The Transmuted Flexible Weibull Distribution

The Transmuted Flexible Weibull Distribution 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 three-parameters model called the transmuted flexible Weibull extension (TFWE) distribution which exhibits bathtub-shaped hazard rate. Some of it’s statistical properties are obtained including the moments, quantile function and median, the mode, the moment generating function and order statistics. 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 an application to real data. Keywords: Generalized Weibull, Flexible Weibull extension distribution, Transmuted Weibull, Transmuted flexible Weibull extension, Reliability function, Hazard function, Moment generating function, Maximum likelihood estimation.

1

Introduction

The Weibull distribution is a highly known distribution due to its utility in modelling lifetime data where the hazard rate function is monotone Weibull [27]. In recent years new classes of distributions were proposed based on modifications of the Weibull distribution to cope with bathtub hazard failure rate Xie and Lai [28]. Among of these, Exponentiated Weibull family, Mudholkar and Srivastava [18], Modified Weibull distribution, Lai et al. [15] and Sarhan and Zaindin [22], Beta-Weibull distribution, Famoye et al. [9], A flexible Weibull extension, Bebbington et al. [4], Extended flexible Weibull, Bebbington et al. [4], Generalized modified Weibull distribution, Carrasco et al. [5], Kumaraswamy Weibull distribution, Cordeiro et al. [6], Beta modified Weibull distribution, Silva et al. [25] and Nadarajah et al. [20], Beta generalized Weibull distribution, Singla et al. [26], A new modified weibull distribution, Almalki and Yuan [2] and Exponentiated modified Weibull extension distribution, Sarhan and Apaloo [23], among others. A good review of these models is presented in Pham and Lai [21] and Murthy et al. [19]. The flexible Weibull (FWE) distribution, Bebbington et al. [4] 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 [4]. A random variable X is said to have the flexible Weibull extension (FWE) distribution with parameters α, β > 0 if it’s probability density function (pdf) is given by β β αx− β x g(x) = α + 2 exp αx − − e , x > 0, (1.1) x x

1

2

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

while the cumulative distribution function (cdf) is given by αx− β x , G(x) = 1 − exp − e

x > 0.

(1.2)

The survival function is given by the equation S(x) = 1 − G(x) = exp and the hazard function is

−e

αx− β x

,

x > 0,

β β h(x) = α + 2 eαx− x . x

(1.3)

(1.4)

Weibull distribution introduced by Weibull [27] is a popular distribution for modeling phenomenon with monotonic failure rates. But this distribution does not provide a good fit to data sets with bathtub shaped or upside-down bathtub shaped (unimodal) failure rates, often encountered in reliability, engineering and biological studies. However, the above distributions sometimes have some respective drawbacks in analyzing lifetime data. An interesting idea of generalizing a distribution, which is know in the literature as transmutation has been used to develop further distribution. A random variable X is said to have a transmuted distribution if its distribution function is given by F (x) = (1 + γ)G(x) − γ[G(x)]2 ,

|γ| ≤ 1,

(1.5)

where G(x) is the cdf base distribution, and γ ∈ [−1, 1] denotes the transmuting parameter. Observe that at γ = 0 we have the distribution of the base random variable. The probability density function corresponding to (1.5) is given by f (x) = g(x)[1 + γ − 2γG(x)],

(1.6)

where g(x) is the corresponding pdf associated with the cdf G(x). Aryall and Tsokos [10] introduced the transmuted Weibull distribution, Hady and Ebraheim [12] studied the exponentiated transmuted Weibull distribution, Pal and Tiensuwan [14] developed the beta transmuted Weibull distribution, Khan and King [13] investigated the transmuted modified Weibull distribution and Ashour and Eltehiwy [11] proposed the transmuted exponentiated modified Weibull distribution. In this paper, we introduce a new distribution depending on Flexible Weibull Extension distribution called The Transmuted Flexible Weibull Extension (TFWE) distribution by using the class of univariate distributions defined above. This paper is organized as follows, we define the cumulative, density, hazard functions and reliability analysis of the Transmuted flexible Weibull extension (TFWE) distribution in Section 2. In Section 3 and 4, we introduced the statistical properties including, quantile function, median, 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. Finally, Section 8 concludes the paper.

2

The Transmuted Flexible Weibull Distribution

In this section, we present and study the transmuted flexible Weibull extension (TFWE) distribution with three parameters α, β > 0, |γ| ≤ 1. Using G(x) and g(x) to obtain the cdf and pdf of Eq. (1.5) and Eq.

3


(1.6) respectively. The cumulative distribution function cdf of the transmuted flexible Weibull extension distribution (TFWE) is given by 2 β β −eαx− x −eαx− x F (x; α, β, γ) = (1 + γ) 1 − e −γ 1−e β β αx− αx− 1 + γe−e x , x > 0, α, β > 0, |γ| ≤ 1. (2.1) F (x; α, β, γ) = 1 − e−e x The pdf corresponding to Eq. (2.1) is given by β β αx− x αx− x β αx− β −e −e xe f (x; α, β, γ) = α + 2 e 1 − γ + 2γe , x

(2.2)

where x > 0 and, α, β > 0 are two additional shape parameters. The survival function S(x), hazard rate function h(x) and reversed- hazard rate function r(x) of X ∼ TFWE (α, β, γ) are given by β β −eαx− x −eαx− x S(x; α, β, γ) = 1 − 1 − e 1 + γe , x > 0, (2.3) β β αx− x αx− x β αx− β −e −e x e 1 − γ + 2γe α + x2 e , (2.4) h(x; α, β, γ) = β β αx− x αx− x −e −e 1− 1−e 1 + γe β β αx− x αx− x β αx− β −e −e x e α + x2 e 1 − γ + 2γe r(x; α, β, γ) = , (2.5) β β αx− x αx− x −e −e 1 + γe 1−e and

Z H(x; α, β, γ) =

x

β β −eαx− x −eαx− x h(u)du = − ln 1 − 1 − e 1 + γe ,

(2.6)

0

respectively, x > 0 and (α, β, γ). Figures (1-6) display the CDF, PDF, survival, hazard rate, reversed hazard rate function and cumulative hazard rate function of the TFWE(α, β, γ) distribution for some parameter values.

Figure1: The cdf of the TFWE for different values of parameters.


Figure 2: The pdf of the TFWE for different values of parameters.

Figure 3: The survival function of the TFWE for different values of parameters.

Figure 4: The hazard rate function of the TFWE for different values of parameters.

4

5


Figure 5: The reversed hazard rate function of the TFWE for different values of parameters.

Figure 6: The cumulative hazard rate function of the TFWE for different values of parameters. From Figures (1–6), we find that the TFWE is unimodal distribution, has increasing, decreasing and constant failure (hazard) rate function.

3

Statistical Properties

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

3.1

Quantile and median

In this subsection, we determine the explicit formulas of the quantile and the median of TFWE distribution. The quantile xq of the TFWE(α, β, γ) is given by F (xq ) = q,

0 < q < 1.

From Eq. (2.1), we have β β 2 αxq − x αxq − x q q −e −e − (1 + γ) 1 − e + q = 0, γ 1−e

(3.1)

(3.2)

6


we obtain xq by solve the following equation αx2q − k(q)xq − β = 0, where k(q) = ln

− ln 1 −

(1 + γ) ±

(3.3)

p (1 + γ)2 − 4qγ . 2γ

So, the simulation of the TFWE random variable is straightforward. Let U be a uniform variate on 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 X= . (3.4) 2α Since the median is 50% quantile then the median of TFWE distribution can be obtained by setting q = 0.5 in Eq. (3.3).

3.2

The mode

In this subsection, we will derive the mode of the TFWE distribution by derivation its pdf with respect to x and equate it to zero. The mode is the solution the following equation with respect to x. f 0 (x) = 0. By substitution PDF from Eq. (2.2) in Eq.(3.5), we have ∂ h(x; α, β, γ) · S(x; α, β, γ) = 0, ∂x 0 2 h (x; α, β, γ) − h (x; α, β, γ) S(x; α, β, γ) = ,

(3.5)

(3.6)

where h(x; α, β, γ) is hazard function of TFWE distribution Eq. (2.4), and S(x; α, β, γ) is the survival function of TFWE Eq. (2.3). It is not possible to get an analytic solution in x to Eq. (3.6) in the general case. It has to be obtained numerically by using methods such as fixed-point or bisection method.

3.3

Skewness and Kurtosis

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

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

(3.7)

and the Moors’ Kurtosis Moors [8] is based on octiles Ku =

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

where q(.) represents quantile function.

(3.8)

7


3.4

The Moments

In this subsection we discuss the rth moment for TFWE distribution. 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 TFWE(α, β, γ) distribution with |γ| ≤ 1, then the rth moments of random variable X, is given by the following ∞ X ∞ X 1 + γ(2i+1 − 1) Γ(r − j + 1) (−1)i+j (i + 1)j β j µr = + βΓ(r − j − 1) . i!j! αr−j−1 (i + 1)r−j−1 α(i + 1)2 0

(3.9)

i=0 j=0

Proof. We start with the well known distribution of the rth moment of the random variable X with probability density function f (x) given by µ0r

Z∞ =

xr f (x; α, β, γ)dx.

(3.10)

0

Substituting from Eq. (2.2) into Eq. (3.10) we get µ0r

Z∞ =

x

r

β β β −eαx− x αx− β −eαx− x x e 1 − γ + 2γe dx, α+ 2 e x

0

Z∞ = (1 − γ)

Z∞ β β β αx− x αx− β β αx− β −e xe dx + 2γ xr α + 2 eαx− x e−2e x dx, x α+ 2 e x x r

0

0

using series expansion of e−e

β αx− x

, we have β αx− x

e−e

=

∞ X (−1)i

i!

i=0

β

ei(αx− x ) ,

we obtain µ0r

= (1 − γ)

Z∞ ∞ X (−1)i i!

i=0

2γ

β β x α + 2 e(i+1)(αx− x ) dx + x r

0

Z∞ ∞ X (−1)i 2i i=0

i!

r

x

β β α + 2 e(i+1)(αx− x ) dx, x

0

Z ∞ ∞ X β (−1)i β i+1 r = 1 + γ(2 − 1) x α + 2 e(i+1)αx e−(i+1) x dx. i! x 0 i=0

β

Using series expansion of e−(i+1) x , we have −(i+1) β x

e

=

∞ X (−1)j (i + 1)j β j j=0

j!

x−j

8


we obtain

µ0r

Z∞ ∞ X ∞ X β (−1)i+j (i + 1)j β j i+1 1 + γ(2 − 1) xr−j α + 2 e(i+1)αx dx = i!j! x i=0 j=0

0

Z∞ ∞ X ∞ X (−1)i+j (i + 1)j β j i+1 r−j r−j−2 = 1 + γ(2 − 1) αx + βx e(i+1)αx dx. i!j! i=0 j=0

0

By using the definition of gamma function in the form, Zwillinger [16], Z ∞ Γ(z) = tz−1 eθt dt, θ, z > 0. θz 0 Finally, we obtain the rth moment of TFWE distribution in the form ∞ X ∞ X βΓ(r − j − 1) (−1)i+j (i + 1)j β j Γ(r − j + 1) 0 i+1 µr = 1 + γ(2 − 1) + . i!j! αr−j (i + 1)r−j+1 αr−j−1 (i + 1)r−j−1 i=0 j=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 TFWE distribution is given by ∞ X ∞ X ∞ X 1 + γ(2i+1 − 1) Γ(r − j + 1) (−1)i+j (i + 1)j β j tr MX (t) = + βΓ(r − j − 1) . i!j!r! αr−j−1 (i + 1)r−j−1 α(i + 1)2 r=0 i=0 i=0 (4.1) Proof. We start with the well known definition of the MX (t) of the random variable X with probability density function f (x) given by Z ∞ etx f (x; α, β, γ)dx. (4.2) MX (t) = 0

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

∞ r Z X t r=0

r!

∞

xr f (x; α, β, γ)dx,

(4.3)

0

Substituting from (3.9) into (4.3), we obtain ∞ X ∞ X ∞ X (−1)i+j (i + 1)j β j tr 1 + γ(2i+1 − 1) Γ(r − j + 1) MX (t) = + βΓ(r − j − 1) . i!j!r! αr−j−1 (i + 1)r−j−1 α(i + 1)2 r=0 i=0 j=0

This completes the proof.

9


5

Order Statistics

In this section, we derive closed form expressions for the PDFs of the rth order statistic of the TFWE distribution. 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 cdf F (x; ϕ) and probability density function pdf 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 TFWE(ϕ) distribution given by Eq. (2.2) and Eq. (2.1) respectively, ϕ = (α, β, γ) 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

[1 − F (x; ϕ)]

=

n−r X i=0

n−r (−1)i [F (x; ϕ)]i . i

(5.2)

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

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

=

n−r X i=0

(−1)i n! i!(r − 1)!(n − r − i)!

f (x; ϕ)[F (x; ϕ)]i+r−1 .

(5.3)

Substituting from Eq. (2.1) and Eq. (2.2) into Eq. (5.3), we obtain the pdf of Xr:n with TFWE(α, β, γ). Relation (5.3) show that fr:n (x; ϕ) is the weighted average of the transmuted flexible Weibull extension distribution withe different shape parameters.

6

Parameters Estimation

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

6.1

Maximum Likelihood Estimation:

Let x1 , x2 , · · · , xn denote a random sample of complete data from the TFWE distribution. The Likelihood function is given as n Y L= f (xi ; α, β, γ), (6.1) i=1

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

n Y i=1

β β αx − β αxi − xβ −eαxi − xi −e i xi i α+ 2 e e 1 − γ + 2γe . xi

10


The log-likelihood function is X X n n n β X αxi − x β β αxi − xβ i −e i + αxi − − L= e ln α + 2 + ln 1 − γ + 2γe . x x i i i=1 i=1 i=1 i=1 n X

(6.2)

The maximum likelihood estimation of the parametres are obtained by differentiating the log-likelihood function L with respect to the parameters α, β and γ and setting the result to zero, we have the following normal equations. ∂L ∂α ∂L ∂β

∂L ∂γ

=

n X i=1

=

n X i=1

=

n X i=1

n

β αxi − x

n

n

αx −

β

i X 2γxi e−e X X x2i e i xi αxi − xβ i − + = 0, x − x e i i β αxi − x αx2i + β i=1 i i=1 1 − γ + 2γe−e i=1

n

n

αx − xβ

Xe i X 1 1 + − xi αx2i + β i=1 xi i=1 2e−e

i

+

n X i=1

β αxi − x

αx −

(6.3)

β

i 2γe−e e i xi = 0, β αxi − x i xi 1 − γ + 2γe−e

(6.4)

β αxi − x i

−1 = 0. β αxi − x i −e 1 − γ + 2γe

(6.5)

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

6.2

Asymptotic confidence bounds

In this section, we derive the asymptotic confidence intervals of these parameters when α, β > 0 and |γ| ≤ 1 as the MLEs of the unknown parameters α, β > 0 and |γ| ≤ 1 can not be obtained in closed forms, by using variance covariance matrix I −1 see Lawless [17], where I −1 is the inverse of the observed information matrix which defined as follows  −1  2  ∂2L ∂2L ˆ cov(ˆ − ∂∂αL2 − ∂α∂β − ∂α∂γ var(ˆ α) cov(ˆ α, β) α, γˆ ) 2  ∂2L ∂2L  ˆ α ˆ ˆ γˆ )  . − ∂∂βL2 − ∂β∂γ I−1 =  − ∂β∂α (6.6) ˆ ) var(β) cov(β,  =  cov(β, 2 2 2 ∂ L ∂ L ∂ L ˆ cov(ˆ γ , α ˆ ) cov(ˆ γ , β) var(ˆ γ ) − ∂γ∂α − ∂γ∂β − ∂γ 2 The second partial derivatives included in I are given as follows. ∂2L ∂α2

= −

∂2L ∂α∂β

= −

n X i=1 n X i=1

n

n

X X x4i αx − β − x2i e i xi + 2 2 (β + αxi ) i=1 i=1 n

n

X αxi − β X x2i xi + e − 2 (β + αxi )2 i=1 i=1

x2i Ai β αx − −e i xi

2 ,

(6.7)

1 − γ + 2γe Ai

β αx − −e i xi

1 − γ + 2γe

2 ,

(6.8)

11


∂2L ∂α∂γ

= −

n X

xi Ei

i=1

∂2L ∂β 2

= −

=

n X

n

∂2L ∂γ 2

= −

αx − xβ

Xe i 1 − (β + αx2i )2 i=1 x2i

n X i=1

(6.9)

1 − γ + 2γe

i=1

∂2L ∂β∂γ

2 ,

β αx − −e i xi

i

−

n X i=1

Ai 2 , β αxi − x i x2i 1 − γ + 2γe−e

Ei xi 1 − γ +

β αxi − x i

(6.11)

2

2 β αxi − x i −e −1 2e

n X

i=1

2γe−e

(6.10)

1−γ+

2γe−e

β αxi − x i

2 ,

(6.12)

where −e

Ai = 2γe

Ei = 2e−e

β αxi − x i

β αxi − x i

e

e

2 β β β αx − αx − αxi − xβ −e i xi −e i xi αxi − xi i 1 − γ + 2γe e − 1 − 2γe e , β β αxi − x αxi − x i i 1 − γ + 2γe−e − γ 2e−e −1 .

αxi − xβ

i

αxi − xβ

i

We can derive the (1 − δ)100% confidence intervals of the parameters α, β and γ, by using variance matrix as in the following forms q p p ˆ ˆ α ˆ ± Z δ var(ˆ α), β ± Z δ var(β), γˆ ± Z δ var(ˆ γ ), 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 a real data set using the TFWE (α, β, γ) model and compare it with the other fitted models like A flexible Weibull extension distributions using Kolmogorov– Smirnov (K–S) statistic, as well as Akaike information criterion(AIC), Akaike [1] and Bayesian information criterion(BIC), Schwarz [24] values. The data have been obtained from Salman Suprawhardana et al. [3], it is for the time between failures (thousands of hours) of secondary reactor pumps, Table 1. Table 1: Time between failures of secondary reactor pumps. 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 Table 2 gives MLEs estimates of parameters of the TFWE and Log-Likelihood, K–S statistic. The statistics AIC , AICC and BIC values of models fitted are in Table 3.


12

Table 2: MLEs of parameters, K–S statistic for secondary reactor pumps. ˆ Model α ˆ βˆ λ γˆ K-S TFWE 0.215 0.231 – - 0.272 0.0794 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 Table 3: Log-likelihood, the corresponding AIC , AICC and BIC values of models fitted. Model L 2L AIC AICC BIC TFWE -31.348 -62.696 68.696 69.9592 72.1025 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 Substituting the MLE’s of the unknown parameters α, β, γ into (6.6), we get estimation of the variance covariance matrix as the following   1.864 × 10−3 −5.666 × 10−4 −4.51 × 10−3  0.018 I0−1 =  −5.666 × 10−4 4.993 × 10−3 −3 −4.51 × 10 0.018 0.189 The approximate 95% two sided confidence intervals of the unknown parameters α, β and γ are [0.13, 0.3], [0.092, 0.37] and [−1.125, 0.581], respectively. To show that the likelihood equation have unique solution, we plot the profiles of the log-likelihood function of α, β and γ in Figures 7, 8 and 9.

Figure 7: The profile of the log-likelihood function of α.


13

Figure 8: The profile of the log-likelihood function of β.

Figure 9: The profile of the log-likelihood function of kγ|. The nonparametric estimate of the survival function using the Kaplan-Meier method and its fitted parametric estimations when the distribution is assumed to be transmuted flexible Weibull extension(TFWE), flexible Weibull (FW), Weibull (W), modified Weibull (MW), reduced additive Weibull (RAW)and extended Weibull (EW) are computed and plotted in Figure 10.


14

Figure 10: The Kaplan-Meier estimate of the survival function for the data. Figures 11 and 12 give the form of the hazard rate and CDF for the T F W E, F W, W, M W, RAW and EW which are used to fit the data after replacing the unknown parameters included in each distribution by their MLE.

Figure 11: The Fitted hazard rate function for the data.


15

Figure 12: The Fitted cumulative distribution function for the data. We find that the TFWE distribution with the four-number of parameters provides a better fit than the previous new modified flexible Weibull extension distribution (FWE) which was the best in Bebbington et al. [4]. It has the largest likelihood, and the smallest KS, AIC, AICC and BIC values among those considered in this paper.

8

Conclusions

A new distribution, based on Transmuted method distribution, has been proposed and its properties 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 Transmuted method, the distribution has flexibility to model the second peak in a distribution. We have shown that the Transmuted flexible Weibull extension distribution fits certain well-known data sets better than existing modifications of the transmuted method of probability distribution.

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