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Annals of Data Science
https://doi.org/10.1007/s40745-018-00188-y
Cubic Transmuted Weibull Distribution: Properties
and Applications
Md. Mahabubur Rahman1,2 · Bander Al-Zahrani1 ·
Muhammad Qaiser Shahbaz1
Received: 18 September 2018 / Revised: 21 November 2018 / Accepted: 13 December 2018
© Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract
In this paper, a cubic transmuted Weibull (C T W ) distribution has been proposed by
using the general family of transmuted distributions introduced by Rahman et al. (Pak
J Stat Oper Res 14:451–469, 2018). We have explored the proposed C T W distribution
in details and have studied its statistical properties as well. The parameter estimation
and inference procedure for the proposed distribution have been discussed. We have
conducted a simulation study to observe the performance of estimation technique.
Finally, we have considered two real-life data sets to investigate the practicality of
proposed C T W distribution.
Keywords Cubic transmutation · Maximum likelihood estimation · Moments · Order
statistics · Reliability analysis · Weibull distribution
1 Introduction
The Weibull distribution, introduced by Waloddi Weibull [2], is a popularly used statistical model in the area of reliability analysis. Several generalizations of the Weibull
distribution are available in the literature which have been obtained by adding shape
parameter(s). Mudholkar et al. [3], generalized the Weibull distribution with application to the analysis of survival data. Pham and Lai [4], describes some recent
B
Muhammad Qaiser Shahbaz
qshahbaz@gmail.com
Md. Mahabubur Rahman
mmriu.stat@gmail.com
Bander Al-Zahrani
bmalzahrani@kau.edu.sa
1
Department of Statistics, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia
2
Department of Statistics, Islamic University, Kushtia, Bangladesh
123
Annals of Data Science
generalization of Weibull distribution including modified Weibull distribution [5, 6],
exponentiated Weibull distribution [7, 8] and inverse Weibull distribution [9].
Shaw and Buckley [11] have proposed quadratic transmuted family of distributions
with cd f
F(x)  (1 + λ)G(x) − λ G 2 (x),
(1)
where λ ∈ [−1, 1] is the transmutation parameter.
Aryal and Tsokos [10], have proposed transmuted Weibull (T W ) distribution by
using the method proposed by Shaw and Buckley [11]. The proposed transmuted
Weibull distribution has wider applicability in reliability analysis. As extension,
Granzotto et al. [12]; Al-Kadim and Mohammed [13] have described two C T W distributions, for capturing the complexity of the data.
The transmuted family of distributions (1) has been recently generalized to cubic
transmuted family by Rahman et al. [1]. The cd f of this cubic transmuted family of
distribution has the form
F(x)  (1 + λ1 )G(x) + (λ2 − λ1 )G 2 (x) − λ2 G 3 (x), x ∈ R,
(2)
where λ1 ∈ [−1, 1], λ2 ∈ [−1, 1] are the transmutation parameters such that −2 ≤
λ1 + λ2 ≤ 1. The cubic transmuted family of distributions (2), introduced by Rahman
et al. [1], and is flexible enough to capture the complexity (bi-modality) of real-life
data sets.
In this paper, we have used the cubic transmuted family (2) and have obtained a
new C T W distribution. The proposed distribution has been studied in detail in the
following.
1.1 Plan of the Paper
The article is structured as follows. The proposed C T W distribution is discussed in
Sect. 2. In Sect. 3, we have explored statistical properties including the moments, generating function, quantile function, random number generation and reliability function
for the C T W distribution along with the distribution of order statistics in Sect. 4. Section 5 contains the parameter estimation and inference for the C T W distribution. In
Sect. 6, we have provided the simulation study to assess the performance of estimation technique along with two real-life applications. Finally, in Sect. 7, we list some
concluding remarks.
2 The New Cubic Transmuted Weibull Distribution
The cd f of Weibull distribution is given by
G(x)  1 − e−(x / λ) , x ∈ [0, ∞),
k
where λ, k ∈ R+ are the scale and shape parameters.
123
(3)
Annals of Data Science
Aryal and Tsokos [10] introduced T W distribution by using (3) in (1) and has the
cd f



k
k
(4)
F(x)  1 − e−(x / λ) 1 + θ e−(x / λ) , x ∈ [0, ∞),
where λ, k ∈ R+ are the scale and shape parameters respectively and θ is the transmutation parameter.
The cd f of C T W distribution is obtained by using (3) in (2) and has following
simple form
F(x)  1 + (λ1 + λ2 − 1)e−(x / λ) − (λ1 + 2λ2 )e−2(x / λ) + λ2 e−3(x / λ) , x ∈ [0, ∞),
(5)
k
k
k
where λ, k ∈ R+ are the scale and shaped parameters including two transmutation
parameters λ1 ∈ [−1, 1] and λ2 ∈ [−1, 1] such that −2 ≤ λ1 + λ2 ≤ 1.
The pd f of C T W distribution can be obtained by differentiating the cd f (5) wr t
x and is given as
f (x) 

k
k
k k−1 −3(x / λ)k 
x e
(1 − λ1 − λ2 )e2(x / λ) + 2(λ1 + 2λ2 )e(x / λ) − 3λ2 , x ∈ [0, ∞),
k
λ
(6)
where λ, k ∈ R+ , λ1 ∈ [−1, 1] and λ2 ∈ [−1, 1] such that −2 ≤ λ1 + λ2 ≤ 1.
2.1 Special Cases
Some special cases for the C T W distribution are listed below:
1. The cd f of C T W distribution (5) provides cd f of C T W distribution proposed by
Al-Kadim and Mohammed [13], for λ2  −λ1  λ.
2. The cd f of C T W distribution (5) turned out to be the cd f of cubic transmuted
exponential distribution as discussed in Rahman et al. [1], for k  1.
3. The cd f of the C T W distribution (5), reduces to the cd f of the T W distribution
(4), for λ2  0.
4. The cd f of the C T W distribution (5) provides cd f of Weibull distribution (3) for
λ1  λ2  0.
Some of the possible shapes for pd f and cd f of the new C T W distribution for
selected values of model parameters k and λ1 setting λ  1 and λ2  −1, are given
in Fig. 1. From the plot we can see that the proposed C T W can be used to model the
bi-modal data.
3 Statistical Properties
Statistical properties of proposed C T W distribution have been discussed in the following subsections.
123
Annals of Data Science
Fig. 1 Density and distribution functions are plotted for the proposed C T W distribution with different values
of model parameters k and λ1 setting λ  1 and λ2  −1
3.1 Moments
The moments play an important role to decide about the shapes of a distribution. In
the following, we have discussed moments of C T W distribution.
Theorem 1 The r th moment of C T W distribution is given as




λr
k + r  r/ k
3
E(X r )  r k Γ
(1 − λ1 − λ2 )2r / k + (λ1 + 2λ2 ) − λ2 2r / k ,
k
6/
or,





λr
k + r  r/ k
E(X )  r k Γ
6 − 3r / k 2r / k − 1 λ1 − 2r / k − 2 · 3r / k + 6r / k λ2 .
k
6/
(7)
r
The mean and variance are given, respectively, as





λ
k + 1  1/ k
E(X )  1 k Γ
6 − 31/ k 21/ k − 1 λ1 − 21/ k − 2 · 31/ k + 61/ k λ2 ,
k
6/
and





λ2
k + 2  2/ k
V (X )  2 k Γ
6 − 32/ k 22/ k − 1 λ1 − 22/ k − 2 · 32/ k + 62/ k λ2
k
6/



2

k
+
1
2
.
61/ k − 31/ k 21/ k − 1 λ1 − 21/ k − 2 · 31/ k + 61/ k λ2
−Γ
k
123
Annals of Data Science
Proof The r th moment is given by

E(X r ) 
x r f (x)dx,
(8)
0
where f (x) is given in (6). Using (6) in (8) and simplifying, the r th moment of C T W
distribution is given as

k
k
k k−1 −3(x / λ)k 
x e
(1 − λ1 − λ2 )e2(x / λ) + 2(λ1 + 2λ2 )e(x / λ) − 3λ2 dx
k
λ
0


λk λr
k
k +r
 k (1 − λ1 − λ2 )
Γ
λ
k
k


k
r
k
1 λ λ
k +r
+ k 2(λ1 + 2λ2 ) r k+1
Γ
λ
k
k
2/


k
r
k
1 λ λ
k +r
− k 3λ2 r k+1
Γ
λ
k
k
3/


k
+
r
λ2
(λ1 + 2λ2 )
− r k
 λr Γ
(1 − λ1 − λ2 ) +
k
2r / k
3/





k + r  r/ k
λr
 r kΓ
6 − 3r / k 2r / k − 1 λ1 − 2r / k − 2 · 3r / k + 6r / k λ2 . (9)
k
6/

E(X r ) 
xr

Mean can be obtained by setting r  1 in (9) and variance is obtained by using the
relation

V (X )  E X 2 − {E(X )}2 ,
where E(X r ) for i  1, 2 are obtained from (9).
One can obtain all other higher moments by using r > 2 in (9).
The mean and variance chart for the proposed C T W distribution with several
combinations of parameters are presented in Tables 1 and 2 respectively.
3.2 Moment Generating Function
Moment generating function is a useful function to obtain moments of random variables. The moment generating function for C T W is given in the following theorem.
Theorem 2 Let X follows the C T W distribution, then the moment generating function,
M X (t), is

M X (t) 
r 0
t r λr
Γ
r ! 6r / k





k + r  r/ k
6 − 3r / k 2r / k − 1 λ1 − 2r / k − 2 · 3r / k + 6r / k λ2 ,
k
(10)
where t ∈ R.
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Annals of Data Science
Table 1 Mean of the C T W distribution for various combinations of the parameters
λ1
K 1
K 5
K  10
λ2
K 1
K 5
K  10
λ3
K 1
K 5
123
λ1  −1
λ1  −0.5
λ1  0
λ1  0.5
λ1  1
λ2  −1
1.833
1.583
1.333
1.083
0.833
λ2  −0.5
1.667
1.417
1.167
0.917
0.667
λ2  0
1.500
1.250
1.000
0.750
0.500
λ2  0.5
1.333
1.083
0.833
0.583

λ2  1
1.167
0.917
0.667


λ2  −1
1.094
1.034
0.975
0.915
0.856
λ2  −0.5
1.065
1.006
0.946
0.887
0.828
λ2  0
1.037
0.978
0.918
0.859
0.799
λ2  0.5
1.009
0.949
0.890
0.830

λ2  1
0.980
0.921
0.862


λ2  −1
1.043
1.012
0.980
0.948
0.916
λ2  −0.5
1.029
0.997
0.966
0.934
0.902
λ2  0
1.015
0.983
0.951
0.919
0.888
λ2  0.5
1.001
0.969
0.937
0.905

λ2  1
0.987
0.955
0.923


λ2  −1
3.667
3.167
2.667
2.167
1.667
λ2  −0.5
3.333
2.833
2.333
1.833
1.333
λ2  0
3.000
2.500
2.000
1.500
1.000
λ2  0.5
2.667
2.167
1.667
1.167

λ2  1
2.333
1.833
1.333


λ2  −1
2.187
2.068
1.950
1.831
1.712
λ2  −0.5
2.131
2.012
1.893
1.774
1.655
λ2  0
2.074
1.955
1.836
1.717
1.599
λ2  0.5
2.017
1.899
1.780
1.661

λ2  1
1.961
1.842
1.723


λ2  −1
2.087
2.023
1.960
1.896
1.832
λ2  −0.5
2.059
1.995
1.931
1.867
1.804
λ2  0
2.030
1.966
1.903
1.839
1.775
λ2  0.5
2.002
1.938
1.874
1.811

λ2  1
1.973
1.910
1.846


λ2  −1
5.500
4.750
4.000
3.250
2.500
λ2  −0.5
5.000
4.250
3.500
2.750
2.000
λ2  0
4.500
3.750
3.000
2.250
1.500
λ2  0.5
4.000
3.250
2.500
1.750

λ2  1
3.500
2.750
2.000


λ2  −1
3.281
3.103
2.924
2.746
2.568
λ2  −0.5
3.196
3.018
2.839
2.661
2.483
λ2  0
3.111
2.933
2.755
2.576
2.398
λ2  0.5
3.026
2.848
2.670
2.491

λ2  1
2.941
2.763
2.585


Annals of Data Science
Table 1 continued
K  10
λ1  −1
λ1  −0.5
λ1  0
λ1  0.5
λ1  1
λ2  −1
3.130
3.035
2.939
2.844
2.748
λ2  −0.5
3.088
2.992
2.897
2.801
2.706
λ2  0
3.045
2.950
2.854
2.758
2.663
λ2  0.5
3.003
2.907
2.811
2.716

λ2  1
2.960
2.864
2.769


Proof The moment generating function is defined as

M X (t)  E[et x ] 
et x f (x)dx,
0

where f (x) is given in (6).
Using the series representation of et x given in Gradshteyn and Ryzhik [14], we
have
∞ ∞
Mx (t) 
0
r 0
tr r
x f (x)dt 
r!

r 0
tr
E(X r ).
r!
(11)
Using E(X r ) from (7) in (11) we have (10).
3.3 Characteristic Function
The characteristic function plays a central role and completely defines its density function. The characteristic function for C T W distribution is given in following theorem.
Theorem 3 Let X have the C T W distribution, then characteristic function, φ X (t), of
X is

φ X (t) 
r 0
where i 
(it)r λr
Γ
r ! 6r / k





k + r  r/ k
6 − 3r / k 2r / k − 1 λ1 − 2r / k − 2 · 3r / k + 6r / k λ2 ,
k

−1 is the imaginary unit and t ∈ R.
Proof The proof is simple.

3.4 Quantile Function
The quantile function xq , of a random variable is inverse of its cd f . The quantile
function for C T W distribution is obtained by solving (5) for x and is obtained as, see
for example Rahman et al. [16],
1
xq  λ{−ln(y)} k ,
(12)
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Annals of Data Science
Table 2 Variance of the C T W distribution for various combinations of the parameters
λ1  −1
λ1
K 1
K 5
K  10
λ2
K 1
K 5
K  10
λ3
K 1
K 5
123
λ1  −0.5
λ1  0
λ1  0.5
λ1  1
1.028
λ2  −1
1.361
1.465
1.444
1.299
λ2  −0.5
1.333
1.354
1.250
1.021
0.667
λ2  0
1.250
1.188
1.000
0.688
0.250
λ2  0.5
1.111
0.965
0.694
0.299

λ2  1
0.917
0.688
0.333


λ2  −1
0.020
0.039
0.051
0.056
0.054
λ2  −0.5
0.024
0.040
0.049
0.050
0.045
λ2  0
0.027
0.039
0.044
0.042
0.034
λ2  0.5
0.027
0.036
0.038
0.033

λ2  1
0.027
0.032
0.031


λ2  −1
0.005
0.011
0.015
0.017
0.017
λ2  −0.5
0.006
0.011
0.014
0.015
0.014
λ2  0
0.007
0.011
0.013
0.013
0.011
λ2  0.5
0.007
0.010
0.012
0.011

λ2  1
0.007
0.009
0.010


4.111
λ2  −1
5.444
5.861
5.778
5.194
λ2  −0.5
5.333
5.417
5.000
4.083
2.667
λ2  0
5.000
4.750
4.000
2.750
1.000
λ2  0.5
4.444
3.861
2.778
1.194

λ2  1
3.667
2.750
1.333


0.216
λ2  −1
0.081
0.157
0.205
0.225
λ2  −0.5
0.097
0.160
0.194
0.200
0.178
λ2  0
0.107
0.156
0.177
0.170
0.134
λ2  0.5
0.110
0.146
0.153
0.132

λ2  1
0.107
0.129
0.123


λ2  −1
0.019
0.043
0.059
0.067
0.067
λ2  −0.5
0.024
0.044
0.057
0.061
0.057
λ2  0
0.027
0.044
0.052
0.053
0.046
λ2  0.5
0.028
0.041
0.047
0.044

λ2  1
0.028
0.038
0.039


9.250
λ2  −1
12.250
13.188
13.000
11.688
λ2  −0.5
12.000
12.188
11.250
9.188
6.000
λ2  0
11.250
10.688
9.000
6.188
2.250
λ2  0.5
10.000
8.688
6.250
2.688

λ2  1
8.250
6.188
3.000


λ2  −1
0.182
0.354
0.461
0.506
0.486
λ2  −0.5
0.219
0.360
0.437
0.451
0.401
λ2  0
0.240
0.351
0.398
0.382
0.302
λ2  0.5
0.247
0.328
0.345
0.298

λ2  1
0.240
0.290
0.277


Annals of Data Science
Table 2 continued
λ1  −1
K  10
λ1  −0.5
λ1  0
λ1  0.5
λ1  1
0.150
λ2  −1
0.043
0.097
0.133
0.151
λ2  −0.5
0.053
0.099
0.127
0.137
0.128
λ2  0
0.060
0.098
0.118
0.119
0.103
λ2  0.5
0.063
0.093
0.105
0.098

λ2  1
0.063
0.085
0.088


where
y−
b

3a
21/3 ξ1
1/3 +


3
2
3a ξ2 + 4ξ1 + ξ2
1/3


3
2
ξ2 + 4ξ1 + ξ2
3(21/3 )a
,
(13)
with ξ1  −b2 + 3ac, ξ2  −2b3 + 9abc − 27a 2 d, a  λ2 , b  −λ1 − 2λ2 ,
c  λ1 + λ2 − 1 and d  1 − q.
The lower quartile, median and upper quartile can be obtained by using q  0.25,
0.50 and 0.75 in (12), respectively.
3.5 Simulating the Random Sample
The quantile function can be used to generate the random data from C T W distribution. The random data from C T W distribution can be obtained by using following
expression, see for example Rahman et al. [16],
(λ1 + λ2 − 1)e−(x / λ) − (λ1 + 2λ2 )e−2(x / λ) + λ2 e−3(x / λ) + 1  u,
k
k
k
on further simplification, we have
1
X  λ{−ln(y)} k ,
(14)
where y is given in (13) with d  1 − u. The random sample from C T W distribution
can be obtained by using (14) for various values of the model parameters λ, k, λ1 and
λ2 .
3.6 Reliability Analysis
The reliability function is simply the complement of distribution function and is defined
as
R(t)  1 − F(t),
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Annals of Data Science
Fig. 2 Reliability and hazard functions are plotted for the proposed C T W distribution with different values
of model parameters k and λ1 setting λ  1 and λ2  −1
and for C T W distribution it is given as
R(t)  (1 − λ1 − λ2 )e−(x / λ) + (λ1 + 2λ2 )e−2(x / λ) − λ2 e−3(x / λ) .
k
k
k
The hazard function is the ratio of the density function to the reliability function
and is given by
k
k
(1 − λ1 − λ2 )e2(x / λ) + 2(λ1 + 2λ2 )e(x / λ) − 3λ2
h(t)  k t k−1 e−3(x / λ)
.
k
k
k
λ
(1 − λ1 − λ2 )e−(x / λ) + (λ1 + 2λ2 )e−2(x / λ) − λ2 e−3(x / λ)
k
k
Figure 2 shows some possible shapes for the reliability and hazard functions of the
CTW distribution with different combination of parameters k, λ1 setting λ  1 and λ2
 − 1.
4 Order Statistics
The pdf of r th order statistic for C T W distribution is given as
k k−1 −3(x / λ)k
n!
x e
(r − 1)! (n − r )! λk


k
k
× (1 − λ1 − λ2 )e2(x / λ) + 2(λ1 + 2λ2 )e(x / λ) − 3λ2


k
k
k r −1
× 1 + (λ1 + λ2 − 1)e−(x / λ) − (λ1 + 2λ2 )e−2(x / λ) + λ2 e−3(x / λ)


k
k
k n−r
× (1 − λ1 − λ2 )e−(x / λ) + (λ1 + 2λ2 )e−2(x / λ) − λ2 e−3(x / λ)
,
f X r :n (x) 
(15)
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Annals of Data Science
where r  1, 2, . . . , n. Using r  1 in (15), we obtain the pd f of smallest order
statistics X 1:n , and is given as

nk k−1 −3(x / λ)k 
2(x / λ)k
(x / λ)k
x
e

λ
+
2(λ
+




λ
)e
)e
(1
1
2
1
2
2
λk


n−1
k
k
k
,
× (1 − λ1 − λ2 )e−(x / λ) + (λ1 + 2λ2 )e−2(x / λ) − λ2 e−3(x / λ)
f X 1:n (x) 
also by using r  n in (15), the pd f of largest order statistics X n:n , is obtain by

nk k−1 −3(x / λ)k 
2(x / λ)k
(x / λ)k
x
e

λ
+
2(λ
+




λ
)e
)e
(1
1
2
1
2
2
λk


k
k
k n−1
.
× 1 + (λ1 + λ2 − 1)e−(x / λ) − (λ1 + 2λ2 )e−2(x / λ) + λ2 e−3(x / λ)
f X n:n (x) 
Note that for λ1  λ2  0, we have the pd f of the r th order statistic for Weibull
distribution, as follows
g X r :n (x) 

k r −1
n!
k k−1 −(n−r +1)(x / λ)k 
x e
; r  1, 2, . . . , k.
1 − e−(x / λ)
k
(r − 1)! (n − r)! λ
The kth order moment of X r :n for C T W distribution is obtained by using
E(X rk:n ) 

0
xrk · f X r :n (x) · dx,
where f X r :n (x) is given in (15).
5 Parameter Estimation and Inference
In this section, we have estimated parameters of the C T W distribution by using maximum likelihood method. Consider a random sample, x1 , x2 , . . . , xn of size n from
C T W distribution. The likelihood function is given by

n
k n

k
k
k n  k−1 −3 i1 (x / λ)  
L  nk ·
xi · e
(1 − λ1 − λ2 )e2(x / λ) + 2(λ1 + 2λ2 )e(x / λ) − 3λ2 ,
λ
n
i1
i1
and the log-likelihood function l  ln(L) is
n
l  n · ln(k) − nk · ln(λ) + (k − 1)
i1
n
+
n
ln(xi ) − 3
i1
x
i
k
λ


k
k
ln (1 − λ1 − λ2 )e2(x / λ) + 2(λ1 + 2λ2 )e(x / λ) − 3λ2 .
(16)
i1
123
Annals of Data Science
The maximum likelihood estimates of λ, k, λ1 and λ2 are obtained by maximizing
the log-likelihood function (16). The derivatives with respect to unknown parameter
are given below
δl
kn
−
+3
δλ
λ
δl

δk
n
i1
n
kxi  xi
λ2 λ
n
k−1
+
i1
k   k−1
k   k−1
2kβ1 xi e2(x / λ) x λ
− 2kβ2 xi e(x / λ) x λ

,
k
k
λ2 β1 e2(x / λ) + 2β2 e(x / λ) − 3λ2
n 
xi k  xi
n
ln
− n · ln(λ) − 3
k
λ
λ
i1






  
k
k   k
k
2β1 e2(x / λ) x λ ln x λ + 2β2 e(x / λ) x λ ln x λ
ln(xi ) +
i1
n
+
β1 e2(x / λ) + 2β2 e(x / λ) − 3λ2
k
i1
δl

δλ1
n
k
2e(x / λ) − e2(x / λ)
,
k
k
β1 e2(x / λ) + 2β2 e(x / λ) − 3λ2
k
i1
,
k
and
δl

δλ2
n
4e(x / λ) − e2(x / λ) − 3
,
k
k
β1 e2(x / λ) + 2β2 e(x / λ) − 3λ2
k
i1
k
where β1  1 − λ1 − λ2 and β2  λ1 + 2λ2 .
δl
δl
δl
 0, δk
 0, δλ
 0 and
Now setting, δλ
1
δl
δλ2
 0, and solving the result-
ing nonlinear system of equations gives the maximum likelihood estimate Θ̂ 


λ̂, k̂, λ̂1 , λ̂2 of Θ  (λ, k, λ1 , λ2 ) . Also as n → ∞, the asymptotic distribu
tion of the M L E  s λ̂ …
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