A New Modification of Modified Weibull Distribution for Modeling Engineering Data

Abstract

This study investigates a novel modification for a modified Weibull distribution called the new modification of modified Weibull distribution. Some distributions related to the NMMWD are given. Some characterization of the NMMWD, including quantiles, Bowley skewness, Moors kurtosis, and moments, are given in closed forms. The hazard rate function of the new distribution takes two distinct shapes, the increasing and the bathtub shapes. Different estimating approaches are investigated utilizing complete data. Three real data sets from the engineering field are analyzed to demonstrate the suggested model’s flexibility in practice. In comparison to certain well-known distributions, the proposed distribution fits the tested data better according to both parametric and non-parametric tests. A simulation study is presented to compare the various estimating approaches using mean square error and average absolute bias.

1. Introduction

Highly elastic statistical distributions that are consistent with data from many real-life problems have become essential for statisticians. Therefore, many statisticians have developed a number of generalizations, modifications, and distributions that are highly elastic and fit well with various real-life data. Consequently, there are many statistical distributions, making it easier for researchers to choose the appropriate distribution to model various data across a wide range of fields.

The Weibull distribution [1] is widely employed in various sectors, including the medical sector, engineering, and industry. It has the limitation of being unable to describe the complex lifetime of some systems due to a lack of bathtub-shaped or unimodal-shaped hazard rate functions. Therefore, several authors have proposed some modifications to the Weibull distribution to overcome this problem.

Many authors have presented extensions to distributions including [2,3,4,5] (for more details, see [6]), with the same cumulative distribution function as [7], which is in the form $F(x;\lambda )=1-e^{-\lambda V\left(x\right)}$, where $V\left(x\right)$ is monotone increasing function. Ref. [6] suggested a new distribution with $V\left(x\right)=\lambda x^{\gamma }(e^{\theta x^{\alpha }}-1)$. Recently, based on the idea given by [7], several authors, including [8,9,10,11,12], proposed and investigated other generalizations and distributions with high flexibility and different properties.

The main objective of this study is to present and investigate a new modification of the Weibull distribution called the new modification of modified Weibull distribution (NMMWD) based on the idea presented in [7], where $V\left(x\right)=\lambda x^{\gamma }{e}^{{\left(\theta x\right)}^{\alpha }}$. This distribution is also a generalization of the modified Weibull distribution.

The aims listed below provide compelling reasons for the significance of the suggested distribution.

• To present and study the various properties and potential uses of a new modification of the modified Weibull distribution.
• To calculate certain characteristics including, quantiles, Bowley skewness, Moors kurtosis, and moments in explicit forms.
• To demonstrate how the hazard rate function for the new distribution might take increasing and bathtub shapes.
• To study several estimation techniques for the unknown parameters of the new distribution using complete data.
• To demonstrate the importance of the new distribution in the field of engineering using different estimation techniques based on complete data.
• To verify the effectiveness of different estimation techniques used to estimate the distribution parameters using a simulation study based on complete data.

This research is presented in seven sections, each section focusing on one of the characteristics of the new distribution. Section 2 presents the basic structure of the NMMWD in detail. Some statistical properties are calculated in Section 3. Section 4 investigates the behavior of the hazard rate function for NMMWD. The different estimation techniques of the parameters of NMMWD are presented in Section 5. In Section 6, the applications to three real data sets are presented to show the importance of the NMMWD in engineering field. A simulation study is performed in Section 7 to evaluate and compare the performance of various estimation techniques. Finally, the main findings of the paper are summarized in Section 8.

2. The New Modification of the Modified Weibull Distribution

The cumulative distribution function (CDF) of NMMWD with parameter vector $\mathsf{\Phi }=(\lambda ,\gamma ,\theta ,\alpha )$ can be given by multiplying the cumulative hazard rate function for Weibull distribution by the component $e^{{\left(\theta x\right)}^{\alpha }}$

$$\begin{array}{c}\hfill F(x;\mathsf{\Phi })=1-e^{-\lambda x^{\gamma }{e}^{{\left(\theta x\right)}^{\alpha }}},x>0,\end{array}$$

(1)

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

Thus the probability density function (PDF), the survival function (SF), and the hazard rate function (HRF) of NMMWD are respectively given by

$$\begin{array}{c}\hfill f(x;\mathsf{\Phi })=\lambda x^{\gamma -1}\left(\gamma +\alpha {\left(\theta x\right)}^{\alpha }\right)e^{{\left(\theta x\right)}^{\alpha }-\lambda x^{\gamma }{e}^{{\left(\theta x\right)}^{\alpha }}},x>0,\end{array}$$

(2)

$$\begin{array}{c}\hfill S(x;\mathsf{\Phi })=e^{-\lambda x^{\gamma }{e}^{{\left(\theta x\right)}^{\alpha }}},x>0,\end{array}$$

(3)

and

$$\begin{array}{c}\hfill h(x;\mathsf{\Phi })=\lambda x^{\gamma -1}{e}^{{\left(\theta x\right)}^{\alpha }}\left(\gamma +\alpha {\left(\theta x\right)}^{\alpha }\right),x>0.\end{array}$$

(4)

Some possible shapes for the PDF of NMMWD are shown in Figure 1. Based on different values of parameters, the PDF of NMMWD can be decreasing, modified bathtub and unimodal. Figure 2a shows the shapes for the CDF of NMMWD and Figure 2b shows the shapes for the survival function of the NMMWD with different values of NMMWD parameters.

Figure 1. The PDFs of NMMWD with different values of parameters.

Figure 2. (a) The CDFs of NMMWD with different paremeter values, (b) the survival functions of NMMWD with different parameter values.

Models Related to NMMWD

Considering the CDF of NMMWD given by (1), some well-known distributions can be obtained as sub-models, as shown in Table 1. Also, we can show that the GIBD (New) can be related to NMMWD as follows:

• GIBD

The SF of GBID can be given as

$$\begin{array}{c}\hfill S(x;\lambda ,\gamma ,c,\alpha ,\eta )=e^{-\lambda x^{\gamma }{(1-c{x}^{\alpha })}^{\eta }},\lambda ,\gamma ,\alpha >0,\eta <0,0<x^{\alpha }<{\displaystyle \frac{1}{c}}.\end{array}$$

(5)

Let $c={\displaystyle \frac{1}{n}}$, $n>0$ and $\eta =-n{\theta }^{\alpha }$, then for $n\to \infty$, it is clear that ${(1-{\displaystyle \frac{x^{\alpha }}{n}})}^{-n{\theta }^{\alpha }}\to e^{{\left(\theta x\right)}^{\alpha }}$. So, the SF of NMMWD can be obtained. The IBD given in [13], which is a special case of GIBD, can be related to MWD, which is a special case of the NMMWD, as shown in [2].

Table 1. The sub-models from NMMWD.

Distribution	$\mathit{\lambda }$	$\mathit{\gamma }$	$\mathit{\theta }$	$\mathit{\alpha }$	CDF	Reference
MWD $(\lambda ,\gamma ,\theta )$	-	-	-	1	$1-e^{-\lambda x^{\gamma }{e}^{\theta x}}$	[2]
WD $(\lambda ,\gamma )$	-	-	0	-	$1-e^{-\lambda x^{\gamma }}$	[1]
ED $\left(\lambda \right)$	-	1	-	0	$1-e^{-\lambda x}$	[14]
RD $\left(c\right)$	${\displaystyle \frac{c}{2}}$	2	-	0	$1-e^{-{\displaystyle \frac{c}{2}}{x}^2}$	[14]
E-VD $\left(\theta \right)$	1	0	-	1	$1-e^{-e^{\theta x}}$
RNMMWD $(\lambda ,\theta ,\alpha )$	-	${\displaystyle \frac{1}{2}}$	-	-	$1-e^{-\lambda \sqrt{x}{e}^{{\left(\theta x\right)}^{\alpha }}}$	New
RMWD $(\theta ,\beta )$	${\displaystyle \frac{\theta }{\beta }}{e}^{-\beta }$	${\displaystyle \frac{1}{2}}$	-	1	$1-e^{-{\displaystyle \frac{\theta }{\beta }}\sqrt{x}{e}^{\theta x-\beta }}$	[9]

Table 1. The sub-models from NMMWD.

Distribution	$\mathit{\lambda }$	$\mathit{\gamma }$	$\mathit{\theta }$	$\mathit{\alpha }$	CDF	Reference
MWD $(\lambda ,\gamma ,\theta )$	-	-	-	1	$1-e^{-\lambda x^{\gamma }{e}^{\theta x}}$	[2]
WD $(\lambda ,\gamma )$	-	-	0	-	$1-e^{-\lambda x^{\gamma }}$	[1]
ED $\left(\lambda \right)$	-	1	-	0	$1-e^{-\lambda x}$	[14]
RD $\left(c\right)$	${\displaystyle \frac{c}{2}}$	2	-	0	$1-e^{-{\displaystyle \frac{c}{2}}{x}^2}$	[14]
E-VD $\left(\theta \right)$	1	0	-	1	$1-e^{-e^{\theta x}}$
RNMMWD $(\lambda ,\theta ,\alpha )$	-	${\displaystyle \frac{1}{2}}$	-	-	$1-e^{-\lambda \sqrt{x}{e}^{{\left(\theta x\right)}^{\alpha }}}$	New
RMWD $(\theta ,\beta )$	${\displaystyle \frac{\theta }{\beta }}{e}^{-\beta }$	${\displaystyle \frac{1}{2}}$	-	1	$1-e^{-{\displaystyle \frac{\theta }{\beta }}\sqrt{x}{e}^{\theta x-\beta }}$	[9]

3. Statistical Properties

This section investigates some statistical properties of NMMWD, including quantiles, Bowley skewness, Moors kurtosis, and moments. The explicit form of the quantile of the NMMWD is derived, which is important for generating random sample data from the NMMWD for simulation studies. It is also important to compute Bowley skewness and Moors kurtosis measures, which are less sensitive to outliers.

3.1. Quantiles

Lemma 1.

Let $\mathbf{x}$ be a random variable with CDF of NMMWD defined by Equation (1). The quantile of NMMWD can be calculated as

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

(6)

where $W(.)$ is the Lambert W function (for more details, see [15]).

Proof. 

Upon using the CDF of NMMWD, set $q=F(x;\mathsf{\Phi })$. Then,

$$\begin{array}{c}\hfill q=1-e^{-\lambda x_q^{\gamma }{e}^{{\left(\theta x_q\right)}^{\alpha }}}.\end{array}$$

(7)

By performing simple algebraic operations on (7), we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{-1}{\lambda }}log(1-q)=x_q^{\gamma }{e}^{{\left(\theta x_q\right)}^{\alpha }},\end{array}$$

(8)

By applying the definition of the Lambert W function on Equation (8), Equation (6) is obtained; see Appendix A. □

3.2. Bowley Skewness and Moors Kurtosis Measures

Assume $x_q$ is the quantile of NMMWD as defined by (8). As a result, the Bowley skewness (BSkew) and Moors kurtosis (Mkur) of NMMWD can be calculated, respectively, as

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

(9)

and

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

(10)

3.3. Moments

Lemma 2.

Let $\mathbf{x}$ be a random variable with PDF of NMMWD defined by Equation (2). Then the $r^{th}$ moment of NMMWD cab be given as

$$\begin{array}{c}\hfill {\mu }_r^{\prime }\left(\mathsf{\Phi }\right)=\sum _{i=0}^{\infty }d\left(i\right){\left({\displaystyle \frac{\gamma }{\alpha {\theta }^{\alpha }}}\right)}^{r/\alpha }\left(\Gamma \left({\displaystyle \frac{r+i\alpha +\gamma }{\gamma }}\right)-\Gamma \left({\displaystyle \frac{r+i\alpha +\gamma }{\gamma }},e^{-{\displaystyle \frac{\gamma }{\alpha }}}{\left({\displaystyle \frac{\gamma {\theta }^{-\alpha }}{\alpha }}\right)}^{\gamma /\alpha }\lambda \right)\right),\end{array}$$

(11)

where ${\displaystyle \frac{r}{\alpha }}$ is considered as natural number, $\Gamma (.)$ is an ordinary gamma function, $\Gamma (.,.)$ is an incomplete gamma function, $d_0\left(b\right)=a_0^b,d_i\left(b\right)={\displaystyle \frac{1}{i{d}_0}}{\sum }_{l=1}^i\left(l\left(b\right)-i+l\right)a_l{d}_{i-l}$ and $a_i={\displaystyle \frac{{(-1)}^i{(i+1)}^i{\left(\frac{\alpha {\theta }^{\alpha }}{\gamma }\right)}^{i+1}{\left(\frac{1}{\lambda }\right)}^{\frac{\alpha (i+1)}{\gamma }}}{(i+1)!}}$ (for more details, see [15]).

Proof. 

Upon using (2), the $r^{th}$ moments of the NMMWD can be written as

$$\begin{array}{c}\hfill {\mu }_r^{\prime }\left(\mathsf{\Phi }\right)={\int }_0^{\infty }{x}^{r}f(x;\mathsf{\Phi })dx={\int }_0^{\infty }{x}^r\lambda x^{\gamma -1}\left(\gamma +\alpha {\left(\theta x\right)}^{\alpha }\right)e^{{\left(\theta x\right)}^{\alpha }-\lambda x^{\gamma }{e}^{{\left(\theta x\right)}^{\alpha }}}dx.\end{array}$$

(12)

Using the substitution $y=\lambda x^{\gamma }{e}^{{\left(\theta x\right)}^{\alpha }}$, reveals that $x={\left({\displaystyle \frac{\gamma W\left(\frac{\alpha {\theta }^{\alpha }{\left({\lambda }^{-1/\gamma }{y}^{1/\gamma }\right)}^{\alpha }}{\gamma }\right)}{{\theta }^{\alpha }\alpha }}\right)}^{1/\alpha }$, where $W(.)$ is the Lambert W function (for more details, see [15,16]). Using $SeriesCoefficient\left[W\right(z),\{z,0,n\left\}\right]$ in Wolfram Mathematica 11, it is easy to show that $W\left({\displaystyle \frac{\alpha {\theta }^{\alpha }{\left({\lambda }^{-1/\gamma }{y}^{1/\gamma }\right)}^{\alpha }}{\gamma }}\right)={\sum }_{i=0}^{\infty }{a}_i{y}^{(i+1){\displaystyle \frac{\alpha }{\gamma }}}$. It is evident by ratio test for convergent that the convergent condition of ${\sum }_{i=0}^{\infty }{a}_i{y}^{(i+1){\displaystyle \frac{\alpha }{\gamma }}}$ can be given as $y<\lambda {\left({\displaystyle \frac{\gamma }{e\alpha {\theta }^{\alpha }}}\right)}^{\gamma /\alpha }$, see [17]. Thus, the Formula (12) reduces to

$$\begin{array}{c}\hfill {\mu }_r^{\prime }\left(\mathsf{\Phi }\right)={\int }_0^{\lambda {\left({\displaystyle \frac{\gamma }{e\alpha {\theta }^{\alpha }}}\right)}^{\gamma /\alpha }}{\left({\displaystyle \frac{\gamma W\left(\frac{\alpha {\theta }^{\alpha }{\left({\lambda }^{-1/\gamma }{y}^{1/\gamma }\right)}^{\alpha }}{\gamma }\right)}{{\theta }^{\alpha }\alpha }}\right)}^{r/\alpha }{e}^{-y}dy.\end{array}$$

(13)

As given in [16], page 17, it is clear that ${\left({\sum }_{i=0}^{\infty }{a}_i{y}^{(i+1){\displaystyle \frac{\alpha }{\gamma }}}\right)}^{{\displaystyle \frac{r}{\alpha }}}={\sum }_{i=0}^{\infty }{d}_i\left({\displaystyle \frac{r}{\alpha }}\right)y^{{\displaystyle \frac{i\alpha +r}{\gamma }}}$. Hence, (13) can be written as

$$\begin{array}{c}\hfill {\mu }_r^{\prime }\left(\mathsf{\Phi }\right)={\int }_0^{\lambda {\left({\displaystyle \frac{\gamma }{e\alpha {\theta }^{\alpha }}}\right)}^{\gamma /\alpha }}{\left({\displaystyle \frac{\gamma }{\alpha {\theta }^{\alpha }}}\right)}^{r/\alpha }\sum _{i=0}^{\infty }{d}_i\left({\displaystyle \frac{r}{\alpha }}\right)y^{{\displaystyle \frac{i\alpha +r}{\gamma }}}{e}^{-y}dy.\end{array}$$

(14)

By calculating the integral (14), the formula in (11) can be easily obtained. □

4. Behavior of the HRF Shapes

This section investigates the behavior of the HRF of NMMWD and displays the NMMWD HRF shapes, such as increasing and bathtub shapes. Considering (4), it is easy to obtain the first derivative of $h(x;\mathsf{\Phi })$ as

$$\begin{array}{c}\hfill h^{\prime }(x;\mathsf{\Phi })=\lambda x^{\gamma -2}{e}^{{\left(\theta x\right)}^{\alpha }}\left({\alpha }^2{\theta }^{2\alpha }{x}^{2\alpha }+\alpha {\theta }^{\alpha }(\alpha +2\gamma -1)x^{\alpha }+(\gamma -1)\gamma \right),x>0.\end{array}$$

(15)

Setting $\alpha =1$ in (15) yields

$$\begin{array}{c}\hfill h^{\prime }(x;\lambda ,\gamma ,\theta )=\lambda x^{\gamma -2}{e}^{\theta x}\left({\theta }^2{x}^2+2\theta \gamma x+\gamma (\gamma -1)\right),x>0\end{array}$$

which is the same formula in [2] for MWD.

Lemma 3.

The HRF of NMMWD can take increasing and bathtub shapes based on the following conditions:

1. 

h(x;Φ) has increasing shaped from zero to infinity if $\gamma >1$.

2. 

h(x;Φ) has increasing shaped from λ to infinity if $\gamma =1$.

3. 

h(x;Φ) has bathtub shaped if $0<\gamma <1$.

Proof. 

For $\lambda >0$, $\theta >0$, and $\alpha >0$, the following results are satisfied.

• It is easy to prove that ${\displaystyle \underset{x\to 0^{+}}{lim}h(x;\mathsf{\Phi })\to 0}$ by L’Hospital’s rule and calculate that ${\displaystyle \underset{x\to \infty }{lim}h(x;\mathsf{\Phi })\to \infty }$ directly for $\gamma >1$. Also, it is clear that $h^{\prime }(x;\mathsf{\Phi })>0$ when $x>0$ and $\gamma >1$. So, $h(x;\mathsf{\Phi })$ has an increasing shape that goes from zero to infinity when $\gamma >1$.
• Upon taking $\gamma =1$, it is easy to prove that ${\displaystyle \underset{x\to 0^{+}}{lim}h(x;\mathsf{\Phi })\to \lambda }$ by L’Hospital’s rule and calculate that ${\displaystyle \underset{x\to \infty }{lim}h(x;\mathsf{\Phi })\to \infty }$ directly. Also, it is clear that $h^{\prime }(x;\mathsf{\Phi })>0$ when $x>0$ and $\gamma =1$. So, $h(x;\mathsf{\Phi })$ has increasing shaped that goes from $\lambda$ to infinity when $\gamma =1$.
• It is evident that ${\displaystyle \underset{x\to 0^{+}}{lim}h(x;\mathsf{\Phi })\to \infty }$ can be calculated by L’Hospital’s rule and ${\displaystyle \underset{x\to \infty }{lim}h(x;\mathsf{\Phi })\to \infty }$ can be calculated directly for $0<\gamma <1$. Also, based on the formula ${\alpha }^2{\theta }^{2\alpha }{x}^{2\alpha }+\alpha {\theta }^{\alpha }(\alpha +2\gamma -1)x^{\alpha }+(\gamma -1)\gamma$, it is clear that $h^{\prime }(x;\mathsf{\Phi })$ has two roots, given as $x_1={\displaystyle \frac{\sqrt{{\alpha }^2+4\alpha \gamma -2\alpha +1}-(\alpha +2\gamma -1)}{2\alpha {\theta }^{\alpha }}}$ and $x_2={\displaystyle \frac{-\sqrt{{\alpha }^2+4\alpha \gamma -2\alpha +1}-(\alpha +2\gamma -1)}{2\alpha {\theta }^{\alpha }}}$. Since $\sqrt{{\alpha }^2+4\alpha \gamma -2\alpha +1}>(\alpha +2\gamma -1)$ for $0<\gamma <1$, so, $h^{\prime }(x;\mathsf{\Phi })$ has a unique root for $x>0$, which is $x_1$. Then, $h^{\prime }(x;\mathsf{\Phi })<0$ when $x<x_1$ and $h^{\prime }(x;\mathsf{\Phi })>0$ when $x>x_1$. Hence, the hazard rate has a bathtub shape.

□

We now explicitly summarize the limiting behavior of the hazard rate function as $x\to 0^{+}$ for the three cases $\gamma >1$, $\gamma =1$, and $0<\gamma <1$ as followss:

$$\begin{array}{c}\hfill \underset{x\to 0^{+}}{lim}h(x;\mathsf{\Phi })=\left\{\begin{array}{cc}0,\hfill & \gamma >1,\mathrm{increasing}\mathrm{HRF}\hfill \\ \lambda ,\hfill & \gamma =1,\mathrm{increasing}\mathrm{HRF}\hfill \\ \infty ,\hfill & 0<\gamma <1,\mathrm{bathtub}\mathrm{HRF}.\hfill \end{array}\right.\end{array}$$

Using different parameter values, as described in Lemma 3, the behavior of HRF of the NMMWD can have increasing and bathtub shapes, as illustrated in Figure 3.

Figure 3. The HRFs of NMMWD with different values of parameters.

5. Different Methods of Estimation

This section discusses five distinct estimation strategies applicable to NMMWD: maximum likelihood estimation (MLE), least square estimation (LSE), weighted least square estimation (WLSE), Cram–von Mises estimation (CVME), and Anderson–Darling estimation (ADE). The new model’s unknown parameters are estimated using these techniques.

5.1. Maximum Likelihood Estimation

Assume $x_1,\dots ,x_n$ is an independent random sample of size n from NMMWD. Then the log-likelihood function can be obtained from Equation (2) as follows:

$$\begin{array}{c}\hfill \ell \left(\mathsf{\Phi }\right)=nlog\left(\lambda \right)-\sum _{i=1}^n\lambda x_i^{\gamma }{e}^{{\left(\theta x_i\right)}^{\alpha }}+\sum _{i=1}^{n}log\left(\gamma +\alpha {\left(\theta x_i\right)}^{\alpha }\right)+\sum _{i=1}^n{\left(\theta x_i\right)}^{\alpha }+(\gamma -1)\sum _{i=1}^{n}log\left(x_i\right).\end{array}$$

(16)

To obtain the MLE of $\mathsf{\Phi }$, the following first partial derivative of (16) with regard to $\lambda$, $\gamma$, $\theta$ and $\alpha$ are calculated.

$$\begin{array}{c}\hfill {\ell }_{\lambda }\left(\mathsf{\Phi }\right)={\displaystyle \frac{n}{\lambda }}-\sum _{i=1}^n{x}_i^{\gamma }{e}^{{\left(\theta x_i\right)}^{\alpha }},\end{array}$$

(17)

$$\begin{array}{c}\hfill {\ell }_{\gamma }\left(\mathsf{\Phi }\right)=-\sum _{i=1}^n\lambda x_i^{\gamma }log\left(x_i\right)e^{{\left(\theta x_i\right)}^{\alpha }}+\sum _{i=1}^n{\displaystyle \frac{1}{\gamma +\alpha {\left(\theta x_i\right)}^{\alpha }}}+\sum _{i=1}^{n}log\left(x_i\right),\end{array}$$

(18)

$$\begin{array}{c}\hfill {\ell }_{\theta }\left(\mathsf{\Phi }\right)=\sum _{i=1}^n{\displaystyle \frac{{\alpha }^2{x}_i{\left(\theta x_i\right)}^{\alpha -1}}{\gamma +\alpha {\left(\theta x_i\right)}^{\alpha }}}-\sum _{i=1}^n\alpha \lambda x_i^{\gamma +1}{e}^{{\left(\theta x_i\right)}^{\alpha }}{\left(\theta x_i\right)}^{\alpha -1}+\sum _{i=1}^n\alpha x_i{\left(\theta x_i\right)}^{\alpha -1},\end{array}$$

(19)

and

$$\begin{array}{c}\hfill {\ell }_{\alpha }\left(\mathsf{\Phi }\right)=\sum _{i=1}^n{\displaystyle \frac{{\left(\theta x_i\right)}^{\alpha }+\alpha {\left(\theta x_i\right)}^{\alpha }log\left(\theta x_i\right)}{\gamma +\alpha {\left(\theta x_i\right)}^{\alpha }}}-\sum _{i=1}^n\lambda x_i^{\gamma }{e}^{{\left(\theta x_i\right)}^{\alpha }}{\left(\theta x_i\right)}^{\alpha }log\left(\theta x_i\right)+\sum _{i=1}^n{\left(\theta x_i\right)}^{\alpha }log\left(\theta x_i\right).\end{array}$$

(20)

5.2. The Parameters $\gamma$, $\theta$, and $\alpha$ Are Known

Setting ${\ell }_{\lambda }\left(\mathsf{\Phi }\right)$ to zero, so (17) reduced to

$$\begin{array}{c}\hfill \widehat{\lambda }={\displaystyle \frac{n}{{\sum }_{i=1}^n{x}_i^{\widehat{\gamma }}{e}^{{\left(\widehat{\theta }{x}_i\right)}^{\widehat{\alpha }}}}}.\end{array}$$

(21)

It is clear that the MLE of $\lambda$ exists and unique which can be calculated directly from (21).

5.3. The Parameters $\lambda$, $\gamma$, $\theta$, and $\alpha$ Are Unknown

Setting the expressions ${\ell }_{\lambda }\left(\mathsf{\Phi }\right)$, ${\ell }_{\gamma }\left(\mathsf{\Phi }\right)$, ${\ell }_{\theta }\left(\mathsf{\Phi }\right)$, and ${\ell }_{\alpha }\left(\mathsf{\Phi }\right)$ to zero and solving them yields the MLE $\widehat{\mathsf{\Phi }}$ of $\mathsf{\Phi }$. Since nonlinear equations are not analytically solvable, a numerical method like the Newton–Raphson algorithm can be used by employing FindRoot in software Wolfram Mathematica v.11.

5.4. Fisher Information Matrix

The asymptotic variance–covariance matrix can be calculated as

$$\begin{array}{c}\hfill \begin{array}{cc}& \left(\begin{array}{cccc}Var\left(\widehat{\lambda }\right)& Cov(\widehat{\lambda },\widehat{\gamma })& Cov(\widehat{\lambda },\widehat{\theta })& Cov(\widehat{\lambda },\widehat{\alpha })\\ Cov(\widehat{\gamma },\widehat{\lambda })& Var\left(\widehat{\gamma }\right)& Cov(\widehat{\gamma },\widehat{\theta })& Cov(\widehat{\gamma },\widehat{\alpha })\\ Cov(\widehat{\theta },\widehat{\lambda })& Cov(\widehat{\theta },\widehat{\gamma })& Var\left(\widehat{\theta }\right)& Cov(\widehat{\alpha },\widehat{\theta })\\ Cov(\widehat{\alpha },\widehat{\lambda })& Cov(\widehat{\alpha },\widehat{\gamma })& Cov(\widehat{\alpha },\widehat{\theta })& Var\left(\widehat{\alpha }\right)\end{array}\right)\hfill \\ \hfill =& {\left(\begin{array}{cccc}-{\ell }_{\lambda \lambda }\left(\mathsf{\Phi }\right)& -{\ell }_{\lambda \gamma }\left(\mathsf{\Phi }\right)& -{\ell }_{\lambda \theta }\left(\mathsf{\Phi }\right)& -{\ell }_{\lambda \alpha }\left(\mathsf{\Phi }\right)\\ -{\ell }_{\gamma \lambda }\left(\mathsf{\Phi }\right)& -{\ell }_{\gamma \gamma }\left(\mathsf{\Phi }\right)& -{\ell }_{\gamma \theta }\left(\mathsf{\Phi }\right)& -{\ell }_{\gamma \alpha }\left(\mathsf{\Phi }\right)\\ -{\ell }_{\theta \lambda }\left(\mathsf{\Phi }\right)& -{\ell }_{\theta \gamma }\left(\mathsf{\Phi }\right)& -{\ell }_{\theta \theta }\left(\mathsf{\Phi }\right)& -{\ell }_{\theta \alpha }\left(\mathsf{\Phi }\right)\\ -{\ell }_{\alpha \lambda }\left(\mathsf{\Phi }\right)& -{\ell }_{\alpha \gamma }\left(\mathsf{\Phi }\right)& -{\ell }_{\alpha \theta }\left(\mathsf{\Phi }\right)& -{\ell }_{\alpha \alpha }\left(\mathsf{\Phi }\right)\end{array}\right)}_{(\widehat{\lambda },\widehat{\gamma },\widehat{\theta },\widehat{\alpha })}^{-1},\hfill \end{array}\end{array}$$

where ${\ell }_{{\mathsf{\Phi }}_i{\mathsf{\Phi }}_j}\left(\mathsf{\Phi }\right)={\displaystyle \frac{{\partial }^2\ell }{\partial {\mathsf{\Phi }}_i{\mathsf{\Phi }}_j}}$, $i,j=1,2,3,4$, see Appendix B. The ACIs for the parameters $\lambda$, $\gamma$, $\theta$, and $\alpha$ based on the asymptotic variance–covariance matrix are given as follows:

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

where $z_{{\displaystyle \frac{\eta }{2}}}$ represents the percentile of the standard normal distribution with right tail probability ${\displaystyle \frac{\eta }{2}}$.

5.5. Least Square and Weighted Least Square Estimations

In a study by [18], LSE and WLSE were used to estimate beta distribution parameters. These approaches will be applied to estimate the parameters of NMMWD. Consider $x_i$, $i=1,\dots ,n$, as the ordered sample of a random sample of size n. The LSE of the NMMWD parameters can be calculated by minimizing the following function

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

with regard to the unknown parameters $\mathsf{\Phi }$ or by solving the following non-linear equations

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial S}{\partial \lambda }}=\sum _{i=1}^{n}2\left(1-e^{-\lambda x_i^{\gamma }{e}^{{\left(\theta x_i\right)}^{\alpha }}}-{\displaystyle \frac{i}{n+1}}\right){\delta }_1(x_i;\mathsf{\Phi })=0,\end{array}$$

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

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

and

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial S}{\partial \gamma }}=\sum _{i=1}^{n}2\left(1-e^{-\lambda x_i^{\gamma }{e}^{{\left(\theta x_i\right)}^{\alpha }}}-{\displaystyle \frac{i}{n+1}}\right){\delta }_4(x_i;\mathsf{\Phi })=0.\end{array}$$

where

$$\begin{array}{c}\hfill {\delta }_1(x_i;\mathsf{\Phi })=x_i^{\gamma }{e}^{e^{{\left(\theta x_i\right)}^{\alpha }}-\lambda x_i^{\gamma }{e}^{{\left(\theta x_i\right)}^{\alpha }}},\end{array}$$

(22)

$$\begin{array}{c}\hfill {\delta }_2(x_i;\mathsf{\Phi })=\lambda \alpha {\theta }^{\alpha -1}{x}_i^{\alpha +\gamma }{e}^{e^{{\left(\theta x_i\right)}^{\alpha }}-\lambda x_i^{\gamma }{e}^{{\left(\theta x_i\right)}^{\alpha }}},\end{array}$$

(23)

$$\begin{array}{c}\hfill {\delta }_3(x_i;\mathsf{\Phi })={\theta }^{\alpha }\lambda log\left(\theta x_i\right)x_i^{\alpha +\gamma }{e}^{e^{{\left(\theta x_i\right)}^{\alpha }}-\lambda x_i^{\gamma }{e}^{{\left(\theta x_i\right)}^{\alpha }}},\end{array}$$

(24)

and

$$\begin{array}{c}\hfill {\delta }_4(x_i;\mathsf{\Phi })=\lambda x_i^{\gamma }log\left(x_i\right)e^{e^{{\left(\theta x_i\right)}^{\alpha }}-\lambda x_i^{\gamma }{e}^{{\left(\theta x_i\right)}^{\alpha }}}.\end{array}$$

(25)

To determine the WLSE of the unknown parameters of NMMWD, minimizes a function

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

with regard to the unknown parameters $\mathsf{\Phi }$ or solve the non linear equations

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

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

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

and

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

where ${\delta }_1(x_i;\mathsf{\Phi })$, ${\delta }_2(x_i;\mathsf{\Phi })$, ${\delta }_3(x_i;\mathsf{\Phi })$ and ${\delta }_4(x_i;\mathsf{\Phi })$ are given by (22), (23), (24), and (25) respectively.

5.6. Cramér–Von Mises Estimation

A study by [19] found that CVME has lower bias compared to other minimum distance estimators. Obtaining the CVME for NMMWD can be done by minimizing the function

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

with regard to the unknown parameters $\mathsf{\Phi }$ or by solving the following non-linear equations

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial C}{\partial \lambda }}=\sum _{i=1}^{r}2\left(1-e^{-\lambda x_i^{\gamma }{e}^{{\left(\theta x_i\right)}^{\alpha }}}-{\displaystyle \frac{2i-1}{2r}}\right){\delta }_1(x_i;\mathsf{\Phi })=0,\end{array}$$

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

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

and

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial C}{\partial \gamma }}=\sum _{i=1}^{r}2\left(1-e^{-\lambda x_i^{\gamma }{e}^{{\left(\theta x_i\right)}^{\alpha }}}-{\displaystyle \frac{2i-1}{2r}}\right){\delta }_4(x_i;\mathsf{\Phi })=0,\end{array}$$

where ${\delta }_1(x_i;\mathsf{\Phi })$, ${\delta }_2(x_i;\mathsf{\Phi })$, ${\delta }_3(x_i;\mathsf{\Phi })$ and ${\delta }_4(x_i;\mathsf{\Phi })$ are given by (22), (23), (24), and (25) respectively.

5.7. Anderson–Darling Estimation

The author of [20] investigated the characteristics of ADE. According to his study, the A-DE of NMMWD can be obtained by minimizing the following function

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

with regard to the unknown parameters $\mathsf{\Phi }$ or by solving the following non-linear equations:

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

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

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

and

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

where ${\delta }_1(x_i;\mathsf{\Phi })$, ${\delta }_2(x_i;\mathsf{\Phi })$, ${\delta }_3(x_i;\mathsf{\Phi })$ and ${\delta }_4(x_i;\mathsf{\Phi })$ are given by (22), (23), (24), and (25) respectively.

6. Application to Real Data

In this section, three engineering real data sets are analyzed to show that the NMMWD offers enhanced flexibility for modeling these data sets better than their sub-models and some well-known distributions. We use the various estimating methods discussed in Section 5 to calculate the estimates of the NMMWD’s unknown parameters for all data sets. Initially, the proposed distribution and its sub-models, in addition to a set of well-known distributions that have bathtub hazard rate shapes summarized in Table 2, can be fitted with each data set using a non-parametric test, namely the K-S test, with its associated p-value. Next, we compare NMMWD with its sub-models and the set of well-known distributions summarized in Table 2 using three parametric tests: AIC, AICc and HQIC. Lower values for AIC, AICc and HQIC imply a better fit. All computations and graphical displays were performed using the Wolfram Mathematica 11.

Table 2. Some well-known distributions with bathtub hazard rate shapes.

Distribution	CDF	References
Add-MWD $(\lambda ,\gamma ,\theta ,\alpha ,\beta )$	$1-e^{e^{-\beta }-\lambda x^{\gamma }{e}^{\theta x}-e^{\alpha x-\beta }}$	[21]
Add-WD $(\lambda ,\gamma ,\theta ,\alpha )$	$1-e^{-{\left(\lambda x\right)}^{\theta }-{\left(\gamma x\right)}^{\alpha }}$	[5]
RNMWD $(\lambda ,\gamma ,\theta )$	$1-e^{-\lambda \sqrt{x}{e}^{\gamma x}-\theta \sqrt{x}}$	[22]
MWED $(\lambda ,\gamma ,\theta )$	$1-e^{\gamma \lambda \left(1-e^{{\left({\displaystyle \frac{x}{\gamma }}\right)}^{\theta }}\right)}$	[3]
NMW3D $(\lambda ,\gamma ,\theta )$	$1-e^{1-e^{\lambda x^{\gamma }{e}^{\theta x}}}$	[10]
MCD$(\lambda ,\gamma ,\theta )$	$1-e^{\theta \left(-e^{x^{\lambda }}-e^{\gamma x}+2\right)}$	[11]

6.1. Electronic Ground Support Equipment Data

The data presented in [23], given by [24], includes 107 observed lifetimes of a unit of electronic ground support equipment. The data set, measured in hours, consists of the following values: 1.0, 6.4, 19.2, 54.2, 88.4, 114.8, 1.2, 6.8, 28.1, 55.6, 89.9, 115.1, 1.3, 6.9, 28.2, 56.4, 90.8, 117.4, 2.0, 7.2, 29.0, 58.3, 91.1, 118.3, 2.4, 7.9, 29.9, 60.2, 91.5, 119.7, 2.9, 8.3, 30.6, 63.7, 92.1, 120.6, 3.0, 8.7, 32.4, 64.6, 97.9, 121.0, 3.1, 9.2, 33.0, 65.3, 100.8, 122.9, 3.3, 9.8, 35.3, 66.2, 102.6, 123.3, 35, 10.2, 36.1, 70.1, 103.2, 124.5, 3.8, 10.4, 40.1, 71.0, 104.0, 125.8, 4.3, 11.9, 42.8, 75.1, 104.3, 126.6, 4.6, 13.8, 43.7, 75.6, 105.0, 127.7, 4.7, 14.4, 44.5, 78.4, 105.8, 128.4, 4.8, 15.6, 50.4, 79.2, 106.5, 129.2, 5.2, 16.2, 51.2, 84.1, 110.7, 129.5, 5.4, 17.0, 52.0, 86.0, 112.6, 129.9, 5.9, 17.5, 53.3, 87.9, 113.5.

Figure 4 shows that this data set first takes a convex shape and subsequently a concave shape, indicating that its hazard rate appears like a bathtub. Table 3 shows the ML estimates for the NMMWD, MWD, WD, RNMMWD, RMWD, Add-MWD, Add-WD, RNMWD, MWED, NMW3D, and MCD, as well as the accompanying K-S distance and the associated p-value, AIC, AICc and HQIC. From Table 3, it is clear that the NMMWD has the lowest AIC, AIC, AICc and HQIC values, indicating the best fit among all distributions. Figure 5 displays the estimated SF and HRF for all distributions. It should be noted that the NMMWD distribution may well model these lifetime data with bathtub-shaped HR. Table 4 summarizes the values of estimates of the un-known parameters of NMMWD and the accompanying K-S distance and the associated p-value for the five different estimation techniques proposed. It is evident from Table 4 that the CVME is the best since it has the lowest value of K-S and largest value of p-value associated to K-S test.

Figure 4. The TTT-plot of the electronic ground support equipment data.

Table 3. The ML estimates of unknown parameters, the K-S with the corresponding p-value, the AIC, the AICc and the HQIC for different models using the electronic ground support equipment data.

Distribution	$\widehat{\mathit{\lambda }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\theta }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\beta }}$	K-S Distance	p-Value	AIC	AICc	HQIC
NMMWD	0.0318	0.7977	0.0077	10.537		0.0754	0.5762	1032.42	1032.81	1036.75
MWD	0.0389	0.5839	0.0110	-		0.1048	0.1903	1064.13	1064.37	1067.38
WD	0.0139	1.0503				0.1159	0.1129	1083.92	1084.04	1086.09
RNMMWD	0.0935	-	0.0081	5.3327		0.0912	0.3359	1043.36	1043.60	1046.61
RMWD			0.0124		0.2035	0.0996	0.2391	1062.76	1062.87	1064.93
Add-MWD	0.0326	0.7805	0.0008	0.1236	15.1346	0.0764	0.5593	1035.50	1036.09	1040.92
Add-WD	0.0136	0.0082	0.9102	14.302		0.0762	0.5641	1034.41	1034.80	1038.74
RNMWD	0.0001	0.0602	0.0930			0.0906	0.3437	1044.48	1044.72	1047.74
MWED	0.0079	43.512	0.7168			0.1242	0.0736	1068.75	1068.80	1071.82
NMW3D	0.0403	0.5474	0.0068	-		0.1112	0.1414	1058.18	1058.41	1061.43
MCD	0.2563	0.0321	0.0333	-		0.0988	0.2473	1056.80	1057.03	1060.05

Figure 5. Estimated SF and HRF of NMMWD for the electronic ground support equipment data.

Table 4. The estimates of NMMWD parameters based on different methods of estimation and the K-S test with the associated p-value for the electronic ground support equipment data.

Methods	$\widehat{\mathit{\lambda }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\theta }}$	$\widehat{\mathit{\alpha }}$	K-S Distance	p-Value
MLE	0.0318	0.7977	0.0077	10.537	0.0754	0.5762
LSE	0.0522	0.6575	0.0076	5.2750	0.0455	0.9799
WLSE	0.0380	0.7485	0.0077	8.3254	0.0647	0.7623
CVME	0.0501	0.6683	0.0076	5.3876	0.0485	0.9629
ADE	0.0424	0.7188	0.0077	7.3367	0.0575	0.8709

The approximate variance–covariance matrix is given by

$$\begin{array}{c}\hfill \left(\begin{array}{cccc}0.0001& 2.6\times {10}^{-7}& -0.0010& -0.0138\\ 2.6\times {10}^{-7}& 1.8\times {10}^{-8}& -4.1\times {10}^{-6}& -0.0001\\ -0.0010& -4.1\times {10}^{-6}& 0.0081& 0.1566\\ -0.0138& -0.0001& 0.1566& 16.1693\end{array}\right)\end{array}$$

Thus the 95% ACIs of $\lambda$, $\gamma$, $\theta$, and $\alpha$ are respectively given by {0.0084, 0.0552}, {0.6213, 0.9742}, {0.0074, 0.0079}, and {2.6556, 18.418}.

6.2. Main Rotor Blade Data

Consider the reliability measures, given by [25], using the MMIR system database collected from October 1995 to September 1999 of a helicopter main rotor blade part code xxx-015-001-107. The data set, measured in hours, consists of the following values: 1634.3, 2094.3, 3318.2, 1100.5, 2166.2, 2317.3, 1100.5, 2956.2, 1081.3, 819.9, 795.5, 1953.5, 1398.3, 795.5, 2418.5, 1181, 204.5, 1485.1, 128.7, 204.5, 2663.7, 1193.6, 1723.2, 1778.3, 254.1, 403.2, 1778.3, 3078.5, 2898.5, 2943.6, 3078.5, 2869.1, 2260, 3078.5, 26.5, 2299.2, 26.5, 26.5, 1655, 26.5, 3180.6, 1683.1, 3265.9, 644.1, 1683.1, 254.1, 1898.5, 2751.4, 2888.3, 3318.2, 2080.2, 1940.1.

Figure 6 shows that this data set first takes a convex shape and subsequently a concave shape, indicating that its hazard rate appears like a bathtub. Table 5 shows the ML estimates for the NMMWD, MWD, WD, RNMMWD, RMWD, Add-MWD, Add-WD, RNMWD, MWED, NMW3D, and MCD, as well as the accompanying K-S distance and the associated p-value, AIC, AICc and HQIC. From Table 5, it is clear that the Add-MWD, RNMMWD, and NMMWD have the lowest AIC, AICc and HQIC values, indicating the best fit among all distributions. Figure 7 displays the estimated SF and HRF for all distributions. It should be noted that the NMMWD distribution may well model these lifetime data with bathtub-shaped HR. Table 6 summarizes the values of estimates of the un-known parameters of NMMWD and the accompanying K-S distance and the associated p-value for the proposed five different estimation techniques. It is evident from Table 6 that the ADE is the best since it has the lowest value of K-S and largest value of p-value associated to K-S test.

Figure 6. The TTT-plot of the main rotor blade data.

Table 5. The ML estimates of unknown parameters, the K-S with the corresponding p-value, the AIC, the AICc and the HQIC for different models using the main rotor blade data.

Distribution	$\widehat{\mathit{\lambda }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\theta }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\beta }}$	K-S Distance	p-Value	AIC	AICc	HQIC
NMMWD	0.0031	0.6826	0.0003	3.0281		0.0707	0.9574	853.764	854.615	856.756
MWD	0.0073	0.4367	0.0007			0.0785	0.9055	857.098	857.598	859.342
WD	0.0001	1.3161			0.1536	0.1571		877.528	877.773	879.024
RNMMWD	0.0096		0.0004	2.2416		0.0730	0.9445	852.662	853.162	854.906
RMWD			0.0007		0.1263	0.0835	0.8616	855.232	855.477	856.728
Add-MWD	0.0052	0.5441	0.0005	0.0076	24.641	0.0596	0.9926	851.072	852.376	854.812
Add-WD	0.0004	0.0002	4.6873	0.6880		0.0711	0.9552	855.869	856.720	858.861
RNMWD	0.0007	0.0013	0.0083			0.0684	0.9679	853.258	853.758	855.502
MWED	0.0002	594.19	0.7116			0.1047	0.6191	861.002	861.502	863.246
NMW3D	0.0070	0.0005	0.4609			0.0741	0.9376	853.962	854.462	856.206
MCD	0.2019	0.0018	0.0057			0.0631	0.9858	853.448	853.948	855.692

Figure 7. Estimated SF and HRF of NMMWD for the main rotor blade data.

Table 6. The estimates of parameters of NMMWD based on different methods of estimation and the K-S test with the associated p-value for the main rotor blade data.

Methods	$\widehat{\mathit{\lambda }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\theta }}$	$\widehat{\mathit{\alpha }}$	K-S Distance	p-Value
MLE	0.0031	0.6826	0.0003	3.0281	0.0707	0.9574
LSE	0.0153	0.3935	0.0005	1.3075	0.0709	0.9563
WLSE	0.0059	0.5781	0.0004	2.1111	0.0540	0.9981
CVME	0.0113	0.4387	0.0005	1.3516	0.0647	0.9815
ADE	0.0066	0.5514	0.0004	1.8780	0.0533	0.9984

The approximate variance–covariance matrix is given by

$$\begin{array}{c}\hfill \left(\begin{array}{cccc}1.7\times {10}^{-5}& 1.5\times {10}^{-7}& -0.0009& -0.0039\\ 1.5\times {10}^{-7}& 3.1\times {10}^{-9}& -9.05\times {10}^{-6}& -0.0001\\ -0.0009& -9.05\times {10}^{-6}& 0.0438& 0.2390\\ -0.039& -0.0001& 0.2390& 2.7836\end{array}\right)\end{array}$$

Thus the 95% ACIs of $\lambda$, $\gamma$, $\theta$, and $\alpha$ are, respectively, given by {0.0, 0.0114}, {0.2723, 1.0928}, {0.0002, 0.0004}, and {0.0, 6.2982}.

6.3. Rear Dump Truck Data

Consider the data set, given by [26] and analyzed by [27], of a 180-ton rear dump truck, which consists of 128 times to failure. The data set, measured in hours, consists of the following values: 78, 158, 331, 381, 523, 620, 664, 1805, 1817, 2068, 3253, 4489, 4725, 4961, 5138, 5200, 5278, 5711, 6400, 6444, 6677, 7999, 8001, 8489, 9000, 9086, 10,262, 10,817, 11,062, 11,082, 11,086, 11,122, 11,534, 12,031, 12,339, 12,733, 13,265, 13,508, 13,673, 13,780, 14,443, 14,501, 14,656, 14,906, 14,983, 15,004, 15,062, 15,072, 15,136, 15,206, 15,247, 15,700, 15,714, 15,972, 16,186, 16,284, 16,329, 16,425, 16,605, 16,723, 16,731, 16,797, 16,859, 17,090, 17,305, 17,484, 17,510, 17,511, 17,536, 17,621, 17,703, 17,809, 17,968, 17,984, 18,175, 18,443, 18,458, 18,667, 18,669, 18,701, 18,723, 18,822, 18,860, 18,922, 18,935, 18,945, 18,960, 18,961, 18,979, 19,013, 19,032, 19,034, 19,169, 19,184, 19,201, 19,416, 19,455, 19,525, 19,595, 19,601, 19,613, 19,643, 19,671, 19,713, 19,785, 19,801, 19,937, 19,990, 20,432, 20,433, 20,434, 20,698, 21,460, 21,543, 21,584, 21,602, 21,645, 21,706, 21,762, 21,867, 21,912, 21,914, 21,937, 21,938, 21,939, 21,951, 21,954, 21,982.

Figure 8 shows that this data set first takes a convex shape and subsequently a concave shape, indicating that its hazard rate appears like a bathtub. Table 7 shows the ML estimates for the NMMWD, MWD, WD, RNMMWD, RMWD, Add-MWD, Add-WD, RNMWD, MWED, NMW3D, and MCD, as well as the accompanying K-S distance and the associated p-value, AIC, AICc and HQIC. From Table 7, it is clear that the NMMWD has the lowest AIC, AIC, AICc and HQIC values, indicating the best fit among all distributions. Figure 9 displays the estimated SF and HRF for all distributions. It should be noted that the NMMWD distribution may well model these lifetime data with bathtub-shaped HR. Table 8 summarizes the values of estimates of the unknown parameters of NMMWD and the accompanying K-S distance and the associated p-value for the proposed five different estimation techniques. It is evident from Table 8 that the CVME is the best since it has the lowest value of K-S and largest value of p-value associated to K-S test.

Figure 8. The TTT-plot of the rear dump truck data.

Table 7. The ML estimates of unknown parameters, the K-S with the corresponding p-value, the AIC, the AICc and the HQIC for different models using the rear dump truck data.

Distribution	$\widehat{\mathit{\lambda }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\theta }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\beta }}$	K-S Distance	p-Value	AIC	AICc	HQIC
NMMWD	0.0003	0.7178	0.0001	4.3144		0.0591	0.7633	2518.76	2519.09	2523.4
MWD	0.0018	0.2773	0.0002			0.1093	0.0941	2554.74	2554.93	2558.21
WD	$4.4\times {10}^{-10}$	2.2197				0.2117	0.00002	2646.99	2647.09	2649.31
RNMMWD	0.0019		0.0001	3.6693		0.0551	0.8317	2519.09	2519.28	2522.57
RMWD			0.0002		0.4341	0.1122	0.0797	2555.07	2555.16	2557.39
Add-MWD	0.0005	0.6193	$3.3\times {10}^{-5}$	0.0005	9.6679	0.0565	0.8092	2521.40	2521.89	2527.19
Add-WD	$1.7\times {10}^{-5}$	0.0001	0.8026	9.1283		0.0549	0.8346	2521.53	2521.85	2526.16
RNMWD	$4.7\times {10}^{-6}$	0.0004	0.0021			0.0531	0.8627	2521.27	2521.46	2524.75
MWED	$5.5\times {10}^{-6}$	5711.4	1.1093			0.1074	0.1046	2563.81	2564.13	2568.44
NMW3D	0.0018	0.3220	0.0002			0.0939	0.2085	2535.68	2535.87	2539.16
MCD	0.1827	0.0004	0.0008			0.0683	0.5891	2523.70	2523.89	2527.17

Figure 9. Estimated SF and estimated HRF of NMMWD for the rear dump truck data.

Table 8. The estimates of NMMWD parameters based on different methods of estimation and the K-S test with the associated p-value for the rear dump truck data.

Methods	$\widehat{\mathit{\lambda }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\theta }}$	$\widehat{\mathit{\alpha }}$	K-S Distance	p-Value
MLE	0.0003	0.7178	0.0001	4.3144	0.0591	0.7633
LSE	0.0008	0.5938	0.0001	3.6969	0.0555	0.8255
WLSE	0.0007	0.6086	0.0001	3.7962	0.0558	0.8203
CVME	0.0006	0.6307	0.0001	3.7809	0.0538	0.8530
ADE	0.0007	0.6074	0.0001	3.4728	0.0574	0.7928

The approximate variance–covariance matrix is given by

$$\begin{array}{c}\hfill \left(\begin{array}{cccc}1.6\times {10}^{-7}& 6.1\times {10}^{-10}& -0.0001& -0.0001\\ 6.1\times {10}^{-10}& 1.1\times {10}^{-11}& -2.9\times {10}^{-7}& -2.6\times {10}^{-6}\\ -0.0001& -2.9\times {10}^{-7}& 0.0236& 0.0641\\ -0.0001& -2.6\times {10}^{-6}& 0.0641& 0.7418\end{array}\right)\end{array}$$

Thus the 95% ACIs of $\lambda$, $\gamma$, $\theta$, and $\alpha$ are, respectively, given by {0.0, 0.0011}, {0.4168, 1.0189}, {0.00004, 0.00006}, and {2.6263, 6.0026}.

7. Simulation Study

To explore the simulation study employing MSE and AAB, the effectiveness of the different estimation methods outlined in Section 5 was assessed. To assess how effectively various strategies work in estimating the unknown parameters of NMMWD, the following steps are employed:

• Three sets of initial values for the parameters $\lambda$, $\gamma$, $\theta$, and $\alpha$ are assigned to the NMMWD. Two sets, $(0.7,0.3,3,0.3)$ and $(0.8,0.2,4,0.5)$, correspond to bathtub-shaped hazard functions, and one set, $(1.5,0.2,2,0.4)$, corresponds to an increasing hazard function.
• Assume sample sizes n = (50, 100, 150, 200, 250, 300, 350, 400) and then generate random samples of these sizes from Equation (6).
• For each method shown in Section 5, calculate the estimate values of the parameters $\lambda$, $\gamma$, $\theta$ and $\alpha$ of the NMMWD.
• Steps 2 and 3 should be repeated N = 1000 times.
• Compute the MSEs and AABs of the parameters $\lambda$, $\gamma$, $\theta$ and $\alpha$ for the five different methods.

All calculations are performed using Wolfram Mathematica 11. The values of MSEs and AABs of the parameters $\lambda$, $\gamma$, $\theta$ and $\alpha$ of the NMMWD for the different methods of estimation are reported in Table 9, Table 10, Table 11, Table 12, Table 13 and Table 14. While studying these tables, we observed several important and desired results. We summarized these results as follows:

Table 9. MSEs and AAB for parameters $\gamma$ and $\lambda$ under proposed methods with varying sample size n based on the initial values of parameters $(\gamma ,\lambda ,\alpha ,\theta )=(0.7,0.3,3,0.3)$.

n	$\mathit{\gamma }$	MLE	LSE	WLSE	CVME	ADE	$\mathit{\lambda }$	MLE	LSE	WLSE	CVME	ADE
50	MSE	0.0325	0.0394	0.0330	0.0382	0.0361	MSE	0.0044	0.0046	0.0045	0.0048	0.0047
	AAB	0.1351	0.1549	0.1401	0.1478	0.1431	AAB	0.0511	0.0519	0.0516	0.0534	0.0527
100	MSE	0.0166	0.0246	0.0192	0.0237	0.0223	MSE	0.0026	0.0030	0.0028	0.0030	0.0028
	AAB	0.0989	0.1186	0.1058	0.1166	0.1093	AAB	0.0397	0.0422	0.0409	0.0427	0.0413
150	MSE	0.0119	0.0172	0.0116	0.0178	0.0146	MSE	0.0020	0.0022	0.0019	0.0022	0.0020
	AAB	0.0838	0.0982	0.0846	0.0998	0.0887	AAB	0.0355	0.0366	0.0350	0.0365	0.0351
200	MSE	0.0097	0.0142	0.0108	0.0146	0.0118	MSE	0.0016	0.0017	0.0015	0.0017	0.0015
	AAB	0.0770	0.0909	0.0805	0.0902	0.0812	AAB	0.0309	0.0324	0.0306	0.0324	0.0302
250	MSE	0.0073	0.0119	0.0076	0.0122	0.0091	MSE	0.0014	0.0015	0.0014	0.0015	0.0014
	AAB	0.0653	0.0798	0.0682	0.0801	0.0701	AAB	0.0290	0.0307	0.0290	0.0305	0.0287
300	MSE	0.0059	0.0099	0.0066	0.0103	0.0077	MSE	0.0011	0.0012	0.0011	0.0012	0.0011
	AAB	0.0599	0.0745	0.0631	0.0756	0.0641	AAB	0.0257	0.0273	0.0259	0.0272	0.0256
350	MSE	0.0056	0.0099	0.0059	0.0096	0.0075	MSE	0.00093	0.0011	0.0009	0.0011	0.0009
	AAB	0.0590	0.0739	0.0601	0.0736	0.0632	AAB	0.0243	0.0255	0.0244	0.0258	0.0242
400	MSE	0.0046	0.0087	0.0048	0.0075	0.0057	MSE	0.00090	0.00103	0.00087	0.00099	0.00087
	AAB	0.0529	0.0682	0.0544	0.0668	0.0559	AAB	0.0233	0.0248	0.0231	0.0244	0.0228

Table 10. MSEs and AABs for parameters $\alpha$ and $\theta$ under proposed methods, for varying sample sizes n based on the initial parameter values $(\gamma ,\lambda ,\alpha ,\theta )=(0.7,0.3,3,0.3)$.

n	$\mathit{\alpha }$	MLE	LSE	WLSE	CVME	ADE	$\mathit{\theta }$	MLE	LSE	WLSE	CVME	ADE
50	MSE	1.1324	1.0609	0.9175	1.2455	0.9325	MSE	0.0028	0.0059	0.0043	0.0056	0.0051
	AAB	0.8203	0.8292	0.7634	0.8814	0.7597	AAB	0.0294	0.0425	0.0387	0.0402	0.0407
100	MSE	0.7759	0.8899	0.7296	0.9556	0.7283	MSE	0.0011	0.0033	0.0016	0.0027	0.0025
	AAB	0.6721	0.7279	0.6595	0.7589	0.6452	AAB	0.0209	0.0304	0.0249	0.0287	0.0278
150	MSE	0.5844	0.7984	0.5496	0.8515	0.5852	MSE	0.0006	0.0014	0.0007	0.0014	0.0016
	AAB	0.5781	0.6842	0.5734	0.7027	0.5802	AAB	0.0177	0.0230	0.0190	0.0225	0.0224
200	MSE	0.4616	0.6491	0.4911	0.6941	0.5057	MSE	0.0004	0.0009	0.0006	0.0008	0.0012
	AAB	0.5106	0.6245	0.5387	0.6356	0.5294	AAB	0.0154	0.0193	0.0169	0.0186	0.0189
250	MSE	0.3267	0.5238	0.3528	0.5585	0.3729	MSE	0.00039	0.0009	0.0004	0.0009	0.0009
	AAB	0.4325	0.5574	0.4639	0.5700	0.4640	AAB	0.0138	0.0185	0.0150	0.0181	0.0167
300	MSE	0.2596	0.4365	0.2921	0.4693	0.3056	MSE	0.00026	0.0007	0.00033	0.0007	0.0006
	AAB	0.3912	0.4995	0.4083	0.5109	0.4064	AAB	0.0125	0.0164	0.0134	0.0163	0.0145
350	MSE	0.2196	0.3756	0.2479	0.3801	0.2878	MSE	0.00025	0.0008	0.00029	0.0006	0.0008
	AAB	0.3585	0.4658	0.3859	0.4729	0.3952	AAB	0.0119	0.0161	0.0127	0.0155	0.0148
400	MSE	0.1778	0.3159	0.1940	0.3045	0.2271	MSE	0.00021	0.00067	0.00024	0.0005	0.00059
	AAB	0.3276	0.4278	0.3453	0.4276	0.3471	AAB	0.0110	0.0147	0.0115	0.0141	0.0129

Table 11. MSEs and AABs for parameters $\gamma$ and $\lambda$ under proposed methods, for varying sample sizes n based on the initial parameter values of parameters $(\gamma ,\lambda ,\alpha ,\theta )=(0.8,0.2,4,0.5)$.

n	$\mathit{\gamma }$	MLE	LSE	WLSE	CVME	ADE	$\mathit{\lambda }$	MLE	LSE	WLSE	CVME	ADE
50	MSE	0.0534	0.0762	0.0585	0.0734	0.0494	MSE	0.0027	0.0031	0.0030	0.0031	0.0038
	AAB	0.1792	0.2227	0.1947	0.2146	0.1754	AAB	0.0412	0.0439	0.0433	0.0436	0.0425
100	MSE	0.0336	0.0521	0.0356	0.0512	0.0329	MSE	0.0018	0.0020	0.0019	0.0020	0.0018
	AAB	0.1404	0.1847	0.1504	0.1796	0.1425	AAB	0.0329	0.0351	0.0342	0.0349	0.0331
150	MSE	0.0233	0.0372	0.0252	0.0380	0.0251	MSE	0.0014	0.0016	0.0015	0.0017	0.0023
	AAB	0.1179	0.1546	0.1237	0.1532	0.1206	AAB	0.0300	0.0318	0.0308	0.0323	0.0318
200	MSE	0.0199	0.0290	0.0216	0.0304	0.0202	MSE	0.0011	0.0012	0.0011	0.0012	0.0011
	AAB	0.1104	0.1334	0.1154	0.1355	0.1113	AAB	0.0265	0.0271	0.0266	0.0273	0.0268
250	MSE	0.0153	0.0298	0.0169	0.0305	0.0158	MSE	0.0009	0.0010	0.0009	0.0009	0.0009
	AAB	0.0968	0.1333	0.1009	0.1327	0.0995	AAB	0.0241	0.0254	0.0243	0.0248	0.0239
300	MSE	0.0113	0.0209	0.0126	0.0216	0.0119	MSE	0.0007	0.0008	0.0008	0.0008	0.0007
	AAB	0.0811	0.1116	0.0869	0.1141	0.0849	AAB	0.0216	0.0228	0.0219	0.0226	0.0216
350	MSE	0.0121	0.0212	0.0131	0.0230	0.0127	MSE	0.0006	0.0007	0.0007	0.0007	0.0006
	AAB	0.0859	0.1123	0.0894	0.1147	0.0885	AAB	0.0201	0.0208	0.0203	0.0209	0.0202
400	MSE	0.0100	0.0197	0.0120	0.0211	0.0115	MSE	0.0006	0.00068	0.00066	0.0007	0.00066
	AAB	0.0781	0.1075	0.0846	0.1095	0.0834	AAB	0.0202	0.0208	0.0205	0.0209	0.0205

Table 12. MSEs and AAB sfor parameters $\alpha$ and $\theta$ under proposed methods, for varying sample sizes n based on the initial parameter values $(\gamma ,\lambda ,\alpha ,\theta )=(0.8,0.2,4,0.5)$.

n	$\mathit{\gamma }$	MLE	LSE	WLSE	CVME	ADE	$\mathbf{\lambda }$	MLE	LSE	WLSE	CVME	ADE
50	MSE	0.7714	0.9187	0.7940	0.9332	0.6979	MSE	0.0022	0.0064	0.0049	0.0059	0.0039
	AAB	0.7191	0.7787	0.7215	0.7763	0.6733	AAB	0.0322	0.0488	0.0454	0.0457	0.0392
100	MSE	0.5837	0.7421	0.5871	0.7666	0.5445	MSE	0.0013	0.0036	0.0022	0.0033	0.0019
	AAB	0.6173	0.7015	0.6249	0.7103	0.5959	AAB	0.0262	0.0375	0.0319	0.0360	0.0297
150	MSE	0.4859	0.6339	0.4885	0.6702	0.4646	MSE	0.0010	0.0021	0.0014	0.0020	0.0015
	AAB	0.5515	0.6386	0.5643	0.6521	0.5442	AAB	0.0233	0.0309	0.0263	0.0305	0.0265
200	MSE	0.4080	0.5346	0.4287	0.5563	0.3944	MSE	0.0007	0.0011	0.0009	0.0011	0.0009
	AAB	0.5086	0.5887	0.5235	0.5965	0.5012	AAB	0.0205	0.0243	0.0223	0.0242	0.0218
250	MSE	0.3333	0.4959	0.3608	0.5197	0.3354	MSE	0.0005	0.0009	0.0006	0.0009	0.0006
	AAB	0.0968	0.5649	0.4824	0.5746	0.4678	AAB	0.0241	0.0233	0.0193	0.0226	0.0190
300	MSE	0.2644	0.4170	0.2909	0.4283	0.2706	MSE	0.0004	0.0008	0.0005	0.0008	0.0005
	AAB	0.4034	0.5131	0.4261	0.5173	0.4103	AAB	0.0159	0.0208	0.0174	0.0204	0.0171
350	MSE	0.2337	0.3852	0.2656	0.4015	0.2511	MSE	0.00038	0.0007	0.0004	0.0007	0.0004
	AAB	0.3802	0.4913	0.4088	0.4989	0.3987	AAB	0.0154	0.0196	0.0164	0.0196	0.0164
400	MSE	0.2062	0.3459	0.2402	0.3583	0.2316	MSE	0.00036	0.00067	0.00046	0.00066	0.00046
	AAB	0.3582	0.4673	0.3865	0.4708	0.3831	AAB	0.0149	0.0193	0.0164	0.0191	0.0165

Table 13. MSEs and AABs for parameters $\gamma$ and $\lambda$ under proposed methods, for varying sample size n based on the initial parameter values $(\gamma ,\lambda ,\alpha ,\theta )=(1.5,0.2,2,0.4)$.

n	$\mathit{\gamma }$	MLE	LSE	WLSE	CVME	ADE	$\mathit{\lambda }$	MLE	LSE	WLSE	CVME	ADE
50	MSE	0.1735	0.3597	0.3089	0.3356	0.2337	MSE	0.0040	0.0038	0.0034	0.0034	0.0030
	AAB	0.3378	0.5279	0.4776	0.5007	0.4115	AAB	0.0515	0.0492	0.0459	0.0454	0.0433
100	MSE	0.1084	0.2458	0.1847	0.2141	0.1457	MSE	0.0027	0.0027	0.0025	0.0024	0.0024
	AAB	0.2548	0.4152	0.3541	0.3759	0.3120	AAB	0.0400	0.0400	0.0382	0.0378	0.0377
150	MSE	0.0945	0.2099	0.1413	0.1836	0.1279	MSE	0.0022	0.0025	0.0022	0.0023	0.0021
	AAB	0.2364	0.3705	0.3007	0.3390	0.2886	AAB	0.0356	0.0382	0.0364	0.0367	0.0361
200	MSE	0.0682	0.1715	0.1035	0.1525	0.0973	MSE	0.0018	0.0024	0.0018	0.0020	0.0017
	AAB	0.2019	0.3341	0.2552	0.3107	0.2476	AAB	0.0327	0.0378	0.0331	0.0342	0.0324
250	MSE	0.0573	0.1477	0.0907	0.1369	0.0829	MSE	0.0017	0.0024	0.0019	0.0021	0.0018
	AAB	0.1876	0.3026	0.2349	0.2883	0.2253	AAB	0.0309	0.0373	0.0335	0.0342	0.0323
300	MSE	0.0534	0.1223	0.0748	0.1126	0.0764	MSE	0.0017	0.0022	0.0017	0.0019	0.0017
	AAB	0.1816	0.2730	0.2123	0.2614	0.2133	AAB	0.0318	0.0352	0.0316	0.0328	0.0312
350	MSE	0.0474	0.1099	0.0687	0.1022	0.0683	MSE	0.0015	0.0021	0.0016	0.0019	0.0016
	AAB	0.1705	0.2561	0.1999	0.2456	0.1994	AAB	0.0299	0.0351	0.0298	0.0333	0.0298
400	MSE	0.0437	0.1052	0.0595	0.0899	0.0638	MSE	0.0014	0.0021	0.0014	0.0018	0.0014
	AAB	0.1617	0.2479	0.1825	0.2314	0.1878	AAB	0.0282	0.0344	0.0283	0.0324	0.0282

Table 14. MSEs and AABs for parameters $\alpha$ and $\theta$ under proposed methods, for varying sample size n based on the initial parameter values $(\gamma ,\lambda ,\alpha ,\theta )=(1.5,0.2,2,0.4)$.

n	$\mathit{\gamma }$	MLE	LSE	WLSE	CVME	ADE	$\mathit{\lambda }$	MLE	LSE	WLSE	CVME	ADE
50	MSE	0.8127	1.2376	0.9871	1.2202	0.8914	MSE	0.0462	0.0744	0.0630	0.0633	0.0517
	AAB	0.7242	0.9022	0.7892	0.8909	0.7252	AAB	0.1471	0.2239	0.1974	0.1999	0.1721
100	MSE	0.4928	0.6551	0.4958	0.6591	0.5048	MSE	0.0331	0.0511	0.0388	0.0405	0.0338
	AAB	0.5519	0.6208	0.5475	0.6394	0.5354	AAB	0.1128	0.1655	0.1412	0.1444	0.1289
150	MSE	0.3835	0.5171	0.3773	0.5198	0.3956	MSE	0.0263	0.0467	0.0332	0.0375	0.0304
	AAB	0.4875	0.5563	0.4777	0.5674	0.4862	AAB	0.0977	0.1527	0.1239	0.1351	0.1199
200	MSE	0.2998	0.4568	0.3114	0.4678	0.3178	MSE	0.0186	0.0397	0.0248	0.0320	0.0231
	AAB	0.4410	0.5348	0.4406	0.5441	0.4373	AAB	0.0813	0.1391	0.1033	0.1243	0.1010
250	MSE	0.2691	0.4365	0.3008	0.4232	0.2998	MSE	0.0178	0.0391	0.0249	0.0329	0.0225
	AAB	0.4231	0.5199	0.4300	0.5134	0.4237	AAB	0.0795	0.1344	0.1010	0.1205	0.0979
300	MSE	0.2887	0.3806	0.2713	0.3971	0.2917	MSE	0.0192	0.0332	0.0213	0.0277	0.0217
	AAB	0.4319	0.4884	0.4148	0.5052	0.4152	AAB	0.0819	0.1203	0.0918	0.1103	0.0931
350	MSE	0.2328	0.3553	0.2501	0.3592	0.2849	MSE	0.0153	0.0308	0.0182	0.0272	0.0199
	AAB	0.3877	0.4815	0.4003	0.4799	0.4123	AAB	0.0732	0.1158	0.0849	0.1069	0.0890
400	MSE	0.2308	0.3606	0.2397	0.3558	0.2785	MSE	0.0154	0.0293	0.0158	0.0247	0.0181
	AAB	0.3923	0.4810	0.3839	0.4815	0.4052	AAB	0.0717	0.1129	0.0785	0.1025	0.0834

• As the generated sample size increases, the AAB and MSE values decrease for all the estimation methods used in this study, indicating that the estimates are consistent.
• Considering the AABs and MSEs, we find that MLE and WLSE are superior to the other estimation techniques used in this study because they consistently produce the lowest AABs and MSEs for all parameters.
• The tables show the stability of the performance of different estimation methods based on MSE and AAB values when changing the data generation mechanism from the bathtub to increasing hazard shapes and from the bathtub to the bathtub hazard shapes with small changes in initial parameters.

8. Conclusions

This paper dealt with a new modification of modified Weibull distribution called the NMMWD. The MWD, RMWD, RNMMWD, and WD are considered its sub-models. Some statistical properties, including quantiles, moments, Bowley skewness, and Moors kurtosis were derived in closed forms. We found that the HRF shapes for this distribution are extremely obvious and can take both increasing and bathtub shapes. Additionally, the inflection point for the HRF bathtub form was easily identified. The classical estimation methods of the NMMWD, including MLE, LSE, WLSE, CVME, and ADE, were investigated. Three engineering data sets were utilized to show that the NMMWD is applicable and more compatible with engineering disciplines than its sub-distributions and some well-known distributions using non-parametric and parametric tests. In particular, we found that the NMMWD performed better than the other distributions, achieving the lowest AIC, AICc and HQIC values in the electronic ground support equipment data and the rear dump truck data. For electronic ground support equipment data, the CVME is the best among the methods based on the p-value corresponding to the K-S test. Further, the ADE is the best for the main rotor blade data. Furthermore, the CVME is the best based on a non-parametric test for the rear dump truck data. The simulation study was also presented to compare the proposed estimation methods using MSEs and AABs. Furthermore, the simulation study demonstrated that MLE and WLSE consistently produced the lowest MSEs and AABs across a range of sample sizes. We also concluded that when changing the data generation mechanism from bathtub to increasing hazard forms and from bathtub to bathtub hazard forms with small changes in the initial parameters, the performance of the different estimation methods remains stable. For future work, these methods can be extended to other censoring schemes. The problem of Bayesian estimation of the unknown parameters under complete and censoring data from the NMMWD can also be investigated.