂ i(T ) is the corresponding estimate. Table 1 gives the AMSEs

for the two methods of estimation.

Table 1: Average Mean Squared Errors for the Method of Least Squares and Maximum Likeli-

hood Method for Estimating θ

Sample size AMSE(LS) AMSE(ML)

10 3.2651 0.9985

15 1.4247 0.4386

25 0.8769 0.1027

50 0.4317 0.0285

100 0.0621 0.0075

250 0.0860 0.0010

100 0.0009 0.0001

The above table shows that as the sample size increases, the average mean squared errors

decrease. This verifies the consistency properties of the estimates. Further, for each sample

size AMSE(LS) > AMSE(M L). Thus, we may conclude that the maximum likelihood method

provides better estimators than the least square method. However, the difference between

AMSE(LS) and AMSE(M L) decrease as the sample size increases.

11. Application of the Exponentiated Transmuted Modified Weibull Distribution

In this section we illustrate the usefulness of the exponentiated transmuted modified Weibull

distribution for modelling real life data. The data set relates to the time-to-failure of 50 de-

vices, and is taken from Aarset [1]:

Table 2: The time-to-failure of 50 devices

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 Weibull (W), modified Weibull (M), transmuted modified Weibull(T) and exponenti-

ated transmuted modified Weibull (E) distributions are fitted to the data and the MLEs of the



M. Pal and M. Tiensuwan / Eur. J. Pure Appl. Math, 8 (2015), 1-14 13

parameters are given in Table 3. The values of the log-likelihood, Kolmogorov-Smirnov statis-

tic (K-S), Akaike information criterion (AIC) and Bayesian information criterion (BIC) for the

different fitted distributions are also given, and show that the ETMW distribution gives a better

fit than the others. The same is also evident from Figure 2, which compares the cumulative

distribution curves of the fitted distributions with that of the empirical distribution.

Table 3: The estimated parameters and the log-likelihood, K-S, AIC, BIC values for the different

fitted distributions

α γ β λ δ -l(θ ) K-S AIC BIC

W 0 0.0268 0.9499 0 1 241.002 0.0750 486.004 489.8280

M 0.012 2.159×10−8 4.014 0 1 230.15 0.0739 466.30 472.0361

T 0.0122 1.0006×10−8 4.1924 0.0747 1 229.06 0.0730 466.12 473.7681

E 0.0056 8.96×10−9 4.2448 -0.4706 0.4553 222.61 0.0655 455.22 464.7801

Figure 2: Comparison of the CDFs of the Fitted Distributions with the Empirical CDF

12. Discussion

In this paper, we introduce a new generalization of the Weibull distribution called expo-

nentiated transmuted modified Weibull distribution and discuss its intrinsic properties. The

distribution is very flexible in the sense that it exhibits both increasing and decreasing failure

rates depending on its parameters.

ACKNOWLEDGEMENTS The authors thank the anonymous referees for their fruitful sug-

gestions, which immensely helped to improve the presentation of the paper.



REFERENCES 14

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