Generalized Mixtures of Exponential Distribution and Associated Inference

Abstract

A new generalization of the exponential distribution, namely the generalized mixture of exponential distribution, is introduced. Some of its basic properties, such as hazard function, moments, order statistics, mean deviation, measures of uncertainly, and reliability probability, are studied. Three different estimation methods are investigated by the maximum likelihood estimator, least-square estimator, and weighted least-square estimator. The performances of the estimators are assessed by simulation studies. Real-world applications of the proposed distribution are explored, and data fitting results show that the new distribution performs better than its competitors.

1. Introduction

Among the parametric models, exponential distribution is perhaps the most widely applied statistical distribution in several fields and plays an important role in the statistical theory of reliability and lifetime analysis. Based on this reason, statisticians have been interested in defining new classes of univariate distributions by adding one or more shape parameters to provide greater flexibility in modeling real data in many applied fields. Gupta and Kundu [1] studied the generalized exponential distribution and used it as an alternative to gamma or Weibull distribution in many situations. Gupta and Kundu [2] used the idea of Azzalini [3], introducing a new class of weighted exponential (WE) distributions, and Kharazmi et al. [4] extended it into the generalized weighted exponential (GWE) distribution. Nadarajah and Haghighi [5] discussed a new two-parameter generalization of the exponential distribution, which had its mode at zero and allowed increasing, decreasing, and constant hazard rates.

On the other hand, generalizations of exponentiated type distributions can be obtained from the class of generalized beta distributions, in particular after the works of Eugene et al. [6] Nadarajah and Kotz [7] introduced the beta exponential distribution, generated from the logit of a beta random variable. Barreto-Souza et al. [8] discussed beta generalized exponential distribution, which includes the beta exponential and generalized exponential distributions as special cases. A generalization of the exponentiated Frenchet distribution, called the beta Frenchet distribution, was studied by Barreto-Souza et al. [9]. Ristic and Balakrishnan [10] proposed the gamma exponential distribution generated by gamma random variables. Being of a similar methodology, many X-family exponential distributions were investigated recently. These include the Weibull exponential (WED) distributions which were introduced by Oguntunde et al. [11], Marshall–Olkin generalized exponential distributions which were defined by Ristic and Kundu [12], Kumaraswamy Marshall–Olkin exponential distributions which were given by George and Thobias [13], and generalized extended exponential-Weibull (GExtEW) distributions which were proposed by Shakhatreh et al. [14].

In this paper, a new class of generalized mixture exponential (GME) distribution is introduced, which has the exponential and WE distributions as its submodels. In order to motivate interest, let us first present the definition of the generalized skew normal distribution introduced by Kumar and Anusree [15]. A random variable Z is said to have a generalized skew normal distribution if its probability density function (pdf) is of the following form,

$$h(z,\lambda ,\beta )=\frac{2}{2+\beta }f\left(z\right)(1+\beta F\left(\lambda z\right)),$$

where $f\left(z\right)=\varphi \left(z\right)$, $F\left(\lambda z\right)=\mathsf{\Phi }\left(\lambda z\right)$, $\lambda \in \Re$ and $\beta >-2$. In fact, the correct values of $\beta$ should be $\beta \ge -1$, which has been discussed in Tian et al. [16].

The rest of the article is organized as follows. The GME distribution is introduced in Section 2. Some important properties of GME distributions, such as cumulative distribution function (cdf), hazard function, mean deviations, order statistics, measure of uncertainly, and reliability probability, are discussed in Section 3. Three different estimation methods are studied in Section 4. Simulations are conducted to investigate and compare the performances of the proposed estimation methods in Section 5. Two real data sets are analyzed for illustrating the usefulness of the proposed GME distribution in Section 6. Some conclusions are presented in Section 7.

2. Generalized Mixture Exponential Distribution

The GME distribution offers more flexible distributions with applications in lifetime modeling, which is defined as follows.

Definition 1.

A random variable X is said to have a GME distribution if its pdf is of the following form,

$$f(x;\lambda ,\alpha ,\beta )=\frac{(\alpha +1)\lambda }{\alpha +1+\alpha \beta }{e}^{-\lambda x}[1+\beta (1-e^{-\alpha \lambda x})],x>0,$$

(1)

where $\alpha >0$ is the scale parameter, $\lambda >0$ and $\beta \ge -1$ are the shape parameters, and we denote it as $X\sim GME(\lambda ,\alpha ,\beta )$.

Remark 1. 

(i) 

For $\beta =0$, or $\alpha \to 0$, or $\alpha \to \infty$, $GME(\lambda ,\alpha ,\beta )$ is reduced into exponential distribution with parameter λ namely, $E\left(\lambda \right)$.

(ii) 

For $\beta =-1$, $GME(\lambda ,\alpha ,\beta )$ is reduced into $E\left(\lambda \right(\alpha +1\left)\right)$.

(iii) 

For $\beta \to \infty$, $GME(\lambda ,\alpha ,\beta )$ is reduced into $WE(\lambda ,\alpha )$.

For different values of $\lambda ,\alpha ,\beta$, the pdfs of $GME(\lambda ,\alpha ,\beta )$ are presented in the Figure 1, which indicate that the GME distribution can generate distributions with various shapes.

Proposition 1.

The cdf, the survival function and the hazard function of $X\sim GME(\lambda ,\alpha ,\beta )$ are given by

$$\begin{array}{ccc}\hfill F(x;\lambda ,\alpha ,\beta )& =& 1+\frac{\beta e^{-(\alpha +1)\lambda x}}{\alpha +1+\alpha \beta }-\frac{(\alpha +1)(\beta +1)e^{-\lambda x}}{\alpha +1+\alpha \beta },\hfill \\ \hfill h(x;\lambda ,\alpha ,\beta )& =& \frac{(\alpha +1)\lambda \left[1+\beta (1-e^{-\lambda \alpha x})\right]}{(\alpha +1+\alpha \beta )+\beta (1-e^{-\lambda \alpha x})},\hfill \\ \hfill S(x;\lambda ,\alpha ,\beta )& =& \frac{(\alpha +1)(\beta +1)e^{-\lambda x}}{\alpha +1+\alpha \beta }-\frac{\beta e^{-(\alpha +1)\lambda x}}{\alpha +1+\alpha \beta }.\hfill \end{array}$$

Proof of Proposition 1. 

According to the Equation (1), we have

$$\begin{array}{cc}\hfill F(x;\lambda ,\alpha ,\beta )& {\displaystyle =\frac{\lambda (\alpha +1)}{\alpha +1+\alpha \beta }\left[(1+\beta ){\int }_0^x{e}^{-\lambda t}dt-\beta {\int }_0^x{e}^{-\lambda \alpha t}dt\right]}\hfill \\ \hfill & =1-e^{-\lambda x}+\frac{\beta }{\alpha +1+\alpha \beta }\left[e^{-\lambda (1+\alpha )x}-e^{-\lambda x}\right].\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill h(x;\lambda ,\alpha ,\beta )& =\frac{f(x;\lambda ,\alpha ,\beta )}{1-F(x;\lambda ,\alpha ,\beta )}=\frac{(\alpha +1)\lambda \left[1+\beta (1-e^{-\lambda \alpha x})\right]}{(\alpha +1+\alpha \beta )+\beta (1-e^{-\lambda \alpha x})},\hfill \end{array}$$

$$\begin{array}{cc}\hfill S(x;\lambda ,\alpha ,\beta )& =1-F(x;\lambda ,\alpha ,\beta )=\frac{(\alpha +1)(\beta +1)e^{-\lambda x}}{\alpha +1+\alpha \beta }-\frac{\beta e^{-(\alpha +1)\lambda x}}{\alpha +1+\alpha \beta }.\hfill \end{array}$$

This ends the proof of Proposition 1. □

Figure 2 shows that the GME distribution produces flexible hazard rate shapes, such as decreasing, increasing, and stable.

3. General Properties of the GME Distribution

In what follows, we discuss various properties associated with the proposed distribution.

Proposition 2.

The shapes of density function of $X\sim GME(\lambda ,\alpha ,\beta )$ can be characterized as follows,

(i) 

$f\left(x\right)$ is monotone decreasing, if $-1\le \beta \le 0$ or $0<\alpha \beta \le 1$,

(ii) 

$f\left(x\right)$ is unimodal, if $\alpha \beta >1$.

Proof of Proposition 2. 

The derivatives of $f\left(x\right)$ are obtained by Equation (1),

$$\begin{array}{c}\hfill f^{\prime }\left(x\right)=\frac{{\lambda }^2(\alpha +1)e^{-\lambda x}}{\alpha +1+\alpha \beta }\left[\frac{\beta (\alpha +1)}{\beta +1}{e}^{-\alpha \lambda x}-1\right].\end{array}$$

(i)

if $-1\le \beta \le 0$, we have ${\displaystyle \frac{{\lambda }^2(\alpha +1)e^{-\lambda x}}{\alpha +1+\alpha \beta }>0}$, and $(\alpha +1)\beta \le 0$, thus,

$${\displaystyle \frac{{\lambda }^2(\alpha +1)e^{-\lambda x}}{\alpha +1+\alpha \beta }\left[\frac{\beta (\alpha +1)}{\beta +1}{e}^{-\alpha \lambda x}-1\right]<0;}$$

if $0<\alpha \beta \le 1$, we get $e^{-\alpha \lambda x}<1$ and ${\displaystyle \frac{\beta (\alpha +1)}{\beta +1}<1}$, thus, ${\displaystyle \left[\frac{\beta (\alpha +1)}{\beta +1}{e}^{-\alpha \lambda x}-1\right]<0}$.

(ii)

Setting $f^{\prime }\left(x\right)=0$, we have ${\displaystyle x_0=-\frac{1}{\alpha \lambda }log\frac{\beta +1}{\beta +\beta \alpha }}$, and $f^{\prime \prime }\left(x_0\right)>0$. Thus, if $\alpha \beta >1$, we have $f\left(x\right)$ is monotone increasing on $0<x<x_0$ and monotone decreasing on $x>x_0$. Thus, $f\left(x\right)$ is unimodal.

This ends the proof of Proposition 2. □

Remark 2.

The two properties in Proposition 2 are actually exclusive.

Proposition 3.

Let $X\sim GME(\lambda ,\alpha ,\beta )$, the moment generating function of X is

$$M_X\left(t\right)=\frac{\lambda (\alpha +1)}{\alpha +1+\alpha \beta }\left(\frac{1+\beta }{\lambda -t}-\frac{\beta }{\lambda \alpha +\lambda -t}\right),t<\lambda .$$

Proof of Proposition 3. 

According to Equation (1) and the definition of moment generating function,

$$\begin{array}{ccc}\hfill M_X\left(t\right)& =& {\displaystyle \frac{\lambda (\alpha +1)}{\alpha +1+\alpha \beta }\left[(1+\beta ){\int }_0^{\infty }{e}^{(t-\lambda )x}dx-\beta {\int }_0^{\infty }{e}^{(t-\lambda -\lambda \alpha )x}dx\right]}\hfill \\ & =& {\displaystyle \frac{\lambda (\alpha +1)}{\alpha +1+\alpha \beta }\left(\frac{1+\beta }{\lambda -t}-\frac{\beta }{\lambda \alpha +\lambda -t}\right),t<\lambda .}\hfill \end{array}$$

This ends the proof of Proposition 3. □

Corollary 1.

Let $X\sim GME(\lambda ,\alpha ,\beta )$, the first four moments of X are

$$\begin{array}{ccc}\hfill E\left[X\right]& =& \frac{{(1+\alpha )}^2(1+\beta )-\beta }{\lambda (\alpha +1+\alpha \beta )(1+\alpha )},E\left[X^2\right]=\frac{2{(1+\alpha )}^3(1+\beta )-2\beta }{{\lambda }^2(\alpha +1+\alpha \beta ){(1+\alpha )}^2},\hfill \\ \hfill E\left[X^3\right]& =& \frac{6{(1+\alpha )}^4(1+\beta )-6\beta }{{\lambda }^3(\alpha +1+\alpha \beta ){(1+\alpha )}^3},E\left[X^4\right]=\frac{24{(1+\alpha )}^5(1+\beta )-24\beta }{{\lambda }^4(\alpha +1+\alpha \beta ){(1+\alpha )}^4}.\hfill \end{array}$$

Proposition 4.

Let $X\sim GME(\lambda ,\alpha ,\beta )$ and $\mu =E\left[X\right]$, then the mean deviation about the mean of X is given by

$$D\left(\mu \right)=\frac{2(\alpha +1+\alpha \beta +\beta )}{\lambda (\alpha +1+\alpha \beta )}{e}^{-\lambda \mu }-\frac{2\beta }{\lambda (\alpha +1+\alpha \beta )(1+\alpha )}{e}^{-\lambda \mu (1+\alpha )}.$$

Proof of Proposition 4. 

According to Equation (1) and $D\left(\mu \right)=E\left[\left|X-\mu \right|\right]$, with $\mu =E\left[X\right]$, we have

$$\begin{array}{cc}\hfill D\left(\mu \right)& ={\int }_0^{\mu }(\mu -x)f\left(x\right)dx+{\int }_{\mu }^{\infty }(x-\mu )f\left(x\right)dx\hfill \\ \hfill & =\mu \left({\int }_0^{\mu }f\left(x\right)dx-{\int }_{\mu }^{\infty }f\left(x\right)dx\right)+\left(-{\int }_0^{\mu }xf\left(x\right)dx+{\int }_{\mu }^{\infty }xf\left(x\right)dx\right)\hfill \\ \hfill & =\mu \left(2F\left(\mu \right)-1\right)+\left(\mu -2{\int }_0^{\mu }xf\left(x\right)dx\right)\hfill \\ \hfill & =\frac{2\beta }{\lambda (\alpha +1+\alpha \beta )(1+\alpha )}\left(1-e^{-\lambda \mu (1+\alpha )}\right)-\frac{2(\alpha +1+\alpha \beta +\beta )}{\lambda (\alpha +1+\alpha \beta )}(1-e^{-\lambda \mu })+2\mu \hfill \\ \hfill & =\frac{2(\alpha +1+\alpha \beta +\beta )}{\lambda (\alpha +1+\alpha \beta )}{e}^{-\lambda \mu }-\frac{2\beta }{\lambda (\alpha +1+\alpha \beta )(1+\alpha )}{e}^{-\lambda \mu (1+\alpha )}.\hfill \end{array}$$

This ends the proof of Proposition 4. □

The entropy of a random variable is a measure of uncertainty, which is an important topic in the fields of communication theory, statistical physics, and probability theory. In the following, we study the entropy measures for $X\sim GME(\lambda ,\alpha ,\beta )$.

Proposition 5.

Let $X\sim GME(\lambda ,\alpha ,\beta )$, then the Shannon entropy, S(x), and Renyi entropy, R(x), of X are given by

$$\begin{array}{ccc}\hfill S\left(x\right)& =& -log\left({\displaystyle \frac{\lambda (\alpha +1)}{\alpha +1+\alpha \beta }}\right)-\left(1+\frac{1}{\alpha }\right)log(1+\beta )+\frac{log\left(\beta \right)}{\alpha }-\frac{log\left(u\right)}{\alpha }-log(1-u),\hfill \\ \hfill R_{\gamma }\left(x\right)& =& \frac{1}{1-\gamma }\left[\gamma log\left({\displaystyle \frac{\lambda (\alpha +1)}{\alpha +1+\alpha \beta }}\right)+\left(\gamma +\frac{\gamma }{\alpha }\right)log(1+\beta )-\frac{\gamma log\left(\beta \right)}{\alpha }-log\left(\lambda \alpha \right)\right.\hfill \\ & & \left.+log\left(B\left(\frac{\beta }{1+\beta };\frac{\gamma }{\alpha },\gamma +1\right)\right)\right],\gamma >0and\gamma \ne 1,\hfill \end{array}$$

respectively, where $B(z;a,b)={\int }_0^z{u}^{a-1}{(1-u)}^{b-1}du$ is the incomplete beta function.

Proof of Proposition 5. 

For any $\gamma >0$ and $\gamma \ne 1$, the Reni entropy of X is defined as

$$\begin{array}{ccc}\hfill R_{\gamma }\left(x\right)& =& \frac{1}{1-\gamma }log\left[{\int }_0^{\infty }{f}^{\gamma }(x;\lambda ,\beta ,\alpha )dx\right]\hfill \\ & =& \frac{1}{1-\gamma }log\left[{\left({\displaystyle \frac{\lambda (\alpha +1)}{\alpha +1+\alpha \beta }}\right)}^{\gamma }{\int }_0^{\infty }{e}^{-\lambda \gamma x}\left(1+\beta {(1-e^{-\lambda \alpha x})}^{\gamma }\right)dx\right]\hfill \\ & =& \frac{1}{1-\gamma }log\left[{\left({\displaystyle \frac{\lambda (\alpha +1)}{\alpha +1+\alpha \beta }}\right)}^{\gamma }{(1+\beta )}^{\gamma +\frac{\gamma }{\alpha }}{\beta }^{-\frac{\gamma }{\alpha }}\right.\hfill \\ & \times & \left.{\int }_0^{\infty }{\left(\frac{\beta }{1+\beta }{e}^{-\lambda \alpha x}\right)}^{\frac{\gamma }{\alpha }}{\left(1-\frac{\beta }{1+\beta }{e}^{-\lambda \alpha x}\right)}^{\gamma }dx\right]\hfill \\ & =& \frac{1}{1-\gamma }\left[\gamma log\left({\displaystyle \frac{\lambda (\alpha +1)}{\alpha +1+\alpha \beta }}\right)+\left(\gamma +\frac{\gamma }{\alpha }\right)log(1+\beta )-\frac{\gamma }{\alpha }log\left(\beta \right)-log\left(\lambda \alpha \right)\right.\hfill \\ & & \left.+log\left(B\left(\frac{\beta }{1+\beta };\frac{\gamma }{\alpha },\gamma +1\right)\right)\right].\hfill \end{array}$$

The Shannon entropy $S\left(x\right)$ is the limiting value of $R_{\gamma }\left(x\right)$ as $\gamma \to 1$ and, thus, the results are obtained.

This ends the proof of Proposition 5. □

In the next proposition, we study the probability that one of the two independent GME random variables exceeds the other, which is named as the reliability probability.

Proposition 6.

Suppose two independent random variables X and Y follow $GME(\lambda ,\alpha ,\beta )$, then the reliability probability is given by

$$\begin{array}{ccc}\hfill P(X>Y)& =& {\displaystyle \frac{{(1+\alpha )}^2{(1+\beta )}^2}{2{(1+\alpha +\alpha \beta )}^2}}+{\displaystyle \frac{\lambda {\beta }^2}{2{(1+\alpha +\alpha \beta )}^2}}\hfill \\ & -& {\displaystyle \frac{\beta (1+\beta ){(1+\alpha )}^2}{(2+\alpha ){(1+\alpha +\alpha \beta )}^2}}-{\displaystyle \frac{\beta (1+\beta )(1+\alpha )}{(2+\alpha ){(1+\alpha +\alpha \beta )}^2}}.\hfill \end{array}$$

Proof of Proposition 6. 

Let $Z=Y-X$ and $X=X$, the joint density function of X and Z is obtained as

$$\begin{array}{cc}\hfill f(x,z;\lambda ,\beta ,\alpha )& =\frac{{\lambda }^2{(1+\alpha )}^2{(1+\beta )}^2}{{(1+\alpha +\alpha \beta )}^2}{e}^{-\lambda (2x+z)}-\frac{{\lambda }^2\beta (1+\beta ){(1+\alpha )}^2}{{(1+\alpha +\alpha \beta )}^2}{e}^{-\lambda \left[(1+\alpha )z+(2+\alpha )x\right]}\hfill \\ \hfill & -\frac{{\lambda }^2\beta (1+\beta ){(1+\alpha )}^2}{{(1+\alpha +\alpha \beta )}^2}{e}^{-\lambda \left[z+(2+\alpha )x\right]}+\frac{{\lambda }^2{\beta }^2{(1+\alpha )}^2}{{(1+\alpha +\alpha \beta )}^2}{e}^{-\lambda \left[(1+\alpha )z+(2+2\alpha )x\right]}.\hfill \end{array}$$

Therefore, the marginal density function of Z is

$$\begin{array}{cc}\hfill f(z;\lambda ,\beta ,\alpha )& {\displaystyle =\frac{{(\alpha +1)}^2{(1+\beta )}^2}{{(1+\alpha +\alpha \beta )}^2}{e}^{\lambda z}+\frac{\lambda {\beta }^2}{2{(1+\alpha +\alpha \beta )}^2}{e}^{\lambda (1+\alpha )z}}\hfill \\ \hfill & -\frac{\beta (1+\beta ){(1+\alpha )}^2}{(2+\alpha ){(1+\alpha +\alpha \beta )}^2}{e}^{\lambda z}-\frac{\beta (1+\beta )(1+\alpha )}{(2+\alpha ){(1+\alpha +\alpha \beta )}^2}{e}^{\lambda (1+\alpha )z}.\hfill \end{array}$$

Thus, the result is obtained by $P(X>Y)=P(Z<0)$.

This ends the proof of Proposition 6. □

Order statistics are fundamental tools in non-parametric statistics and inference. In what follows, we derive an expression for the density function of the $r^{th}$ order statistic in a random sample size $n\ge r$ from the GME distribution.

Proposition 7.

Suppose $X_1,X_2,\cdots ,X_n$ is a random sample from $GME(\lambda ,\alpha ,\beta )$. Let $X_{1:n}\le X_{2:n}\le \cdots \le X_{n:n}$ denote the corresponding order statistics. Then the pdf and cdf of $r^{th}$ order statistic, $X_{r:n}$, $1\le r\le n$, are respectively,

$$\begin{array}{ccc}\hfill f_{r:n}\left(x\right)& =& \frac{n!}{(r-1)!(n-r)!}{\left[{\displaystyle \frac{(1+\alpha )\lambda }{\alpha +1+\alpha \beta }(e^{-\lambda (1+\alpha )x}-e^{-\lambda x})-e^{-\lambda x}+1}\right]}^{r-1}\hfill \\ & \times & {\left[{\displaystyle \frac{(1+\alpha )\lambda }{\alpha +1+\alpha \beta }(e^{-\lambda x}-e^{-\lambda (1+\alpha )x})+e^{-\lambda x}}\right]}^{n-r}\frac{(\alpha +1)\lambda }{\alpha +1+\alpha \beta }{e}^{-\lambda x}[1+\beta (1-e^{-\alpha \lambda x})],\hfill \\ \hfill F_{r:n}\left(x\right)& =& {\displaystyle \sum _{l=r}^n\sum _{u=0}^{n-r}{(-1)}^u\left(\genfrac{}{}{0pt}{}{n}{l}\right)\left(\genfrac{}{}{0pt}{}{n-r}{u}\right){\left[{\displaystyle \frac{(1+\alpha )\lambda }{\alpha +1+\alpha \beta }(e^{-\lambda (1+\alpha )x}-e^{-\lambda x})-e^{-\lambda x}+1}\right]}^{l+u}.}\hfill \end{array}$$

Proof of Proposition 7. 

It is well known that the pdf and cdf of $X_{r:n}$, $1\le r\le n$ are given by

$$\begin{array}{ccc}\hfill f_{r:n}\left(x\right)& =& \frac{n!}{(r-1)!(n-r)!}{\left[F\left(x\right)\right]}^{r-1}{[1-F\left(x\right)]}^{n-r}f\left(x\right),\hfill \\ \hfill F_{r:n}\left(x\right)& =& \sum _{l=r}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right){\left[F\left(x\right)\right]}^l{[1-F\left(x\right)]}^{n-l},\hfill \end{array}$$

respectively. Thus, the results are derived straightly from Equation (1) and Proposition 1.

This ends the proof of Proposition 7. □

Proposition 8.

Let $X\sim GME(\lambda ,\alpha ,\beta )$, the quantile function of GME distribution, $x_q$, wherein $0<q<1$, can be obtained by solving the following equation,

$$\beta e^{-(\alpha +1)\lambda x_q}-(\alpha +1)(\beta +1)e^{-\lambda x_q}=(q-1)(1+\alpha +\alpha \beta ).$$

(2)

We can see from Equation (2) that there is no closed form of the solution in $x_q$ and, thus, we have to use numerical techniques to obtain the quantile.

The mean residual life (MRL) function plays a very important role in reliability engineering, survival analysis and many other fields. It represents the period from time t till the time of failure, and the MRL also represents the expected additional life length for a unit.

Proposition 9.

Let $X\sim GME(\lambda ,\alpha ,\beta )$, the MRL function of GME distribution, defined as ${\mu }_X\left(t\right)$, is given by

$${\displaystyle {\mu }_X\left(t\right)=\frac{{(1+\alpha )}^2(1+\beta )e^{-\lambda t}-\beta e^{-(1+\alpha )\lambda t}}{{(1+\alpha )}^2(1+\beta )\lambda e^{-\lambda t}-(1+\alpha )\lambda \beta e^{-(1+\alpha )\lambda t}},t>0.}$$

Proof of Proposition 9. 

For $t>0$, we have

$$\begin{array}{ccc}\hfill {\mu }_X\left(t\right)=E(X-t|X>t)& =& {\displaystyle \frac{{\int }_t^{\infty }S(x;\lambda ,\alpha ,\beta )dx}{S(t;\lambda ,\alpha ,\beta )}}\hfill \\ & =& \frac{{\displaystyle {\int }_t^{\infty }\left[\frac{(\alpha +1)(\beta +1)e^{-\lambda x}}{\alpha +1+\alpha \beta }-\frac{\beta e^{-(\alpha +1)\lambda x}}{\alpha +1+\alpha \beta }\right]dx}}{S(t;\lambda ,\alpha ,\beta )},\hfill \end{array}$$

where $S(·)$ is survival function of GME distribution. We know that

$$\begin{array}{ccc}\hfill {\int }_t^{\infty }\left[\frac{(\alpha +1)(\beta +1)e^{-\lambda x}}{\alpha +1+\alpha \beta }-\frac{\beta e^{-(\alpha +1)\lambda x}}{\alpha +1+\alpha \beta }\right]dx& =& {\displaystyle {\int }_t^{\infty }\frac{(\alpha +1)(\beta +1)e^{-\lambda x}}{\alpha +1+\alpha \beta }dx-{\int }_t^{\infty }\frac{\beta e^{-(\alpha +1)\lambda x}}{\alpha +1+\alpha \beta }dx}\hfill \\ & =& {\displaystyle \frac{{(1+\alpha )}^2(1+\beta )e^{-\lambda t}-\beta e^{-(1+\alpha )\lambda t}}{(1+\alpha +\alpha \beta )(1+\alpha )\lambda }.}\hfill \end{array}$$

Thus, the result is obtained.

This ends the proof of Proposition 9. □

4. Methods of Estimation

In this section, we consider the methods of maximum likelihood, least squares, and weighted least squares to estimate the unknown parameters, $\theta =(\lambda ,\alpha ,\beta )$, of the GME distribution. Suppose $x_1,x_2,\cdots ,x_n$ is a random sample from $GME(\lambda ,\alpha ,\beta )$.

4.1. Maximum Likelihood Estimator

The method of maximum likelihood is the most frequently used method for parameter estimation. According to the Equation (1), the likelihood function is calculated as

$$L(\lambda ,\alpha ,\beta |x_1,\cdots ,x_n)={\left(\frac{(\alpha +1)\lambda }{\alpha +1+\alpha \beta }\right)}^n{e}^{-\lambda {\displaystyle \sum _{i=1}^n}{x}_i}\prod _{i=1}^n[1+\beta (1-e^{-\alpha \lambda x_i})].$$

The log-likelihood function is given by

$$\begin{array}{cc}\hfill \ell (\lambda ,\alpha ,\beta |x_1,\cdots ,x_n)& =n[log(\alpha +1)+log\left(\lambda \right)-log(\alpha +1+\alpha \beta )]-\lambda \sum _{i=1}^n{x}_i\hfill \\ \hfill & +\sum _{i=1}^{n}log[1+\beta (1-e^{-\alpha \lambda x_i})].\hfill \end{array}$$

(3)

We denote the first partial derivatives of (3) by ${\ell }_{\lambda }$, ${\ell }_{\alpha }$ and ${\ell }_{\beta }$. Setting ${\ell }_{\lambda }=0$, ${\ell }_{\alpha }=0$, and ${\ell }_{\beta }=0$, we have

$$\begin{array}{ccc}\hfill {\ell }_{\lambda }& =& \frac{n}{\lambda }-\sum _{i=1}^n{x}_i+\sum _{i=1}^n\frac{\beta \alpha x_i{e}^{-\alpha \lambda x_i}}{1+\beta (1-e^{-\alpha \lambda x_i})}=0,\hfill \\ \hfill {\ell }_{\alpha }& =& \frac{n}{\alpha +1}-\frac{n(1+\beta )}{\alpha +1+\alpha \beta }+\sum _{i=1}^n\frac{\beta \lambda x_i{e}^{-\alpha \lambda x_i}}{1+\beta (1-e^{-\alpha \lambda x_i})}=0,\hfill \\ \hfill {\ell }_{\beta }& =& -\frac{n\alpha }{\alpha +1+\alpha \beta }+\sum _{i=1}^n\frac{1-e^{-\alpha \lambda x_i}}{1+\beta (1-e^{-\alpha \lambda x_i})}=0.\hfill \end{array}$$

The maximum likelihood estimator (MLE) $\widehat{\theta }$ of the unknown parameters $\theta$ can be obtained by optimizing the log-likelihood function with respect to the involved parameters. Due to the non-linearity of these equations, the MLEs of parameters can be obtained numerically. These estimators can be easily obtained by using the functions from the statistical software R.

Fisher information is helpful to get the reference priors for the model parameters. In the following, we observe that the Fisher information is given by

$$\begin{array}{c}\hfill I\left(\theta \right)=-E\left[\begin{array}{ccc}{\ell }_{\lambda \lambda }& {\ell }_{\lambda \alpha }& {\ell }_{\lambda \beta }\\ {\ell }_{\alpha \lambda }& {\ell }_{\alpha \alpha }& {\ell }_{\alpha \beta }\\ {\ell }_{\beta \lambda }& {\ell }_{\beta \alpha }& {\ell }_{\beta \beta }\end{array}\right],\end{array}$$

where

$$\begin{array}{ccc}\hfill {\ell }_{\lambda \lambda }& =& -\frac{n}{{\lambda }^2}-\sum _{i=1}^n\frac{\beta (1+\beta ){\alpha }^2{x}_i^2{e}^{-2\alpha \lambda x_i}}{{[1+\beta (1-e^{-\alpha \lambda x_i})]}^2},\hfill \\ \hfill {\ell }_{\alpha \alpha }& =& -\frac{n}{{(\alpha +1)}^2}+\frac{n{(1+\beta )}^2}{{(1+\alpha +\alpha \beta )}^2}-\sum _{i=1}^n\frac{\beta (1+\beta ){\lambda }^2{x}_i^2{e}^{-2\alpha \lambda x_i}}{{[1+\beta (1-e^{-\alpha \lambda x_i})]}^2},\hfill \\ \hfill {\ell }_{\beta \lambda }& =& \sum _{i=1}^n\frac{\alpha x_i{e}^{-\alpha \lambda x_i}}{{[1+\beta (1-e^{-\alpha \lambda x_i})]}^2}={\ell }_{\lambda \beta },\hfill \\ \hfill {\ell }_{\beta \alpha }& =& \sum _{i=1}^n\frac{\lambda x_i{e}^{-\alpha \lambda x_i}}{{[1+\beta (1-e^{-\alpha \lambda x_i})]}^2}={\ell }_{\alpha \beta },\hfill \\ \hfill {\ell }_{\lambda \alpha }& =& \sum _{i=1}^n\frac{\beta (1+\beta )(1-\alpha \lambda x_i)x_i{e}^{-\alpha \lambda x_i}-{\beta }^2{x}_i{e}^{-2\alpha \lambda x_i}}{[1+\beta {(1-e^{-\alpha \lambda x_i})]}^2}={\ell }_{\alpha \lambda },\hfill \\ \hfill {\ell }_{\beta \beta }& =& \frac{n{\alpha }^2}{{(1+\alpha +\alpha \beta )}^2}-\sum _{i=1}^n\frac{{(1-e^{-\alpha \lambda x_i})}^2}{{[1+\beta (1-e^{-\alpha \lambda x_i})]}^2}.\hfill \end{array}$$

4.2. Least-Square Estimator

Suppose $F\left(x_{\left(j\right)}\right)$ denotes the distribution function of the ordered random variables $x_{\left(1\right)}<\cdots <x_{\left(n\right)}$. Denote the following function

$$h(\lambda ,\alpha ,\beta )=\sum _{i=1}^n{\left[F(x_{\left(i\right)};\lambda ,\alpha ,\beta )-\frac{i}{n+1}\right]}^2,$$

(4)

where ${\displaystyle F(x;\lambda ,\alpha ,\beta )=\frac{\beta }{\alpha +1+\alpha \beta }[e^{-\lambda (1+\alpha )x}-e^{-\lambda x}]-e^{-\lambda x}+1}$, and the least-square estimator (LS) of $\theta$ can be obtained by minimizing $h(\lambda ,\alpha ,\beta )$. Therefore, $\widehat{\theta }$ can be obtained by solving the following equations,

$$\begin{array}{ccc}& & \frac{\partial h(\lambda ,\alpha ,\beta )}{\partial \lambda }=\sum _{i=1}^n\left\{2Q_i\left\{\frac{\beta }{\alpha +1+\alpha \beta }\left[-(1+\alpha )C_i+B_i\right]+B_i\right\}\right\}=0,\hfill \\ & & \frac{\partial h(\lambda ,\alpha ,\beta )}{\partial \alpha }=\sum _{i=1}^n\left\{2Q_i\left\{\frac{-(1+\beta )}{{(\alpha +1+\alpha \beta )}^2}[e^{-\lambda (1+\alpha )X_{\left(i\right)}}-e^{-\lambda X_{\left(i\right)}}]-\frac{\beta \lambda }{\alpha +1+\alpha \beta }{C}_i\right\}\right\}=0,\hfill \\ & & \frac{\partial h(\lambda ,\alpha ,\beta )}{\partial \beta }=\sum _{i=1}^n\left\{2Q_i\left\{\frac{1-\alpha }{{(\alpha +1+\alpha \beta )}^2}\left[e^{-\lambda (1+\alpha }\right)X_{\left(i\right)}-e^{-\lambda X_{\left(i\right)}}]\right\}\right\}=0,\hfill \end{array}$$

where ${\displaystyle Q_i=\frac{\beta }{\alpha +1+\alpha \beta }[e^{-\lambda (1+\alpha )x_{\left(i\right)}}-e^{-\lambda x_{\left(i\right)}}]-e^{-\lambda x_{\left(i\right)}}+1-\frac{i}{n+1}}$, $B_i=x_{\left(i\right)}{e}^{-\lambda x_{\left(i\right)}}$ and

$C_i=x_{\left(i\right)}{e}^{-\lambda (1+\alpha )x_{\left(i\right)}}$.

4.3. Weighted Least-Square Estimator

The weighted least-square estimator (WLS) is an extension of LS and proposed by Swain et al. [17], which studied the WLS is obtained by minimizing the function,

$$W(\lambda ,\alpha ,\beta )=\sum _{i=1}^n\frac{{(n+1)}^2(n+2)}{i(n-i+1)}{\left[F(x_{\left(i\right)};\lambda ,\alpha ,\beta )-\frac{i}{n+1}\right]}^2,$$

where $F(·)$ function has been given in Equation (4). Therefore, the WLS of $\theta$ can be obtained by

$$\begin{array}{cc}\hfill \frac{\partial W(\lambda ,\alpha ,\beta )}{\partial \lambda }& =\sum _{i=1}^n\left\{\frac{2{(n-1)}^2(n+2)}{i(n-i+1)}{Q}_i\left\{\frac{\beta }{\alpha +1+\alpha \beta }\left[-(1+\alpha )C_i+B_i\right]+B_i\right\}\right\}=0,\hfill \\ \hfill \frac{\partial W(\lambda ,\alpha ,\beta )}{\partial \alpha }& =\sum _{i=1}^n\left\{\frac{2{(n-1)}^2(n+2)}{i(n-i+1)}{Q}_i\right\{\frac{-(1+\beta )}{{(\alpha +1+\alpha \beta )}^2}[e^{-\lambda (1+\alpha )x_{\left(i\right)}}-e^{-\lambda x_{\left(i\right)}}]\hfill \\ \hfill & -\frac{\beta \lambda }{\alpha +1+\alpha \beta }{C}_i\left\}\right\}=0,\hfill \\ \hfill \frac{\partial W(\lambda ,\alpha ,\beta )}{\partial \beta }& =\sum _{i=1}^n\left\{\frac{2{(n-1)}^2(n+2)}{i(n-i+1)}{Q}_i\left\{\frac{1-\alpha }{{(\alpha +1+\alpha \beta )}^2}[e^{-\lambda (1+\alpha )x_{\left(i\right)}}-e^{-\lambda x_{\left(i\right)}}]\right\}\right\}=0,\hfill \end{array}$$

where $Q_i,B_i$ and $C_i$, $i=1,\cdots ,n$, are defined as above.

5. Simulation Studies

In this section, we assess the performance of the estimation methods proposed in the previous section by conducting several simulations for different sample sizes and values of the parameter.

As indicated in Proposition 1, the $F(x;\lambda ,\alpha ,\beta )$ there is used to generate pseudo-random numbers from the GME distribution. This technique is called the inverse transform method, which consists of the following steps:

(i)

Generate a random number u from the standard uniform distribution in the interval [0,1].

(ii)

Apply the numerical techniques to solve the equation $F\left(x\right)=u$ with given $\lambda ,\alpha ,\beta$.

We take the sample size $n=50$, $100,200,300,400,500,1000$ for each simulation, and each sample was replicated $N=1000$ times. The values of parameter $\theta =(1,5,-0.5),(2,1,1.5)$, and $(3,10,5)$ are considered, respectively. All the results were computed using the R programming. The evaluation of the estimators are performed based on the average bias and the standard error (SE) for each single parameter, where $Bias\left({\widehat{\theta }}_j\right)=\frac{1}{N}{\sum }_{i=1}^N({\widehat{\theta }}_j^{\left(i\right)}-{\theta }_j)$, $SE\left({\widehat{\theta }}_j\right)=\frac{1}{N}{\sum }_{i=1}^N{({\widehat{\theta }}_j^{\left(i\right)}-{\theta }_j)}^2$, and ${\theta }_j$ is the $j^{th}$ component of $\theta$. Moreover, the overall bias and mean squared error (MSE) of $\widehat{\theta }$ are also considered, where the $Bias\left(\widehat{\theta }\right)={\sum }_{j=1}^{3}Bias\left({\widehat{\theta }}_j\right)$, ${\displaystyle MSE\left(\widehat{\theta }\right)=\frac{1}{N}\sum _{i=1}^N\left|\right|{\widehat{\theta }}^{\left(i\right)}-\theta {\left|\right|}^2}$, and $\left|\right|·\left|\right|$ is the Euclidean norm. The simulation results for different scenarios are given in the

Table 1. $\lambda =1$, $\alpha =5$, $\beta =-0.5$.

Sample Size	Method	$\widehat{\mathit{\lambda }}\left(\mathit{SE}\right)$	$\widehat{\mathit{\alpha }}\left(\mathit{SE}\right)$	$\widehat{\mathit{\beta }}\left(\mathit{SE}\right)$	$\mathit{MSE}$
n = 50	MLE	0.9337 (0.3743)	5.0838 (2.4378)	−0.1080 (1.7229)	1.5273
	LS	0.9747 (0.2605)	4.7614 (2.3440)	−0.4964 (0.2255)	0.9491
	WLS	0.9162 (0.2344)	5.0376 (1.1337)	−0.4444 (0.7600)	0.7204
n = 100	MLE	0.8927 (0.3087)	5.7791 (3.1581)	−0.5004 (0.8806)	1.4852
	LS	1.0136 (0.2050)	4.6547 (2.2373)	−0.4550 (0.2004)	0.8934
	WLS	0.9433 (0.1481)	4.6418 (0.9328)	−0.4858 (0.4072)	0.5307
n = 200	MLE	0.9407 (0.2054)	5.3528 (2.7663)	−0.5606 (0.2314)	1.0774
	LS	0.9954 (0.1574)	4.7508 (2.2406)	−0.4717 (0.1863)	0.8692
	WLS	0.9646 (0.1087)	4.9761 (0.9436)	−0.49608 (0.1881)	0.4350
n = 300	MLE	0.9780 (0.1157)	4.9851 (2.1748)	−0.5229 (0.1454)	0.8116
	LS	1.0224 (0.1291)	5.4474 (2.1971)	−0.4555 (0.1605)	0.8504
	WLS	0.9654 (0.0982)	4.5041 (0.8171)	−0.5127 (0.1298)	0.4120
n = 400	MLE	0.9731 (0.1137)	4.0055 (1.4628)	−0.5137 (0.1432)	0.6742
	LS	1.0158 (0.1180)	5.1742 (1.9422)	−0.4639 (0.1519)	0.7472
	WLS	0.9980 (0.0733)	5.0675 (0.9835)	−0.4948 (0.1078)	0.4018
n = 500	MLE	0.9841 (0.0970)	4.8382 (1.9489)	−0.5135 (0.1202)	0.7242
	LS	1.0023 (0.1071)	4.9641 (1.7818)	−0.4779 (0.1345)	0.6818
	WLS	0.9821 (0.0610)	4.9304 (0.9001)	−0.5062 (0.0849)	0.3783
n = 1000	MLE	0.9938 (0.0705)	4.9315 (1.6172)	−0.5020 (0.0932)	0.5937
	LS	1.0060 (0.0763)	4.9750 (1.7033)	−0.4797 (0.1029)	0.6367
	WLS	0.9894 (0.0501)	4.8743 (0.8206)	−0.5019 (0.0635)	0.3545

,

Table 2. $\lambda =2$, $\alpha =1$, $\beta =1.5$.

Sample Size	Method	$\widehat{\mathit{\lambda }}\left(\mathit{SE}\right)$	$\widehat{\mathit{\alpha }}\left(\mathit{SE}\right)$	$\widehat{\mathit{\beta }}\left(\mathit{SE}\right)$	$\mathit{MSE}$
n = 50	MLE	1.4458 (0.9097)	1.2917 (0.5589)	1.6460 (3.2067)	1.6314
	LS	1.8709 (0.3747)	1.4399 (0.5717)	2.0283 (2.1736)	1.1183
	WLS	1.9537 (0.3217)	0.9440 (0.6706)	2.2223 (1.8281)	0.9885
n = 100	MLE	1.7026 (0.7886)	1.3034 (0.5729)	1.9781 (2.6490)	1.3911
	LS	1.9554 (0.2942)	1.2025 (0.6341)	2.3221 (2.0580)	0.9073
	WLS	1.9519 (0.2279)	0.9008 (0.6066)	1.9902 (1.6569)	0.8606
n = 200	MLE	1.7535 (0.6525)	1.3555 (0.5795)	1.6087 (1.8850)	1.0782
	LS	1.9669 (0.2506)	1.1475 (0.6229)	1.9986 (1.5482)	0.7287
	WLS	1.9578 (0.1269)	1.0596 (0.4735)	1.5661 (0.6906)	0.4392
n = 300	MLE	1.8438 (0.5467)	1.2813 (0.5962)	1.6082 (1.5177)	0.9151
	LS	1.8927 (0.1585)	1.6745 (0.4603)	1.3887 (0.8766)	0.5973
	WLS	1.9835 (0.1121)	1.0563 (0.4295)	1.5963 (0.6744)	0.4141
n = 400	MLE	1.8684 (0.4775)	1.3238 (0.5647)	1.5057 (1.2785)	0.8075
	LS	1.9413 (0.1308)	1.3760 (0.5588)	1.5542 (0.8849)	0.5718
	WLS	1.9638 (0.0916)	1.1273 (0.3971)	1.4569 (0.3923)	0.3095
n = 500	MLE	1.9002 (0.4561)	1.2733 (0.5697)	1.6020 (1.1967)	0.7662
	LS	2.0143 (0.1602)	1.0437 (0.5819)	1.9070 (0.9159)	0.5142
	WLS	1.9573 (0.0739)	1.0416 (0.3624)	1.5810 (0.5662)	0.3489
n = 1000	MLE	1.9189 (0.3525)	1.2974 (0.5674)	1.5166 (0.8654)	0.6228
	LS	1.9928 (0.1392)	1.1979 (0.6193)	1.7116 (0.7494)	0.5268
	WLS	1.9700 (0.0613)	1.1250 (0.29170	1.4208 (0.2599)	0.2280

and

Table 3. $\lambda =3$, $\alpha =10$, $\beta =5$.

Sample Size	Method	$\widehat{\mathit{\lambda }}\left(\mathit{SE}\right)$	$\widehat{\mathit{\alpha }}\left(\mathit{SE}\right)$	$\widehat{\mathit{\beta }}\left(\mathit{SE}\right)$	$\mathit{MSE}$
n = 50	MLE	3.0424 (0.4181)	9.7778 (2.4919)	5.6122 (3.8162)	2.2299
	LS	3.0549 (0.5084)	9.9521 (2.3594)	4.4837 (4.1413)	2.3605
	WLS	3.0281 (0.4395)	10.1081 (2.0774)	5.0283 (3.9031)	2.1545
n = 100	MLE	3.0380 (0.3025)	9.8275 (2.4712)	5.5335 (3.4914)	2.0887
	LS	3.0416 (0.3537)	9.8668 (2.2719)	5.0004 (4.0233)	2.2335
	WLS	2.9993 (0.3392)	10.1385 (2.0807)	4.6623 (3.5851)	2.0216
n = 200	MLE	3.0172 (0.2143)	9.5225 (2.1802)	5.4294 (3.2236)	1.9055
	LS	3.0208 (0.2368)	9.7251 (2.2322)	4.9982 (3.6205)	2.0550
	WLS	3.0072 (0.2337)	10.3121 (2.2217)	4.6433 (2.9485)	1.8293
n = 300	MLE	3.0239 (0.1718)	9.5911 (2.2102)	5.5706 (3.1488)	1.8680
	LS	3.0251 (0.2052)	9.5669 (1.9457)	5.3041 (3.7496)	2.0098
	WLS	3.0066 (0.1789)	10.052 (2.0334)	4.6414 (2.7485)	1.6785
n = 400	MLE	3.0222 (0.1455)	9.3730 (1.9607)	5.4066 (2.9740)	1.7391
	LS	3.0285 (0.1623)	9.5468 (2.1307)	5.1461 (3.3433)	1.9214
	WLS	3.0104 (0.1522)	9.9920 (1.9927)	4.6140 (2.4915)	1.5730
n = 500	MLE	3.0166 (0.1405)	9.3607 (1.9663)	5.3163 (2.8919)	1.6961
	LS	3.0163 (0.1561)	9.4042 (1.9839)	4.9983 (3.4092)	1.9053
	WLS	3.0071 (0.1374)	10.0114 (1.9581)	4.7762 (2.4242)	1.5292
n = 1000	MLE	3.0139 (0.0984)	9.3293 (1.1715)	5.1518 (2.4282)	1.4799
	LS	3.0235 (0.1120)	9.2747 (1.8501)	5.1674 (3.4079)	1.8668
	WLS	3.0136 (0.1032)	9.7242 (1.6997)	4.9825 (2.2380)	1.3759

.

From Table 1, Table 2 and Table 3, we find that the SE of all three estimator decrease as the sample size n increases and all estimators will tend to more accuracy when n is large. In addition, the estimated values obtained by the three estimator are close to the true values. Furthermore, the plots of bias of the simulated estimators of $\lambda$, $\alpha$, and $\beta$, corresponding with different sample size n, are shown in Figure 3, Figure 4 and Figure 5, respectively.

From Figure 3, Figure 4 and Figure 5, we observe that the magnitude of bias of all estimators tends to zero as n grows, which means these estimators are asymptotically unbiased and consistent for the parameters. Thus, these estimator techniques perform well for estimating the parameters in the GME distribution. For further studying, we draw the plots of overall bias and MSE for $\widehat{\theta }$ in Figure 6 and Figure 7.

Figure 6 and Figure 7 show the bias and MSE of $\widehat{\theta }$, and we can find that as n increases, the bias of $\widehat{\theta }$ towards to zero, and the WLS always has the smallest value of MSE. The LS estimators have the largest MSE among the three considered estimators. Thus, we can conclude that WLS can be chosen as a more reliable estimator for the GME distribution.

6. Real Data Analysis

In this section, we use the weighted least-square estimator to analyze two real data sets for investigating the advantage of proposed GME distribution and compare it with some other distributions, including the exponential distribution distribution, WED distribution, GExtEW distribution, and GWE distribution, where the pdfs are given as follows.

(1)

Exponential distribution: $E\left(\lambda \right)$

$$f_E\left(x\right)=\lambda e^{-\lambda x},x\ge 0,\lambda >0.$$

(2)

Weibull exponential distribution: $WED(\lambda ,\alpha ,\beta )$

$$f_{WED}\left(x\right)=\alpha \beta \left(\lambda e^{-\lambda x}\right)\left[\frac{{(1-e^{-\lambda x})}^{\beta -1}}{{\left(e^{-\lambda x}\right)}^{\beta +1}}\right]exp\left\{-\alpha {\left[\frac{1-e^{-\lambda x}}{e^{-\lambda x}}\right]}^{\beta }\right\},x>0,\alpha ,\beta ,\lambda >0.$$

(3)

Generalized extended exponential-Weibull distribution: $GExtEW(\lambda ,\alpha ,\beta ,r,c)$

$$f_{GExtEW}\left(x\right)=c\alpha (r\beta x^{r-1}+\lambda ){(\beta x^r+\lambda x)}^{c-1}{e}^{-{(\beta x^r+\lambda x)}^c},x>0,c,\beta ,\lambda >0.r\in (0,\infty )\setminus \left\{1\right\}.$$

(4)

Generalized weighted exponential distribution: $GWE(\lambda ,\alpha ,k)$

$$f_{GWE}\left(x\right)=\frac{\alpha }{B(1/\alpha ,k+1)}\lambda e^{-\lambda x}{(1-e^{-\lambda \alpha x})}^k,x>0,\alpha ,\lambda >0,k\in {\mathbf{Z}}^{+}.$$

6.1. Data Set 1: Waiting Times

This data set represents the waiting times (in minutes) before the service of 100 bank customers, which has been previously used by Ghitany et al. [18]. It can be seen in Appendix A.1 of Appendix A.

Table 4. Data set 1: Comparison between the GME, E, WED, GExtEW, and GWE by using different criteria.

	$\mathit{GME}(\mathit{\lambda },\mathit{\alpha },\mathit{\beta })$	$\mathit{E}\left(\mathit{\lambda }\right)$	$\mathit{WED}(\mathit{\lambda },\mathit{\alpha },\mathit{\beta })$	$\mathit{GExtEW}(\mathit{\lambda },\mathit{\alpha },\mathit{\beta },\mathit{r},\mathit{c})$	$\mathit{GWE}(\mathit{\lambda },\mathit{\alpha },\mathit{k})$
$\lambda$	0.1554	0.0883	0.0234	0.0807	0.1284
$\alpha$	0.8202	-	5.3703	3.9509	6.2681
$\beta$	124.9639	-	1.3486	0.0812	-
r	-	-	-	1.7231	-
c	-	-	-	0.4711	-
k	-	-	-	-	4
-loglike	317.3592	329.9063	321.8420	317.1620	318.4602
AIC	640.7184	661.8126	649.6839	644.3241	642.9205
BIC	648.5339	664.4177	657.4995	657.3499	650.7360

shows the parameter estimator results of the GME, E, WED, GExtEW, and GWE distributions for these data. The corresponding minus log-likelihood, Akaike information criterion (AIC), and Bayesian information criterion (BIC) are also presented. From Table 4, we find that the GME has the smallest values of all criteria for comparing all other distributions.

Figure 8 shows the fitted models for data set 1. The first subgraph of Figure 8 shows the fitted densities to the data set histogram and some estimated distributions, and the second subgraph displays the empirical distribution function for the data set and the estimated distributions. Both figures reveal that the GME distribution provides a qualified fit for the data set.

6.2. Date Set 2: Survival Times

The second data set represents the survival times of 121 patients with breast cancer obtained from a large hospital in a period from 1929 to 1938. This data set has recently been studied by Lee [19] and Tahir et al. [20]. The data set can be seen in Appendix A.2 of Appendix A. We compare the GME distribution with the E, WED, GExtEW, and GWE distributions. The estimated value of the parameters, AIC and BIC statistics of these distributions are listed in

Table 5. Data set 2: Comparison between the GME, E, WED, GExtEW, and GWE by using different criteria.

	$\mathit{GME}(\mathit{\lambda },\mathit{\alpha },\mathit{\beta })$	$\mathit{E}\left(\mathit{\lambda }\right)$	$\mathit{WED}(\mathit{\lambda },\mathit{\alpha },\mathit{\beta })$	$\mathit{GExtEW}(\mathit{\lambda },\mathit{\alpha },\mathit{\beta },\mathit{r},\mathit{c})$	$\mathit{GWE}(\mathit{\lambda },\mathit{\alpha },\mathit{k})$
$\lambda$	0.0311	0.0197	0.0114	0.0280	0.0237
$\alpha$	0.6384	-	1.2263	1.8338	11.8562
$\beta$	7.5142	-	1.0035	0.0069	-
r	-	-	-	0.8196	-
c	-	-	-	0.9454	-
k	-	-	-	-	3
-loglike	578.9263	585.5995	580.1301	580.6417	590.0420
AIC	1163.8530	1173.1990	1166.2600	1171.2830	1186.0850
BIC	1172.2400	1175.9950	1174.6480	1185.2620	1194.4720

. It can be seen that GME distribution provides the best fit among these competing models. Figure 9 displays the fitted pdfs and cdfs of the GME, E, WED, GExtEW, and GWE distributions for data set 2, and suggests that the fit of the GME distribution is reasonable.

7. Conclusions

In this paper, we introduce a new lifetime distribution, GME distribution, and propose several statistical properties of it. As it is not feasible to compare these methods theoretically, we have studied several simulations to identify the most efficient estimation method for GME distribution. The simulation results show that weighted least-square estimator (WLS) is the best performing estimator in terms of MSE. That is, the weighted least-square estimation method is more feasible for estimating parameters in the GME distribution. Finally, two real data sets were analyzed to indicate the importance and flexibility of GME distribution in comparison to some existing lifetime distributions. In the future, the development of properties and proper estimation procedure of the bivariate model and multivariate generalization will be of interest, and more work is needed along that direction.