# The Exponential Flexible Weibull Extension Distribution

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

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura
35516, Egypt.

###### Abstract

This paper is devoted to study a new three- parameters model called the
Exponential Flexible Weibull extension (EFWE) distribution which exhibits
bathtub-shaped hazard rate. Some of it’s statistical properties are obtained
including ordinary and incomplete moments, quantile and 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 derived. We illustrate the usefulness of the proposed model by applications
to real data.

Keywords: Exponential distribution; Flexible Weibull distribution; Exponential
Weibull; Reliability; Hazard function; Moments; Maximum likelihood estimation.

2010 MSC: 60E05, 62N05, 62H10

##  1 Introduction

The Weibull distribution (WD) introduced by Weibull [23], is a popular
distribution for modeling lifetime data where the hazard rate function is
monotone. Recently appeared new classes of distributions were based on
modifications of the Weibull distribution (WD) to provide a good fit to data
set with bathtub hazard failure rate Xie and Lai [21]. Among of these,
Modified Weibull (MW) distribution, Lai et al. [9] and Sarhan and Zaindin
[17], Beta-Weibull (BW)distribution, Famoye et al. [6], Beta modified Weibull
(BMW)distribution, Silva et al. [20] and Nadarajah et al. [14], Kumaraswamy
Weibull (KW) distribution, Cordeiro et al. [5], Generalized modified Weibull
(GMW) distribution, Carrasco et al. [4] and Exponentiated modified Weibull
extension (EMWE) distribution, Sarhan and Apaloo [18], among others. A good
review of these models is presented in Pham and Lai [15] and Murthy et al.
[13].

The Flexible Weibull (FWE) distribution, Bebbington et al. [3] 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 [3]. A random variable $X$ is
said to have the Flexible Weibull Extension (FWE) distribution with parameters
$\alpha,\beta>0$ if it’s probability density function (pdf) is given by

$g(x)=(\alpha+\frac{\beta}{x^{2}})e^{\alpha
x-\frac{\beta}{x}}\exp\left\\{-e^{\alpha x-\frac{\beta}{x}}\right\\},\quad
x>0,$ (1.1)

while the cumulative distribution function (cdf) is given by

$G(x)=1-\exp\left\\{-e^{\alpha x-\frac{\beta}{x}}\right\\},\quad x>0.$ (1.2)

The survival function is given by the equation

$S(x)=1-G(x)=\exp\left\\{-e^{\alpha x-\frac{\beta}{x}}\right\\},\quad x>0,$
(1.3)

and the hazard function is

$h(x)=(\alpha+\frac{\beta}{x^{2}})e^{\alpha x-\frac{\beta}{x}}.$ (1.4)

In this article, a new generalization of the Flexible Weibull Extension (FWE)
distribution called exponential flexible Weibull extension (EFWE) distribution
is derived. Using the exponential generator applied to the odds ratio
$\frac{1}{1-G(x)}$, such as the exponential Pareto distribution by AL-Kadim
and Boshi [7], exponential lomax distribution by El-Bassiouny et al. [10]. If
$G(x)$ is the baseline cumulative distribution function (cdf) of a random
variable, with probability density function (pdf) $g(x)$ and the exponential
cumulative distribution function is

$F(x;\lambda)=1-e^{-\lambda x},\quad x\geq 0,\quad\lambda\geq 0.$ (1.5)

Based on this density, by replacing $x$ with ratio $\frac{1}{1-G(x)}$. The cdf
of exponential generalized distribution is defined by (see AL-Kadim and Boshi
[7] and El-Bassiouny et al. [10])

$\displaystyle F(x)$ $\displaystyle=$
$\displaystyle\int\limits_{0}^{\frac{1}{1-G(x)}}\lambda e^{-\lambda t}dt$
(1.6) $\displaystyle=$ $\displaystyle
1-\exp\left\\{-\lambda\left[\frac{1}{1-G(x}\right]\right\\},\quad x\geq
0,\quad\lambda\geq 0,$

where $G(x)$ is a baseline cdf. Hence the pdf corresponding to Eq. (1.6) is
given by

$f(x)=\frac{\lambda\cdot
g(x)}{\left[1-G(x)\right]^{2}}\cdot\exp\left\\{-\lambda\left[\frac{1}{1-G(x)}\right]\right\\}.$
(1.7)

This paper is organized as follows, we define the cumulative, density and
hazard functions of the exponential flexible Weibull extension (EFWE)
distribution in Section 2. In Sections 3 and 4, we introduce the statistical
properties including , quantile function 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, we introduce the
conclusions in Section 8.

##  2 The Exponential Flexible Weibull Extension Distribution

In this section we study the three parameters Exponential Flexible Weibull
Extension (EFWE) distribution. Using $G(x)$ Eq. (1.2) and $g(x)$ Eq. (1.1) in
Eq. (1.6) and Eq. (1.7) to obtained the cdf and pdf of EFWE distribution. The
cumulative distribution function cdf of the Exponential Flexible Weibull
Extension distribution (EFWE) is given by

$F(x;\alpha,\beta,\lambda)=1-\exp\left\\{-\lambda e^{e^{\alpha
x-\frac{\beta}{x}}}\right\\},\;x>0,\;\alpha,\beta,\lambda>0.$ (2.1)

The pdf corresponding to Eq. (2.1) is given by

$f(x;\alpha,\beta,\lambda)=\lambda\left(\alpha+\frac{\beta}{x^{2}}\right)e^{\alpha
x-\frac{\beta}{x}}e^{e^{\alpha x-\frac{\beta}{x}}}\exp\left\\{-\lambda
e^{e^{\alpha x-\frac{\beta}{x}}}\right\\},$ (2.2)

where $x>0$ and $,\alpha,\beta>0$ are two additional shape parameters.

The survival function $S(x)$, hazard rate function $h(x)$, reversed- hazard
rate function $r(x)$ and cumulative hazard rate function $H(x)$ of $X\sim
EFWE(\alpha,\beta,\lambda)$ are given by

$S(x;\alpha,\beta,\lambda)=1-F(x;\alpha,\beta,\lambda)=\exp\left\\{-\lambda
e^{e^{\alpha x-\frac{\beta}{x}}}\right\\},\;x>0,$ (2.3)
$h(x;\alpha,\beta,\lambda)=\lambda\left(\alpha+\frac{\beta}{x^{2}}\right)e^{\alpha
x-\frac{\beta}{x}}e^{e^{\alpha x-\frac{\beta}{x}}},$ (2.4)
$r(x;\alpha,\beta,\lambda)=\frac{\lambda\left(\alpha+\frac{\beta}{x^{2}}\right)e^{\alpha
x-\frac{\beta}{x}}e^{e^{\alpha x-\frac{\beta}{x}}}\exp\left\\{-\lambda
e^{e^{\alpha x-\frac{\beta}{x}}}\right\\}}{1-\exp\left\\{-\lambda e^{e^{\alpha
x-\frac{\beta}{x}}}\right\\}},$ (2.5)
$H(x;\alpha,\beta,\lambda)=\int_{0}^{x}h(u)du=\lambda\exp\left\\{e^{\alpha
x-\frac{\beta}{x}}\right\\},$ (2.6)

respectively, $x>0$ and $\alpha,\beta,\lambda>0$.

Figures 1–6 display the cdf, pdf, survival function, hazard rate function,
reversed hazard rate function and cumulative hazard rate function of the
EFWE($\alpha$, $\beta$, $\lambda$) distribution for some parameter values.

Figure 1: The cdf of the EFWE distribution for different values of parameters.

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

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

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

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

Figure 6: The cumulative hazard rate function of the EFWE distribution for
different values of parameters.

##  3 Statistical Properties

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

###  3.1 Quantile and simulation

The quantile $x_{q}$ of the EFWE ($\alpha,\beta,\lambda$) distribution random
variable is given by

$F(x_{q})=q,\quad 0<q<1.$ (3.1)

Using the distribution function of EFWE distribution, from (2.1), we have

$\alpha x_{q}^{2}-k(q)x_{q}-\beta=0,$ (3.2)

where

$k(q)=\ln\left\\{\ln\left(-\frac{\ln(1-q)}{\lambda}\right)\right\\}.$ (3.3)

So, the simulation of the EFWE distribution random variable is
straightforward. Let $U$ be a uniform variate on the unit interval $(0,1)$.
Thus, by means of the inverse transformation method, we consider the random
variable $X$ given by

$X=\frac{k(u)\pm\sqrt{k(u)^{2}+4\alpha\beta}}{2\alpha}.$ (3.4)

Since the median is $50$ % quantile then by setting $q=0.5$ in Eq. (3.2), can
be obtained the median $M$ of EFWE distribution.

###  3.2 The Mode of EFWE

In this subsection, we will derive the mode of the
EFWE($\alpha,\beta,\lambda$) distribution by deriving its pdf with respect to
$x$ and equal it to zero thus the mode of the EFWE($\alpha,\beta,\lambda$)
distribution can be obtained as a nonnegative solution of the following
nonlinear equation

$\displaystyle\lambda e^{\alpha x-\frac{\beta}{x}}e^{e^{\alpha
x-\frac{\beta}{x}}}\exp\left\\{-\lambda e^{e^{\alpha
x-\frac{\beta}{x}}}\right\\}\left[-\frac{2\beta}{x^{3}}+\left(\alpha+\frac{\beta}{x^{2}}\right)\left(1+e^{\alpha
x-\frac{\beta}{x}}-\lambda e^{e^{\alpha x-\frac{\beta}{x}}}\right)\right]=0$
(3.5)

From Figure LABEL:fig2, the pdf for EFWE distribution has only one peak, It is
a unimodal distribution, so the above equation has only one solution. It is
not possible to get an explicit solution of (3.5) in the general case.
Numerical methods should be used such as bisection or fixed-point method to
solve it. Some values of median and mode for various values of parameters
$\alpha,\beta$ and $\lambda$ calculated in Table 1.

Table 1: The median and mode for EFWE($\alpha,\beta,\lambda$)

$\alpha$ | $\beta$ | $\lambda$ | median | mode
---|---|---|---|---
0.015 | 0.381 | 0.076 | 53.3576 | 10.6657
0.158 | 0.158 | 0.273 | 0.801066 | 1.96923
0.700 | 1.000 | 0.150 | 1.537340 | 1.87122
1.000 | 0.700 | 0.130 | 1.132920 | 1.35312
1.000 | 0.800 | 0.200 | 1.009750 | 1.27259
1.200 | 1.000 | 0.100 | 1.228750 | 1.38465

###  3.3 The Skewness and Kurtosis

The analysis of the variability Skewness and Kurtosis on the shape parameters
$\alpha,\beta$ can be investigated based on quantile measures. The short
comings of the classical Kurtosis measure are well-known. The Bowely’s
skewness based on quartiles is given by, Kenney and Keeping [8]

$S_{k}=\frac{q_{(0.75)}-2q_{(0.5)}+q_{(0.25)}}{q_{(0.75)}-q_{(0.25)}},$ (3.6)

and the Moors Kurtosis based on quantiles, Moors [12]

$K_{u}=\frac{q_{(0.875)}-q_{(0.625)}-q_{(0.375)}+q_{(0.125)}}{q_{(0.75)}-q_{(0.25)}},$
(3.7)

where $q_{(.)}$ represents quantile function.

###  3.4 The Moments

In this subsection we discuss the $rth$ moment for EFWE 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 EFWE $(\alpha,\beta,\lambda)$ distribution, then the $r$th moments
of random variable $X$, is given by the following

$\mu_{r}^{{}^{\prime}}=\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}\frac{(-1)^{i+k}\lambda^{i+1}\beta^{k}(i+1)^{j}(j+1)^{k}}{i!j!k!}\left[\frac{\Gamma(r-k+1)}{\alpha^{r-k}(j+1)^{r-k+1}}+\frac{\beta\Gamma(r-k-1)}{\alpha^{r-k-1}(j+1)^{r-k-1}}\right].$
(3.8)

###### Proof.

We start with the well known distribution of the $r$th moment of the random
variable $X$ with probability density function $f(x)$ given by

$\mu_{r}^{{}^{\prime}}=\int\limits_{0}^{\infty}x^{r}f(x;\alpha,\beta,\lambda)dx.$
(3.9)

Substituting from Eq. (2.2) into Eq. (3.9) we get

$\mu_{r}^{{}^{\prime}}=\int_{0}^{\infty}x^{r}\lambda\left(\alpha+\frac{\beta}{x^{2}}\right)e^{\alpha
x-\frac{\beta}{x}}e^{e^{\alpha x-\frac{\beta}{x}}}\exp\left\\{-\lambda
e^{e^{\alpha x-\frac{\beta}{x}}}\right\\}dx,$

the expansion of $\exp\left\\{-\lambda e^{e^{\alpha
x-\frac{\beta}{x}}}\right\\}$ is

$e^{-\lambda e^{e^{\alpha
x-\frac{\beta}{x}}}}=\sum_{i=0}^{\infty}\frac{(-1)^{i}\lambda^{i}}{i!}e^{ie^{\alpha
x-\frac{\beta}{x}}},$

then we get

$\mu_{r}^{{}^{\prime}}=\sum_{i=0}^{\infty}\frac{(-1)^{i}\lambda^{i+1}}{i!}\int_{0}^{\infty}x^{r}\left(\alpha+\frac{\beta}{x^{2}}\right)e^{\alpha
x-\frac{\beta}{x}}e^{(i+1)e^{\alpha x-\frac{\beta}{x}}}dx,$

using series expansion of $e^{(i+1)e^{\alpha x-\frac{\beta}{x}}}$,

$e^{(i+1)e^{\alpha
x-\frac{\beta}{x}}}=\sum_{j=0}^{\infty}\frac{(i+1)^{j}}{j!}e^{j(\alpha
x-\frac{\beta}{x})},$

we obtain

$\displaystyle\mu_{r}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\frac{(-1)^{i}\lambda^{i+1}(i+1)^{j}}{i!j!}\int_{0}^{\infty}x^{r}\left(\alpha+\beta
x^{-2}\right)e^{(j+1)(\alpha x-\frac{\beta}{x})}dx,$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\frac{(-1)^{i}\lambda^{i+1}(i+1)^{j}}{i!j!}\int_{0}^{\infty}x^{r}\left(\alpha+\beta
x^{-2}\right)e^{(j+1)\alpha x}e^{-(j+1)\frac{\beta}{x}}dx,$

using series expansion of $e^{-(j+1)\frac{\beta}{x}}$,

$e^{-(j+1)\frac{\beta}{x}}=\sum_{k=0}^{\infty}\frac{(-1)^{k}(j+1)^{k}\beta^{k}}{k!}x^{-k},$

we have

$\displaystyle\mu_{r}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}\frac{(-1)^{i+k}\lambda^{i+1}\beta^{k}(i+1)^{j}(j+1)^{k}}{i!j!k!}\int_{0}^{\infty}x^{r-k}\left(\alpha+\frac{\beta}{x^{2}}\right)e^{(j+1)\alpha
x}dx,$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}\frac{(-1)^{i+k}\lambda^{i+1}\beta^{k}(i+1)^{j}(j+1)^{k}}{i!j!k!}\left[\int_{0}^{\infty}\alpha
x^{r-k}e^{(j+1)\alpha x}dx\right.$ $\displaystyle\left.+\int_{0}^{\infty}\beta
x^{r-k-2}e^{(j+1)\alpha x}dx\right],$

by using the definition of gamma function ( Zwillinger [22]), in the form,

$\Gamma(z)=x^{z}\int_{0}^{\infty}e^{tx}t^{z-1}dt,\quad z,x,>0.$

Finally, we obtain the $r$th moment of EFWE distribution in the form

$\mu_{r}^{{}^{\prime}}=\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}\frac{(-1)^{i+k}\lambda^{i+1}\beta^{k}(i+1)^{j}(j+1)^{k}}{i!j!k!}\times\left[\frac{\Gamma(r-k+1)}{\alpha^{r-k}(j+1)^{r-k+1}}+\frac{\beta\Gamma(r-k-1)}{\alpha^{r-k-1}(j+1)^{r-k-1}}\right].$

This completes the proof.

∎

##  4 The Moment Generating Function

The moment generating function (mgf) $M_{X}(t)$ 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 EFWE distribution is given by

$\displaystyle M_{X}(t)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}\sum_{r=0}^{\infty}\frac{(-1)^{i+k}\lambda^{i+1}\beta^{k}(i+1)^{j}(j+1)^{k}t^{r}}{i!j!k!r!}\times$
(4.1) $\displaystyle\hskip
85.35826pt\left[\frac{\Gamma(r-k+1)}{\alpha^{r-k}(j+1)^{r-k+1}}+\frac{\beta\Gamma(r-k-1)}{\alpha^{r-k-1}(j+1)^{r-k-1}}\right].$

###### Proof.

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

$M_{X}(t)=\int\limits_{0}^{\infty}e^{tx}f(x)dx.$ (4.2)

using series expansion of $e^{tx}$, we obtain

$\displaystyle M_{X}(t)$ $\displaystyle=$
$\displaystyle\sum_{r=0}^{\infty}\frac{t^{r}}{r!}\int_{0}^{\infty}x^{r}f(x)dx.$
(4.3) $\displaystyle=$
$\displaystyle\sum_{r=0}^{\infty}\frac{t^{r}}{r!}\mu_{r}^{{}^{\prime}}.$

Substituting from Eq. (3.8) into Eq. (4.3) we obtain the moment generating
function (mgf) of EFWE distribution in the form

$\displaystyle M_{X}(t)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}\sum_{r=0}^{\infty}\frac{(-1)^{i+k}\lambda^{i+1}\beta^{k}(i+1)^{j}(j+1)^{k}t^{r}}{i!j!k!r!}\times$
$\displaystyle\hskip
85.35826pt\left[\frac{\Gamma(r-k+1)}{\alpha^{r-k}(j+1)^{r-k+1}}+\frac{\beta\Gamma(r-k-1)}{\alpha^{r-k-1}(j+1)^{r-k-1}}\right].$

This completes the proof. ∎

##  5 Order Statistics

In this section, we derive closed form expressions for the probability density
function of the $r$th order statistic of the EFWE distribution. Let
$X_{1:n},X_{2:n},\cdots,X_{n:n}$ denote the order statistics obtained from a
random sample $X_{1}$, $X_{2}$, $\cdots$, $X_{n}$ which taken from a
continuous population with cumulative distribution function cdf $F(x;\varphi)$
and probability density function pdf $f(x;\varphi)$, then the probability
density function of $X_{r:n}$ is given by

$f_{r:n}(x;\varphi)=\frac{1}{B(r,n-r+1)}\left[F(x;\varphi)\right]^{r-1}\left[1-F(x;\varphi)\right]^{n-r}f(x;\varphi),$
(5.1)

where $f(x;\varphi)$, $F(x;\varphi)$ are the pdf and cdf of EFWE
$(\alpha,\beta,\lambda)$ distribution given by Eq. (2.2) and Eq. (2.1)
respectively, $\varphi=(\alpha,\beta,\lambda)$ and $B(.,.)$ is the Beta
function, also we define first order statistics
$X_{1:n}=\min(X_{1},X_{2},\cdots,X_{n})$, and the last order statistics as
$X_{n:n}=\max(X_{1},X_{2},\cdots,X_{n})$. Since $0<F(x;\varphi)<1$ for $x>0$,
we can use the binomial expansion of $[1-F(x;\varphi)]^{n-r}$ given as follows

$\left[1-F(x;\varphi)\right]^{n-r}=\sum_{i=0}^{n-r}\begin{pmatrix}n-r\\\
i\end{pmatrix}(-1)^{i}[F(x;\varphi)]^{i}.$ (5.2)

Substituting from Eq. (5.2) into Eq. (5.1), we obtain

$f_{r:n}(x;\varphi)=\frac{1}{B(r,n-r+1)}f(x;\varphi)\sum_{i=0}^{n-r}\begin{pmatrix}n-r\\\
i\end{pmatrix}(-1)^{i}\left[F(x;\varphi)\right]^{i+r-1}.$ (5.3)

Substituting from Eq. (2.1) and Eq. (2.2) into Eq. (5.3), we obtain

$f_{r:n}(x;\varphi)=\sum_{i=0}^{n-r}\frac{(-1)^{i}n!}{i!(r-1)!(n-r-i)!}\left[F(x,\varphi)\right]^{i+r-1}f(x;\varphi).$
(5.4)

Relation (5.4) shows that $f_{r:n}(x;\varphi)$ is the weighted average of the
Exponential Flexible Weibull Extension distribution withe different shape
parameters.

##  6 Parameters Estimation

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

###  6.1 Maximum likelihood estimation

Let $x_{1},x_{2},\cdots,x_{n}$ denote a random sample of complete data from
the EFWE distribution. The Likelihood function is given as

$L=\prod_{i=1}^{n}f(x_{i};\alpha,\beta,\lambda),$ (6.1)

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

$L=\prod_{i=1}^{n}\lambda\left(\alpha+\frac{\beta}{x^{2}}\right)e^{\alpha
x-\frac{\beta}{x}}e^{e^{\alpha x-\frac{\beta}{x}}}\exp\left\\{-\lambda
e^{e^{\alpha x-\frac{\beta}{x}}}\right\\}.$

The log-likelihood function is

$\mathcal{L}=n\ln(\lambda)+\sum_{i=1}^{n}\ln(\alpha+\frac{\beta}{x_{i}^{2}})+\sum_{i=1}^{n}\left(\alpha
x_{i}-\frac{\beta}{x_{i}}\right)+\sum_{i=1}^{n}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}-\lambda\sum_{i=1}^{n}e^{e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}.$ (6.2)

The maximum likelihood estimation of the parameters are obtained by
differentiating the log-likelihood function $\mathcal{L}$ with respect to the
parameters $\alpha,\beta$ and $\lambda$ and setting the result to zero, we
have the following normal equations.

$\displaystyle\frac{\partial\mathcal{L}}{\partial\alpha}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\frac{x_{i}^{2}}{\beta+\alpha
x_{i}^{2}}+\sum_{i=1}^{n}x_{i}+\sum_{i=1}^{n}x_{i}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}-\lambda\sum_{i=1}^{n}x_{i}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}e^{e^{\alpha x_{i}-\frac{\beta}{x_{i}}}}=0,$ (6.3)
$\displaystyle\frac{\partial\mathcal{L}}{\partial\beta}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\frac{1}{\beta+\alpha
x_{i}^{2}}-\sum_{i=1}^{n}\frac{1}{x_{i}}-\sum_{i=1}^{n}\frac{1}{x_{i}}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}+\lambda\sum_{i=1}^{n}\frac{1}{x_{i}}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}e^{e^{\alpha x_{i}-\frac{\beta}{x_{i}}}}=0,$ (6.4)
$\displaystyle\frac{\partial\mathcal{L}}{\partial\lambda}$ $\displaystyle=$
$\displaystyle\frac{n}{\lambda}-\sum_{i=1}^{n}e^{e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}=0.$ (6.5)

The MLEs can be obtained by solving the nonlinear equations previous,
(6.3)-(6.5), numerically for $\alpha,\beta$ and $\lambda$.

###  6.2 Asymptotic confidence bounds

In this section, we derive the asymptotic confidence intervals when
$\alpha,\beta>0$ and $\lambda>0$ as the MLEs of the unknown parameters
$\alpha,\beta>0$ and $\lambda>0$ can not be obtained in closed forms, by using
variance covariance matrix $I^{-1}$ see Lawless [11], where $I^{-1}$ is the
inverse of the observed information matrix which defined as follows

$\displaystyle\mathbf{I^{-1}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}-\frac{\partial^{2}\mathcal{L}}{\partial\alpha^{2}}&-\frac{\partial^{2}\mathcal{L}}{\partial\alpha\partial\beta}&-\frac{\partial^{2}\mathcal{L}}{\partial\alpha\partial\lambda}\\\
-\frac{\partial^{2}\mathcal{L}}{\partial\beta\partial\alpha}&-\frac{\partial^{2}\mathcal{L}}{\partial\beta^{2}}&-\frac{\partial^{2}\mathcal{L}}{\partial\beta\partial\lambda}\\\
-\frac{\partial^{2}\mathcal{L}}{\partial\lambda\partial\alpha}&-\frac{\partial^{2}\mathcal{L}}{\partial\lambda\partial\beta}&-\frac{\partial^{2}\mathcal{L}}{\partial\lambda^{2}}\end{array}\right)^{-1}=\left(\begin{array}[]{ccc}var(\hat{\alpha})&cov(\hat{\alpha},\hat{\beta})&cov(\hat{\alpha},\hat{\lambda})\\\
cov(\hat{\beta},\hat{\alpha})&var(\hat{\beta})&cov(\hat{\beta},\hat{\lambda})\\\
cov(\hat{\lambda},\hat{\alpha})&cov(\hat{\lambda},\hat{\beta})&var(\hat{\lambda})\end{array}\right).$
(6.12)

The second partial derivatives included in $I$ are given as follows.

$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial\alpha^{2}}$
$\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}\frac{x_{i}^{4}}{\left(\beta+\alpha
x_{i}^{2}\right)^{2}}+\sum_{i=1}^{n}x_{i}^{2}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}-\lambda\sum_{i=1}^{n}x_{i}^{2}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}e^{e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}\left[1+e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}\right],$ (6.13)
$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial\alpha\partial\beta}$
$\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}\frac{x_{i}^{2}}{\left(\beta+\alpha
x_{i}^{2}\right)^{2}}-\sum_{i=1}^{n}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}+\lambda\sum_{i=1}^{n}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}e^{e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}\left[1+e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}\right],$ (6.14)
$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial\alpha\partial\lambda}$
$\displaystyle=$ $\displaystyle-\sum_{i=1}^{n}x_{i}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}e^{e^{\alpha x_{i}-\frac{\beta}{x_{i}}}},$ (6.15)
$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial\beta^{2}}$
$\displaystyle=$ $\displaystyle-\sum_{i=1}^{n}\frac{1}{\left(\beta+\alpha
x_{i}^{2}\right)^{2}}+\sum_{i=1}^{n}\frac{1}{x_{i}^{2}}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}-\lambda\sum_{i=1}^{n}\frac{1}{x_{i}^{2}}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}e^{e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}\left[1+e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}\right],$
$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial\beta\partial\lambda}$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{n}\frac{1}{x_{i}}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}e^{e^{\alpha x_{i}-\frac{\beta}{x_{i}}}},$ (6.17)
$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial\lambda^{2}}$
$\displaystyle=$ $\displaystyle-\frac{n}{\lambda^{2}}.$ (6.18)

We can derive the $(1-\delta)100\%$ confidence intervals of the parameters
$\alpha,\beta$ and $\lambda$, by using variance matrix as in the following
forms

$\hat{\alpha}\pm
Z_{\frac{\delta}{2}}\sqrt{var(\hat{\alpha})},\quad\hat{\beta}\pm
Z_{\frac{\delta}{2}}\sqrt{var(\hat{\beta})},\quad\hat{\lambda}\pm
Z_{\frac{\delta}{2}}\sqrt{var(\hat{\lambda})},$

where $Z_{\frac{\delta}{2}}$ is the upper $(\frac{\delta}{2})$-th percentile
of the standard normal distribution.

##  7 Application

In this section, we present the analysis of a real data set using the EFWE
$(\alpha,\beta,\lambda)$ model and compare it with the other fitted models
like A flexible Weibull extension (FWE) distribution, Weibull distribution
(WD) , linear failure rate distribution (LFRD), exponentiated Weibull
distribution(EWD), generalized linear failure rate distribution (GLFRD) and
exponentiated flexible Weibull distribution (EFWD) using Kolmogorov Smirnov
(K-S) statistic, as well as Akaike information criterion (AIC), [2], Akaike
Information Citerion with correction (AICC) and Bayesian information criterion
(BIC) [19] values.

Consider the data have been obtained from Aarset [1], and widely reported in
many literatures. It represents the lifetimes of 50 devices, and also, possess
a bathtub-shaped failure rate property, Table 2.

Table 2: Life time of 50 devices, see Aarset [1].

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 |

Table 3 gives MLEs of parameters of the EFWE distribution and K-S Statistics.
The values of the log-likelihood functions, AIC, AICC and BIC are in Table 4.

Table 3: MLEs and K–S of parameters for Aarset data [1].

Model | MLE of the parameters | K-S | P-value
---|---|---|---
FW($\alpha,\beta$ ) |  $\hat{\alpha}$ = 0.0122, $\hat{\beta}$ = 0.7002 | 0.4386 |  4.29 $\times 10^{-9}$
W($\alpha,\beta$ ) |  $\hat{\alpha}$ = 44.913, $\hat{\beta}$ = 0.949 | 0.2397 | 0.0052
LFR (a, b) |  $\hat{a}$ = 0.014, $\hat{b}$ = 2.4 $\times 10^{-4}$ | 0.1955 | 0.0370
EW($\alpha,\beta,\gamma$) |  $\hat{\alpha}$ = 91.023, $\hat{\beta}$ = 4.69, $\hat{\gamma}$ = 0.164 | 0.1841 | 0.0590
GLFR(a, b, c) |  $\hat{a}$ = 0.0038, $\hat{b}$ = 3.04 $\times 10^{-4}$, $\hat{c}$ = 0.533 | 0.1620 | 0.1293
EFW($\alpha,\beta,\theta$ ) |  $\hat{\alpha}$ = 0.0147, $\hat{\beta}$ = 0.133, $\hat{\theta}$ = 4.22 | 0.1433 | 0.2617
EFWE($\alpha,\beta,\lambda$ ) |  $\hat{\alpha}$ = 0.015, $\hat{\beta}$ = 0.381, $\hat{\lambda}$ = 0.076 | 0.13869 | 0.2719

Table 4: Log-likelihood, AIC, AICC and BIC values of models fitted for Aarset
data [1].

Model | $\mathcal{L}$ |  -2 $\mathcal{L}$ | AIC | AICC | BIC
---|---|---|---|---|---
FW($\alpha,\beta$) | -250.810 | 501.620 | 505.620 | 505.88 | 509.448
W($\alpha,\beta$ ) | -241.002 | 482.004 | 486.004 | 486.26 | 489.828
LFR (a, b) | -238.064 | 476.128 | 480.128 | 480.38 | 483.952
EW($\alpha,\beta,\gamma$ ) | -235.926 | 471.852 | 477.852 | 478.37 | 483.588
GLFR(a, b, c) | -233.145 | 466.290 | 472.290 | 472.81 | 478.026
EFW($\alpha,\beta,\theta$ ) | -226.989 | 453.978 | 459.979 | 460.65 | 465.715
EFWE($\alpha,\beta,\lambda$ ) | -224.832 | 449.664 | 455.664 | 456.19 | 461.400

We find that the EFWE distribution with the three-number of parameters
provides a better fit than the previous new modified a flexible Weibull
extension distribution(FWE) which was the best in Bebbington et al. [3]. It
has the largest likelihood, and the smallest K-S, AIC, AICC and BIC values
among those considered in this paper.

Substituting the MLE’s of the unknown parameters $\alpha,\beta,\lambda$ into
(6.12), we get estimation of the variance covariance matrix as the following

$I_{0}^{-1}=\left(\begin{array}[]{rrrr}1.11\times 10^{-6}&-1.175\times
10^{-5}&-2.187\times 10^{-5}\\\ -1.175\times 10^{-5}&0.021&3.275\times
10^{-4}\\\ -2.187\times 10^{-5}&3.275\times 10^{-4}&5.469\times
10^{-4}\end{array}\right)$

The approximate 95% two sided confidence intervals of the unknown parameters
$\alpha,\beta$ and $\lambda$ are $\left[0.013,0.017\right]$,
$\left[0.1,0.662\right]$ and $\left[0.03,0.122\right]$, respectively.

To show that the likelihood equation have unique solution, we plot the
profiles of the log-likelihood function of $\alpha,\beta$ and $\lambda$ in
Figures 7 and 8.

Figure 7: The profile of the log-likelihood function of $\alpha,\beta$.

Figure 8: The profile of the log-likelihood function of $\lambda$.

The nonparametric estimate of the survival function using the Kaplan-Meier
method and its fitted parametric estimations when the distribution is assumed
to be $FW,W,LFR,EW,GLFR,EFW$ and $EFWE$ are computed and plotted in Figure 9.

Figure 9: The Kaplan-Meier estimate of the survival function for the data.

Figures 10 and 11 give the form of the hazard rate and CDF for the
$FW,W,LFR,EW,GLFR,EFW$ and $EFWE$ which are used to fit the data after
replacing the unknown parameters included in each distribution by their MLE.

Figure 10: The Fitted hazard rate function for the data.

Figure 11: The Fitted cumulative distribution function for the data.

##  8 Conclusions

A new distribution, based on exponential generalized distributions, has been
proposed and its properties are 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 Weibull generator
component, the distribution has flexibility to model the second peak in a
distribution. We have shown that the exponential flexible Weibull extension
EFWE distribution fits certain well-known data sets better than existing
modifications of the exponential generalized family of probability
distribution.

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