A Novel Exponentiated Generalized Weibull Exponential Distribution: Properties, Estimation, and Regression Model

Abstract

The exponential distribution is one of the most popular models for fitting lifetime data. This study proposes a novel generalization of the exponential distribution, referred to as the exponentiated generalized Weibull exponential, for the modeling of lifetime data. This new distribution is a member of a family that combines two well-known distribution families: the exponentiated generalized family and the T-X family. It has five parameters, allowing it to fit data that exhibit increasing, decreasing, bathtub, upside-down bathtub, S-shaped, J-shaped and reversed-J hazard rates. Some mathematical and statistical properties of the newly suggested distribution are derived and the estimation of its parameters is studied using the method of maximum likelihood. Different simulation studies have been applied to evaluate the parameter estimation. Four lifetime datasets are analyzed to investigate the superiority of the proposed exponentiated generalized Weibull exponential distribution. A regression model based on the proposed distribution is then developed for both complete and censored samples, and its performance is assessed on two real datasets. The new distribution and its associated regression model are empirically demonstrated to be practically useful.

1. Introduction

Statistical distributions can be considered important tools for describing and fitting a variety of data in different disciplines. They are commonly used in survival analysis to reflect the duration of time until an event of interest occurs, such as the lifetime, survival time, or failure time. A lifetime dataset includes events such as deaths, the occurrence of diseases, or machine failures, and fitting a given lifetime dataset with an adequate probability distribution is one of the fundamental requirements for accurate statistical inference. Modeling lifetime data plays an essential role in fields such as engineering, medical sciences, biology, and management. Despite the fact that numerous probability models which can be used to describe lifetime data have been proposed over the past few decades, a majority of classical distributions are not suitable for fitting the complex shapes of lifetime data. Therefore, the need to develop new distributions with more flexibility to better fit lifetime data and accommodate different datasets has been emphasized in recent years. Particularly, several attempts have been made to extend the classical models and generate new families of distributions that generalize existing distributions. This goal can be accomplished using a variety of generalization techniques, such as compounding two or more families or adding extra parameter(s) to a baseline probability distribution. For example, the exponentiated method was introduced in [1], which generalizes a distribution by adding an extra shape parameter to its probability distribution function. Some exponentiated distributions include the exponentiated Weibull [2], exponentiated gamma [3], exponentiated Lomax [4], exponentiaeted Chen [5], exponentiated additive Weibull [6], exponentiated Teissier [7], exponentiated power Ishita [8], exponentiated Ailamujia [9], exponentiated arctan exponential [10], and exponentiated Gompertz Marshall Olkin Weibull [11] distributions.

In ref. [12], the exponentiated generalized (EG) class of distributions was proposed, which generalizes the exponentiated class by adding another shape parameter. This class can be obtained for a base cumulative distribution function (cdf) $G\left(x\right)$ and probability density function (pdf) $g\left(x\right)$ as follows

$$F\left(x\right)={\left\{1-{\left[1-G\left(x\right)\right]}^a\right\}}^b,$$

(1)

with the corresponding pdf

$$f\left(x\right)=abg\left(x\right){\left[1-G\left(x\right)\right]}^{a-1}{\left\{1-{\left[1-G\left(x\right)\right]}^a\right\}}^{b-1},$$

where $a>0$ and $b>0$ are shape parameters. The EG method has been applied in various studies to introduce new distributions, such as the exponentiated generalized Gumbel [13], exponentiated generalized Fréchet [14], exponentiated generalized Lindley [15], exponentiated generalized Kumaraswamy [16], exponentiated generalized extended Gompertz [17], exponentiated generalized inverted Gompertz [18], exponentiated generalized alpha power exponential [19], exponentiated generalized exponentiated exponential [20], exponentiated generalized Ramos Louzada distribution [21], exponentiated generalized Burr XII [22], and exponentiated generalized Topp Leone exponential [23] distributions.

In addition, a new method to generate families of distributions was developed in [24] called the transformed-transformer (T-X) family of distributions. This family of distributions can be defined for any base pdf g and cdf G by assuming a continuous generator random variable (RV) T defined on $[a,b]$ with a pdf $r\left(t\right)$ and a cdf $R\left(t\right)$, where the cdf of this generalized family of distributions for an RV X is defined as

$$F\left(x\right)={\int }_{r_n}^{W\left(G\right(x\left)\right)}r\left(t\right)dt=R\left(W\left(G\left(x\right)\right)\right),$$

(2)

with corresponding pdf

$$f\left(x\right)=\left\{{\displaystyle \frac{d}{dx}}W\left(G\left(x\right)\right)\right\}r\left(W\left(G\left(x\right)\right)\right),$$

(3)

where $r_n$ is a real number and $W\left[G\right(x\left)\right]$ is a function of the cdf for the base RV X. For instance, considering the Weibull G family, also referred to as the Weibull X family (see [25,26]), in which an RV X has baseline cdf $G\left(x\right)$, the cdf and pdf for the Weibull G distribution with two extra parameters c and $\beta$ can be obtained as follows

$$F\left(x\right)={\int }_0^{-log(1-G(x\left)\right)}{\displaystyle \frac{c}{\beta }}{\left({\displaystyle \frac{t}{\beta }}\right)}^{c-1}{e}^{-{\left({\displaystyle \frac{t}{\beta }}\right)}^c}dt=1-e^{-{\left({\displaystyle \frac{-log(1-G(x\left)\right)}{\beta }}\right)}^c},$$

(4)

$$f\left(x\right)={\displaystyle \frac{c}{\beta }}\left({\displaystyle \frac{g\left(x\right)}{1-G\left(x\right)}}\right){\left({\displaystyle \frac{-log(1-G(x\left)\right)}{\beta }}\right)}^{c-1}{e}^{-{\left({\displaystyle \frac{-log(1-G(x\left)\right)}{\beta }}\right)}^c}.$$

The Weibull G family defined in Equation (4) has been applied to define many statistical distribution, such as the Weibull Pareto in [27], Weibull Rayleigh in [28], Weibull gamma in [29], Weibull exponential in [30], and Weibull Lindley in [31].

The exponential (E) distribution has become one of the most commonly applied models for modeling lifetimes over the years; however, it cannot provide an adequate parametric fit for some applications due to its restriction to fitting data with a constant hazard rate, which is rare for most practical lifetime data [32]. Thus, several studies have developed different generalizations of the E distribution to increase its capability to model various types of data. Modifications of the E distribution include the exponentiated exponential (EE) distribution discussed in [33], the beta exponential (BE) distribution introduced in [34,35], the exponentiated Kumaraswamy exponential in [36], exponentiated generalized exponential (EGE) distribution studied in [37], alpha power exponential distribution proposed in [38], odd generalized exponential exponential in [39], Marshall Olkin logistic exponential distribution in [40], and extended odd Weibull exponential studied in [41].

In this paper, the exponentiated generalized Weibull exponential (EGWE) distribution is proposed as a generalization of the E distribution, with the aim of enhancing its flexibility in modeling diverse hazard rate behaviors. It is constructed by integrating two generator families—the EG and a T-X approach (namely the Weibull G family)—to provide a more flexible and adequate model for a wide range of datasets. The EG family, introduced in [12], is well known for its ability to improve tail flexibility. An attractive feature of this class is the inclusion of two additional parameters, which allow control over the tail weights and add flexibility to the center of the distribution, making it widely applicable in engineering and biological fields. In contrast, the Weibull G family provides a powerful baseline for modeling diverse hazard rate shapes, including increasing, decreasing, bathtub, and upside-down bathtub patterns [25]. Integrating these two families, the proposed model inherits the strengths of both approaches, yielding a distribution which is capable of capturing complex lifetime data patterns and a wide range of hazard rate shapes with greater tractability compared with many existing alternatives.

Moreover, in many practical applications, particularly in lifetime and reliability studies, it is essential not only to model the response distribution itself, but also to assess the effects of observed covariates on its characteristics. Therefore, regression models that incorporate explanatory variables into the parameters of a probability distribution have become powerful tools in modern statistics. For this reason, it is useful to investigate the advantages of the EGWE distribution in the regression context, where the objective is to describe the relationships between a response variable and a set of covariates.

The reminder of this paper organized as follows. In Section 2, a new five-parameter generalization of the E distribution is introduced, while a useful representation of its density is suggested in Section 3. Some statistical properties of the proposed model are discussed in Section 4. Estimation of the parameters of the new model is discussed in Section 5, which is illustrated through simulation studies in Section 6. In Section 7, the usefulness and adequacy of the proposed model are investigated through its application to four lifetime datasets. The regression models are presented and applied to a real dataset in Section 8. The statistical software [42] was used to perform all the computations and generate the plots presented in this paper.

2. Exponentiated Generalized Weibull Exponential Distribution

If an RV X has an E distribution with rate parameter $\lambda >0$, then the pdf and cdf respectively have the following forms

$$g(x;\lambda )=\lambda e^{-\lambda x};G(x;\lambda )=1-e^{-\lambda x};x>0.$$

Substituting $g(x;\lambda )$ and $G(x;\lambda )$ into Equation (4), we obtain the Weibull exponential (WE) distribution. Then, combining this member of the Weibull G family with the exponentiated generalized family in Equation (1), the new five-parameter EGWE distribution is constructed. Its pdf and cdf are respectively given as follows

$$f(x;a,b,c,\beta ,\lambda )={\displaystyle \frac{abc\lambda }{\beta }}{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^{c-1}{e}^{-a{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c}{\left[1-e^{-a{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c}\right]}^{b-1};x>0;a,b,c,\beta ,\lambda >0,$$

(5)

$$F(x;a,b,c,\beta ,\lambda )={\left[1-e^{-a{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c}\right]}^b;x>0.$$

(6)

Some characteristics of interest of the EGWE distribution are the survival and hazard functions, which can be derived respectively as follows

$$S(x;a,b,c,\beta ,\lambda )=1-{\left[1-e^{-a{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c}\right]}^b,$$

(7)

and

$$h(x;a,b,c,\beta ,\lambda )={\displaystyle \frac{{\displaystyle \frac{abc\lambda }{\beta }}{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^{c-1}{e}^{-a{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c}{\left[1-e^{-a{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c}\right]}^{b-1}}{1-{\left[1-e^{-a{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c}\right]}^b}}.$$

(8)

Some possible shapes of the EGWE density and hazard rate functions for appropriately chosen parameter values are shown in Figure 1. The pdf of the EGWE is uni-modal and can take right-skewed, left-skewed, symmetrical, and reversed-J shapes. Additionally, the flexibility of the EGWE can be clearly seen with respect its to hazard rate functions, which exhibit increasing, decreasing, bathtub, upside-down bathtub, S-shaped, J-shaped, and reversed-J shapes. Moreover, to provide a quantitative view of its flexibility,

Table 1. Skewness and kurtosis of the EGWE distribution for selected parameter values.

a	b	c	$\mathit{\beta }$	$\mathit{\lambda }$	Skewness	Kurtosis
0.7	0.8	0.7	2.3	0.7	3.7943	25.2583
7.5	8.4	1.4	2.3	0.8	0.8497	4.2573
2.7	4.6	0.5	1.9	3.6	4.6672	46.7914
8.2	0.5	4.6	1.1	0.3	−0.0896	2.4648
2.1	2.8	1.2	2.4	3.2	1.0775	4.5906
5.4	0.4	1.9	0.7	0.3	1.1459	4.1250

reports the skewness and kurtosis values for selected parameter combinations, further confirming the ability of the EGWE to capture a wide range of asymmetry and tail behaviors.

Some special cases and extensions of the E distribution can be obtained from the EGWE in Equation (6) as follows

• For $a=1$, $b=1$, $c=1$, and $\beta =1$, the E distribution with parameter $\lambda$ is obtained.
• For $a=1$ and $b=1$, the Weibull distribution with shape parameter c and a scale parameter ${\displaystyle \frac{\lambda }{\beta }}$ is obtained.
• For $a=1$, $\beta =1$, and $c=1$, the EE distribution in [33] with parameters $(b,\lambda )$ is obtained.
• For $a=1$ and $\beta =1$, the exponentiated Weibull distribution in [43] is obtained.
• For $c=1$ and $\beta =1$, the EGE distribution in [37] with parameters $(a,b,\lambda )$ is obtained.
• For $a=1$ and $b=1$, the WE distribution in [24] with parameters $(c,\beta ,\lambda )$ is obtained.
• For $a=1$ in [44], the proposed EGWE is obtained.

3. Useful Representation

For a real non-integer number $d>0$ and $\left|z\right|<1$, the generalized binomial series can be defined as

$${(1-z)}^{d-1}=\sum _{l=0}^{\infty }{(-1)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{d-1}{l}}\right)y^l.$$

(9)

Using Equation (9), a useful expansions for the pdf in Equation (5) can be obtained as follows

$$f(x;a,b,c,\beta ,\lambda )=\sum _{l=0}^{\infty }{(-1)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{b-1}{l}}\right){\displaystyle \frac{abc\lambda }{\beta }}{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^{c-1}{e}^{-a(l+1){\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c}.$$

(10)

4. Statistical Properties

4.1. Quantile Function

The quantile function of the EGWE distribution for $0<u<1$ can be obtained by inverting the cdf in Equation (6) as follows

$$u={\left[1-e^{-a{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c}\right]}^b.$$

Solving for x yields the quantile function

$$x={\displaystyle \frac{\beta }{\lambda }}{\left({\displaystyle \frac{-1}{a}}log\left(1-u^{{\displaystyle \frac{1}{b}}}\right)\right)}^{{\displaystyle \frac{1}{c}}}.$$

(11)

This closed-form expression facilitates the generation of random samples using the inverse transform method.

4.2. Moments

If X is an RV following the EGWE distribution with parameters $a,b,c,\beta$, and $\lambda$, then its $r^{th}$ moments

$$E\left(x_r\right)={\mu }_r^{\prime }={\int }_0^{\infty }{x}^{r}f(x;a,b,c,\beta ,\lambda )dx,$$

can be derived using the pdf expansion in Equation (10) as follows

$${\mu }_r^{\prime }=\sum _{l=0}^{\infty }{(-1)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{b-1}{l}}\right)abc{\left({\displaystyle \frac{\lambda }{\beta }}\right)}^c{\int }_0^{\infty }{x}^{r+c-1}{e}^{-a(l+1){\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c}dx.$$

To simplify the integration, the substitution $u=a(l+1){\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c$ is applied, transforming the integral into a gamma-type form. Then, the $r^{th}$ moments can be defined as

$${\mu }_r^{\prime }=\sum _{l=0}^{\infty }{(-1)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{b-1}{l}}\right)ab{\left({\displaystyle \frac{\beta }{\lambda }}\right)}^r{\left({\displaystyle \frac{1}{a(l+1)}}\right)}^{{\displaystyle \frac{r}{c}}+1}\mathrm{\Gamma }\left({\displaystyle \frac{r}{c}}+1\right),$$

(12)

where $\mathrm{\Gamma }\left(a\right)={\int }_0^{\infty }{x}^{a-1}{e}^{-x}dx$ is the gamma function.

4.3. Moment Generating Function

The moment generating function of an RV X can be obtained as

$$E\left(e^{zx}\right)=M_X\left(z\right)={\int }_0^{\infty }{e}^{zx}f(x;a,b,c,\beta ,\lambda )dx.$$

Using the power series expansion for the exponential function

$$e^{zx}=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k{x}^k}{k!}},$$

(13)

the moment generating function, $E\left(e^{zx}\right)=M_X\left(z\right)$, of the EGWE can be obtained as

$$M_X\left(z\right)=\sum _{k=0}^{\infty }\sum _{l=0}^{\infty }{\displaystyle \frac{{(-1)}^l}{k!}}{z}^k\left({\displaystyle \genfrac{}{}{0pt}{}{b-1}{l}}\right)ab{\left({\displaystyle \frac{\beta }{\lambda }}\right)}^k{\left({\displaystyle \frac{1}{a(l+1)}}\right)}^{{\displaystyle \frac{k}{c}}+1}\mathrm{\Gamma }\left({\displaystyle \frac{k}{c}}+1\right).$$

(14)

4.4. Incomplete Moments

The $s^{th}$ incomplete moments of an RV X that has an EGWE model with parameters $a,b,c,\beta$, and $\lambda$ has the following form

$$m_x\left(t\right)={\int }_0^t{x}^{s}f(x;a,b,c,\beta ,\lambda )dx.$$

It follows from Equation (10) that

$$m_s\left(t\right)=\sum _{l=0}^{\infty }{(-1)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{b-1}{l}}\right)abc{\left({\displaystyle \frac{\lambda }{\beta }}\right)}^c{\int }_0^t{x}^{s+c-1}{e}^{-a(l+1){\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c}dx.$$

(15)

Thus, after assuming $u=a(l+1){\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c$, the $s^{th}$ incomplete moments obtained as

$$m_s\left(t\right)=\sum _{l=0}^{\infty }{(-1)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{b-1}{l}}\right)ab{\left({\displaystyle \frac{\beta }{\lambda }}\right)}^s{\left({\displaystyle \frac{1}{a(l+1)}}\right)}^{{\displaystyle \frac{s}{c}}+1}\mathrm{\Gamma }\left({\displaystyle \frac{s}{c}}+1,a(l+1){\left({\displaystyle \frac{\lambda t}{\beta }}\right)}^c\right),$$

(16)

where $\mathrm{\Gamma }\left(d,t\right)={\int }_0^t{x}^{d-1}{e}^{-x}dx$ is the incomplete gamma function.

4.5. Rényi Entropy

An entropy can be defined as a variation or uncertainty measure of an RV X. The Rényi entropy of an RV X with pdf f is defined as

$$I_{\delta }\left(x\right)={\displaystyle \frac{1}{1-\delta }}log\left({\int }_{-\infty }^{\infty }{\left[f\left(x\right)\right]}^{\delta }dx\right)\delta >0,\delta \ne 1.$$

Then, by raising Equation (5) to the power $\delta$ and applying Equation (9), the Rényi entropy is obtained as

$$I_{\delta }\left(x\right)={\displaystyle \frac{1}{1-\delta }}log\left(\sum _{l=0}^{\infty }{(-1)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{\delta (b-1)}{l}}\right){\left(abc\right)}^{\delta }{\left({\displaystyle \frac{\lambda }{\beta }}\right)}^{\delta c}{\int }_0^{\infty }{x}^{\delta (c-1)}{e}^{-a(\delta +l){\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c}dx\right).$$

To evaluate the above integral, the method of integration by substitution is applied with the substitution $u=a(\delta +l){\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c$. With this substitution, the Rényi Entropy of EGWE($a,b,c,\beta ,\lambda$) can be expressed in a gamma-type form as follows

$$\begin{array}{cc}\hfill I_{\delta }\left(x\right)=& {\displaystyle \frac{\delta }{1-\delta }}log\left(ab\right)+log\left({\displaystyle \frac{\beta }{c\lambda }}\right)+\hfill \\ \hfill & {\displaystyle \frac{1}{1-\delta }}log\left(\sum _{l=0}^{\infty }{(-1)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{\delta (b-1)}{l}}\right){\left({\displaystyle \frac{1}{a(\delta +l)}}\right)}^{\delta \left(1-{\displaystyle \frac{1}{c}}\right)+{\displaystyle \frac{1}{c}}}\mathrm{\Gamma }\left(\delta \left(1-{\displaystyle \frac{1}{c}}\right)+{\displaystyle \frac{1}{c}}\right)\right).\hfill \end{array}$$

(17)

4.6. Order Statistics

The pdf of the $o^{th}$ order statistic for a random sample $x_1,x_2,\dots ,x_n$ from an EGWE distribution with parameters $a,b,c,\beta$, and $\lambda$, can be obtained as

$$\begin{array}{cc}\hfill f_{X_O}(x;a,b,c,\beta ,\lambda )={\displaystyle \frac{n!}{(o-1)!(n-o)!}}& f(x;a,b,c,\beta ,\lambda ){\left[F(x;a,b,c,\beta ,\lambda )\right]}^{o-1}\times \hfill \\ \hfill & {\left[1-F(x;a,b,c,\beta ,\lambda )\right]}^{n-i}.\hfill \end{array}$$

It follows that

$$\begin{array}{cc}\hfill f_{X_O}(x;a,b,c,\beta ,\lambda )={\displaystyle \frac{n!}{(o-1)!(n-o)!}}\sum _{l=0}^{\infty }\sum _{v=0}^{n-o}& {(-1)}^{v+l}\left({\displaystyle \genfrac{}{}{0pt}{}{n-o}{v}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{b(v+o)-1}{l}}\right)\hfill \\ \hfill & {\displaystyle \frac{abc\lambda }{\beta }}{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^{c-1}{e}^{-a(l+1){\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c}.\hfill \end{array}$$

(18)

5. Estimation of Parameters

The method of maximum likelihood estimation is applied to estimate the unknown parameters of the EGWE distribution. Suppose that a random sample of size n is obtained from the EGWE distribution with parameters $\mathrm{\Theta }=\{a,b,c,\beta ,\lambda \}$. Then, the log-likelihood, ℓ, for $\mathrm{\Theta }$ is given by

$$\begin{array}{cc}\hfill \mathcal{l}=& nloga+nlogb+nlogc+nlog\lambda -nlog\beta \hfill \\ \hfill & +(c-1)\sum _{i=0}^{n}log\left({\displaystyle \frac{\lambda x}{\beta }}\right)-a\sum _{i=0}^n{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c+(b-1)\sum _i^{n}log\left(1-e^{-a{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c}\right).\hfill \end{array}$$

(19)

Assuming that

$$\zeta (x;a,c,\beta ,\lambda )={\displaystyle \frac{e^{-a{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c}}{1-e^{-a{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c}}}={\left(e^{a{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c}-1\right)}^{-1},$$

the corresponding score vector, $U\left(\mathrm{\Theta }\right)=\left({\displaystyle \frac{\partial \mathcal{l}}{\partial a}},{\displaystyle \frac{\partial \mathcal{l}}{\partial b}},{\displaystyle \frac{\partial \mathcal{l}}{\partial c}},{\displaystyle \frac{\partial \mathcal{l}}{\partial \beta }},{\displaystyle \frac{\partial \mathcal{l}}{\partial \lambda }}\right)$, the components of which are obtained by taking the first partial derivatives of the log-likelihood function in Equation (19) with respect to each model’s parameter, can be given as

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial \mathcal{l}}{\partial a}}={\displaystyle \frac{n}{a}}-\sum _{i=1}^n{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c\left[1-(b-1)\zeta (x;a,c,\beta ,\lambda )\right],\hfill \\ \hfill & {\displaystyle \frac{\partial \mathcal{l}}{\partial b}}={\displaystyle \frac{n}{b}}+\sum _{i=1}^{n}log\left[1-e^{-a{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c}\right],\hfill \\ \hfill & {\displaystyle \frac{\partial \mathcal{l}}{\partial c}}={\displaystyle \frac{n}{c}}+\sum _{i=1}^{n}log\left({\displaystyle \frac{\lambda x}{\beta }}\right)\left[1-a{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c+a(b-1){\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c\zeta (x;a,c,\beta ,\lambda )\right],\hfill \\ \hfill & {\displaystyle \frac{\partial \mathcal{l}}{\partial \beta }}={\displaystyle \frac{-n}{\beta }}-{\displaystyle \frac{n(c-1)}{\beta }}+{\displaystyle \frac{ac}{\beta }}\sum _{i=1}^n{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c\left[1-(b-1)\zeta (x;a,c,\beta ,\lambda )\right],\hfill \\ \hfill & {\displaystyle \frac{\partial \mathcal{l}}{\partial \lambda }}={\displaystyle \frac{n}{\lambda }}+{\displaystyle \frac{n(c-1)}{\lambda }}-{\displaystyle \frac{ac}{\lambda }}\sum _{i=1}^n{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c\left[1-(b-1)\zeta (x;a,c,\beta ,\lambda )\right].\hfill \end{array}$$

(20)

The maximum likelihood estimator (MLEs) of the EGWE parameters are then the solution of the system of nonlinear equations given in Equation (20). It is clear that these equations are nonlinear and need to be solved numerically using an optimization technique to find the MLEs of the EGWE parameters; namely, $\widehat{\mathrm{\Theta }}=\{\widehat{a},\widehat{b},\widehat{c},\widehat{\beta },\widehat{\lambda \}}$. In this paper, a quasi-Newton method called “BFGS”, which was published simultaneously in 1970 by Broyden, Fletcher, Goldfarb, and Shanno, is applied to obtain the MLEs. Additionally, to facilitate convergence of the estimation procedure, the exponential rate parameter was initialized as $1/\overline{x}$, where $\overline{x}$ denotes the sample mean, while the remaining parameters were set to simple starting values within their admissible ranges.

6. Simulation Study

Monte Carlo simulation studies were conducted to assess and evaluate the performance of the MLEs for the five parameters of the EGWE distribution. The simulation was perfermed over several iterations equal to $nsim=1000$, for four different sample sizes n. In each iteration, we generated a random sample of size n with the following combinations for the true parameters ${\mathrm{\Theta }}_{tr}$.

$$\begin{array}{cc}\hfill & \mathbf{Case}\mathbf{I}:a=3.3,b=0.6,c=1.1,\beta =4.3,\lambda =3.0.\hfill \\ \hfill & \mathbf{Case}\mathbf{II}:a=5.0,b=1.2,c=0.8,\beta =2.5,\lambda =3.6.\hfill \\ \hfill & \mathbf{Case}\mathbf{III}:a=3.4,b=0.4,c=1.3,\beta =0.5,\lambda =0.8.\hfill \end{array}$$

The MLEs of the parameters of the EGWE were obtained using optim in R with the “Broyden–Fletcher–Goldfarb–Shann” maximization method. These MLEs $\widehat{\mathrm{\Theta }}$ were evaluated using two accuracy measures—the bias $Bias\left(\widehat{\theta }\right)$, and the root mean squared error (RMSE)— which can be calculated as follows

$$bias\left(\widehat{\theta }\right)={\displaystyle \frac{{\sum }_{i=1}^{nsim}{\widehat{\theta }}_i}{nsim}}-{\theta }_{tr},$$

(21)

and

$$RMSE\left(\widehat{\theta }\right)=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^{nsim}{({\widehat{\theta }}_i-{\theta }_{tr})}^2}{nsim}}}.$$

(22)

Table 2. Simulation study: EGWE parameter estimates, together with bias, and RMSE, for three different cases and different sample sizes.

		Case I			Case II			Case III
		MLE	Bias	RMSE	MLE	Bias	RMSE	MLE	Bias	RMSE
$n=25$	a	3.8304	0.5304	0.9561	5.6457	0.6457	1.2918	3.5945	0.1945	0.5675
	b	1.2520	0.6520	3.2481	3.5196	2.3196	10.5700	0.9861	0.5861	4.1433
	c	1.9426	0.8426	1.8225	1.5328	0.7328	1.7611	2.1497	0.8497	1.6991
	$\beta$	4.2165	−0.0835	0.4208	2.3085	−0.1915	0.4664	0.6906	0.1906	0.3922
	$\lambda$	3.5154	0.5154	1.0332	4.2455	0.6455	1.2440	1.4549	0.6549	1.1340
$n=50$	a	3.5934	0.2934	0.7838	5.3644	0.3644	1.0432	3.5533	0.1533	0.4868
	b	0.7486	0.1486	0.8655	1.9173	0.7173	3.7844	0.4952	0.0952	0.6847
	c	1.4530	0.3530	0.9739	1.0041	0.2041	0.6572	1.6703	0.3703	0.9592
	$\beta$	4.2923	−0.0077	0.3547	2.4617	−0.0383	0.4292	0.6531	0.1531	0.3878
	$\lambda$	3.3112	0.3112	0.7342	4.0611	0.4611	1.0468	1.1756	0.3756	0.8140
$n=100$	a	3.4483	0.1483	0.4988	5.1423	0.1423	0.6679	3.4810	0.0810	0.4048
	b	0.6550	0.0550	0.3769	1.3811	0.1811	0.9960	0.4289	0.0289	0.2320
	c	1.2349	0.1349	0.4669	0.8773	0.0773	0.2977	1.4695	0.1695	0.5617
	$\beta$	4.2895	−0.0105	0.2869	2.4388	−0.0612	0.2807	0.5728	0.0728	0.2754
	$\lambda$	3.1446	0.1446	0.5104	3.7813	0.1813	0.5648	0.9771	0.1771	0.5670
$n=500$	a	3.3228	0.0228	0.2089	5.0172	0.0172	0.3722	3.4401	0.0401	0.2055
	b	0.6053	0.0053	0.1153	1.2320	0.0320	0.2751	0.4035	0.0035	0.0775
	c	1.1229	0.0229	0.1509	0.8092	0.0092	0.1013	1.3313	0.0313	0.1907
	$\beta$	4.2895	−0.0105	0.1518	2.4819	−0.0181	0.1887	0.5304	0.0304	0.1719
	$\lambda$	3.0150	0.0150	0.1716	3.6317	0.0317	0.2190	0.8548	0.0548	0.2973

shows the results for the MLEs of the parameters of the EGWE, along with their corresponding average bias and RMSE. Generally, it can be seen from Table 2 that both the bias and RMSE decrease as the sample size n increases. This behavior is consistent with the asymptotic properties of the MLEs, which become more accurate and stable as more information is available from larger samples.

7. Applications

Several studies have applied the E distribution and its extensions to lifetime data, including [45,46,47,48,49,50,51]. In this section, four lifetime datasets are analyzed to illustrate the flexibility of EGWE. In particular the EGWE is assessed in comparison with the following competitive models

• The E distribution, with cdf
  $F(x;\lambda )=1-e^{-\lambda x}$.
• The EE distribution in [33], with cdf
  $F(x;\lambda ,a)={\left[1-e^{-\lambda x}\right]}^a$.
• The EGE distribution in [37], with cdf
  $F(x;\lambda ,a,b)={\left[1-e^{-a\lambda x}\right]}^b$.
• The WE distribution (a member of the Weibull G family) in [24], with cdf
  $F(x;\lambda ,c,\beta )=1-e^{-{\left({\displaystyle \frac{\lambda x}{\beta }}\right)}^c}$.
• The BE distribution in [34], with cdf
  $F(x;\lambda ,A,B)=I_{1-e^{-\lambda x}}(A,B)$,
  $\mathrm{where}{I}_y(A,B)={\displaystyle \frac{BET{A}_y(A,B)}{BETA(A,B)}},BET{A}_y(A,B)={\int }_0^y{w}^{A-1}{(1-w)}^{B-1}dw$.

The function “fitdistr” in the “MASS” package in R was used to fit the E distribution, while the “Newdistns” package [52] was applied for fitting of the EE, EGE, WE, and EGWE distributions. Some measures of goodness of fit, such as the Akaike information criterion (AIC) ([53]) and its small-sample correction, the corrected Akaike information criterion (AICc) ([54]), as well as the Kolmogorov–Smirnov (K-S) test statistic and its p-value were calculated to evaluate the fitting performance among the applied distributions. Generally, the model with the lowest of these measures or the largest log-likelihood, ℓ and p-value is regarded as having the best fit to the data.

7.1. The First Dataset

The first dataset obtained from [55], contains data reflecting the number of successive failures for the air conditioning system of a jet airplanes, given as the following values: 194, 413, 90, 74, 55, 23, 97, 50, 359, 50, 130, 487, 57, 102, 15, 14, 10, 57, 320, 261, 51, 44, 9, 254,493, 33, 18, 209, 41, 58, 60, 48, 56, 87, 11, 102, 12, 5, 14, 14, 29, 37, 186, 29, 104, 7, 4, 72, 270, 283, 7, 61, 100, 61, 502, 220, 120, 141, 22, 603, 35, 98, 54, 100, 11, 181, 65, 49, 12, 239, 14, 18, 39, 3, 12, 5, 32, 9, 438, 43, 134, 184, 20, 386, 182, 71, 80, 188, 230, 152, 5, 36, 79, 59, 33, 246, 1, 79, 3, 27, 201, 84, 27, 156, 21, 16, 88, 130, 14, 118, 44, 15, 42, 106, 46, 230, 26, 59, 153, 104, 20, 206, 5, 66, 34, 29, 26, 35, 5, 82, 31, 118, 326, 12, 54, 36, 34, 18, 25, 120, 31, 22, 18, 216, 139, 67, 310, 3, 46, 210, 57, 76, 14, 111, 97, 62, 39, 30, 7, 44, 11, 63, 23, 22, 23, 14, 18, 13, 34, 16, 18, 130, 90, 163, 208, 1, 24, 70, 16, 101, 52, 208, 95, 62, 11, 191, 14, 71.

7.2. The Second Dataset

We next considered 40 observations of time to failure data for turbochargers obtained from [56], with the following values: 1.6, 2, 2.6, 3, 3.5, 3.9, 4.5, 4.6, 4.8, 5, 5.1, 5.3, 5.4, 5.6, 5.8, 6, 6, 6.1, 6.3, 6.5, 6.5, 6.7, 7, 7.1, 7.3, 7.3, 7.3, 7.7, 7.7, 7.8, 7.9, 8, 8.1, 8.3, 8.4, 8.4, 8.5, 8.7, 8.8, 9.

7.3. The Third Dataset

The real data comprising the third dataset were obtained in a study of 40 patients with blood cancer (leukemia) from one of the Ministry of Health Hospitals in Saudi Arabia. These data were studied in [57] and are given as: 0.315, 0.496, 0.616, 1.145, 1.208, 1.263, 1.414, 2.025, 2.036, 2.162, 2.211, 2.37, 2.532, 2.693, 2.805, 2.91, 2.912, 3.192, 3.263, 3.348, 3.348, 3.427, 3.499, 3.534, 3.767, 3.751, 3.858, 3.986, 4.049, 4.244, 4.323, 4.381, 4.392, 4.397, 4.647, 4.753, 4.929, 4.973, 5.074, 5.381.

7.4. The Fourth Dataset

This dataset represents the survival times (life times in years) of 119 patients from a random sample of patients attending Bolgatanga Regional Hospital in the Upper East region of Ghana. This dataset was obtained from [58] and comprises the following values: 71, 5, 39, 62, 52, 71, 38, 56, 35, 69, 34, 71, 66, 70, 52, 37, 35, 71, 73, 19, 74, 74, 75, 51, 76, 49, 19, 76, 78, 76, 76, 49, 47, 48, 48, 46, 46, 46, 41, 40, 43, 45, 47, 47, 44, 45, 46, 42, 43, 42, 20, 28, 26, 60, 27, 24, 29, 60, 25, 60, 69, 36, 69, 69, 68, 68, 67, 67, 67, 52, 35, 66, 55, 66, 61, 61, 64, 64, 65, 65, 63, 63, 62, 39, 62, 62, 62, 59, 59, 59, 58, 58, 58, 18, 57, 57, 56, 56, 37, 53, 53, 53, 53, 54, 54, 66, 17, 50, 75, 51, 38, 52, 66, 4, 52, 55, 19, 58, 73.

Table 3. MLEs (with SE in parentheses) and goodness-of-fit for various models fitted on the first dataset.

Model	MLEs	ℓ	AIC	AICc	K-S	p-Value
E	$\widehat{\lambda }=0.0109$	−1038.2480	2078.4970	2078.5180	0.0845	0.1368
	(0.0008)
EE	$\widehat{a}=0.9147\widehat{\lambda }=0.0102$	−1037.7510	2079.5030	2079.5670	0.0732	0.2654
	(0.0869)       (0.0010)
EGE	$\widehat{a}=0.4262\widehat{b}=0.9107$	−1037.7500	2081.5000	2081.6300	0.0728	0.2722
	(0.6170)       (0.0867)
	$\widehat{\lambda }=0.0240$
	(0.0346)
WE	$\widehat{c}=0.9134\widehat{\beta }=1.3489$	−1036.7520	2079.5050	2079.6350	0.0578	0.5555
	(0.0505)       (1.1443)
	$\widehat{\lambda }=0.0154$
	(0.0130)
BE	$\widehat{A}=0.9237\widehat{B}=1.1398$	−1037.7390	2081.4770	2081.6080	0.0733	0.2653
	(0.0868)       (0.5976)
	$\widehat{\lambda }=0.0090$
	(0.0048)
EGWE	$\widehat{a}=2.1185\widehat{b}=3.2505$	−1033.1490	2076.2980	2076.6280	0.0447	0.8473
	(8.4796)       (1.5611)
	$\widehat{c}=0.5174\widehat{\beta }=2.3963$
	(0.1092)       (17.1033)
	$\widehat{\lambda }=0.0280$
	(0.0466)

,

Table 4. MLEs (with SE in parentheses) and goodness-of-fit for various models fitted on the second dataset.

Model	MLEs	ℓ	AIC	AICc	K-S	p-Value
E	$\widehat{\lambda }=0.1599$	−113.3193	228.6385	228.7438	0.3631	0.0001
	(0.0253)
EE	$\widehat{a}=9.5115\widehat{\lambda }=0.4498$	−90.1427	184.2853	184.6097	0.1542	0.2973
	(2.8950)       (0.0578)
EGE	$\widehat{a}=0.6163\widehat{b}=9.5148$	−90.1427	186.2853	186.952	0.1542	0.2975
	(11.4663)       (2.8962)
	$\widehat{\lambda }=0.7300$
	(13.5817)
WE	$\widehat{c}=3.8721\widehat{\beta }=23.0323$	−82.4753	170.9506	171.6172	0.1074	0.7458
	(0.5175)       (1.1286)
	$\widehat{\lambda }=3.3276$
	(0.0805)
BE	$\widehat{A}=7.8424\widehat{B}=11.703$	−87.5174	181.0347	181.7014	0.1269	0.5396
	(1.7605)       (11.9902)
	$\widehat{\lambda }=0.0847$
	(0.0718)
EGWE	$\widehat{a}=0.3301\widehat{b}=0.1402$	−78.0985	166.197	167.9617	0.0666	0.9943
	(0.0489)       (0.0234)
	$\widehat{c}=18.1585\widehat{\beta }=3.0926$
	(0.0466)       (0.0450)
	$\widehat{\lambda }=0.3809$
	(0.0105)

,

Table 5. MLEs with (SE in parentheses) and goodness-of-fit for various models fitted on the third dataset.

Model	MLEs	ℓ	AIC	AICc	K-S	p-Value
E	$\widehat{\lambda }=0.3184$	−85.7781	173.5563	173.6616	0.3002	0.0015
	(0.0503)
EE	$\widehat{a}=3.5189\widehat{\lambda }=0.6141$	−74.9624	153.9249	154.2492	0.1612	0.2494
	(0.8757)       (0.0910)
EGE	$\widehat{a}=0.5494\widehat{b}=3.5189$	−74.9624	155.9249	156.5915	0.1612	0.2494
	(24.3012)       (0.8757)
	$\widehat{\lambda }=1.1178$
	(49.4421)
WE	$\widehat{c}=2.4994\widehat{\beta }=0.8469$	−69.558	145.1159	145.7826	0.1184	0.6290
	(0.3371)       (13.2653)
	$\widehat{\lambda }=0.2407$
	(3.7702)
BE	$\widehat{A}=3.4393\widehat{B}=13.9238$	−73.5721	153.1443	153.8109	0.1588	0.2657
	(0.7364)       (49.5981)
	$\widehat{\lambda }=0.0726$
	(0.2398)
EGWE	$\widehat{a}=0.1911\widehat{b}=0.1335$	−64.9851	139.9701	141.7348	0.0617	0.9980
	(0.1609)       (0.0228)
	$\widehat{c}=12.1065\widehat{\beta }=3.0955$
	(0.1060)       (0.1852)
	$\widehat{\lambda }=0.7123$
	(0.0446)

and

Table 6. MLEs (with SE in parentheses) and goodness-of-fit for various models fitted on the fourth dataset.

Model	MLEs	ℓ	AIC	AICc	K-S	p-Value
E	$\widehat{\lambda }=0.0191$	−590.1748	1182.3500	1182.3840	0.3610	0
	(0.0017)
EE	$\widehat{a}=7.0200\widehat{\lambda }=0.0480$	−530.9825	1065.9650	1066.0680	0.1607	0.0043
	(1.1285)       (0.0036)
EGE	$\widehat{a}=0.1360\widehat{b}=7.0133$	−530.9824	1067.9650	1068.1730	0.1607	0.0043
	(0.0473)       (1.1269)
	$\widehat{\lambda }=0.3528$
	(0.1198)
WE	$\widehat{c}=3.7307\widehat{\beta }=25.0525$	−502.8310	1011.6620	1011.8710	0.0854	0.3500
	(0.2884)       (0.7119)
	$\widehat{\lambda }=0.4329$
	(0.0054)
BE	$\widehat{A}=6.6624\widehat{B}=6.0211$	−522.8252	1051.6500	1051.8590	0.1413	0.0173
	(0.8673)       (2.3122)
	$\widehat{\lambda }=0.0150$
	(0.0044)
EGWE	$\widehat{a}=0.1743\widehat{b}=0.2213$	−492.0572	994.1143	994.6453	0.0446	0.9717
	(0.0138)       (0.0221)
	$\widehat{c}=11.4245\widehat{\beta }=10.2318$
	(0.0153)       (0.0147)
	$\widehat{\lambda }=0.1671$
	(0.0030)

display summaries of the MLEs of the parameters, as well as the log-likelihoods, AIC, AICc, and K-S statistic and its p-value for each model. It can be observed that the EGWE achieved the best performance, according to its maximum log-likelihood and p-value, and its lowest AIC and AICc, when compared with the other fitted distributions. The histograms of the data with plots of the estimated pdfs, as well as the fitted cdfs vs. empirical values for the datasets are displayed in Figure 2, Figure 3, Figure 4 and Figure 5 for each model. Comparing the pdf and cdf of the proposed distribution with those of the compared distributions, the EGWE distribution can be seen to more closely mimic the empirical pdf and cdf.

The likelihood ratio test (LRT) is a statistical test that is used to compare two nested models, where one model is a special case of the other. It evaluates whether the improvement in fit achieved by the more general model is statistically significant when compared with its simpler sub-model. The LRT statistic is defined as

$$D=2({\mathcal{l}}_{\mathrm{EGWE}}-{\mathcal{l}}_{\mathrm{reduced}}),$$

where ℓ denotes the maximized log-likelihood; here, ${\mathcal{l}}_{\mathrm{EGWE}}$ is the maximized log-likelihood of the full five-parameter EGWE model, and ${\mathcal{l}}_{\mathrm{reduced}}$ is the maximized log-likelihood of the corresponding nested sub-model. The degrees of freedom (df) correspond to the difference in the number of parameters between the EGWE (5) and each sub-model: E (df = 4), EE (df = 3), EGE (df = 2), and WE (df = 2). Each entry in

Table 7. LRT results comparing EGWE with its nested sub-models across four datasets.

Model	First Dataset	Second Dataset	Third Dataset	Fourth Dataset
E	$10.2006$$\left(0.0372\right)$	$70.3980$$\left(0.0000\right)$	$41.5344$$\left(0.0000\right)$	$196.3780$$\left(0.0000\right)$
EE	$9.2064$$\left(0.0267\right)$	$24.0448$$\left(0.0000\right)$	$19.9030$$\left(0.0002\right)$	$77.9934$$\left(0.0000\right)$
EGE	$9.2034$$\left(0.0100\right)$	$24.0448$$\left(0.0000\right)$	$19.9030$$\left(0.0000\right)$	$77.9932$$\left(0.0000\right)$
WE	$7.2084$$\left(0.0272\right)$	$8.7100$$\left(0.0128\right)$	$9.0942$$\left(0.0106\right)$	$21.6904$$\left(0.0000\right)$

reports the LRT statistic D with the corresponding p-value in parentheses. A small p-value indicates that the additional parameters of the EGWE distribution provide a statistically significant improvement in model fit. All reported p-values are less than $0.05$, indicating statistically significant improvements in fit. Cases where the p-value appears as 0 are due to numerical precision limits, and should be interpreted as $p<{10}^{-16}$.

For the purposes of comparison, the EGWE distribution was also compared with three alternative distributions constructed using generator families available in the “Newdistns” package [52], including the Beta Weibull exponential (BWE) [59], Kumaraswamy Weibull exponential (KWE) [60], and Exponentiated Kumaraswamy Weibull (EKW) [61] distributions. Although the EKW is not a direct generalization of the exponential distribution, it was included due to its similar parameter structure and flexibility. A performance comparison is provided in

Table 8. Comparison of EGWE with comparable five-parameter distributions across four datasets.

Dataset	Model	ℓ	AIC	AICc	K-S	p-Value
First dataset	EGWE	−1033.1490	2076.2980	2076.628	0.0447	0.8473
	BWE	−1033.1590	2076.3180	2076.647	0.0449	0.8434
	KWE	−1033.2060	2076.4120	2076.7420	0.0469	0.8033
	EKW	−1033.3820	2076.7650	2077.0940	0.0454	0.8336
Second dataset	EGWE	−78.0985	166.1970	167.9617	0.0666	0.9943
	BWE	−78.2736	166.5472	168.3119	0.0728	0.9839
	KWE	−82.1965	174.3931	176.1578	0.1077	0.7418
	EKW	−81.0747	172.1494	173.9141	0.0989	0.8285
Third dataset	EGWE	−64.9851	139.9701	141.7348	0.0617	0.9980
	BWE	−65.4759	140.9518	142.7165	0.0717	0.9863
	KWE	−69.2655	148.5310	150.2957	0.1359	0.4514
	EKW	65.2224	140.4448	142.2095	0.0687	0.9916
Fourth dataset	EGWE	−492.0572	994.1143	994.6453	0.0446	0.9717
	BWE	−500.5021	1011.0040	1011.5350	0.0995	0.1897
	KWE	−500.2737	1010.548	1011.0780	0.0868	0.3308
	EKW	−504.3558	1018.7120	1019.2420	0.1183	0.0716

, based on which it can be seen that the proposed EGWE distribution consistently outperformd the other five-parameter distributions in terms of ℓ, AIC, AICc, K-S, and p-value across all datasets. These results provide strong evidence of the superiority for the EGWE model.

8. Exponentiated Generalized Weibull Exponential Distribution Regression Model

A new parametric regression model based on the proposed EGWE distribution is introduced, analogously to the formulation of E regression as a generalized linear model with a log link function, such that its scale parameter depends on covariates through the log link function. This approach is commonly used in lifetime modeling and survival analysis ([32,62]). This method relies on allowing one of the parameters of the EGWE to vary in response to a set of covariates. Specifically, to develop the EGWE regression model, we assume that, for $i=1,2,\dots ,n$, the response $X_i$ follows the EGWE distribution with the parameters $(a,b,c,\beta ,{\lambda }_i)$, where ${\lambda }_i$ is related to the independent variables through a log link function as follows

$$log\left({\lambda }_i\right)={\mathit{w}}_{\mathit{i}}^{\mathit{T}}\mathit{\alpha },$$

(23)

where ${\mathit{w}}_{\mathit{i}}^{\mathit{T}}=(w_{i1},\dots ,w_{ip})$ represents the vector of covariates associated with the $i^{th}$ observation, and $\mathit{\alpha }={({\alpha }_0,\dots ,{\alpha }_p)}^T$ represents the corresponding regression coefficient vector. Other parameters in the proposed distribution can be kept constant or also be modeled as functions of covariates as needed.

The probability of $X|\mathit{W}$ from Equation (5) is represented as

$$f\left(x_i\right|{\mathit{w}}_{\mathit{i}})={\displaystyle \frac{abc{e}^{{\mathit{w}}_{\mathit{i}}^{\mathit{T}}{\mathit{\alpha }}_{\mathit{i}}}}{\beta }}{\left({\displaystyle \frac{e^{{\mathit{w}}_{\mathit{i}}^{\mathit{T}}{\mathit{\alpha }}_{\mathit{i}}}{x}_i}{\beta }}\right)}^{c-1}{e}^{-a{\left({\displaystyle \frac{e^{{\mathit{w}}_{\mathit{i}}^{\mathit{T}}{\mathit{\alpha }}_{\mathit{i}}}{x}_i}{\beta }}\right)}^c}{\left[1-e^{-a{\left({\displaystyle \frac{e^{{\mathit{w}}_{\mathit{i}}^{\mathit{T}}{\mathit{\alpha }}_{\mathit{i}}}{x}_i}{\beta }}\right)}^c}\right]}^{b-1},$$

(24)

with the survival function

$$S\left(x_i\right|{\mathit{w}}_{\mathit{i}})=1-{\left[1-e^{-a{\left({\displaystyle \frac{e^{{\mathit{w}}_{\mathit{i}}^{\mathit{T}}{\mathit{\alpha }}_{\mathit{i}}}{x}_i}{\beta }}\right)}^c}\right]}^b.$$

(25)

In studies of survival and reliability, it is common that not all lifetimes are observed; rather, some observations are censored, meaning that the event has not yet occurred. In particular, right censoring occurs when the actual lifetime exceeds the observed time, resulting in only partial knowledge of the response. In such cases, the contribution of each subject to the likelihood is given by $f\left(x_i\right)$ if the event is observed $({\delta }_i=1)$, and by $S\left(x_i\right)$ if the observation is censored $({\delta }_i=0)$. Thus, both full and censored samples can be analyzed within the same estimation framework. Then, to estimate the regression model parameters, the maximum likelihood technique can be applied by maximizing the logarithm of the following likelihood function

$$\begin{array}{cc}\hfill \mathcal{l}(a,b,c,\beta ,\mathit{\alpha })=& \sum _{i=1}^n{\delta }_{i}log\left\{{\displaystyle \frac{abc{e}^{{\mathit{w}}_{\mathit{i}}^{\mathit{T}}\mathit{\alpha }}}{\beta }}{\left({\displaystyle \frac{e^{{\mathit{w}}_{\mathit{i}}^{\mathit{T}}\mathit{\alpha }}{x}_i}{\beta }}\right)}^{c-1}{e}^{-a{\left({\displaystyle \frac{e^{{\mathit{w}}_{\mathit{i}}^{\mathit{T}}\mathit{\alpha }}{x}_i}{\beta }}\right)}^c}{\left[1-e^{-a{\left({\displaystyle \frac{e^{{\mathit{w}}_{\mathit{i}}^{\mathit{T}}\mathit{\alpha }}{x}_i}{\beta }}\right)}^c}\right]}^{b-1}\right\}\hfill \\ \hfill & +\sum _{i=1}^n\left(1-{\delta }_i\right)log\left\{1-{\left[1-e^{-a{\left({\displaystyle \frac{e^{{\mathit{w}}_{\mathit{i}}^{\mathit{T}}\mathit{\alpha }}{x}_i}{\beta }}\right)}^c}\right]}^b\right\}.\hfill \end{array}$$

Two applications are considered to illustrate the performance of the EGWE regression model. The first application is based on a complete dataset without censoring, while the second application involves right-censored survival data. Through this dual analysis, applicability of the regression framework to both fully observed and censored sampling schemes can be demonstrated.

8.1. Application 1: Regression Model for Complete Sample

First, the data considered in [58] were used to model the relationship between gender and survival times (in years) until the onset of hypertension in 119 random patients attending the Bolgatanga Regional Hospital in the Upper East Region of Ghana. For each of the following lifetimes values, the gender (1 = male, 0 = female) is shown in brackets: 71(1), 5(1), 39(1), 62(1), 52(0), 71(0), 38(0), 56(1), 35(1), 69(1), 34(1), 71(1), 66(0), 70(1), 52(0), 37(0), 35(0), 71(1), 73(1), 19(0), 74(0), 74(1), 75(1), 51(0), 76(1), 49(0), 19(1), 76(0), 78(1), 76(0), 76(0), 49(1), 47(1), 48(0), 48(0), 46(0), 46(1), 46(1), 41(0), 40(0), 43(1), 45(0), 47(0), 47(0), 44(0), 45(1), 46(1), 42(1), 43(0), 42(0), 20(1), 28(0), 26(0), 60(0), 27(1), 24(0), 29(0), 60(1), 25(1), 60(1), 69(1), 36(1), 69(0), 69(1), 68(0), 68(0), 67(1), 67(0), 67(0), 52(0), 35(0), 66(0), 55(0), 66(1), 61(1), 61(0), 64(0), 64(0), 65(0), 65(0), 63(1), 63(1), 62(0), 39(1), 62(0), 62(0), 62(0), 59(1), 59(0), 59(1), 58(0), 58(0), 58(0), 18(1), 57(0), 57(0), 56(0), 56(0), 37(1), 53(0), 53(0), 53(0), 53(1), 54(1), 54(1), 66(0), 17(0), 50(0), 75(0), 51(0), 38(0), 52(1), 66(0), 4(1), 52(0), 55(0), 19(1), 58(1), 73(0). These data were used in [58] to assess the log harmonic mixture Weibull Weibull in comparison with the log Marshal Olkin Weibull Weibull. The dependent variable, namely, time to the onset of hypertension ($x_i$), was modeled with gender ($w_i$) as a covariate. The EGWE regression model, was obtained by considering that the dependent variable $x_i$ follows the EGWE distribution in Equation (24), in which the parameter $\lambda$ is linked to the covariate $w_i$ as follows ${\lambda }_i=e^{{\alpha }_0+{\alpha }_1{w}_{i1}}$. The parameter estimates and the goodness-of-fit measures for the EGWE regression model are presented in

Table 9. MLEs (with SE in parentheses) and goodness-of-fit for the EGWE regression model on the complete sample.

Model	MLEs	ℓ	acAIC	AICc	K-S	p-Value
EGWE	$\widehat{a}=0.0991\widehat{b}=0.2216$	−491.9093	995.8187	996.5687	0.0569	0.8352
	(0.0140)       (0.0216)
	$\widehat{c}=11.3824\widehat{\beta }=7.9761$
	(0.0157)       (0.0156)
	$\widehat{{\alpha }_0}=-1.978\widehat{{\alpha }_1}=-0.0171$
	(0.0253)    (0.0272)

.

The performance of the EGWE regression model was compared with the log harmonic mixture Weibull Weibull and the log Marshal Olkin Weibull Weibull from [58]. It was also compared with exponential regression, which resulted in $AIC=1184.253$, $AICc=1184.356$, and a very small p-value in the K-S test (i.e, <0.01). Compared with other regression models, the EGWE regression model mostly had the best measurements, which suggests that it fits the data better than the other models.

In addition, Cox–Snell residual analysis [63] was applied to evaluate the adequacy of the EGWE regression model. The Cox–Snell residual is defined as $r_i=-log\left(S\left(x_i\right|w_i)\right);i=1,\dots ,n$, where S is the survival function defined in Equation (25). In the case of an adequate model, the residual should follow a standard exponential distribution. First, a K-S test was performed, resulting in a p-value of $0.9041$, indicating that the Cox–Snell residuals were consistent with the standard exponential distribution. Moreover, the Kaplan–Meier survival plot of the residuals in Figure 6 was observed to closely follow the theoretical exponential survival curve, supporting the adequacy of the fitted model. The observed survival probabilities, obtained from the Kaplan–Meier estimate of the Cox–Snell residuals, were also compared with the expected probabilities from the E(1) distribution in the P–P plot shown in Figure 6. The strong alignment of the points along the diagonal further confirms that the EGWE regression model provides an excellent fit to the data.

8.2. Application 2: Regression Model for Censored Sample

The second dataset, obtained from [64], was used to assess the performance of the EGWE regression model on censored samples. These data contains the survival times (in days) of 137 patients with advanced, inoperable lung cancer undergoing chemotherapy treatment in the Veteran’s Administration’s Lung Cancer Trial, as reported in [65]. These data can be found in the “survival” package of the R programming language under the name “veteran”. The survival response variable, i.e, the survival times $x_i$, was modeled in relation to age $w_{i1}$ and the type of treatment $w_{i2}$ (0 = standard, 1 = test). In [64], these data were used to test the log cosine Topp Leone Weibull against the generalized inverse Topp Leone Weibull and a log exponentiated Topp Leone Weibull regression models. Accordingly, the EGWE regression model was derived by considering the dependent variable $x_i$ to follow the EGWE distribution in Equation (24), in which the parameter $\lambda$ is linked to the covariate $w_i$ as follows ${\lambda }_i=e^{{\alpha }_0+{\alpha }_1{w}_{i1}+{\alpha }_2{w}_{i2}}$.

The parameter estimates and the goodness-of-fit measures for the EGWE regression model are presented in

Table 10. MLEs with SE in parentheses and goodness-of-fit for the EGWE regression model on the censored sample.

Model	MLEs	ℓ	AIC	AICc	K-S	p-Value
EGWE	$\widehat{a}=0.0108\widehat{b}=2.4800$	−746.3648	1506.73	1505.376	0.0661	0.5870
	(0.0047)       (1.4239)
	$\widehat{c}=0.5366\widehat{\beta }=0.0352$
	(0.1487)       (0.0733)
	$\widehat{{\alpha }_0}=1.1125\widehat{{\alpha }_1}=0.0049\widehat{{\alpha }_2}=0.0445$
	(3.9221)    (0.0118)    (0.2323)

.

The performance of the EGWE regression model was compared with the log cosine Topp Leone Weibull and generalized inverse Topp Leone Weibull and the log exponentiated Topp Leone Weibull regression models from [64]. Comparing the EGWE regression model to the other regression models, it obtained nearly the best measurements, suggesting that it has a better fit to the data than the other models.

In addition, in the Cox–Snell residual analysis, the p-value for the K-S test was $0.9296$. Figure 7 shows a Kaplan–Meier survival plot that closely matches the theoretical exponential survival curve, supporting the good fit of the model. A P-P plot is shown in Figure 7, comparing the observed survival probabilities with those predicted by the E(1) distribution. The strong alignment of the points along the diagonal further confirms that the EGWE regression model provides an excellent fit to the data.

9. Conclusions

In this paper, we defined a new generalization of the E distribution called the EGWE. This new model has five parameters that can capture data distributions presenting varied shapes (e.g., right-skewed, left-skewed, symmetric, and reversed-J shapes), as well as different hazard rate trends (e.g., increasing, decreasing, bathtub, upside-down bathtub, S-shaped, J-shaped, and reversed-J). Some of the properties of the EGWE, including its quantiles, moments, moment-generating function, incomplete moments, Rényi entropy, and order statistics, were derived, and the parameters of the distribution were estimated using the method of maximum likelihood. To illustrate the performance of these estimated parameters, different simulation studies with varying sample sizes were conducted, and the results indicated the appropriateness of the maximum likelihood method for estimation. The utility and flexibility of the novel distribution in modeling lifetime data were assessed by means of its application to four datasets. The EGWE model obtained the largest log-likelihood values and p-value in the K-S test, as well as the smallest AIC and AICc values when compared with other distributions. Thus, it can be concluded that the EGWE distribution better fit the considered data than other competitive models. Furthermore, a regression model associated with the EGWE distribution was developed for both complete and censored lifetime data. This model was estimated via maximum likelihood, and its adequacy was assessed on two real datasets: one comprising censored samples and the other complete samples. The results indicate that the EGWE regression framework provides a flexible and effective method for modeling lifetime data with covariates. A parameter-linked regression structure for the EGWE distribution was demonstrated to be useful in this paper; however, further extensions may enhance its practical utility. Particularly, a log-location-scale regression model formulation for the EGWE could be considered in future research. Generally, it can be concluded that the EGWE distribution and its regression model have the potential to serve as alternative models for lifetime datasets, which can have a number of varying characteristics.