A Novel Adaptable Weibull Distribution and Its Applications

Abstract

This work proposes a novel extension for a new extended Weibull distribution. Some statistical properties of the proposed distribution are studied including quantile, moments, skewness, and kurtosis. The hazard rate function of the new distribution has certain elastic qualities, allowing it to take increasing, upside-down bathtub, and modified upside-down bathtub shapes commonly observed in medical contexts. Different methods of estimation are studied using complete data. Two real data sets from the medical field are analyzed to demonstrate that the proposed model has adaptability in practice. In comparison to some well-known distributions, the suggested distribution fits the tested data better based on both parametric and non-parametric statistical criteria. A simulation study is presented to compare the obtained estimates based on mean square error and average absolute bias.

1. Introduction

The provided statistical lifetime distributions play a significant role in both the quality of the statistical analysis procedures and in the realization and forecasting of real-world phenomena. The statistician’s choice of model is made easier by the availability of many lifetime distributions that are available to them. Thus, the generalization of the existing lifetime distributions has become urgently necessary due to their flexibility because the data of many significant problems in engineering, medicine, and other areas are incompatible with or poorly fit by the available lifetime distributions. It is important and challenging to select the best statistical model for data analysis. As a result, several authors have worked very hard in recent years to propose new families that extend current distributions and offer adaptable classes for modeling data in many different areas.

In several disciplines, including medicine and reliability engineering, the Weibull distribution [1] is one of the most often utilized distributions. The complicated lifetime of some systems cannot be modeled by the Weibull distribution because it does not possess the property of bathtub-shaped or unimodal-shaped hazard rate functions. Several extensions to the Weibull distribution have been developed to address this issue.

Numerous publications, including [2,3,4,5,6], proposed several Weibull distribution extensions with more adaptable characteristics.

Additionally, the Weibull distribution also has two important extensions, namely, the modified Weibull distribution (MWD) [7] and new extended Weibull distribution (NEWD) [8], which are extremely significant in numerous applications. For the MWD’s HRF, only increasing or bathtub shapes are available, while the NEWD’s HRF only permits the use of increasing or upside-down bathtub shapes. Recently, many publications proposed several extension of distributions with HRFs with upside-down bathtub and modified upside-down bathtub shapes, including [9,10,11,12].

This study’s primary goal is to introduce a new extension known as the Modification for New Extended Weibull distribution, which has more flexible qualities than the New Extended Weibull distribution.

The following goals give a significant justification for the importance of the proposed distribution.

• To present a novel lifetime distribution by extending the new extended Weibull distribution and discuss its advantages and potential uses.
• To calculate the formula of the quantile in closed form and study the numerical results of the rth moment of the proposed distribution.
• To show the hazard rate function of the new distribution can take not only the upside-down bathtub shape like NEWD but also the modified upside-down bathtub shape commonly observed in medical contexts.
• To investigate several techniques of estimation for the new model’s unknown parameters based on complete data.
• Using the ideas and techniques presented in this study, the significance of this new model in the medical field is demonstrated under complete data.
• When applied to the clinical data, the new distribution demonstrates a superior fit compared to its sub-models and some well-known distributions.
• The effectiveness of the various estimation strategies used to estimate the distribution parameters is investigated using a simulation study.

The article is divided into seven sections, each focusing on a particular topic of the study. In Section 2, the construction of the GNEWD is thoroughly detailed. In Section 3, some statistical properties are studied. The behavior of the HRF of the GNEWD is presented in Section 4. Section 5 is dedicated to studying the different techniques of estimation for the new model. Two applications are presented to demonstrate the practical relevance and effectiveness of the proposed methods based on real-life data in Section 6. A simulation study is carried out in Section 7 to evaluate the performance of the proposed estimators.

2. The New Modification for Weibull Distribution

The CDF of the GNEWD with parameter vector $\Theta =(\lambda ,\gamma ,a,b)$ is obtained by multiplying the cumulative hazard rate function for the Weibull distribution by the component $e^{-a{x}^{-b}}$

$$\begin{array}{c}\hfill F(x;\Theta )=1-e^{-\lambda x^{\gamma }{e}^{-a{x}^{-b}}},x>0,\end{array}$$

(1)

where $\lambda >0$ is the scale parameter, and $\gamma ,a,b>0$ are the shape parameters.

Accordingly, the PDF, SF, and HRF of the GNEWD are given, respectively, as follows.

$$\begin{array}{c}\hfill f(x;\Theta )=\lambda x^{\gamma -1}{e}^{-\lambda x^{\gamma }{e}^{-a{x}^{-b}}}{e}^{-a{x}^{-b}}[\gamma +ab{x}^{-b}],x>0,\end{array}$$

(2)

$$\begin{array}{c}\hfill S(x;\Theta )=e^{-\lambda x^{\gamma }{e}^{-a{x}^{-b}}},x>0,\end{array}$$

(3)

and

$$\begin{array}{c}\hfill h(x;\Theta )=\lambda x^{\gamma -1}{e}^{-a{x}^{-b}}[\gamma +ab{x}^{-b}],x>0.\end{array}$$

(4)

Some possible shapes for the PDF of the GNEWD are exhibited in Figure 1. Based on different values of parameters, the PDF of the GNEWD can be unimodal with a small and long right tail. Figure 2a exhibits the shapes for the CDF of the GNEWD and Figure 2b exhibits the shapes for the survival function of the GNEWD with different values of parameters of the GNEWD.

From Equation (1), some well-known distributions can be obtained as in

Table 1. The sub-models from GNEWD.

Distribution	$\mathit{\lambda }$	$\mathit{\gamma }$	a	b	CDF	Reference
NEWD $(\lambda ,\gamma ,a)$	-	-	-	1	$1-e^{-\lambda x^{\gamma }{e}^{-a{x}^{-1}}}$	[8]
WD $(\lambda ,\gamma )$	-	-	0	0	$1-e^{-\lambda x^{\gamma }}$	[1]
ED $\left(\lambda \right)$	-	1	0	0	$1-e^{-\lambda x}$	[13]
RD $\left(c\right)$	${\displaystyle \frac{c}{2}}$	2	0	0	$1-e^{-{\displaystyle \frac{c}{2}}{x}^2}$	[13]
RGNEWD $(\lambda ,a,b)$	-	${\displaystyle \frac{1}{2}}$	-	-	$1-e^{-\lambda \sqrt{x}{e}^{-a{x}^{-b}}}$	New
RNEWD $(\lambda ,a)$	-	${\displaystyle \frac{1}{2}}$	-	1	$1-e^{-\lambda \sqrt{x}{e}^{-a{x}^{-1}}}$	New

.

3. Some Statistical Properties

In this section, an explicit form of the GNEWD’s quantile function is presented. This is required for generating random data samples from the proposed distribution to conduct the simulation study. It is also important to calculate the Bowley skewness and Moors kurtosis measures which are used to demonstrate a distribution’s asymmetry and tail behavior, respectively, utilizing quantile-based techniques that are less sensitive to outliers. Further, the formula for the moments of the GNEWD is studied.

3.1. Quantiles

Lemma 1. 

From (1), it is evident that the quantile function of the GNEWD can be given as follows:

$$\begin{array}{c}\hfill x_q={\left({\displaystyle \frac{ab}{\gamma W\left(\frac{ab{\left(-\frac{log(1-q)}{\lambda }\right)}^{-\frac{b}{\gamma }}}{\gamma }\right)}}\right)}^{1/b},0<q<1,\end{array}$$

(5)

where $W\left(.\right)$ is the LambertW function [for more details, see [14]].

Proof 1. 

Let $q=F(x;\Theta )$; then

$$\begin{array}{c}\hfill q=1-e^{-\lambda x_q^{\gamma }{e}^{-a{x}_q^{-b}}},\end{array}$$

by applying some algebraic calculation, we get

$$\begin{array}{c}\hfill x_q^{\gamma }{e}^{-a{x}_q^{-b}}={\displaystyle \frac{-log(1-q)}{\lambda }},\end{array}$$

using the definition of the LambertW function, the result is satisfied. □

Let $x_q$ be the quantile of the GNEWD which is given by (5). Then the Bowley skewness and Moors kurtosis of the GNEWD can be given, respectively, as follows:

$$\begin{array}{c}\hfill BSkew={\displaystyle \frac{x_{0.75}+x_{0.25}-2x_{0.5}}{x_{0.75}-x_{0.25}}},\end{array}$$

(6)

and

$$\begin{array}{c}\hfill Mkur={\displaystyle \frac{x_{0.875}-x_{0.625}+x_{0.375}-x_{0.125}}{x_{0.75}-x_{0.25}}}.\end{array}$$

(7)

3.2. Moments

Moments are important in statistical analysis, both theoretically and practically. They provide crucial insights into probability distribution behavior by quantifying key properties like central tendency, variability, skewness, and kurtosis. Upon using (2), the rth moments of the GNEWD can be given as

$$\begin{array}{c}\hfill {\mu }_r^{\prime }(\Theta )={\int }_0^{\infty }{x}^r\lambda x^{\gamma -1}{e}^{-\lambda x^{\gamma }{e}^{-a{x}^{-b}}}{e}^{-a{x}^{-b}}[\gamma +ab{x}^{-b}]dx.\end{array}$$

(8)

The integration in (8) cannot be calculated analytically. So, numerical method can be used to compute (8) using NIntegrate in Wolfram Mathematica 11.

The measurements of variance $\left(Var\right(\Theta \left)\right)$, skewness $\left(Sk\right(\Theta \left)\right)$, and kurtosis $\left(Kur\right(\Theta \left)\right)$ can be calculated, respectively, by using the first four ordinary moments of the GNEWD as:

$$\begin{array}{c}\hfill Var(\Theta )={\mu }_2^{\prime }(\Theta )-{\mu }_1^{\prime }{(\Theta )}^2,\end{array}$$

(9)

$$\begin{array}{c}\hfill Sk(\Theta )={\displaystyle \frac{{\mu }_3^{\prime }(\Theta )-3{\mu }_1^{\prime }(\Theta ){\mu }_2^{\prime }(\Theta )+2{\mu }_1^{\prime }{(\Theta )}^3}{Var{(\Theta )}^{3/2}}},\end{array}$$

(10)

and

$$\begin{array}{c}\hfill Kur(\Theta )={\displaystyle \frac{{\mu }_4^{\prime }(\Theta )-4{\mu }_1^{\prime }(\Theta ){\mu }_3^{\prime }(\Theta )+6{\mu }_1^{\prime }{(\Theta )}^2{\mu }_2^{\prime }(\Theta )-3{\mu }_1^{\prime }{(\Theta )}^4}{Var{(\Theta )}^2}}.\end{array}$$

(11)

Some numerical values of the first four ordinary moments, variance, skewness, and kurtosis of the GNEWD are calculated and listed in

Table 2. The first four ordinary moments, variance, skewness, and kurtosis of the GNEWD using different values of a and b under fixed values of $\lambda$ and $\gamma$.

($\mathit{\lambda }$, $\mathit{\gamma }$)	a	b	${\mathit{\mu }}_1^{\prime }\left(\mathsf{\Theta }\right)$	${\mathit{\mu }}_2^{\prime }\left(\mathsf{\Theta }\right)$	${\mathit{\mu }}_3^{\prime }\left(\mathsf{\Theta }\right)$	${\mathit{\mu }}_4^{\prime }\left(\mathsf{\Theta }\right)$	$\mathbf{Var}\left(\mathsf{\Theta }\right)$	$\mathbf{Sk}\left(\mathsf{\Theta }\right)$	$\mathbf{Kur}\left(\mathsf{\Theta }\right)$
(1, 1)	0.5	0.5	1.4473	3.5886	12.6798	58.1681	1.4939	0.0713	7.4830
	0.5	1	1.3832	3.023	9.3733	38.0833	1.1099	−0.4485	8.0729
	0.5	5	1.3376	2.4506	6.5653	24.7723	0.6614	−1.6275	14.5145
	0.5	10	1.3468	2.4392	6.4857	24.5254	0.6253	−1.8741	16.0129
	0.5	20	1.3556	2.4494	6.4888	24.5103	0.6118	−2.0510	16.5705
	2	0.5	2.8849	11.9064	63.9485	422.034	3.5836	−2.2241	5.5169
	6	0.5	7.4456	69.6021	777.718	10052.8	14.1651	−6.8316	4.096
	12	0.5	15.9514	301.24	6515.96	157785.	46.7928	−11.9994	3.5161
(3, 3)	0.5	0.5	0.7481	0.622	0.5605	0.5387	0.0624	−26.7671	2.7259
	0.5	1	0.7673	0.6421	0.5761	0.5475	0.0533	−36.5933	2.7415
	0.5	5	0.8789	0.7881	0.7212	0.6741	0.0156	−348.682	3.5212
	0.5	10	0.9291	0.8697	0.8207	0.7814	0.0065	−1521.18	5.5792
	0.5	20	0.963	0.9301	0.9013	0.8767	0.0027	−6198.52	12.7795
	2	0.5	1.1428	1.4105	1.8511	2.556	0.1045	−44.1751	2.7403
	6	0.5	2.2807	5.4702	13.6825	35.4733	0.2686	−85.3607	2.8100
	12	0.5	4.2216	18.4909	83.5819	388.294	0.6690	−137.725	2.8969

. From Table 2, one can show that the following results are satisfied:

• Under fixed values of $(\lambda ,\gamma )=(1,1)$,
  • As b increases with fixed a, the mean and variance of the GNEWD decrease.
  • As a increases with fixed b, the mean and variance of the GNEWD increase.
  • As b increases with fixed a, the skewness of the GNEWD decreases.
  • As a increases with fixed b, the skewness of the GNEWD decreases.
  • As b increases with fixed a, the kurtosis of the GNEWD increases.
  • As a increases with fixed b, the kurtosis of the GNEWD decreases.
• Under fixed values of $(\lambda ,\gamma )=(3,3)$,
  • As b increases with fixed a, the mean of the GNEWD increases and variance of the GNEWD decreases.
  • As a increases with fixed b, the mean and variance of the GNEWD increase.
  • As b increases with fixed a, the skewness of the GNEWD decreases.
  • As a increases with fixed b, the skewness of the GNEWD decreases.
  • As b increases with fixed a, the kurtosis of the GNEWD increases.
  • As a increases with fixed b, the kurtosis of the GNEWD increases.

4. Some Attributes of the HRF

This section deals with some characterization of the HRF. The different shapes for the HRF of the GNEWD including increasing, upside-down bathtub, and modified upside-down bathtub shapes are shown in this section. From (4), one can show that the first derivative of $h\left(x\right)$ can be written as

$$\begin{array}{c}\hfill h^{\prime }(x;\Theta )=\lambda x^{\gamma -2}{e}^{-a{x}^{-b}}[\gamma (\gamma -1)+ab(-b+2\gamma -1)x^{-b}+a^2{b}^2{x}^{-2b}],x>0.\end{array}$$

(12)

If $b=1$, Equation (12) reduces to

$$\begin{array}{c}\hfill h^{\prime }(x;\lambda ,\gamma ,a)=\lambda x^{\gamma -4}{e}^{-{\displaystyle \frac{a}{x}}}[\gamma (\gamma -1)x^2+2a(\gamma -1)x+a^2],x>0\end{array}$$

which is the same for NEWD as in [8].

Remark 1. 

For $\lambda >0$, $b>0$, and $a>0$, the following limits are satisfied:

1. 

${\displaystyle \underset{x\to 0^{+}}{lim}\left(\lambda x^{\gamma -1}{e}^{-a{x}^{-b}}\left(\gamma +ab{x}^{-b}\right)\right)\to 0}$ for $0<\gamma <1$.

2. 

${\displaystyle \underset{x\to \infty }{lim}\left(\lambda x^{\gamma -1}{e}^{-a{x}^{-b}}\left(\gamma +ab{x}^{-b}\right)\right)\to 0}$ for $0<\gamma <1$.

3. 

${\displaystyle \underset{x\to 0^{+}}{lim}\left(\lambda x^{\gamma -1}{e}^{-a{x}^{-b}}\left(\gamma +ab{x}^{-b}\right)\right)\to 0}$ for $\gamma =1$.

4. 

${\displaystyle \underset{x\to \infty }{lim}\left(\lambda x^{\gamma -1}{e}^{-a{x}^{-b}}\left(\gamma +ab{x}^{-b}\right)\right)\to \lambda }$ for $\gamma =1$.

5. 

${\displaystyle \underset{x\to 0^{+}}{lim}\left(\lambda x^{\gamma -1}{e}^{-a{x}^{-b}}\left(\gamma +ab{x}^{-b}\right)\right)\to 0}$ for $\gamma >1$.

6. 

${\displaystyle \underset{x\to \infty }{lim}\left(\lambda x^{\gamma -1}{e}^{-a{x}^{-b}}\left(\gamma +ab{x}^{-b}\right)\right)\to \infty }$ for $\gamma >1$.

It is easy to show that ${\displaystyle \underset{x\to 0^{+}}{lim}h(x;\Theta )\to 0}$ for all values of $\gamma >0$ using L’Hospital’s rule. Further, the value of the limit ${\displaystyle \underset{x\to \infty }{lim}h(x;\Theta )}$ can be obtained directly as shown in Remark 1.

Lemma 2. 

For $\lambda >0$ and $a>0$, the HRF take the following shapes:

• h(x;Θ) has increasing shape from zero to infinity if $\gamma >1$ and $b\le 1$.
• h(x;Θ) has increasing shape from 0 to λ if $\gamma =1$ and $b\le 1$.
• h(x;Θ) has an upside-down bathtub shape if $0<\gamma <1$ and $b>0$.
• h(x;Θ) has a modified upside-down bathtub shape if $1<\gamma \le {\displaystyle \frac{1+2b+b^2}{4b}}$ and $b>1$.

Proof 2. 

Equation (12) can be rewritten as

$$\begin{array}{c}\hfill h^{\prime }(x;\Theta )=\lambda x^{\gamma -2b-2}{e}^{-a{x}^{-b}}[\gamma (\gamma -1)x^{2b}+ab(-b+2\gamma -1)x^b+a^2{b}^2].\end{array}$$

(13)

Upon using (13) with the results in Remark 1, one can derived the following results:

• For $\gamma >1$ and $b\le 1$, it is clear that $h^{\prime }(x;\Theta )>0$ for all $x>0$. Then $h(x;\Theta )$ has an increasing shape that goes from zero to ∞ as $x\to \infty$.
• For $\gamma =1$ and $b\le 1$, it is clear that $h^{\prime }(x;\Theta )>0$ for all $x>0$. Then $h(x;\Theta )$ has an increasing shape that goes from zero to $\lambda$ as $x\to \infty$.
• For $0<\gamma <1$ and $b>0$, it is clear that $h^{\prime }(x;\Theta )$ has two roots based on the expression $\gamma (\gamma -1)x^2+2a(\gamma -1)x+a^2$ which are given as follows: $x_1={\left({\displaystyle \frac{ab(b-2\gamma +1)-ab\sqrt{b^2-4b\gamma +2b+1}}{2\gamma (\gamma -1)}}\right)}^{1/b}$ and $x_2={\left({\displaystyle \frac{ab(b-2\gamma +1)+ab\sqrt{b^2-4b\gamma +2b+1}}{2\gamma (\gamma -1)}}\right)}^{1/b}$. Under the two conditions $0<\gamma <1$ and $b>0$, it is evident that $x_1>0$ and $x_2<0$ because $\sqrt{b^2-4b\gamma +2b+1}>(b-2\gamma +1)$. So, $h^{\prime }(x;\Theta )$ has a unique root for $x>0$ which is $x_1$. Additionally, $h^{\prime }(x;\Theta )>0$ when $x<x_1$ and $h^{\prime }(x;\Theta )<0$ when $x>x_1$. This means that the hazard rate function has an upside-down bathtub shape that converges to 0 when x approaches 0 or ∞.
• For $1<\gamma \le {\displaystyle \frac{1+2b+b^2}{4b}}$ and $b>1$, it is clear that $h^{\prime }(x;\Theta )$ has two roots, $x_1$ and $x_2$. Under the two conditions $1<\gamma \le {\displaystyle \frac{1+2b+b^2}{4b}}$ and $b>1$, it is clear that $\sqrt{b^2-4b\gamma +2b+1}<(b-2\gamma +1)$, $x_1>0$, $x_2>0$, and $x_2>x_1$. So, there are two change points. Also, $h^{\prime }(x;\Theta )>0$ when $x<x_1$ and $x>x_2$ and $h^{\prime }(x;\Theta )<0$ when $x>x_1$ and $x<x_2$ (using $Reduce$ in Wolfram Mathematica). This means that the hazard rate function has a modified upside-down bathtub shape.

□

Upon using different values of parameters as shown in the previous two lemmas, the behavior of the HRF of the GNEWD can be increasing, upside-down bathtub, and modified upside-down bathtub as shown in Figure 3.

5. Different Methods of Estimation

This section outlines five specific estimation techniques relevant to the GNEWD: MLE, LSE, WLSE, CVME, and ADE. These methods are utilized to estimate the unknown parameters of the new model.

5.1. Maximum Likelihood Estimation

Suppose that $x_1,\dots ,x_n$ is an independent random sample of size n from the GNEWD. From Equation (2), the log-likelihood function can be obtained as

$$\begin{array}{c}\hfill \ell (\Theta )=nlog\left(\lambda \right)-\sum _{i=1}^n\lambda x_i^{\gamma }{e}^{-a{x}_i^{-b}}+\sum _{i=1}^{n}log\left(ab{x}_i^{-b}+\gamma \right)-\sum _{i=1}^{n}a{x}_i^{-b}+(\gamma -1)\sum _{i=1}^{n}log\left(x_i\right).\end{array}$$

(14)

By taking the first derivative (${\ell }_{\Theta }(\Theta )={\displaystyle \frac{\partial \ell }{\partial \Theta }}$) of (14) with respect to $\lambda$, $\gamma$, a, and b we get

$$\begin{array}{c}\hfill {\ell }_{\lambda }(\Theta )={\displaystyle \frac{n}{\lambda }}-\sum _{i=1}^n{x}_i^{\gamma }{e}^{-a{x}_i^{-b}},\end{array}$$

$$\begin{array}{c}\hfill {\ell }_{\gamma }(\Theta )=\sum _{i=1}^{n}log\left(x_i\right)-\sum _{i=1}^n\lambda x_i^{\gamma }log\left(x_i\right)e^{-a{x}_i^{-b}}+\sum _{i=1}^n{\displaystyle \frac{1}{ab{x}_i^{-b}+\gamma }},\end{array}$$

$$\begin{array}{c}\hfill {\ell }_a(\Theta )=-\sum _{i=1}^n\lambda \left(-e^{-a{x}_i^{-b}}\right)x_i^{\gamma -b}+\sum _{i=1}^n{\displaystyle \frac{b{x}_i^{-b}}{ab{x}_i^{-b}+\gamma }}-\sum _{i=1}^n{x}_i^{-b},\end{array}$$

and

$$\begin{array}{c}\hfill {\ell }_b(\Theta )=-\sum _{i=1}^{n}a\lambda log\left(x_i\right)e^{-a{x}_i^{-b}}{x}_i^{\gamma -b}+\sum _{i=1}^n{\displaystyle \frac{a{x}_i^{-b}-ab{x}_i^{-b}log\left(x_i\right)}{ab{x}_i^{-b}+\gamma }}-\sum _{i=1}^n-a{x}_i^{-b}log\left(x_i\right).\end{array}$$

5.2. The Parameters $\lambda$, $\gamma$, a, and b Are Unknown

The MLE $\widehat{\Theta }$ of $\Theta$ is given by solving the four normal equations ${\ell }_{\lambda }(\Theta )=0$, ${\ell }_{\gamma }(\Theta )=0$, ${\ell }_a(\Theta )=0$, and ${\ell }_b(\Theta )=0$. Since these equations cannot be solved analytically, they are solved numerically by employing $FindRoot$ in the Wolfram Mathematica 11 software. The initial values for the parameters can be obtained by fitting special GNEWD sub-models using $FindDistributionParameters$ in the Wolfram Mathematica 11 software.

5.3. Fisher Information Matrix

Since the computation of the Fisher information matrix (given by taking the expectation of the second derivative of (14)) is very difficult, it seems appropriate to approximate these expected values by their MLEs. Then, the asymptotic variance–covariance matrix is given as follows [see, [15]]:

$$\begin{array}{ccc}\hfill \left(\begin{array}{cccc}Var\left(\widehat{\lambda }\right)& Cov(\widehat{\lambda },\widehat{\gamma })& Cov(\widehat{\lambda },\widehat{a})& Cov(\widehat{\lambda },\widehat{b})\\ Cov(\widehat{\gamma },\widehat{\lambda })& Var\left(\widehat{\gamma }\right)& Cov(\widehat{\gamma },\widehat{a})& Cov(\widehat{\gamma },\widehat{b})\\ Cov(\widehat{a},\widehat{\lambda })& Cov(\widehat{a},\widehat{\gamma })& Var\left(\widehat{a}\right)& Cov(\widehat{b},\widehat{a})\\ Cov(\widehat{b},\widehat{\lambda })& Cov(\widehat{b},\widehat{\gamma })& Cov(\widehat{b},\widehat{a})& Var\left(\widehat{b}\right)\end{array}\right)& =& \\ \hfill {\left(\begin{array}{cccc}-{\ell }_{\lambda \lambda }(\Theta )& -{\ell }_{\lambda \gamma }(\Theta )& -{\ell }_{\lambda a}(\Theta )& -{\ell }_{\lambda b}(\Theta )\\ -{\ell }_{\gamma \lambda }(\Theta )& -{\ell }_{\gamma \gamma }(\Theta )& -{\ell }_{\gamma a}(\Theta )& -{\ell }_{\gamma b}(\Theta )\\ -{\ell }_{a\lambda }(\Theta )& -{\ell }_{a\gamma }(\Theta )& -{\ell }_{aa}(\Theta )& -{\ell }_{ab}(\Theta )\\ -{\ell }_{b\lambda }(\Theta )& -{\ell }_{b\gamma }(\Theta )& -{\ell }_{ba}(\Theta )& -{\ell }_{bb}(\Theta )\end{array}\right)}_{(\widehat{\lambda },\widehat{\gamma },\widehat{a},\widehat{b})}^{-1},\end{array}$$

(15)

where ${\ell }_{{\theta }_i{\theta }_j}(\Theta )={\displaystyle \frac{{\partial }^2\ell }{\partial {\theta }_i{\theta }_j}}$, $i,j=1,2,3,4$, see Appendix A. Accordingly, the ACIs based on the asymptotic variance–covariance matrix for the parameters $\lambda$, $\gamma$, a, and b are, respectively, given as follows:

$$\widehat{\lambda }\pm z_{{\displaystyle \frac{\alpha }{2}}}\sqrt{Var\left(\widehat{\lambda }\right)}, \widehat{\gamma }\pm z_{{\displaystyle \frac{\alpha }{2}}}\sqrt{Var\left(\widehat{\gamma }\right)}, \widehat{a}\pm z_{{\displaystyle \frac{\alpha }{2}}}\sqrt{Var\left(\widehat{a}\right)}and \widehat{b}\pm z_{{\displaystyle \frac{\alpha }{2}}}\sqrt{Var\left(\widehat{b}\right),}$$

where $z_{{\displaystyle \frac{\alpha }{2}}}$ is the percentile of the standard normal distribution with right tail probability ${\displaystyle \frac{\alpha }{2}}$. Even if the parameters are positive, the ACIs can yield a negative lower bound. In this case, the negative numbers could be substituted by zero.

5.4. Least Square and Weighted Least Square Estimations

The study [16] presents the LSE and WLSE for estimating the parameters of the beta distribution. These methods will estimate the parameters of the GNEWD. For this purpose, take $x_i$, $i=1,\dots ,n$ as the ordered sample of a random sample of size n. Then the LSE of the parameters of the GNEWD can be obtained by minimizing the following function:

$$\begin{array}{c}\hfill S=\sum _{i=1}^n{\left(F(x_i;\Theta )-{\displaystyle \frac{i}{n+1}}\right)}^2\end{array}$$

with respect to the unknown parameters $\Theta$ or by solving the following non-linear equations:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial S}{\partial \lambda }}=\sum _{i=1}^{n}2\left(1-e^{-\lambda x_i^{\gamma }{e}^{-a{x}_i^{-b}}}\right){\delta }_1(x_i;\Theta )=0,\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial S}{\partial \theta }}=\sum _{i=1}^{n}2\left(1-e^{-\lambda x_i^{\gamma }{e}^{-a{x}_i^{-b}}}\right){\delta }_2(x_i;\Theta )=0,\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial S}{\partial \alpha }}=\sum _{i=1}^{n}2\left(1-e^{-\lambda x_i^{\gamma }{e}^{-a{x}_i^{-b}}}\right){\delta }_3(x_i;\Theta )=0,\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial S}{\partial \gamma }}=\sum _{i=1}^{n}2\left(1-e^{-\lambda x_i^{\gamma }{e}^{-a{x}_i^{-b}}}\right){\delta }_4(x_i;\Theta )=0,\end{array}$$

where

$$\begin{array}{c}\hfill {\delta }_1(x_i;\Theta )=x_i^{\gamma }{e}^{-\lambda x_i^{\gamma }{e}^{-a{x}_i^{-b}}-a{x}_i^{-b}},\end{array}$$

(16)

$$\begin{array}{c}\hfill {\delta }_2(x_i;\Theta )=\lambda x_i^{\gamma }log\left(x_i\right)e^{-\lambda x_i^{\gamma }{e}^{-a{x}_i^{-b}}-a{x}_i^{-b}},\end{array}$$

(17)

$$\begin{array}{c}\hfill {\delta }_3(x_i;\Theta )=\lambda x_i^{\gamma -b}\left(-e^{-\lambda x_i^{\gamma }{e}^{-a{x}_i^{-b}}-a{x}_i^{-b}}\right),\end{array}$$

(18)

and

$$\begin{array}{c}\hfill {\delta }_4(x_i;\Theta )=a\lambda log\left(x_i\right)x_i^{\gamma -b}{e}^{-\lambda x_i^{\gamma }{e}^{-a{x}_i^{-b}}-a{x}_i^{-b}}.\end{array}$$

(19)

In order to determine the WLSE of the unknown parameters of the GNEWD, minimize the following function

$$\begin{array}{c}\hfill W=\sum _{i=1}^n{\displaystyle \frac{(n+2){(n+1)}^2}{i(n-i+1)}}{\left(F(x_i;\Theta )-{\displaystyle \frac{i}{n+1}}\right)}^2\end{array}$$

with respect to the unknown parameters $\Theta$ or solve the non-linear equations

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial W}{\partial \lambda }}=\sum _{i=1}^{n}2{\displaystyle \frac{(n+2){(n+1)}^2}{i(n-i+1)}}\left(1-e^{-\lambda x_i^{\gamma }{e}^{-a{x}_i^{-b}}}\right){\delta }_1(x_i;\Theta )=0,\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial W}{\partial \theta }}=\sum _{i=1}^{n}2{\displaystyle \frac{(n+2){(n+1)}^2}{i(n-i+1)}}\left(1-e^{-\lambda x_i^{\gamma }{e}^{-a{x}_i^{-b}}}\right){\delta }_2(x_i;\Theta )=0,\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial W}{\partial \alpha }}=\sum _{i=1}^{n}2{\displaystyle \frac{(n+2){(n+1)}^2}{i(n-i+1)}}\left(1-e^{-\lambda x_i^{\gamma }{e}^{-a{x}_i^{-b}}}\right){\delta }_3(x_i;\Theta )=0,\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial W}{\partial \gamma }}=\sum _{i=1}^{n}2{\displaystyle \frac{(n+2){(n+1)}^2}{i(n-i+1)}}\left(1-e^{-\lambda x_i^{\gamma }{e}^{-a{x}_i^{-b}}}\right){\delta }_4(x_i;\Theta )=0.\end{array}$$

where ${\delta }_1(x_i;\Theta )$, ${\delta }_2(x_i;\Theta )$, ${\delta }_3(x_i;\Theta )$, and ${\delta }_4(x_i;\Theta )$ are given by (16), (17), (18), and (19), respectively.

5.5. Cramér Von-Mises Estimation

In the study [17], it was shown that the bias of CVME is smaller than the other minimum distance estimator. The CVME of the GNEWD is obtained by minimizing the following function:

$$\begin{array}{c}\hfill C={\displaystyle \frac{1}{12r}}+\sum _{i=1}^r{\left(F(x_i;\Theta )-{\displaystyle \frac{2i-1}{2r}}\right)}^2\end{array}$$

with respect to the unknown parameters $\Theta$ or by solving the following non-linear equations:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial C}{\partial \lambda }}=\sum _{i=1}^{r}2\left(1-e^{-\lambda x_i^{\gamma }{e}^{-a{x}_i^{-b}}}\right){\delta }_1(x_i;\Theta )=0,\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial C}{\partial \theta }}=\sum _{i=1}^{r}2\left(1-e^{-\lambda x_i^{\gamma }{e}^{-a{x}_i^{-b}}}\right){\delta }_2(x_i;\Theta )=0,\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial C}{\partial \alpha }}=\sum _{i=1}^{r}2\left(1-e^{-\lambda x_i^{\gamma }{e}^{-a{x}_i^{-b}}}\right){\delta }_3(x_i;\Theta )=0,\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial C}{\partial \gamma }}=\sum _{i=1}^{r}2\left(1-e^{-\lambda x_i^{\gamma }{e}^{-a{x}_i^{-b}}}\right){\delta }_4(x_i;\Theta )=0,\end{array}$$

where ${\delta }_1(x_i;\Theta )$, ${\delta }_2(x_i;\Theta )$, ${\delta }_3(x_i;\Theta )$, and ${\delta }_4(x_i;\Theta )$ are given by (16), (17), (18), and (19), respectively.

5.6. Anderson Darling Estimation

Another study [18], studied the properties of ADE. From the results given there, the A-DE of the GNEWD is obtained by minimizing the following function:

$$\begin{array}{c}\hfill A=-r-\sum _{i=1}^r{\displaystyle \frac{(2i-1)}{r}}\left(log\left(F(x_i;\Theta )\right)+log(1-F(x_{r-i+1};\Theta )\right)),\end{array}$$

with respect to the unknown parameters $\Theta$ or by solving the following non-linear equations:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial A}{\partial \lambda }}=-\sum _{i=1}^r{\displaystyle \frac{(2i-1)}{r}}\left({\displaystyle \frac{{\delta }_1(x_i;\Theta )}{1-e^{-\lambda x_i^{\gamma }{e}^{-a{x}_i^{-b}}}}}-{\displaystyle \frac{{\delta }_1(x_{r-i+1};\Theta )}{e^{-\lambda x_{r-i+1}^{\gamma }{e}^{-a{x}_{r-i+1}^{-b}}}}}\right)=0,\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial A}{\partial \theta }}=-\sum _{i=1}^r{\displaystyle \frac{(2i-1)}{r}}\left({\displaystyle \frac{{\delta }_2(x_i;\Theta )}{1-e^{-\lambda x_i^{\gamma }{e}^{-a{x}_i^{-b}}}}}-{\displaystyle \frac{{\delta }_2(x_{r-i+1};\Theta )}{e^{-\lambda x_{r-i+1}^{\gamma }{e}^{-a{x}_{r-i+1}^{-b}}}}}\right)=0,\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial A}{\partial \alpha }}=-\sum _{i=1}^r{\displaystyle \frac{(2i-1)}{r}}\left({\displaystyle \frac{{\delta }_3(x_i;\Theta )}{1-e^{-\lambda x_i^{\gamma }{e}^{-a{x}_i^{-b}}}}}-{\displaystyle \frac{{\delta }_3(x_{r-i+1};\Theta )}{e^{-\lambda x_{r-i+1}^{\gamma }{e}^{-a{x}_{r-i+1}^{-b}}}}}\right)=0,\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial A}{\partial \gamma }}=-\sum _{i=1}^r{\displaystyle \frac{(2i-1)}{r}}\left({\displaystyle \frac{{\delta }_4(x_i;\Theta )}{1-e^{-\lambda x_i^{\gamma }{e}^{-a{x}_i^{-b}}}}}-{\displaystyle \frac{{\delta }_4(x_{r-i+1};\Theta )}{e^{-\lambda x_{r-i+1}^{\gamma }{e}^{-a{x}_{r-i+1}^{-b}}}}}\right)=0,\end{array}$$

where ${\delta }_1(x_i;\Theta )$, ${\delta }_2(x_i;\Theta )$, ${\delta }_3(x_i;\Theta )$, and ${\delta }_4(x_i;\Theta )$ are given by (16), (17), (18), and (19), respectively.

6. Application to Real Data

In this section, three clinical real data sets are analyzed to show that the generalized new extended Weibull distribution (GNEWD) offers enhanced flexibility for modeling these data sets better than their sub-models and some well-known distributions. For every data set, the NGNEWD can be compared, using a non-parametric test like the Kolmogorov–Smirnov test (K-S) and parametric tests such as the Akaike information criterion (AIC) and Bayesian information criterion (BIC), with its sub models (NEWD, RGNEWD, RNEWD, and WD) and also with the following well-known distributions summarized in

Table 3. Some unimodal well-known distributions.

Distribution	CDF	Reference
GMWD $(\lambda ,\gamma ,a,b)$	${(1-e^{-\lambda x^{\gamma }{e}^{ax}})}^b$	[19]
GIGWD $(\lambda ,\gamma ,a,b)$	$1-{(1-e^{-\gamma {({\displaystyle \frac{\lambda }{x}})}^a})}^b$	[20]
GIWD $(\lambda ,\gamma ,a)$	$e^{-\gamma {\left({\displaystyle \frac{\lambda }{x}}\right)}^a}$	[21]
EWD $(\lambda ,\gamma ,a)$	${\left(1-e^{-\lambda x^{\gamma }}\right)}^a$	[2]
M-O WD $(\lambda ,\gamma ,a)$	$1-{\displaystyle \frac{a{e}^{-\lambda x^{\gamma }}}{1-(1-a)e^{-\lambda x^{\gamma }}}}$	[22]
Logistic NHD $(\lambda ,\gamma ,a)$	${\displaystyle \frac{{\left({(\lambda x+1)}^{\gamma }-1\right)}^a}{{\left({(\lambda x+1)}^{\gamma }-1\right)}^a+1}}$	[23]
IGLED $(\lambda ,\gamma ,a)$	$e^{-{\left({\displaystyle \frac{\gamma }{2x^2}}+{\displaystyle \frac{\lambda }{x}}\right)}^a}$	[24]
IWD $(\lambda ,\gamma )$	$e^{-\lambda x^{-\gamma }}$	[25]

.

6.1. Liver Cancer Data

The data presented in [26], and analyzed in [27,28,29], includes records for 39 liver cancer patients. These patients were treated at Elminia Cancer Center, which operates under the Ministry of Health in Egypt. The cases were registered in the year 1999. The data set contains the lifetimes of the patients, measured in days, being 10, 14, 14, 14, 14, 14, 15, 17, 18, 20, 20, 20, 20, 20, 23, 23, 24, 26, 30, 30, 31, 40, 49, 51, 52, 60, 61, 67, 71, 74, 75, 87, 96, 105, 107, 107, 107, 116, 150.

For this data,

Table 4. The ML estimates of unknown parameters, the K-S with the corresponding p-value, the AIC, and the BIC for different models using the liver cancer data.

Distribution	$\widehat{\mathit{\lambda }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{a}}$	$\widehat{\mathit{b}}$	K-S Distance	p-Value	AIC	BIC
GNEWD $(\lambda ,\gamma ,a,b)$	0.0163	1.0701	42749.2	4.0287	0.0891	0.9160	368.81	375.46
NEWD $(\lambda ,\gamma ,a)$	0.1098	0.7196	26.3715	-	0.1350	0.4756	374.52	379.51
WD $(\lambda ,\gamma )$	0.0039	1.3942	-	-	0.1657	0.2347	378.56	381.89
RGNEWD $(\lambda ,a,b)$	0.2497	-	91.8778	1.4109	0.1081	0.7521	374.73	379.72
RNEWD $(\lambda ,a)$	0.3175	-	33.6649	-	0.1271	0.5542	373.05	376.37
GMWD $(\lambda ,\gamma ,a,b)$	19.7618	0.0490	0.0004	$1.5379\times {10}^{10}$	0.1304	0.5209	375.76	382.41
GIGWD $(\lambda ,\gamma ,a,b)$	5.0609	10.4434	1.3711	1.2049	0.1179	0.6493	377.64	384.29
GIWD $(\lambda ,\gamma ,a)$	4.2936	14.7942	1.5307	-	0.1133	0.6986	375.65	380.64
EWD $(\lambda ,\gamma ,a)$	6.3119	0.1458	25387.1	-	0.1242	0.5846	375.29	380.29
M-O WD $(\lambda ,\gamma ,a)$	0.0002	1.8812	0.1801	-	0.1373	0.4546	378.16	383.15
Logistic NHD $(\lambda ,\gamma ,a)$	0.7379	0.2122	5.2217	-	0.1270	0.5554	379.49	384.49
IGLED $(\lambda ,\gamma ,a)$	23.1730	75.0697	1.4495	-	0.1121	0.7112	375.56	380.55
IWD $(\lambda ,\gamma )$	137.63	1.5306	-	-	0.1133	0.6986	373.65	376.98

provides a summary of the numerical results of the parametric and non-parametric tests based on the MLEs of all comparison distributions. At a level of significance of $\alpha =0.05$, it is possible to demonstrate that all distributions fit the liver cancer data based on the p-value related to the K-S test. Since the GNEWD has the lowest value based on the parametric tests AIC and BIC, it is clear from Table 4 that it fits this data better than other distributions. Based on the liver cancer data, Figure 4 displays the log-likelihood function profiles for each parameter. The approximate 95% CIs for $\lambda$, $\gamma$, a, and b are [0, 0.0439], [0, 296,323], [1.6289, 6.4285], and [0.6842, 1.4559], respectively, derived using the asymptotic variance–covariance matrix based on the MLEs for this data. Additionally,

Table 5. Unknown parameter estimates using different approaches and the K-S test with the associated p-value for the liver cancer data.

Methods	$\widehat{\mathit{\lambda }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{a}}$	$\widehat{\mathit{b}}$	K-S Distance	p-Value
MLE	0.0163	1.0701	42749.2	4.0287	0.0891	0.9160
LSE	0.0279	0.9156	$2.1\times {10}^6$	5.4809	0.0887	0.9187
WLSE	0.0188	1.0191	$7.01\times {10}^7$	6.8139	0.0771	0.9745
CVME	0.0254	0.9428	$3.3\times {10}^6$	5.6172	0.0807	0.9614
ADE	0.0227	0.9771	86756.89	4.3104	0.0717	0.9881

provides the values of K-S along with the matching p-value for the liver cancer data, as well as all estimations of the unknown parameters based on the various approaches that were taken into consideration in an earlier section. Table 5 makes it clear that the ADE is the best approach for the liver cancer data when the K-S test is used with the associated p-value.

6.2. Plasma Concentrations of Indomethicin Data

Consider the following data set from [30] and analyzed by [31], which includes 66 observations of plasma concentrations of indomethicin (mcg/mL). The data are 1.50, 0.94, 0.78, 0.48, 0.37, 0.19, 0.12, 0.11, 0.08, 0.07, 0.05, 2.03, 1.63, 0.71, 0.70, 0.64, 0.36, 0.32, 0.20, 0.25, 0.12, 0.08, 2.72, 1.49, 1.16, 0.80, 0.80, 0.39, 0.22, 0.12, 0.11, 0.08, 0.08, 1.85, 1.39, 1.02, 0.89, 0.59, 0.40, 0.16, 0.11, 0.10, 0.07, 0.07, 2.05, 1.04, 0.81, 0.39, 0.30, 0.23, 0.13, 0.11, 0.08, 0.10, 0.06, 2.31, 1.44, 1.03, 0.84, 0.64, 0.42, 0.24, 0.17, 0.13, 0.10, 0.09.

For this data,

Table 6. The ML estimates of unknown parameters, the K-S with the corresponding p-value, the AIC, and the BIC for different models using the plasma concentrations of indomethicin data.

Distribution	$\widehat{\mathit{\lambda }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{a}}$	$\widehat{\mathit{b}}$	K-S Distance	p-Value	AIC	BIC
GNEWD $(\lambda ,\gamma ,a,b)$	1.7033	0.7700	0.0001	3.6783	0.0815	0.7730	45.43	54.18
NEWD $(\lambda ,\gamma ,a)$	1.9787	0.6145	0.1092	-	0.1322	0.1992	56.33	62.90
WD $(\lambda ,\gamma )$	1.6857	0.9546	-	-	0.1348	0.1817	66.51	70.89
RGNEWD $(\lambda ,a,b)$	1.6526	-	0.0012	2.7340	0.1064	0.4433	52.06	58.63
RNEWD $(\lambda ,a)$	2.0325	-	0.1332	-	0.1173	0.3243	55.19	59.57
GMWD $(\lambda ,\gamma ,a,b)$	24.8199	0.0324	0.0251	$2.02768\times {10}^{10}$	0.1327	0.1956	58.66	67.42
GIGWD $(\lambda ,\gamma ,a,b)$	1.1312	0.8082	0.5515	3.3338	0.1348	0.1817	64.32	73.08
GIWD $(\lambda ,\gamma ,a)$	0.5715	0.3212	1.0196	-	0.1148	0.3494	62.92	69.49
EWD $(\lambda ,\gamma ,a)$	8.2427	0.1495	660.96	-	0.1299	0.2147	61.73	68.29
M-O WD $(\lambda ,\gamma ,a)$	0.7083	1.2691	0.1985	-	0.1327	0.1956	65.05	71.62
Logistic NHD $(\lambda ,\gamma ,a)$	0.3594	5.5097	1.0471	-	0.13604	0.1737	68.28	74.85
IGLED $(\lambda ,\gamma ,a)$	0.1696	0.0047	0.9568	-	0.1119	0.3793	62.52	69.08
IWD $(\lambda ,\gamma )$	0.1816	1.0196	-	-	0.1148	0.3494	60.92	65.29

provides a summary of the numerical results of the parametric and non-parametric tests based on the MLEs of all comparison distributions. At a level of significance of $\alpha =0.05$, it is possible to demonstrate that all distributions fit the plasma concentrations of the indomethicin data based on the p-value related to the K-S test. Since the GNEWD has the lowest value based on the parametric tests AIC and BIC, it is clear from Table 6 that it fits this data better than other distributions. Based on the plasma concentrations of the indomethicin data, Figure 5 displays the log-likelihood function profiles for each parameter. The approximate 95% CIs for $\lambda$, $\gamma$, a, and b are [1.2896, 2.1171], [0.5794, 0.9607], [0, 0.0004], and [1.9355, 5.4211], respectively, derived using the asymptotic variance–covariance matrix based on the MLEs for this data. The values of K-S and the corresponding p-value for the plasma concentrations of the indomethicin data are also shown in

Table 7. Unknown parameter estimates using different approaches and the K-S test with the associated P-value for the plasma concentrations of indomethicin data.

Methods	$\widehat{\mathit{\lambda }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{a}}$	$\widehat{\mathit{b}}$	K-S Distance	p-Value
MLE	1.7033	0.7700	0.0001	3.6783	0.0815	0.7730
LSE	1.5353	0.6966	$1.2\times {10}^{-6}$	5.2577	0.0529	0.9925
WLSE	1.6049	0.7416	$1.5\times {10}^{-6}$	5.1512	0.0638	0.9513
CVME	1.5569	0.7097	$8.7\times {10}^{-7}$	5.4223	0.0558	0.9863
ADE	1.6088	0.7343	$1.4\times {10}^{-5}$	4.2803	0.0675	0.9244

, along with all estimations of the unknown parameters based on the different methodologies that were considered in the previous section. When the K-S test is employed with the corresponding p-value, it is evident from Table 7 that the LSE is the optimal method for the plasma concentrations of the indomethicin data.

7. Simulation Study

The performance of various estimating techniques, as detailed in Section 5, is examined for the simulation study using the MSE and the AAB. In order to evaluate how well different approaches work in estimating the unknown parameters of the GNEWD, the following procedure is used:

• Choose different values for the initial values of the parameters of the proposed distribution.
• Generate random samples from the inverse CDF of the GNEWD with size n = (20, 60, 100, 150, 200, 250, 300).
• Calculate the estimates for all the varying methods shown in Section 5.
• Repeat Steps 2 and 3 N = 2000 times.
• Calculate the MSEs and AABs.

All numerical results obtained from the simulation study for the different estimation methods are presented in

Table 8. MSEs and AAB for parameter $\gamma$ under proposed methods with varying sample size n.

n	$(\mathit{\gamma },\mathit{\lambda },\mathit{b},\mathit{a})$		MLE	LSE	WLSE	CVME	ADE
20	$(0.8,0.4,1.5,0.2)$	MSE	0.0357	0.0805	0.1174	0.0578	0.036
		AAB	0.1421	0.1703	0.1832	0.1531	0.1455
	$(0.8,0.4,3.5,0.2)$	MSE	0.0285	0.0629	0.1249	0.0506	0.0381
		AAB	0.1314	0.2059	0.2019	0.1776	0.1532
60	$(0.8,0.4,1.5,0.2)$	MSE	0.0135	0.0255	0.0216	0.0205	0.0168
		AAB	0.0914	0.1232	0.1154	0.1107	0.0994
	$(0.8,0.4,3.5,0.2)$	MSE	0.012	0.0237	0.0175	0.0201	0.0145
		AAB	0.0839	0.1227	0.1031	0.1119	0.0922
100	$(0.8,0.4,1.5,0.2)$	MSE	0.0105	0.0191	0.0146	0.0154	0.0116
		AAB	0.0792	0.1052	0.093	0.0955	0.0815
	$(0.8,0.4,3.5,0.2)$	MSE	0.0089	0.0143	0.0097	0.013	0.0088
		AAB	0.0709	0.093	0.0761	0.0886	0.0717
150	$(0.8,0.4,1.5,0.2)$	MSE	0.0083	0.0154	0.01	0.0128	0.0091
		AAB	0.0686	0.0929	0.0764	0.0856	0.0729
	$(0.8,0.4,3.5,0.2)$	MSE	0.0074	0.0093	0.0062	0.0087	0.0059
		AAB	0.0623	0.0743	0.0607	0.0719	0.0584
200	$(0.8,0.4,1.5,0.2)$	MSE	0.0083	0.0139	0.0087	0.0119	0.008
		AAB	0.0672	0.0888	0.0710	0.0831	0.0678
	$(0.8,0.4,3.5,0.2)$	MSE	0.0062	0.0067	0.0043	0.0065	0.0043
		AAB	0.0569	0.0628	0.0505	0.0618	0.0499
250	$(0.8,0.4,1.5,0.2)$	MSE	0.0072	0.0124	0.2579	0.0112	0.0063
		AAB	0.0603	0.0821	0.0605	0.0784	0.0602
	$(0.8,0.4,3.5,0.2)$	MSE	0.0054	0.0060	0.0036	0.0053	0.0040
		AAB	0.0536	0.0584	0.0468	0.0555	0.0472
300	$(0.8,0.4,1.5,0.2)$	MSE	0.0073	0.0111	0.0057	0.0099	0.0058
		AAB	0.0601	0.0789	0.0584	0.0755	0.0580
	$(0.8,0.4,3.5,0.2)$	MSE	0.0048	0.0045	0.0028	0.0045	0.0032
		AAB	0.0497	0.0508	0.0412	0.0505	0.0415

,

Table 9. MSEs and AAB for parameter $\lambda$ under proposed methods with varying sample size n.

n	$(\mathit{\gamma },\mathit{\lambda },\mathit{b},\mathit{a})$		MLE	LSE	WLSE	CVME	ADE
20	$(0.8,0.4,1.5,0.2)$	MSE	0.0361	0.0481	0.055	0.0267	0.0324
		AAB	0.1351	0.1355	0.1493	0.1106	0.1194
	$(0.8,0.4,3.5,0.2)$	MSE	0.0247	0.0693	0.0679	0.0464	0.0396
		AAB	0.1131	0.1809	0.1763	0.1512	0.1378
60	$(0.8,0.4,1.5,0.2)$	MSE	0.0244	0.0409	0.0372	0.0242	0.0251
		AAB	0.1026	0.1211	0.1215	0.0987	0.1005
	$(0.8,0.4,3.5,0.2)$	MSE	0.0109	0.0273	0.0169	0.0211	0.0145
		AAB	0.0727	0.1078	0.0898	0.097	0.0826
100	$(0.8,0.4,1.5,0.2)$	MSE	0.018	0.0385	0.0267	0.0214	0.019
		AAB	0.0882	0.1125	0.1018	0.0912	0.0857
	$(0.8,0.4,3.5,0.2)$	MSE	0.0081	0.015	0.0079	0.0122	0.0081
		AAB	0.0600	0.0782	0.0631	0.0729	0.0614
150	$(0.8,0.4,1.5,0.2)$	MSE	0.0155	0.0343	0.0176	0.0205	0.0147
		AAB	0.0761	0.1054	0.0824	0.088	0.0756
	$(0.8,0.4,3.5,0.2)$	MSE	0.0063	0.0094	0.0051	0.0081	0.55
		AAB	0.0537	0.0625	0.0507	0.0597	0.0492
200	$(0.8,0.4,1.5,0.2)$	MSE	0.016	0.0337	0.0146	0.0218	0.0136
		AAB	0.0747	0.1019	0.0747	0.0871	0.0714
	$(0.8,0.4,3.5,0.2)$	MSE	0.0048	0.006	0.003	0.0059	0.0035
		AAB	0.0476	0.0504	0.0412	0.0495	0.0415
250	$(0.8,0.4,1.5,0.2)$	MSE	0.0147	0.0317	0.0087	0.0237	0.0094
		AAB	0.0669	0.0954	0.0613	0.0860	0.0622
	$(0.8,0.4,3.5,0.2)$	MSE	0.0043	0.0058	0.0024	0.0049	0.0030
		AAB	0.0444	0.0466	0.0375	0.0439	0.0386
300	$(0.8,0.4,1.5,0.2)$	MSE	0.0163	0.0259	0.0075	0.0188	0.0082
		AAB	0.0680	0.0918	0.0585	0.0820	0.0594
	$(0.8,0.4,3.5,0.2)$	MSE	0.0039	0.0041	0.0019	0.0041	0.0030
		AAB	0.0411	0.0403	0.0334	0.0404	0.0345

,

Table 10. MSEs and AAB for parameter b under proposed methods with varying sample size n.

n	$(\mathit{\gamma },\mathit{\lambda },\mathit{b},\mathit{a})$		MLE	LSE	WLSE	CVME	ADE
20	$(0.8,0.4,1.5,0.2)$	MSE	0.9414	0.7407	0.6926	0.8787	0.5949
		AAB	0.8356	0.6813	0.6555	0.7473	0.5953
	$(0.8,0.4,3.5,0.2)$	MSE	1.2476	2.2261	2.1038	1.9285	1.3228
		AAB	0.8986	1.163	1.1444	1.0923	0.9141
60	$(0.8,0.4,1.5,0.2)$	MSE	0.5766	0.6468	0.4778	0.7173	0.3704
		AAB	0.5967	0.6453	0.5452	0.6682	0.4679
	$(0.8,0.4,3.5,0.2)$	MSE	0.8035	1.4601	1.1329	1.3874	0.9445
		AAB	0.7293	0.975	0.8483	0.9559	0.7774
100	$(0.8,0.4,1.5,0.2)$	MSE	0.4457	0.5812	0.3948	0.6397	0.289
		AAB	0.5157	0.6074	0.4958	0.6305	0.418
	$(0.8,0.4,3.5,0.2)$	MSE	0.682	1.1735	0.7642	1.1746	0.7257
		AAB	0.6617	0.8764	0.7048	0.8784	0.6773
150	$(0.8,0.4,1.5,0.2)$	MSE	0.3807	0.5273	0.3308	0.5468	0.2521
		AAB	0.4662	0.576	0.4463	0.581	0.3926
	$(0.8,0.4,3.5,0.2)$	MSE	0.5742	0.9648	0.5999	0.9791	0.55
		AAB	0.5884	0.7868	0.6126	0.7879	0.5798
200	$(0.8,0.4,1.5,0.2)$	MSE	0.3187	0.4978	0.3027	0.5225	0.2321
		AAB	0.4248	0.5545	0.4209	0.5614	0.3758
	$(0.8,0.4,3.5,0.2)$	MSE	0.5063	0.8208	0.4900	0.8965	0.4902
		AAB	0.5600	0.7350	0.5643	0.7631	0.5523
250	$(0.8,0.4,1.5,0.2)$	MSE	0.2545	0.4694	0.2579	0.4884	0.2083
		AAB	0.3805	0.5427	0.3929	0.5493	0.3608
	$(0.8,0.4,3.5,0.2)$	MSE	0.4222	0.7409	0.4245	0.7082	0.4339
		AAB	0.5031	0.6881	0.5248	0.6766	0.5164
300	$(0.8,0.4,1.5,0.2)$	MSE	0.2419	0.4412	0.2352	0.4563	0.2006
		AAB	0.3648	0.5323	0.3740	0.5361	0.3480
	$(0.8,0.4,3.5,0.2)$	MSE	0.3625	0.6478	0.3650	0.6650	0.3847
		AAB	0.4614	0.6343	0.4776	0.6409	0.4717

and

Table 11. MSEs and AAB for parameter a under proposed methods with varying sample size n.

n	$(\mathit{\gamma },\mathit{\lambda },\mathit{b},\mathit{a})$		MLE	LSE	WLSE	CVME	ADE
20	$(0.8,0.4,1.5,0.2)$	MSE	0.134	0.0754	0.092	0.054	0.0711
		AAB	0.2545	0.1824	0.2003	0.1605	0.1768
	$(0.8,0.4,3.5,0.2)$	MSE	0.0885	0.1579	0.1526	0.1321	0.1019
		AAB	0.1954	0.2749	0.2675	0.2535	0.2159
60	$(0.8,0.4,1.5,0.2)$	MSE	0.0853	0.0989	0.0906	0.066	0.0672
		AAB	0.1948	0.2084	0.2019	0.1768	0.1729
	$(0.8,0.4,3.5,0.2)$	MSE	0.0423	0.097	0.0655	0.0839	0.0622
		AAB	0.1349	0.2065	0.1758	0.1952	0.1663
100	$(0.8,0.4,1.5,0.2)$	MSE	0.064	0.097	0.0768	0.0644	0.0625
		AAB	0.1697	0.2032	0.1845	0.1754	0.1632
	$(0.8,0.4,3.5,0.2)$	MSE	0.0322	0.0621	0.0356	0.0551	0.0411
		AAB	0.1162	0.163	0.1293	0.1555	0.1319
150	$(0.8,0.4,1.5,0.2)$	MSE	0.055	0.0922	0.0573	0.0656	0.0522
		AAB	0.1538	0.1982	0.1597	0.1741	0.1518
	$(0.8,0.4,3.5,0.2)$	MSE	0.0256	0.0421	0.0225	0.0398	0.0257
		AAB	0.1006	0.1309	0.1027	0.1284	0.1031
200	$(0.8,0.4,1.5,0.2)$	MSE	0.0547	0.0918	0.0505	0.0694	0.0492
		AAB	0.1521	0.1951	0.1509	0.1764	0.1481
	$(0.8,0.4,3.5,0.2)$	MSE	0.0202	0.0301	0.0142	0.0316	0.0184
		AAB	0.0904	0.1128	0.0863	0.1158	0.0899
250	$(0.8,0.4,1.5,0.2)$	MSE	0.0462	0.0868	0.0336	0.0714	0.0404
		AAB	0.1337	0.1845	0.1275	0.1721	0.1357
	$(0.8,0.4,3.5,0.2)$	MSE	0.0183	0.0284	0.0119	0.0247	0.0173
		AAB	0.0828	0.1057	0.0790	0.1009	0.0850
300	$(0.8,0.4,1.5,0.2)$	MSE	0.0521	0.0790	0.0310	0.0636	0.0368
		AAB	0.1365	0.1850	0.1245	0.1708	0.1310
	$(0.8,0.4,3.5,0.2)$	MSE	0.0169	0.0203	0.0086	0.0210	0.0156
		AAB	0.0777	0.0906	0.0693	0.0907	0.0760

. Upon examination of these tables, several notable patterns and findings emerge. Specifically, Table 8, Table 9, Table 10 and Table 11 illustrate the following:

• The MSEs and the AABs for all methods of estimation decrease by increasing the sample size n.
• The MLE and A-DE are superior to other estimating techniques based on MSE and AAB.

8. Conclusions

In this paper, a new extension of the Weibull distribution was proposed. It extended the new extended Weibull distribution by introducing an additional shape parameter, allowing it to capture more complex hazard rate patterns, including upside-down bathtub, and modified upside-down bathtub behaviors commonly observed in medical fields. The quantile of the GNEWD was calculated in closed form while the formula of moments could not be given in closed form. The estimation of the unknown parameters for the GNEWD using several estimating techniques, such as MLE, LSE, WLSE, CVME, and ADE, was introduced in this study. The GNEWD was applied to two real data sets in order to demonstrate how broadly it can be used in medical fields. The use of the parametric and non-parametric tests gives the GNEWD a competitive advantage when modeling lifetime data. Additionally, the simulation study was presented to compare the proposed estimation techniques employing MSEs and AABs. For future work, additional properties for the GNEWD can be studied including reliability analysis and entropy. Further, we will try to extend the proposed methods using different censoring schemes. Furthermore, the unknown parameters can be estimated by Bayesian estimation under complete and censoring data.