# The Odd Generalized Exponential Gompertz Distribution

M. A. El-Damcese1, Abdelfattah Mustafa2,111Corresponding author:
<EMAIL_ADDRESS>
B. S. El-Desouky2 and M. E. Mustafa2
1Tanta University, Faculty of Science, Mathematics Department, Egypt.
2Department of Mathematics, Faculty of Science, Mansoura University, Mansoura
35516, Egypt.

###### Abstract

In this paper we propose a new lifetime model, called the odd generalized
exponential gompertz distribution, We obtained some of its mathematical
properties. Some structural properties of the new distribution are studied.
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: Gompertz distribution; Hazard function; Moments; Maximum likelihood
estimation.

2010 MSC: 60E05, 62F10, 62F12, 62N02, 62N05.

##  1 Introduction

In the analysis of lifetime data we can use the Gompertz, exponential and
generalized exponential distributions. It is known that the exponential
distribution have only constant hazard rate function where as Gompertz, and
generalized exponential distributions can have only monotone (increasing in
case of Gompertz and increasing or decreasing in case of generalized
exponential distribution) hazard rate. These distributions are used for
modelling the lifetimes of components of physical systems and the organisms of
biological populations. The Gompertz distribution has received considerable
attention from demographers and actuaries. Pollard and Valkovics [14] were the
first to study the Gompertz distribution, they both defined the moment
generating function of the Gompertz distribution in terms of the incomplete or
complete gamma function and their results are either approximate or left in an
integral form. Later, Marshall and Olkin [10] described the negative Gompertz
distribution; a Gompertz distribution with a negative rate of aging parameter.

Recently, a generalization of the Gompertz distribution based on the idea
given in [2] was proposed by [3] this new distribution is known as generalized
Gompertz (GG) distribution which includes the exponential (E), generalized
exponential (GE), and Gompertz (G) distributions. A new generalization of th
Gompertz (G) distribution which results of the application of the Gompertz
distribution to the Beta generator proposed by [4], called the Beta-Gompertz
(BG) distribution which introduced by [8]. On the other hand the two-parameter
exponentiated exponential or generalized exponential distribution (GE)
introduced by [2]. This distribution is a particular member of the
exponentiated Weibull (EW) distribution introduced by [11]. The GE
distribution is a right skewed unimodal distribution, the density function and
hazard function of the exponentiated exponential distribution are quite
similar to the density function and hazard function of the Gamma distribution.
Its applications have been wide-spread as model to power system equipment,
rainfall data, software reliability and analysis of animal behavior.

Recently [15] proposed a new class of univariate distributions called the odd
generalized exponential (OGE) family and studied each of the OGE-Weibull
(OGE-W) distribution, the OGE-Fréchet (OGE-Fr) distribution and the OGE-Normal
(OGE-N) distribution. This method is flexible because of the hazard rate
shapes could be increasing, decreasing, bathtub and upside down bathtub.

In this article we present a new distribution from the exponentiated
exponential distribution and gompertz distribution called the Odd Generalized
Exponential-Gompertz (OGE-G) distribution using new family of univariate
distributions proposed by [15].

A random variable $X$ is said to have generalized exponential (GE)
distribution with parameters $\alpha,\beta$ if the cumulative distribution
function (CDF) is given by

$F(x)=\left(1-e^{-\alpha x}\right)^{\beta},\;x>0,\,\alpha>0,\beta>0.$ (1.1)

The Odd Generalized Exponential family by [15] is defined as follows. Let
$G(x;\epsilon)$ is the CDF of any distribution depends on parameter $\epsilon$
and thus the survival function is $\overline{G}(x,\epsilon)=1-G(x;\epsilon)$,
then the CDF of OGE-family is defined by replacing $x$ in CDF of GE in
equation (1.1) by $\frac{G(x;\epsilon)}{\overline{G}(x,\epsilon)}$ to get

$F(x;\alpha,\epsilon,\beta)=\left[1-e^{-\alpha\frac{G(x;\epsilon)}{\overline{G}(x,\epsilon)}}\right]^{\beta},\;x>0,\alpha>0,\epsilon>0,\beta>0.$
(1.2)

This paper is outlined as follows. In Section 2, we define the cumulative
distribution function, density function, reliability function and hazard
function of the Odd Generalized exponential-Gompertz (OGE-G) distribution. In
Section 3, we introduce the statistical properties include, the quantile
function, the mode, the median and the moments. Section 4 discusses the
distribution of the order statistics for (OGE-G) distribution. Moreover,
maximum likelihood estimation of the parameters is determined in Section 5.
Finally, an application of OGE-G using a real data set is presented in Section
6.

##  2 The OGE-G Distribution

###  2.1 OGE-G specifications

In this section we define new four parameters distribution called Odd
Generalized Exponential-Gompertz distribution with parameters
$\alpha,\lambda,c,\beta$ written as OGE-G($\Theta$), where the vector $\Theta$
is defined by $\Theta=(\alpha,\lambda,c,\beta)$.

A random variable $X$ is said to have OGE-G with parameters
$\alpha,\lambda,c,\beta$ if its cumulative distribution function given as
follows

$F(x;\Theta)=\left\\{1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}\right\\}^{\beta},\;x>0,\alpha,\lambda,c,\beta>0,$
(2.1)

where $\alpha,\lambda,c$ are scale parameters and $\beta$ is shape parameter.

###  2.2 PDF and hazard rate

If a random variable $X$ has CDF in (2.1) then the corresponding probability
density function is

$f(x;\Theta)=\alpha\beta\lambda
e^{cx}e^{\frac{\lambda}{c}(e^{cx}-1)}e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}\left\\{1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}\right\\}^{\beta-1},\;$
(2.2)

where $x>0,\alpha,\lambda,c,\beta>0.$

A random variable $X\thicksim OGE-G(\Theta)$ has survival function in the form

$S(x)=1-\left\\{1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}\right\\}^{\beta}.$
(2.3)

The failure rate function of $OGE-G(\Theta$) is given by

$h(x)=\frac{f(x)}{S(x)}=\frac{\alpha\beta\lambda
e^{cx}e^{\frac{\lambda}{c}(e^{cx}-1)}e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}\left\\{1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}\right\\}^{\beta-1}}{1-\left\\{1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}\right\\}^{\beta}}.$
(2.4)

Figure 1: The CDF of various OGE-G distributions for some values of the
parameters.

Figure 2: The pdf’s of various OGE-G distributions for some values of the
parameters.

Figure 3: The hazard of various OGE-G distributions for some values of the
parameters.

##  3 Statistical properties

In this section, we study some statistical properties of OGE-G, especially
quantile, median, mode and moments.

###  3.1 Quantile and median of OGE-G

The quantile of OGE-G($\Theta$) distribution is given by using

$F(x_{q})=q.$ (3.1)

Substituting from (2.1) into (3.1), $x_{q}$ can be obtained as

$x_{q}=\frac{1}{c}\ln\left\\{1+\frac{c}{\lambda}\ln\left[1-\frac{1}{\alpha}\ln(1-q^{\frac{1}{\beta}})\right]\right\\},\;0<q<1.$
(3.2)

Setting $q=\frac{1}{2}$ in (3.2), we obtain the median of
OGE-G($\alpha,\lambda,c,\beta$) distribution as follows

$Med=\frac{1}{c}\ln\left\\{1+\frac{c}{\lambda}\ln\left[1-\frac{1}{\alpha}\ln(1-(\frac{1}{2})^{\frac{1}{\beta}})\right]\right\\}.$
(3.3)

###  3.2 The mode of OGE-G

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

$\displaystyle\left[c+\lambda e^{cx}-\alpha\lambda
e^{cx}e^{\frac{\lambda}{c}(e^{cx}-1)}\right]\left\\{1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}\right\\}+$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\alpha\lambda(\beta-1)e^{cx}e^{\frac{\lambda}{c}(e^{cx}-1)}e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}=0.$
(3.4)

It is not possible to get an explicit solution of (3.2) in the general case.
Numerical methods should be used such as bisection or fixed-point method to
solve it.

###  3.3 The moments

Moments are necessary and 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). In this subsection, we will derive the rth moments of the
OGE-G($\Theta$) distribution as infinite series expansion.

######  Theorem 3.1.

If $X\sim OGE-G(\Theta)$, where $\Theta=(\alpha,\lambda,c,\beta)$, then the
rth moment of $X$ is given by

$\mu_{r}^{{}^{\prime}}=\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{j}\sum_{\ell=0}^{\infty}\sum_{m=0}^{\ell}(-1)^{i+j+k+m}\begin{pmatrix}\beta-1\\\
i\end{pmatrix}\begin{pmatrix}\ell\\\ m\end{pmatrix}\begin{pmatrix}j\\\
k\end{pmatrix}\frac{\alpha^{j+1}\beta\lambda^{\ell+1}(i+1)^{j}(j-k+1)^{\ell}r!}{c^{\ell+r+1}j!\ell!(\ell-m+1)^{r+1}}.$

###### Proof.

The rth moment of the random variable $X$ with pdf $f(x)$ is defined by

$\mu_{r}^{{}^{\prime}}=\int_{0}^{\infty}x^{r}f(x)dx.$ (3.5)

Substituting from (2.2) into (3.5), we get

$\mu_{r}^{{}^{\prime}}=\int_{0}^{\infty}x^{r}\alpha\beta\lambda
e^{cx}e^{\frac{\lambda}{c}(e^{cx}-1)}e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}\left\\{1-e^{\mathbf{-\alpha}\left[\mathbf{e}^{\frac{\lambda}{c}(e^{cx}-1)}\mathbf{-1}\right]}\right\\}^{\beta-1}dx.$
(3.6)

Since $0<1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}<1$ for
$x>0$, we have

$\left\\{1-e^{\mathbf{-\alpha}\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}\right\\}^{\beta-1}=\sum_{i=0}^{\infty}\begin{pmatrix}\beta-1\\\
i\end{pmatrix}(-1)^{i}e^{-\alpha
i\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}.$ (3.7)

Substituting from (3.7) into (3.6), we obtain

$\mu_{r}^{{}^{\prime}}=\sum_{i=0}^{\infty}\alpha\beta\lambda(-1)^{i}\begin{pmatrix}\beta-1\\\
i\end{pmatrix}\int_{0}^{\infty}x^{r}e^{cx}e^{\frac{\lambda}{c}(e^{cx}-1)}e^{-\alpha(i+1)\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}dx.$

Using the series expansion of
$e^{-\alpha(i+1)(e^{\frac{\lambda}{c}(e^{cx}-1)}-1)}$, we get

$\mu_{r}^{{}^{\prime}}=\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\alpha\beta\lambda(-1)^{i+j}\begin{pmatrix}\beta-1\\\
i\end{pmatrix}\frac{\alpha^{j}(i+1)^{j}}{j!}\int_{0}^{\infty}x^{r}e^{cx}e^{\frac{\lambda}{c}(e^{cx}-1)}(e^{\frac{\lambda}{c}(e^{cx}-1)}-1)^{j}dx.$

Using binomial expansion of $(e^{\frac{\lambda}{c}(e^{cx}-1)}-1)^{j}$, we get

$\mu_{r}^{{}^{\prime}}=\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{j}(-1)^{i+j+k}\begin{pmatrix}\beta-1\\\
i\end{pmatrix}\begin{pmatrix}j\\\
k\end{pmatrix}\frac{\alpha^{j+1}\beta\lambda(i+1)^{j}}{j!}\int_{0}^{\infty}x^{r}e^{cx}e^{\frac{\lambda}{c}(j-k+1)(e^{cx}-1)}dx.$

Using series expansion of $e^{\frac{\lambda}{c}(j-k+1)(e^{cx}-1)}$, we get

$\displaystyle\mu_{r}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{j}\sum_{\ell=0}^{\infty}(-1)^{i+j+k}\begin{pmatrix}\beta-1\\\
i\end{pmatrix}\begin{pmatrix}j\\\
k\end{pmatrix}\frac{\alpha^{j+1}\beta\lambda^{\ell+1}(i+1)^{j}(j-k+1)^{\ell}}{c^{\ell}j!\ell!}\times$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\int_{0}^{\infty}x^{r}e^{cx}(e^{cx}-1)^{\ell}dx,$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{j}\sum_{\ell=0}^{\infty}\sum_{m=0}^{\ell}(-1)^{i+j+k+m}\begin{pmatrix}\beta-1\\\
i\end{pmatrix}\begin{pmatrix}\ell\\\ m\end{pmatrix}\begin{pmatrix}j\\\
k\end{pmatrix}\frac{\alpha^{j+1}\beta\lambda^{\ell+1}(i+1)^{j}(j-k+1)^{\ell}}{c^{\ell}j!\ell!}\times$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\int_{0}^{\infty}x^{r}e^{c(\ell-m+1)x}dx.$

By using the definition of gamma function in the form

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

Thus we obtain the moment of OGE-G($\Theta$ ) as follows

$\mu_{r}^{{}^{\prime}}=\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{j}\sum_{\ell=0}^{\infty}\sum_{m=0}^{\ell}(-1)^{i+j+k+m}\begin{pmatrix}\beta-1\\\
i\end{pmatrix}\begin{pmatrix}\ell\\\ m\end{pmatrix}\begin{pmatrix}j\\\
k\end{pmatrix}\frac{\alpha^{j+1}\beta\lambda^{\ell+1}(i+1)^{j}(j-k+1)^{\ell}r!}{c^{\ell+r+1}j!\ell!(\ell-m+1)^{r+1}}.$

This completes the proof. ∎

##  4 Order Statistic

Let $X_{1},X_{2},\cdots,X_{n}$ be a simple random sample of size $n$ from
OGE-G($\Theta$) distribution with cumulative distribution function
$F(x;\Theta)$ and probability density function $f(x;\Theta)$ given by (2.1)
and (2.2) respectively. Let $X_{1:n}\leq X_{2:n}\leq\cdots\leq X_{n:n}$ denote
the order statistics obtained from this sample. The probability density
function of $X_{r:n}$ is given by

$f_{r:n}(x,\Theta)=\frac{1}{B(r,n-r+1)}\left[F(x,\Theta)\right]^{r-1}\left[1-F(x,\Theta)\right]^{n-r}f(x;\Theta),$
(4.1)

where $f(x;\Theta)$ and $F(x,\Theta)$ are the pdf and cdf of OGE-G($\Theta$)
distribution given by (2.1) and (2.2) respectively and $B(.,.)$ is the beta
function. Since $0<F(x,\Theta)<1$ for $x>0$, we can use the binomial expansion
of $\left[1-F(x,\Theta)\right]^{n-r}$ given as follows

$\left[1-F(x,\Theta)\right]^{n-r}=\sum_{i=0}^{n-r}\begin{pmatrix}n-r\\\
i\end{pmatrix}(-1)^{i}\left[F(x,\Theta)\right]^{i}.$ (4.2)

Substituting from (4.2) into (4.1), we have

$f_{r:n}(x,\Theta)=\frac{1}{B(r,n-r+1)}f(x;\Theta)\sum_{i=0}^{n-r}\begin{pmatrix}n-r\\\
i\end{pmatrix}(-1)^{i}\left[F(x,\Theta)\right]^{i+r-1}.$ (4.3)

Substituting from (2.1) and (2.2) into (4.3), we obtain

$f_{r:n}(x;\alpha,\lambda,c,\beta)=\sum_{i=0}^{n-r}\frac{(-1)^{i}n!}{i!(r-1)!(n-r-i)!(r+i)}f(x,\alpha,\lambda,c,(r+i)\beta)$
(4.4)

Thus $f_{r:n}(x;\alpha,\lambda,c,\beta)$ defined in (4.4) is the weighted
average of the OGE-G distribution with different shape parameters.

##  5 Estimation and Inference

Now, we determine the maximum-likelihood estimators (MLE’s) of the OGE-G
parameters.

###  5.1 Maximum likelihood estimators

Let $X_{1},X_{2},\cdots,X_{n}$ be a random sample of size $n$ from
OGE-G($\Theta$), where $\Theta=(\alpha,\lambda,c,\beta)$, then the likelihood
function $\mathcal{L}$ of this sample is defined as

$\mathcal{L}=\prod_{i=1}^{n}f(x_{i};\alpha,\lambda,c,\beta).$ (5.1)

Substituting from (2.2) into (4.3), we get

$\mathcal{L}=\prod_{i=1}^{n}\alpha\beta\lambda
e^{cx_{i}}e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}-1\right]}\left\\{1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}\right\\}^{\beta-1}.$

The log-likelihood function is given as follow

$\displaystyle L$ $\displaystyle=$ $\displaystyle
n\ln(\alpha)+n\ln(\beta)+n\ln(\lambda)+c\sum_{i=0}^{n}x_{i}+\frac{\lambda}{c}\sum_{i=0}^{n}(e^{cx_{i}}-1)-\alpha\sum_{i=0}^{n}(e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}-1)$
(5.2)
$\displaystyle+(\beta-1)\sum_{i=0}^{n}\ln\left\\{1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}\right\\}.$

The log-likelihood can be maximized either directly or by solving the
nonlinear likelihood equations obtained by differentiating equation (5.2) with
respect to $\alpha,\lambda,c$ and $\beta$. The components of the score vector
$U(\Theta)=\left(\frac{\partial L}{\partial\alpha},\frac{\partial
L}{\partial\lambda},\frac{\partial L}{\partial c},\frac{\partial
L}{\partial\beta}\right)$ are given by

$\displaystyle\frac{\partial L}{\partial\beta}$ $\displaystyle=$
$\displaystyle\frac{n}{\beta}+\sum_{i=1}^{n}\ln\left\\{1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}\right\\},$
(5.3) $\displaystyle\frac{\partial L}{\partial\alpha}$ $\displaystyle=$
$\displaystyle\frac{n}{\alpha}-\sum_{i=1}^{n}(e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}-1)-(\beta-1)\sum_{i=1}^{n}\frac{1-e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}}{e^{\alpha\left[e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}-1\right]}-1},$
(5.4) $\displaystyle\frac{\partial L}{\partial\lambda}$ $\displaystyle=$
$\displaystyle\frac{n}{\lambda}+\frac{1}{c}\sum_{i=1}^{n}(e^{cx_{i}}-1)-\frac{\alpha}{c}\sum_{i=1}^{n}(e^{cx_{i}}-1)e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}+\frac{\alpha(\beta-1)}{c}\sum_{i=1}^{n}\frac{\left(e^{cx_{i}}-1\right)e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}}{e^{\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}-1},$
$\displaystyle\frac{\partial L}{\partial c}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}x_{i}-\sum_{i=1}^{n}\tau(x_{i},\lambda,c)-\alpha\sum_{i=1}^{n}\tau(x_{i},\lambda,c)e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}+(\beta-1)\alpha\sum_{i=1}^{n}\frac{\tau(x_{i},\lambda,c)e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}}{e^{\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}-1},$

where
$\tau(x_{i},\lambda,c)=-\frac{\lambda}{c^{2}}(e^{cx_{i}}-1)+\frac{\lambda}{c}x_{i}e^{cx_{i}}.$

The normal equations can be obtained by setting the above non-linear equations
(5.3)-(5.1) to zero. That is, the normal equations take the following form

$\displaystyle\frac{n}{\beta}+\sum_{i=1}^{n}\ln\left\\{1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}\right\\}=0,$
(5.7)
$\displaystyle\frac{n}{\alpha}-\sum_{i=1}^{n}(e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}-1)-(\beta-1)\sum_{i=1}^{n}\frac{1-e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}}{e^{\alpha\left[e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}-1\right]}-1}=0,$
(5.8)
$\displaystyle\frac{n}{\lambda}+\frac{1}{c}\sum_{i=1}^{n}(e^{cx_{i}}-1)-\frac{\alpha}{c}\sum_{i=1}^{n}(e^{cx_{i}}-1)e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}+\frac{\alpha(\beta-1)}{c}\sum_{i=1}^{n}\frac{\left(e^{cx_{i}}-1\right)e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}}{e^{\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}-1}=0,$
(5.9)
$\displaystyle\sum_{i=1}^{n}x_{i}-\sum_{i=1}^{n}\tau(x_{i},\lambda,c)-\alpha\sum_{i=1}^{n}\tau(x_{i},\lambda,c)e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}+(\beta-1)\alpha\sum_{i=1}^{n}\frac{\tau(x_{i},\lambda,c)e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}}{e^{\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}-1}=0.$

The normal equations do not have explicit solutions and they have to be
obtained numerically. From equation (5.7) the MLEs of $\beta$ can be obtained
as follows

$\hat{\beta}=\frac{-n}{\sum_{i=1}^{n}\ln\left\\{1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}\right\\}}.$
(5.11)

Substituting from (5.11) into (5.8), (5.1), and (5.1), we get the MLEs of
$\alpha,\lambda,c$ by solving the following system of non-linear equations

$\displaystyle\frac{n}{\hat{\alpha}}-\sum_{i=1}^{n}(e^{\frac{\hat{\lambda}}{\hat{c}}(e^{\hat{c}x_{i}}-1)}-1)-(\hat{\beta}-1)\sum_{i=1}^{n}\frac{1-e^{\frac{\hat{\lambda}}{\hat{c}}(e^{\hat{c}x_{i}}-1)}}{e^{\hat{\alpha}\left[e^{\frac{\hat{\lambda}}{\hat{c}}(e^{\hat{c}x_{i}}-1)}-1\right]}-1}=0,$
(5.12)
$\displaystyle\frac{n}{\hat{\lambda}}+\frac{1}{\hat{c}}\sum_{i=1}^{n}(e^{\frac{\hat{\lambda}}{\hat{c}}(e^{\hat{c}x_{i}}-1)}-1)-\frac{\hat{\alpha}}{\hat{c}}\sum_{i=1}^{n}(e^{\hat{c}x_{i}}-1)e^{\frac{\hat{\lambda}}{\hat{c}}(e^{\hat{c}x_{i}}-1)}+\frac{\hat{\alpha}(\hat{\beta}-1)}{\hat{c}}\sum_{i=1}^{n}\frac{(e^{\hat{c}x_{i}}-1)e^{\frac{\hat{\lambda}}{\hat{c}}(e^{\hat{c}x_{i}}-1)}}{e^{\hat{\alpha}\left[e^{\frac{\hat{\lambda}}{\hat{c}}(e^{\hat{c}x_{i}}-1)}-1\right]}-1}=0,$
(5.13)
$\displaystyle\sum_{i=1}^{n}x_{i}-\sum_{i=1}^{n}\tau(x_{i},\hat{\lambda},\hat{c})-\hat{\alpha}\sum_{i=1}^{n}\tau(x_{i},\hat{\lambda},\hat{c})e^{\frac{\hat{\lambda}}{\hat{c}}(e^{\hat{c}x_{i}}-1)}+(\hat{\beta}-1)\hat{\alpha}\sum_{i=1}^{n}\frac{\tau(x_{i},\hat{\lambda},\hat{c})e^{\frac{\hat{\lambda}}{\hat{c}}(e^{\hat{c}x_{i}}-1)}}{e^{\hat{\alpha}\left[e^{\frac{\hat{\lambda}}{\hat{c}}(e^{\hat{c}x_{i}}-1)}-1\right]}-1}=0,$

where
$\tau(x_{i},\hat{\lambda},\hat{c})=-\tfrac{\hat{\lambda}}{\hat{c}^{2}}({\small
e}^{\hat{c}x_{i}}-1)+\tfrac{\hat{\lambda}}{\hat{c}}x_{i}{\small
e}^{\hat{c}x_{i}}.$

These equations cannot be solved analytically and statistical software can be
used to solve the equations numerically. We can use iterative techniques such
as Newton Raphson type algorithm to obtain the estimate $\hat{\beta}$.

###  5.2 Asymptotic confidence bounds

In this subsection, we derive the asymptotic confidence intervals of the
unknown parameters $\alpha,\lambda,c,\beta$ when $\alpha>0,\lambda>0,c>0$ and
$\beta>0$. The simplest large sample approach is to assume that the
MLEs($\alpha,\lambda,c,\beta$) are approximately multivariate normal with mean
($\alpha,\lambda,c,\beta$) and covariance matrix $I_{0}^{-1}$, where
$I_{0}^{-1}$ is the inverse of the observed information matrix which defined
as follows

$\displaystyle I_{0}^{-1}$ $\displaystyle=$
$\displaystyle-\left[\begin{array}[]{cccc}\frac{\partial^{2}L}{\partial\alpha^{2}}&\frac{\partial^{2}L}{\partial\lambda\partial\alpha}&\frac{\partial^{2}L}{\partial
c\partial\alpha}&\frac{\partial^{2}L}{\partial\beta\partial\alpha}\\\
\frac{\partial^{2}L}{\partial\alpha\partial\lambda}&\frac{\partial^{2}L}{\partial\lambda^{2}}&\frac{\partial^{2}L}{\partial
c\partial\lambda}&\frac{\partial^{2}L}{\partial\beta\partial\lambda}\\\
\frac{\partial^{2}L}{\partial\alpha\partial
c}&\frac{\partial^{2}L}{\partial\lambda\partial
c}&\frac{\partial^{2}L}{\partial
c^{2}}&\frac{\partial^{2}}{\partial\beta\partial c}\\\
\frac{\partial^{2}}{\partial\alpha\partial\beta}&\frac{\partial^{2}}{\partial\lambda\partial\beta}&\frac{\partial^{2}L}{\partial
c\partial\beta}&\frac{\partial^{2}}{\partial\beta^{2}}\end{array}\right]^{-1}$
(5.19) $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccc}var(\hat{\alpha})&cov(\hat{\lambda},\hat{\alpha})&cov(\hat{c},\hat{\alpha})&cov(\hat{\beta},\hat{\alpha})\\\
cov(\hat{\alpha},\hat{\lambda})&var(\hat{\lambda})&cov(\hat{c},\hat{\lambda})&cov(\hat{\beta},\hat{\lambda})\\\
cov(\hat{\alpha},\hat{c})&cov(\hat{\lambda},\hat{c})&var(\hat{c})&cov(\hat{\beta},\hat{c})\\\
cov(\hat{\alpha},\hat{\beta})&cov(\hat{\lambda},\hat{\beta})&cov(\hat{c},\hat{\beta})&var(\hat{\beta})\end{array}\right].$
(5.24)

The second partial derivatives included in $I_{0}^{-1}$ are given as follows

$\displaystyle\frac{\partial^{2}L}{\partial\beta^{2}}$ $\displaystyle=$
$\displaystyle\frac{-n}{\beta^{2}},$
$\displaystyle\frac{\partial^{2}L}{\partial\alpha\partial\beta}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\frac{\left(e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}-1\right)e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}-1\right]}}{1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}-1\right]}},$
$\displaystyle\frac{\partial^{2}L}{\partial\lambda\partial\beta}$
$\displaystyle=$
$\displaystyle\frac{\alpha}{c}\sum_{i=1}^{n}\frac{\left(e^{cx_{i}}-1\right)A_{i}}{1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}-1\right]}},$
$\displaystyle\frac{\partial^{2}L}{\partial c\partial\beta}$ $\displaystyle=$
$\displaystyle\alpha\sum_{i=1}^{n}\frac{A_{i}B_{i}}{1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}-1\right]}},$
$\displaystyle\frac{\partial^{2}L}{\partial\alpha^{2}}$ $\displaystyle=$
$\displaystyle\frac{-n}{\alpha^{2}}-(\beta-1)\sum_{i=1}^{n}\frac{\left(e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}-1\right)^{2}e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}-1\right]}}{\left\\{1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}\right\\}^{2}},$
$\displaystyle\frac{\partial^{2}L}{\partial\lambda\partial\alpha}$
$\displaystyle=$
$\displaystyle\frac{-1}{c}\sum_{i=1}^{n}\left(e^{cx_{i}}-1\right)e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}+\frac{(\beta-1)}{c}\sum_{i=1}^{n}\frac{\left(e^{cx_{i}}-1\right)A_{i}C_{i}}{\left\\{1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}\right\\}^{2}},$
$\displaystyle\frac{\partial^{2}L}{\partial c\partial\alpha}$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}B_{i}e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}+(\beta-1)\sum_{i=1}^{n}\frac{A_{i}B_{i}C_{i}}{\left\\{1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}\right\\}^{2}},$
$\displaystyle\frac{\partial^{2}L}{\partial\lambda^{2}}$ $\displaystyle=$
$\displaystyle\frac{-n}{\lambda^{2}}-\frac{\alpha}{c^{2}}\sum_{i=1}^{n}\left(e^{cx_{i}}-1\right)^{2}e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}+\frac{\alpha{\small(\beta-1)}}{c^{2}}\sum_{i=1}^{n}\frac{\left(e^{cx_{i}}-1\right)^{2}A_{i}D_{i}}{\left\\{1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}\right\\}^{2}},$
$\displaystyle\frac{\partial^{2}L}{\partial c\partial\lambda}$
$\displaystyle=$
$\displaystyle\frac{1}{\lambda}\sum_{i=1}^{n}B_{i}-\frac{\alpha}{c}\sum_{i=1}^{n}\left[\frac{-1}{c}(e^{cx_{i}}-1)+x_{i}e^{cx_{i}}+B_{i}\left(e^{cx_{i}}-1\right)\right]e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}$
$\displaystyle+\frac{(\beta-1)\alpha}{\lambda}\sum_{i=1}^{n}\frac{\left[1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}-1\right]}+\frac{\lambda}{c}(e^{cx_{i}}-1)D_{i}\right]A_{i}B_{i}}{\left\\{1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}\right\\}^{2}},$

$\displaystyle\frac{\partial^{2}L}{\partial c^{2}}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}E_{i}-\alpha\sum_{i=1}^{n}\left[E_{i}+B_{i}^{2}\right]e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}+\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle(\beta-1)\alpha\sum_{i=1}^{n}\frac{\left[B_{i}^{2}D_{i}+E_{i}\left(1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}-1\right]}\right)\right]A_{i}}{\left\\{1-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]}\right\\}^{2}},$

where
$A_{i}=e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]},\qquad
B_{i}=-\frac{\lambda}{c^{2}}(e^{cx_{i}}-1)+\frac{\lambda}{c}(x_{i}e^{cx_{i}}),$

$C_{i}=1-\alpha(e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}-1)-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]},\qquad
D_{i}=1-\alpha
e^{\frac{\lambda}{c}(e^{cx_{i}}-1)}-e^{-\alpha\left[e^{\frac{\lambda}{c}(e^{cx}-1)}-1\right]},$
$E_{i}=\frac{2\lambda}{c^{3}}\left(e^{cx_{i}}-1\right)-\frac{2\lambda}{c^{2}}x_{i}e^{cx_{i}}+\frac{\lambda}{c}x_{i}^{2}e^{cx_{i}}.$

The asymptotic $(1-\gamma)100\%$ confidence intervals of $\alpha,\lambda,c$
and $\beta$ are $\hat{\alpha}\pm
z_{\frac{\gamma}{2}}\sqrt{var(\hat{\alpha})},\;\hat{\lambda}\pm
z_{\frac{\gamma}{2}}\sqrt{var(\hat{\lambda})},\;\hat{c}\pm
z_{\frac{\gamma}{2}}\sqrt{var(\hat{c})}$ and $\hat{\beta}\pm
z_{\frac{\gamma}{2}}\sqrt{var(\hat{\beta})})$ respectively, where
$z_{\frac{\gamma}{2}}$ is the upper ( $\frac{\gamma}{2}$)th percentile of the
standard normal distribution.

##  6 Data Analysis

In this section we perform an application to real data to illustrate that the
OGE-G can be a good lifetime model, comparing with many known distributions
such as the Exponential, Generalized Exponential, Gompertz, Generalized
Gompertz and Beta Gompertz distributions (ED, GE, G, GG, BG), see [7, 6, 3,
8].

Consider the data have been obtained from Aarset [1], and widely reported in
some literatures, see [2, 3, 9, 12, 13]. It represents the lifetimes of 50
devices, and also, possess a bathtub-shaped failure rate property, Table 1.

Table 1: The data from 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

Based on some goodness-of-fit measures, the performance of the OGE-G
distribution is compared with others five distributions: E, GE, G, GG, and BG
distributions. The MLE’s of the unknown parameters for these distributions are
given in Tables 2 and 3. Also, the values of the log-likelihood functions
(-$L$), the statistics K-S (Kolmogorov-Smirnov), AIC (Akaike Information
Criterion), the statistics AICC (Akaike Information Citerion with correction)
and BIC (Bayesian Information Criterion) are calculated for the six
distributions in order to verify which distribution fits better to these data.

Table 2: MLE’s of parameters for Aarset data.

Model |  $\hat{\alpha}$ |  $\hat{\beta}$ |  $\hat{\lambda}$ |  $\hat{c}$ |  -L
---|---|---|---|---|---
E | 0.0219 | – | – | – | 241.0896
GE | 0.0212 | 0.9012 | – | – | 240.3855
G | – | – | 0.00970 | 0.0203 | 235.3308
GG | – | 0.2625 | 0.00010 | 0.0828 | 222.2441
BG | 0.2158 | 0.2467 | 0.00030 | 0.0882 | 220.6714
OGE-G | 0.0400 | 0.1940 | 0.000345 | 0.0780 | 215.9735

Table 3: The AIC, CAIC, BIC and K-S values for Aarset data.  Model AIC AICC
BIC K-S p-value(K-S) E 484.1792 484.2625 486.0912 0.19110 0.0519 GE 484.7710
485.0264 488.5951 0.19400 0.0514 G 474.6617 475.1834 482.3977 0.16960 0.1123
GG 450.4881 451.0099 456.2242 0.14090 0.2739 BG 449.3437 450.2326 456.9918
0.13220 0.3456 OGE-G 423.9470 424.8359 447.5951 0.13205 0.3476

Based on Tables 2 and 3, it is shown that OGE-G$(\alpha,\lambda,c,\beta)$
model is the best among of those distributions because it has the smallest
value of (K-S), AIC, CAIC and BIC test.

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

$I_{0}^{-1}=\left[\begin{array}[]{rrrr}1.5022\times 10^{-3}&2.7638\times
10^{-7}&-1.4405\times 10^{-4}&9.6331\times 10^{-4}\\\ 2.7638\times
10^{-7}&4.4773\times 10^{-8}&-1.7837\times 10^{-6}&2.785\,8\times 10^{-6}\\\
-1.4405\times 10^{-4}&-1.7837\times 10^{-6}&8.4428\times 10^{-5}&-1.9127\times
10^{-4}\\\ 9.6331\times 10^{-4}&2.7858\times 10^{-6}&-1.9127\times
10^{-4}&1.5309\times 10^{-3}\end{array}\right]$

The approximate 95% two sided confidence intervals of the unknown parameters
$\alpha,\lambda,c$ and $\beta$ are
$\left[0,0.116\right],\;\left[0,0.001\right],\;\left[0.060,0.096\right],\;\left[0.116,0.270\right]$,
respectively.

To show that the likelihood equation have unique solution, we plot the
profiles of the log-likelihood function of $\alpha,\lambda,\beta$ and $c$ in
Figures 4-5.

Figure 4: The profile of the log-likelihood function of $\alpha,\lambda$.

Figure 5: The profile of the log-likelihood function of $c,\beta$.

The nonparametric estimate of the survival function using the Kaplan-Meier
method and its fitted parametric estimations when the distribution is assumed
to be $ED,GED,GD,GGD$ and $OGE-GD$ are computed and plotted in Figure 6.

Figure 6: The Kaplan-Meier estimate of the survival function.

Figure 7, gives the form of the hazard rate for the $ED,GED,GD,GGD,BGD$ and
$OGE-GD$ which are used to fit the data after replacing the unknown parameters
included in each distribution by their MLE.

Figure 7: The failure rate function for the data.

##  7 Conclusions

In this paper, we propose a new model, called the Odd Generalized Exponential-
Gompertz (OGE-G) distribution and studied its different properties. Some
statistical properties of this distribution have been derived and discussed.
The quantile, median, mode and moments of OGE-G are derived in closed forms.
The distribution of the order statistics are discussed. Both point and
asymptotic confidence interval estimates of the parameters are derived using
the maximum likelihood method and we obtained the observed Fisher information
matrix. We use application on set of real data to compare the OGE-G with other
known distributions such as Exponential (E), Generalized Exponential (GE),
Gompertz (G), Generalized Gompertz (GG) and Beta-Gompertz (BG). Applications
on set of real data showed that the OGE-G is the best distribution for fitting
these data sets compared with other distributions considered in this article.

## References

  * [1] Aarset, M. V. (1987). How to identify a bathtub hazard rate. IEEE Transactions on Reliability, 36(1), 106–108.
  * [2] Abu-Zinadah, H.H. and Aloufi, A.S. (2014). Some Characterizations of the Exponentiated Gompertz Distribution. International Mathematical Forum, 9(30), 1427–1439.
  * [3] El-Gohary, A., Alshamrani, A. and Al-Otaibi, A.N. (2013). The generalized Gompertz distribution. Applied Mathematical Modelling, 37(1), 13–24.
  * [4] Eugene, N., Lee, C. and Famoye, F. (2002). Beta-normal distribution and its applications. Communications in Statistics-Theory and Methods, 31(4), 497–512.
  * [5] Gupta, R.D. and Kundu, D. (1999). Generalized exponential distribution. Australian and New Zealand Journal of Statistics, 41, 173–-188.
  * [6] Gupta, R.D. and Kundu, D. (2001). Generalized exponential distribution: different method of estimations. Journal of Statistical Computation and Simulation, 69(4), 315–337.
  * [7] Gupta, R.D. and Kundu, D. (2007). Generalized exponential distribution: Existing results and some recent developments. Journal of Statistical Planning and Inference, 137(11), 3537–3547.
  * [8] Jafari, A.A., Tahmasebi, S. and Alizadeh, M. (2014). The beta-Gompertz distribution. Revista Colombiana de Estadĺstica, 37(1), 141–158.
  * [9] Lenart, A. (2014). The moments of the Gompertz distribution and maximum likelihood estimation of its parameters. Scandinavian Actuarial Journal, 2014(3), 255–277.
  * [10] Marshall, A.W. and Olkin, I. (2007). Life Distributions. Structure of Nonparametric, Semiparametric and Parametric Families. Springer, New York.
  * [11] Mudholkar, G.S. and Srivastava, D.K., (1993). Exponentiated Weibull family for analyzing bathtub failure data. IEEE Transactions on Reliability, 42, 299-–302.
  * [12] Nadarajah, S. and Kotz, S. (2006). The beta exponential distribution. Reliability engineering & system safety, 91(6), 689–697.
  * [13] Pasupuleti, S.S. and Pathak, P. (2010). Special form of Gompertz model and its application. Genus, 66(2), 95–125.
  * [14] Pollard, J.H. and Valkovics, E.J., (1992). The Gompertz distribution and its applications. Genus, 48(3-4),15–28.
  * [15] Tahir, M.H., Cordeiro, G.M., Alizadeh, M., Mansoor, M., Zubair, M. and Hamedani, G. G. (2015). The odd generalized exponential family of distributions with applications. Journal of Statistical Distributions and Applications, 2(1), 1–28.