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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

[email protected]

Md. Mahabubur Rahman

[email protected]

Bander Al-Zahrani

[email protected]

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.

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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

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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

λ 1

K 1

K 5

K 10

λ 2

K 1

K 5

K 10

λ 3

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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Table 2 Variance of the C T W distribution for various combinations of the parameters

λ1 −1

λ 1

K 1

K 5

K 10

λ 2

K 1

K 5

K 10

λ 3

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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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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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

+

2λ

−

3λ

−

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

+

2λ

−

3λ

−

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 i 1 (x / λ)

L nk ·

xi · e

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

λ

n

i 1

i 1

and the log-likelihood function l ln(L) is

n

l n · ln(k) − nk · ln(λ) + (k − 1)

i 1

n

+

n

ln(xi ) − 3

i 1

x

i

k

λ

k

k

ln (1 − λ1 − λ2 )e2(x / λ) + 2(λ1 + 2λ2 )e(x / λ) − 3λ2 .

(16)

i 1

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

i 1

n

kxi xi

λ2 λ

n

k−1

+

i 1

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

i 1

k

k k

k

2β1 e2(x / λ) x λ ln x λ + 2β2 e(x / λ) x λ ln x λ

ln(xi ) +

i 1

n

+

β1 e2(x / λ) + 2β2 e(x / λ) − 3λ2

k

i 1

δl

δλ1

n

k

2e(x / λ) − e2(x / λ)

,

k

k

β1 e2(x / λ) + 2β2 e(x / λ) − 3λ2

k

i 1

,

k

and

δl

δλ2

n

4e(x / λ) − e2(x / λ) − 3

,

k

k

β1 e2(x / λ) + 2β2 e(x / λ) − 3λ2

k

i 1

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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