# Journal of Statistical Theory and Applications

Volume 17, Issue 1, March 2018, Pages 77 - 90

# Odd Generalized Exponential Flexible Weibull Extension Distribution

Authors
Abdelfattah Mustafaabdelfatah_mustafa@yahoo.com
Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt
Beih S. El-Desoukyb_desouky@yahoo.com
Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt
Shamsan AL-Garashshamsan_algarash@hotmail.com
Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt
Received 21 December 2016, Accepted 29 December 2017, Available Online 31 March 2018.
DOI
https://doi.org/10.2991/jsta.2018.17.1.6How to use a DOI?
Keywords
Odd Generalized Exponential Family; Flexible Weibull Extension distribution; Generalized Weibull; Odd Generalized Exponential Flexible Weibull distribution; Maximum likelihood estimation
Abstract

In this article we introduce a new four - parameters model called the odd generalized exponential flexible Weibull extension (OGE-FWE) distribution which exhibits bathtub-shaped hazard rate. Some of it’s statistical properties are obtained including ordinary and incomplete moments, quantile and mode, the moment generating functions, reliability and order statistics. The method of maximum likelihood is used for estimating the model parameters and the observed Fisher’s information matrix is given. Moreover, we give the advantage of the OGE-FWE distribution by an application using real data.

Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

## 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, [27]. Recently new classes of distributions were proposed based on modifications of the Weibull distribution to provide a good fit to data set with bathtub hazard failure rate, see [25]. Among of these, Exponentiated Weibull family, [14], Modified Weibull distribution, [11, 19], Beta-Weibull distribution, [8], A flexible Weibull extension, [4], Extended flexible Weibull, [4], Generalized modified Weibull distribution, [5], Kumaraswamy Weibull distribution, [6], Beta modified Weibull distribution, [16,22], Beta generalized Weibull distribution, [23], A new modified weibull distribution, [2] and Exponentiated modified Weibull extension distribution, [20], among others. A good review of these models is presented in [3,15,17].

The flexible Weibull extension (FWE) distribution, [4] has many applications in life testing experiments, reliability analysis, applied statistics and clinical studies. For more details on this distribution, see [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

g(x)=(α+βx2)exp(αxβxeαxβx),x>0,
while the cumulative distribution function (cdf) is given by
G(x)=1exp{eαxβx},x>0.
The survival function is given by the equation
S(x)=1G(x)=exp{eαxβx},x>0,
and the hazard rate function is
h(x)=(α+βx2)eαxβx.
Gupta and Kundu [9] proposed a generalization of the exponential distribution named as Generalized Exponential (GE) distribution. The GE distribution with parameters ϑ, γ > 0, has the following distribution function
F(x;ϑ,γ)=(1eϑx)γ,x>0,ϑ>0,γ>0.
Recently, a new class of univariate continuous distributions named as the odd generalized exponential (OGE) class introduced in [7, 24]. This class is flexible because of the hazard rate shapes could be increasing, decreasing, bathtub and upside down bathtub. The odd generalized exponential (OGE) class is defined as follows.

If G(x), x > 0 is cumulative distribution function (cdf) of a random variable X, then the corresponding survival function is G¯(x)=1G(x) and the probability density function is g(x), then we define the cdf of the OGE class by replacing x in the distribution function of generalized exponential (GE) distribution given in equation (1.5) by G(x)G¯(x) leading to

F(x;ϑ,γ)=[1exp{ϑG(x)G¯(x)}]γ,x>0,ϑ>0,γ>0.
The probability density function corresponding to (1.6) is given by
f(x;ϑ,γ)=ϑγg(x)G¯(x)2exp{ϑG(x)G¯(x)}[1exp{ϑG(x)G¯(x)}]γ1,
where x > 0, ϑ > 0, γ > 0. In this article we present a new distribution depending on flexible Weibull extension distribution referred to as the odd generalized exponential flexible Weibull extension (OGE-FWE) distribution by using the class of univariate distributions defined above.

This paper can be organized as follows, we define the cumulative, density and hazard functions of the odd generalized exponential flexible Weibull extension (OGE-FWE) distribution in Section 2. In Sections 3, we present some statistical properties including, quantile function and median, the mode, rth moment, skewness and kurtosis. In Sections 4, we introduce the 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. We use real data sets and analyzed it by an application in Section 7 and the results are compared with existing distributions. Finally, we present a conclusion in Section 8.

## 2. The Odd Generalized Exponential Flexible Weibull Extension Distribution

In this section we studied the four parameters odd generalized exponential flexible Weibull extension OGE-FWE (ϑ, γ, α, β) distribution. Using G(x) from Eq. (1.2) and g(x) from Eq. (1.1) to obtain the cdf and pdf of Eqs. (1.6) and (1.7), respectively. The cumulative distribution function cdf of the odd generalized exponential flexible Weibull extension distribution (OGE-FWE) is given by

F(x;ϑ,γ,α,β)=[1eϑ(eeαxβx1)]γ,x>0,ϑ,γ,α,β>0,
The pdf corresponding to Eq. (2.1) is given by
f(x;ϑ,γ,α,β)=ϑγ(α+βx2)eαxβx+eαxβxeϑ(eeαxβx1)[1eϑ(eeαxβ2)1]γ1,
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 ~ OGE-FWE (ϑ, γ, α, β) are given by

S(x;ϑ,γ,α,β)=1[1eϑ(eeαxβx1)]γ,x>0,
h(x;ϑ,γ,α,β)=ϑγ(α+βx2)eαxβx+eαxβxeϑ(eeαxβx1)[1eϑ(eeαxβx1)]γ11[1eϑ(eeαxβx1)]γ,
r(x;ϑ,γ,α,β)=ϑγ(α+βx2)eαxβx+eαxβxeϑ(eeαxβx1)1eϑ(eeαxβx1),
respectively, x > 0 and ϑ, γ, α, β > 0.

Figures (13) display the cdf, pdf, survival, hazard rate and reversed hazard rate function of the OGE-FWE (ϑ, γ, α,β) distribution for some parameter values.

## 3. Statistical Properties

In this section, we will study some statistical properties for the OGE-FWE distribution, specially quantile function and simulation median, the mode, moments, skewness and kurtosis.

## 3.1. Quantile and median

We determine the explicit formulas of the quantile and simulation median of the OGE-FWE distribution. The quantile xq of the OGE-FWE(ϑ, γ, α, β) distribution is given by

F(xq)=P[xqq]=q,0<q<1.
From Eq. (2.1), we have
[1eϑ(eeαxqβxq1)]γ=q,
we obtain xq by solving the following equation.
αxq2k(q)xqβ=0,
where
k(q)=ln{ln[1ln(1q1γ)ϑ]}.
So, the simulation of the OGE-FWE random variable is straightforward. If U is a uniform random variable on unit interval (0, 1). Then, by means of the inverse transformation method, we can obtain the random variable X as follows
X=k(u)±k(u)2+4αβ2α.
Since the median of OGE-FWE distribution can be obtain by setting q = 0.5 in Eq. (3.3).

## 3.2. The mode

In this subsection, we can obtain the mode of the OGE-FWE distribution by differentiating its probability density function pdf with respect to x and equaling it to zero. The mode is the solution the following equation

f(x)=0.
Since
f(x;ϑ,γ,α,β))=h(x;ϑ,γ,α,β)S(x;ϑ,γ,α,β).
Then from Eq. (3.5), we have
[h(x;ϑ,γ,α,β)h2(x;ϑ,γ,α,β)]S(x;ϑ,γ,α,β)=0,
where h(x; ϑ, γ, α, β) is hazard rate function of OGE-FWE distribution Eq. (2.4), and S(x; ϑ, γ, α, β) is survival function of OGE-FWE Eq. (2.3).

It is difficult to get an analytic solution in x to Eq. (3.6) in the general case. So, it has to be obtained by numerically methods.

## 3.3. Skewness and Kurtosis

In this subsection, we obtain the skewness and kurtosis based on quantile measures. The Moors Kurtosis is given by, [13]

Ku=q(0.875)q(0.625)q(0.375)+q(0.125)q(0.75)q(0.25),
and the Bowely’s skewness based on quartiles is given by, [10]
Sk=q(0.75)2q(0.5)+q(0.25)q(0.75)q(0.25),
where q(.) is quantile function.

## 3.4. The Moments

We derive the rth moment for the OGE-FWE distribution in Theorem (3.1)

### Theorem 3.1.

If X has OGE-FWE (ϑ, γ, α, β) distribution, then The rth moments of random variable X, is given by

μr=i=0j=0k=0=0m=0(1)i+j+k+mγϑj+1(i+1)j(jk+1)βmj!!m!(+1)r2m1αrm1×(γ1i)(jk)[Γ(rm+1)α(+1)2+βΓ(rm1)].

### 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

μr=0xrf(x;ϑ,γ,α,β)dx.
Substituting from Eq. (2.2) into Eq. (3.10) we get
μr=0xrϑγ(α+βx2)eαxβxeeαxβxeϑ(eeαxβx1)[1eϑ(eeαxβx1)]γ1dx,
since 0<[1eϑ(eeαxβx1)]<1 for x > 0, the binomial series expansion of [1eϑ(eeαxβx1)]γ1 yields
[1eϑ(eeαxβx1)]γ1=i=0(1)i(γ11)eϑi(eeαxβx1),
then we get
μr=i=0(1)i(γ1i)ϑγ0xr(α+βx2)eαxβxeeαxβxeϑ(i+1)(eeαxβx1)dx,
using series expansion
eϑ(i+1)(eeαxβx1)=j=0(1)jϑj(i+1)jj!(eeαxβx1)j,
we obtain
μr=i=0j=0(1)i+j(γ1i)γϑj+1(i+1)jj!0xr(α+βx2)eαxβxeeαxβx[eeαxβx1]jdx.
Using series expansion
(eeαxβx1)j=k=0(1)k(jk)e(jk)eαxβx,
hence
μr=i=0j=0k=0(γ1i)(jk)(1)i+j+kγϑj+1(i+1)jj!0xr(α+βx2)eαxβxe(jk+1)eαxβxdx,
using series expansion
e(jk+1)eαxβx==0(jk+1)e(αxβx)!,
we obtain
μr=i=0j=0k=0=0(γ1i)(jk)(1)i+j+kΓϑj+1(i+1)j(jk+1)j!!0xr(α+βx2)e(+1)(αxβx)dx.
Using series expansion
e(+1)βx=m=0(β)m(+1)mm!xm
gives
μr=i=0j=0k=0=0m=0(1)i+j+k+mγϑj+1(i+1)j(jk+1)βm(+1)mj!!m!(γ1i)(jk)×0xrm(α+βx2)e(+1)αxdx,i=0j=0k=0=0m=0(1)i+j+k+mγϑj+1(i+1)j(jk+1)βm(+1)mj!!m!(γ1i)(jk)×[0αxrme(+1)αxdx+0βxrm2e(+1)αxdx],
by using the definition of gamma function in the form, [26],
Γ(z)=xz0etxtz1dt,z,x,>0.
Finally, we obtain the rth moment of OGE-FWE distribution in the form
μr=i=0j=0k=0=0m=0(1)i+j+k+mγϑj+1(i+1)j(jk+1)βm(+1)mj!!m!(γ1i)(jk)×[Γ(rm+1)αrm(+1)rm+1+βΓ(rm1)αrm1(+1)rm1].
This completes the proof.

## 4. The Moment Generating Function

The moment generating function (mgf) of the OGE-FWE distribution is given by Theorem (4.1).

### Theorem 4.1.

The moment generating function (mgf) of OGE-FWE distribution is given by

MX(t)=r=0i=0j=0k=0=0m=0(1)i+j+k+mγϑj+1(i+1)j(jk+1)βmtrj!!m!r!αrm1(+1)r2m1×(γ1i)(jk)[Γ(rm+1)α(+1)2+βΓ(rm1)].

### Proof.

The moment generating function MX(t) of the random variable X with probability density function f(x) is given by

MX(t)=0etxf(x;ϑ,γ,α,β)dx,
Using series expansion of etx, we obtain
MX(t)=r=0trr!0xrf(x)dx=r=0trr!μr
Substituting from (3.9) into (4.3), we obtain
MX(t)=r=0i=0j=0k=0=0m=0(1)i+j+k+mγϑj+1(i+1)j(jk+1)βmtrj!!m!r!αrm1(+1)r2m1×(γ1i)(jk)[Γ(rm+1)α(+1)2+βΓ(rm1)].
This completes the proof.

## 5. Order Statistics

In this section, we derive closed form expressions for the PDFs of the rth order statistic of the OGE-FWE 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 pdf of Xr:n is given as follows

fr:n(x;ϕ)=1B(r,nr+1)[F(x;ϕ)]r1[1F(x;ϕ)]nrf(x;ϕ),
where f(x; φ) and F(x; φ) are the pdf and cdf of OGE-FWE(φ) distribution given by Eq. (2.2) and Eq. (1.7) respectively, φ = (ϑ, γ, α, β) 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 as follows
[1F(x;ϕ)]nr=i=0nr(nri)(1)i[F(x;ϕ)]i.
Substituting from Eq. (5.2) into Eq. (5.1), we obtain
fr:n(x;ϑ,γ,α,β)=i=0nr(1)in!i!(r1)!(nri)!f(x;ϕ)[F(x;ϕ)]i+r1.
Substituting from Eq. (2.1) and Eq. (2.2) into Eq. (5.3), we obtain the probability density function for rth order statistics.

Relation (5.3) show that fr:n(x; φ) is the weighted average of the OGE-FWE distribution with different shape parameters.

## 6. Parameters Estimation

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

## 6.1. Maximum Likelihood Estimation:

Let x1, x2,··· ,xn denote a random sample of complete data from the OGE-FWE distribution. The Likelihood function is given as

L=i=1nf(xi;ϑ,γ,α,β),
substituting from (2.2) into (6.1), we have
L=i=1nϑγ(α+βxi2)eαxiβxieeαxiβxieϑ[eeαxiβxi1][1eϑ[eeαxiβxi1]]γ1.
The log-likelihood function is
=nln(ϑγ)+i=1nln(α+βxi2)+i=1n(α+βxi2)+i=1neαxiβxiϑi=1n(eeαxiβxi1)+(γ1)i=1nln[1eϑ(eeαxiβxi1)].
The maximum likelihood estimation of the parameters (ϑ, γ, α, β) are obtained by differentiated the log-likelihood function with respect to the parameters ϑ, γ, α and β and setting the result to zero, we have the following normal equations.
ϑ=nϑi=1n(eeαxiβxi1)+(γ1)i=1neeαxiβxi1eϑ(eeαxiβxi1)1=0
γ=nγ+i=1nln(1eϑ(eeαxiβxi1))=0
α=i=1nxi2β+αxi2+i=1nxi+i=1nxieαxiβxiϑi=1nxi𝒟i+ϑ(γ1)i=1nxi𝒟ieϑ(eeαxiβxi1)1=0
β=i=1n1β+αxi2i=1n1xii=1n1xieαxiβxi+ϑi=1n𝒟ixiϑ(γ1)i=1n𝒟ixi[eϑ(eeαxiβxi1)1]=0,
where 𝒟i=exp{αxiβxi+eαxiβxi}. The MLEs can be obtained by solving the equations, (6.3)(6.6), numerically for ϑ, γ, α and β.

## 6.2. Asymptotic confidence bounds

In this section, we derive the asymptotic confidence intervals of these parameters when ϑ, γ, α > 0 and β > 0 as the MLEs of the unknown parameters ϑ, γ, α > 0 and β > 0 can not be obtained in closed forms, by using variance covariance matrix I 1 see [12], where I 1 is the inverse of the observed information matrix which defined as follows

I1=(2ϑ22ϑγ2aα2ϑβ2γϑ2γ22γα2γβ2αϑ2αγ2α22αβ2βϑ2βγ2βα2β2)1=(var(ϑ^)cov(ϑ^,γ^)cov(ϑ^,α^)cov(ϑ^,β^)cov(γ^,ϑ^)var(γ^)cov(γ^,α^)cov(γ^,β^)cov(α^,ϑ^)cov(α^,γ^)var(α^)cov(α^,β^)cov(β^,ϑ^)cov(β^,γ^)cov(β^,α^)var(β^)),
where
2ϑ2=nϑ2(γ1)i=1neϑ(eeαxiβxi1)𝒜i2,2ϑγ=i=1n𝒜i
2ϑα=i=1nxi𝒟i+(γ1)i=1nxii,2ϑβ=i=1n𝒟ixi(γ1)i=1nixi
2γ2=nγ2,2γα=ϑi=1nxi𝒟ieϑ(eeαxiβxi1)1
2γβ=ϑi=1n𝒟ixi[eϑ(eeαxiβxi1)1]
2α2=i=1nxi4(β+αxi2)2+i=1nxi2eαxiβxiϑi=1nxi2𝒟i[eαxiβxi+1]+ϑ(γ1)i=1nxi2𝒦i
2αβ=i=1nxi2(β+αxi2)2i=1neαxiβxi+ϑi=1n𝒟i[eαxiβxi+1]ϑ(γ1)i=1n𝒦i
2β2=i=1nxi2(β+αxi2)2+i=1neαxiβxixi2ϑi=1n𝒟i[eαxiβxi+1]xi2+ϑ(γ1)i=1n𝒦ixi2
where
𝒜i=[eeαxiβxi1][eϑ(eeαxiβxi1)1]1,
i=𝒟i[eϑ(eeαxiβxi1)(1ϑ(eeαxiβxi1))1][eϑ[eeαxiβxi1]1]2,
𝒦i=𝒟i[(eϑ(eeαxiβxi1)1)(eαxiβxi+1)ϑ𝒟ieϑ(eeαxiβxi1)][eϑ(eeαxiβxi1)1]2.
We can obtain the (1 − δ)100% confidence intervals of the parameters ϑ, γ, α and β by using variance matrix as in the following forms
ϑ^±Zδ2var(ϑ^),γ^±Zδ2var(γ^),α^±Zδ2var(α^),β^±Zδ2var(β^),
where Zδ2 is the upper (δ2)-th percentile of the standard normal distribution.

## 7. Application

In this section, we will analysis of a real data set using the OGE-FWE (ϑ, γ, α, β) 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 Information Citerion with correction (AICC), Bayesian information criterion (BIC), Hannan-Quinn information criterion (HQIC) and Schwarz information criterion (SIC) values, [21]. The data have been obtained from [18], it is for the time between failures (thousands of hours) of secondary reactor pumps, Table 1.

 2.16 0.746 0.402 0.954 0.491 6.56 4.992 0.347 0.15 0.358 0.101 1.359 3.465 1.06 0.614 1.921 4.082 0.199 0.605 0.273 0.07 0.062 5.32
Table 1.

Time between failures of secondary reactor pumps.

Table 2 gives MLEs of parameters of the OGE-FWE and K-S Statistics. The values of the log-likelihood functions, AIC, AICC, BIC, HQIC, and SIC are in Table 3.

Model α^ β^ λ^ ϑ^ γ^ K-S
OGE-FWE 0.2380 2.0700 0.069 0.113 0.0760
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 2.

MLEs and K–S of parameters for secondary reactor pumps.

Model L AIC AICC BIC HQIC SIC
OGE-FEW −29.2980 66.5960 68.8182 71.1380 10.5590 71.1380
Flexible Weibull −83.3424 170.6848 171.2848 172.9558 12.5416 172.9558
Weibull −85.4734 174.9468 175.5468 177.2178 12.5915 177.2178
Modified Weibull −85.4677 176.9354 178.1986 180.3419 12.6029 180.3419
Reduced additive Weibull −86.0728 178.1456 179.4088 181.5521 12.6168 181.5521
Extended Weibull −86.6343 179.2686 180.5318 182.6751 12.6296 182.6751
Table 3.

Log-likelihood, AIC, AICC, BIC, HQIC and SIC values of models fitted.

Substituting the MLEs of the unknown parameters ϑ, γ, α, β into (6.7), we obtain estimation of the variance covariance matrix as the following

I01=(6.773×1034.189×1043.945×1031.65×1034.189×1044.308×1044.871×1040.0133.945×1034.871×1043.343×1030.0171.65×1030.0130.0170.806).
The approximate 95% two sided confidence intervals of the unknown parameters ϑ, γ, α and β are [0, 0.230], [0.072, 0.154], [0.125, 0.351] and [0.31, 3.83], respectively.

The nonparametric estimate of the survival function S(x) using the Kaplan-Meier method and its fitted parametric estimations when the distribution is assumed to be OGE-FWE, FW, W, MW, RAW and EW are computed and plotted in Figure 4.

Figure 5 gives the form of the hazard rate h(x) and cumulative density function cdf for the OGE-FWE, FW, W, MW, RAW and EW which are used to fit the data after the unknown parameters included in each distribution are replaced by their MLEs.

We find that the OGE-FWE distribution with the four - parameters provides a better fit than the previous new modified a flexible Weibull extension distribution(FWE) which was the best in [4]. It has the largest likelihood, and the smallest AIC, AICC, BIC, HQIC and SIC values among those considered in this paper.

## 8. Conclusions

We proposed a new distribution, based on odd generalized exponential family distributions, this distribution is named the odd generalized exponential flexible Weibull extension OGE-FWE distribution. Some its statistical properties are studied. The maximum likelihood method is used for estimating the parameters model. Finally, we introduce an application using real data. We show that the OGE-FWE distribution fits certain well known data sets better than existing modifications of the Weibull and flexible Weibull extension distributions.

Journal
Journal of Statistical Theory and Applications
Volume-Issue
17 - 1
Pages
77 - 90
Publication Date
2018/03/31
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
https://doi.org/10.2991/jsta.2018.17.1.6How to use a DOI?
Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

TY  - JOUR
AU  - Abdelfattah Mustafa
AU  - Beih S. El-Desouky
AU  - Shamsan AL-Garash
PY  - 2018
DA  - 2018/03/31
TI  - Odd Generalized Exponential Flexible Weibull Extension Distribution
JO  - Journal of Statistical Theory and Applications
SP  - 77
EP  - 90
VL  - 17
IS  - 1
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.2018.17.1.6
DO  - https://doi.org/10.2991/jsta.2018.17.1.6
ID  - Mustafa2018
ER  -