Symmetric and Asymmetric Expansion of the Weibull Distribution: Features and Applications to Complete, Upper Record, and Type-II Right-Censored Data

Abstract

This paper introduces a new continuous lifetime model called the Odd Flexible Weibull-Weibull (OFW-W) distribution, which features three parameters. The new model is capable of modeling both symmetric and asymmetric datasets, regardless of whether they are positively or negatively skewed. Its hazard rate functions can exhibit various behaviors, including increasing, decreasing, unimodal, or bathtub-shaped. The key characteristics of the OFW-W model are discussed, including the quantile function, median, reliability and hazard rate functions, kurtosis and skewness, mean waiting (residual) lifetimes, moments, and entropies. The unknown parameters of the model are estimated using eight different techniques. A comprehensive simulation study evaluates the performance of these estimators based on bias, mean squared error (MSE), and mean relative error (MRE). The practical usefulness of the OFW-W distribution is demonstrated through four real datasets from the fields of engineering and medicine, including complete data, upper record data, and type-II right-censored data. Comparisons with five other lifetime distributions reveal that the OFW-W model exhibits superior flexibility and capability in fitting various data types, highlighting its advantages and improvements. In conclusion, we anticipate that the OFW-W model will prove valuable in various applications, including human health, environmental studies, reliability theory, actuarial science, and medical sciences, among others.

1. Introduction

There is no doubt that preserving human health at all stages of life is a precious treasure and an urgent necessity due to its utmost importance in preventing diseases, improving immunity in individuals, and increasing the number of healthy people. To determine the health trends and pathways necessary to maintain human health and provide required health services, such as accurate diagnosis, availability of vaccines and medicines, and ensuring their safety, many researchers have resorted to studying statistical distributions in various fields related to human health. Engineering is a major necessity in our daily lives because it helps us analyze and understand natural phenomena and solve many practical problems. It is also a fundamental pillar in many industries, construction projects, safety systems, and technological advancements that include computers, the Internet, equipment, devices, and more. Probability distributions are crucial and highly effective in describing various natural phenomena, including rain, earthquakes, and hurricanes. Their significance extends to modeling different types of data in various fields, such as medicine, biology, actuarial science, engineering, economics, insurance, and more. To adequately represent these phenomena, researchers must explore new life distributions that offer flexibility, efficiency, and a strong capacity to accurately describe natural events and derive precise results that aid in the decision-making process.

Continuous probability distributions are essential for modeling a diverse array of phenomena in which the variable can assume any value within a specified range. Understanding continuous probability distributions and their derivatives is vital for accurately modeling and analyzing data across multiple disciplines. Using these distributions enables researchers to derive meaningful insights and make well-informed decisions based on statistical analysis. Among these common distributions, the normal distribution is fundamental in lifetime situations across various fields because of its unique properties and the valuable insights it offers. In addition, the exponential distribution is frequently used to model the time until an event occurs, such as equipment failure, customer arrivals, or the time until a patient relapses. The Weibull (W) distribution is one of the most widely used continuous probability distributions, applicable across a variety of fields, including communication systems, wind speed data analysis, electrical and industrial engineering, survival and failure analysis, insurance, hydrology, and medicine. It was initially introduced by Weibull [1] to model and characterize the distribution of yield power and intensity in various materials. Due to its ability to model varying failure times, the Weibull distribution has been applied in a wide range of other applications, including the fracture strength of glass, failure of carbon fiber composites, pitting corrosion in pipes, fatigue life of steel, brittleness in concrete materials, adhesive wear in metals, and others, as referenced in Weibull [2].

The Weibull distribution is appropriate for modeling monotonic hazard rates, which are significant and effective in reliability analysis, system design, maintenance planning, and other areas. Incorporating non-monotonic failure rates into these fields leads to more effective strategies for improving system performance, managing risk, and extending the life of components. In recent years, many researchers and scientists have focused on making numerous modifications and variations of the Weibull distribution to enhance its adaptability for describing non-monotonic failure rates, for example, Ezeah et al. [3], S. Mastor et al. [4], El-Din et al. [5], Ikbal et al. [6], Ahmad et al. [7], Al-Aqtash et al. [8], Cordeiro et al. [9], Silva et al. [10], Alzaatreh et al. [11], Bourguignon et al. [12], Afify et al. [13], Jehhan et al. [14], and El-Morshedy et al. [15,16]. The cumulative distribution function (CDF) of a non-negative random variable Y that has the W distribution with one parameter $\gamma >0$ is given as:

$$G\left(y\right)=1-e^{-y^{\gamma }};y>0,\gamma >0.$$

(1)

Also, the probability density function (PDF) of Y is given as:

$$g\left(y\right)=\gamma y^{y-1}{e}^{-y^{\gamma }};y>0,\gamma >0.$$

(2)

Bebbington et al. [17] suggested a new continuous lifetime model consisting of two parameters, titled the Flexible Weibull Extension (FWE) distribution. It describes various classes of hazard (failure) rate functions (HRFs), such as increasing HRFs, decreasing HRFs, and bathtub-shaped HRFs. The FWE is suitable for several applications and fields, including clinical surveys, lifetime experiments, and survival analysis.

Various extensions of the FWE model have been suggested and offered in recent years, such as El-Gohary et al. [18], El-Damcese et al. [19], El-Morshedy et al. [20], Ahmad and Iqbal [21], Ahmad and Hussain [22], Sangun and Jiwhan [23], Kamal and Ismail [24], Abubakari et al. [25], and Javed et al. [26], among others. Recently, El-Morshedy and Eliwa [27] studied a new flexible family of distributions named the Odd Flexible Weibull-H (OFW-H) family, which can be utilized to model symmetric, negatively skewed, and positively skewed datasets. The CDF and PDF of the OFW-H family are defined, respectively, as:

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

(3)

and

$$f\left(x\right)={\displaystyle \frac{\alpha G{\left(x\right)}^2+\beta \overline{G}{\left(x\right)}^2}{G{\left(x\right)}^2\overline{G}{\left(x\right)}^2}}g\left(x\right)e^{\alpha {\displaystyle \frac{G\left(x\right)}{\overline{G}\left(x\right)}}-\beta {\displaystyle \frac{\overline{G}\left(x\right)}{G\left(x\right)}}}{e}^{-e^{\alpha {\displaystyle \frac{G\left(x\right)}{\overline{G}\left(x\right)}}-\beta {\displaystyle \frac{\overline{G}\left(x\right)}{G\left(x\right)}}}};x>0;\alpha ,\beta >0.$$

(4)

where $g\left(x\right)$ and $G\left(x\right)$ are the PDF and CDF of a non-negative random variable X. In addition, $\alpha$ and $\beta$ are two scale parameters that enhance the model’s flexibility. In this article, we present a new flexible expansion of the Weibull distribution, named the odd flexible Weibull-Weibull distribution (OFW-W), which has three parameters, i.e., $\alpha ,\beta ,$ and $\gamma$, based on Equations (3) and (4). The new model is characterized by a hazard rate function (HRF) that is increasing, decreasing, unimodal, and bathtub-shaped. Additionally, the OFW-W can be used to describe both symmetric and asymmetric datasets.

Our paper is organized as follows. In Section 2, we present the OFW-W model. Several statistical features of the OFW-W are introduced in Section 3. The unknown OFW-W parameters will be estimated in Section 4 using eight different estimation techniques. In Section 5, we will perform a simulation study to analyze the performance of the OFW-W model estimators. We will analyze four engineering and medical applications in Section 6 to demonstrate and clarify the fitting capability of the new flexible model. Finally, in Section 7, we provide a brief conclusion summarizing the obtained results.

2. Odd Flexible Weibull-Weibull Distribution

The OFW-W distribution is a novel statistical model that combines the properties of the Weibull distribution with increased flexibility to more effectively capture varying data patterns. By incorporating extra parameters, this distribution can model a wider range of shapes, making it suitable for datasets with diverse behaviors. It stands out by providing greater adaptability compared to traditional Weibull distributions and their modifications. Unlike other Weibull variants, this distribution provides enhanced flexibility, enabling more effective modeling of complex datasets. These features highlight the significance of the OFW-W distribution in advancing statistical methodologies and enhancing the precision of data analysis.

In this part, we will define the OFW-W model in terms of the CDF and PDF obtained by substituting from Equations (1) and (2) into Equations (3) and (4). Consider X be a random variable that follow the OFW-W with the three parameters $\alpha , \beta ,$ and $\gamma$, say $X\sim OFW-W(\alpha , \beta , \gamma )$, then the CDF and PDF of X are expressed, respectively, as:

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

(5)

and

$$f\left(x\right)=\gamma x^{\gamma -1}{e}^{x^{\gamma }}[\beta {(e^{x^{\gamma }}-1)}^{-2}+\alpha ]e^{\alpha (e^{x^{\gamma }}-1)-\beta {(e^{x^{\gamma }}-1)}^{-1}}{e}^{-e^{\alpha (e^{x^{\gamma }}-1)-\beta {(e^{x^{\gamma }}-1)}^{-1}}}; x>0, \alpha , \beta , \gamma >0.$$

(6)

Moreover, the reliability function (SF) of X is found by the formula:

$$R\left(x\right)=e^{-e^{\alpha (e^{x^{\gamma }}-1)-\beta {(e^{x^{\gamma }}-1)}^{-1}}}.$$

(7)

The HRF and reversed HRF of $X\sim OFW-W(\alpha , \beta , \gamma )$ are formulated as:

$$h\left(x\right)=\gamma x^{\gamma -1}{e}^{x^{\gamma }}[\beta {(e^{x^{\gamma }}-1)}^{-2}+\alpha ]e^{\alpha (e^{x^{\gamma }}-1)-\beta {(e^{x^{\gamma }}-1)}^{-1}}$$

(8)

and

$$r\left(x\right)={\displaystyle \frac{\gamma x^{\gamma -1}{e}^{x^{\gamma }}[\beta {(e^{x^{\gamma }}-1)}^{-2}+\alpha ]e^{\alpha (e^{x^{\gamma }}-1)-\beta {(e^{x^{\gamma }}-1)}^{-1}}}{e^{\alpha (e^{x^{\gamma }}-1)-\beta {(e^{x^{\gamma }}-1)}^{-1}}-1}}.$$

(9)

The HRF and PDF plots for the OFW-W model with some specified values of $\alpha , \beta ,$ and $\gamma$ are shown in Figure 1. It is apparent that OFW-W has an increasing HRF, decreasing HRF, and bathtub-shaped curve HRF. Also, the PDF takes several graphs depending upon its parameters, and they are unimodally and bimodally shaped curves.

3. Main Features and Properties

3.1. The Quantiles and Median

Assume $X\sim OFW-W(\alpha , \beta , \gamma )$, then the quantile, say $x_q$, of the OFW-W model can be obtained in an explicit formula as:

$$x_q={\left\{ln\left({\displaystyle \frac{\sqrt{{\left[ln(-ln(1-q\left)\right)\right]}^2+4\alpha \beta }+ln(-ln(1-q))+2\alpha }{2\alpha }}\right)\right\}}^{{\displaystyle \frac{1}{\gamma }}}; 0<q<1.$$

(10)

If we put $q=0.5$ in Equation (10), we will obtain the median of the OFW-W, say $Med\left(X\right)$, as:

$$Med\left(X\right)={\left\{ln\left({\displaystyle \frac{\sqrt{{\left[ln(-ln(0.5\left)\right)\right]}^2+4\alpha \beta }+ln(-ln\left(0.5\right))+2\alpha }{2\alpha }}\right)\right\}}^{{\displaystyle \frac{1}{\gamma }}}.$$

(11)

3.2. The Mode

The mode of $X\sim OFW-W(\alpha , \beta , \gamma )$ is computed if we solve the non-linear equation below with regard to x:

$$\gamma x^{\gamma }{e}^{x^{\gamma }}[\alpha +\beta {(e^{x^{\gamma }}-1)}^{-2}]\left[1-e^{\alpha (e^{x^{\gamma }}-1)-\beta {(e^{x^{\gamma }}-1)}^{-1}}\right]-{\displaystyle \frac{2\beta \gamma x^{\gamma }{e}^{x^{\gamma }}{(e^{x^{\gamma }}-1)}^{-3}}{\alpha +\beta {(e^{x^{\gamma }}-1)}^{-2}}}+\gamma (x^{\gamma }+1)-1=0.$$

(12)

Equation (12) can be solved numerically to get the mode of the OFW-W.

3.3. The $r^{th}$ Moment

Assume $X\sim OFW-W(\alpha , \beta , \gamma )$, then we can obtain the $r^{th}$ moment for X using:

$${\mu }^{\left(r\right)}=E\left(X^r\right)=\underset{0}{\overset{\infty }{\int }}{x}^{r}f\left(x\right)dx.$$

(13)

By substituting Equation (6) in Equation (13), we get the ${\mu }^{\left(r\right)}$ in the following formula:

$$\begin{array}{ccc}\hfill {\mu }^{\left(r\right)}& =& \sum _{i=0}^{\infty }\sum _{j=0}^{\infty }\sum _{n=0}^j\left({\displaystyle \genfrac{}{}{0pt}{}{j}{n}}\right){\displaystyle \frac{{(-1)}^{i+m+n}{(i+1)}^j{\alpha }^{j-n}{\beta }^n\Gamma \left(1+\frac{r}{\gamma }\right)}{i!j!}}\hfill \\ & & \times \left\{\alpha \sum _{m=0}^{j-2n}\left({\displaystyle \genfrac{}{}{0pt}{}{j-2n}{m}}\right){\left({\displaystyle \frac{1}{2n+m-(j+1)}}\right)}^{1+{\displaystyle \frac{r}{\gamma }}}+\beta \sum _{m=0}^{j-2(n+1)}\left({\displaystyle \genfrac{}{}{0pt}{}{j-2(n+1)}{m}}\right){\left({\displaystyle \frac{1}{2n+m-(j-1)}}\right)}^{1+{\displaystyle \frac{r}{\gamma }}}\right\}.\hfill \end{array}$$

(14)

3.4. Probability Weighted Moment (PWM)

Assume $X\sim OFW-W(\alpha ,\beta ,\gamma )$, then the PWM of X, say ${\xi }_{s,r}$, is given by the following formula:

$${\xi }_{s,r}=E\left(X^s{F}^r\left(x\right)\right)=\underset{0}{\overset{\infty }{\int }}{x}^s{F}^r\left(x\right)f\left(x\right)dx.$$

(15)

Substituting Equation (5) in Equation (15), we find that:

$$\begin{array}{ccc}\hfill {\xi }_{s,r}& =& \sum _{\ell =0}^r\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }\sum _{n=0}^j\left({\displaystyle \genfrac{}{}{0pt}{}{r}{\ell }}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{j}{n}}\right){\displaystyle \frac{{(-1)}^{\ell +i+m+n}{(1+\ell )}^i{(i+1)}^j{\alpha }^{j-n}{\beta }^n\Gamma \left(1+\frac{s}{\gamma }\right)}{i!j!}}\hfill \\ & & \times \left\{\alpha \sum _{m=0}^{j-2n}\left({\displaystyle \genfrac{}{}{0pt}{}{j-2n}{m}}\right){\left({\displaystyle \frac{1}{2n+m-(j+1)}}\right)}^{1+{\displaystyle \frac{s}{\gamma }}}+\beta \sum _{m=0}^{j-2(n+1)}\left({\displaystyle \genfrac{}{}{0pt}{}{j-2(n+1)}{m}}\right){\left({\displaystyle \frac{1}{2n+m-(j-1)}}\right)}^{1+{\displaystyle \frac{s}{\gamma }}}\right\}.\hfill \end{array}$$

(16)

3.5. Moment Generating Function

Assume $X\sim OFW-W(\alpha ,\beta ,\gamma )$, then the moment generating function for X, say $M_X\left(t\right),$ is computed using:

$$M_X\left(t\right)=E\left(e^{tx}\right)=\sum _{r=0}^{\infty }{\displaystyle \frac{t^r}{r!}}\underset{0}{\overset{\infty }{\int }}{x}^{r}f(x;\alpha ,\beta ,\gamma )dx=\sum _{r=0}^{\infty }{\displaystyle \frac{t^r}{r!}}{\mu }^{\left(r\right)}.$$

(17)

From Equation (14) in Equation (17), we will obtain:

$$\begin{array}{ccc}\hfill M_X\left(t\right)& =& \sum _{r=0}^{\infty }\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }\sum _{n=0}^j\left({\displaystyle \genfrac{}{}{0pt}{}{j}{n}}\right){\displaystyle \frac{{(-1)}^{i+m+n}{(i+1)}^j{\alpha }^{j-n}{\beta }^n{t}^r\Gamma \left(1+\frac{r}{\gamma }\right)}{r!i!j!}}\hfill \\ & & \times \left\{\alpha \sum _{m=0}^{j-2n}\left({\displaystyle \genfrac{}{}{0pt}{}{j-2n}{m}}\right){\left({\displaystyle \frac{1}{2n+m-(j+1)}}\right)}^{1+{\displaystyle \frac{r}{\gamma }}}+\beta \sum _{m=0}^{j-2(n+1)}\left({\displaystyle \genfrac{}{}{0pt}{}{j-2(n+1)}{m}}\right){\left({\displaystyle \frac{1}{2n+m-(j-1)}}\right)}^{1+{\displaystyle \frac{r}{\gamma }}}\right\}.\hfill \end{array}$$

(18)

3.6. Skewness and Kurtosis

To investigate the efficiency and influence of the shape and scale parameters for the OFW-W model on the kurtosis (say $K_u$) and skewness (say $S_k$), we can use Equation (10) that represent the quantiles of the OFW-W. Kenney and Keeping [28] introduced the Bowley skewness which is found by:

$$S_k={\displaystyle \frac{q\left(0.25\right)-2q\left(0.5\right)+q\left(0.75\right)}{-q\left(0.25\right)+q\left(0.75\right)}}.$$

(19)

Furthermore, Moors [29] introduced the Moors kurtosis, which is found by:

$$K_u={\displaystyle \frac{q\left(0.375\right)-\left[q\right(0.125)+q(0.625\left)\right]+q\left(0.875\right)}{-q\left(0.25\right)+q\left(0.75\right)}},$$

(20)

where $q(.)$ is the quantile function of the variable X. The plots of $S_k$ and $K_u$ are given, in Figure 2, for various choices of $\alpha$ and $\beta$ as $\gamma =2$. Figure 3 presents the graphs for some other values of $\gamma$ and $\beta$ when $\alpha =2$. These figures reveal that the OFW-W is adequate for describing the negatively and positively skewed data.

3.7. The Entropies

In some various fields, like engineering, computer science, probability theory, and others, the entropy is used to appreciate the variation of uncertainty concerning with X whose the PDF $f\left(x\right)$. The Rényi entropy of the variable X, say $I_{\rho }\left(X\right)$, can be found by:

$$I_{\rho }\left(X\right)={\displaystyle \frac{1}{-\rho +1}}log\underset{0}{\overset{\infty }{\int }}{f}^{\rho }\left(x\right)dx;\rho \in (0,\infty )-\left\{1\right\}.$$

(21)

If $X\sim OFW-W(\alpha ,\beta ,\gamma )$, then $I_{\rho }\left(X\right)$ will take the formula:

$$I_{\rho }\left(X\right)={\displaystyle \frac{1}{-\rho +1}}log\left({\displaystyle \genfrac{}{}{0pt}{}{{\gamma }^{\rho -1}{\displaystyle \sum _{\ell =0}^{\rho }}{\displaystyle \sum _{i=0}^{\infty }}{\displaystyle \sum _{j=0}^{\infty }}{\displaystyle \sum _{n=0}^j}{\displaystyle \sum _{m=0}^{j-2(\ell +n)}}\left(\genfrac{}{}{0pt}{}{j}{n}\right)\left(\genfrac{}{}{0pt}{}{\rho }{\ell }\right)\left(\genfrac{}{}{0pt}{}{j-2(\ell +n)}{m}\right)\Gamma \left(\frac{(\gamma -1)(\rho -1)}{\gamma }\right)}{\times \frac{{(-1)}^{i+m+n}{(i+\rho )}^j{\rho }^i{\alpha }^{\rho +j-(\ell +n)}{\beta }^{\ell +n}}{i!j!}{\left(\frac{-1}{j+\rho -m-2(\ell +n)}\right)}^{\frac{\rho (1+\gamma )+1}{\gamma }}}}\right).$$

(22)

The $\rho$-entropy of the variable X, say $V_{\rho }\left(X\right)$, is obtained by:

$$V_{\rho }\left(X\right)={\displaystyle \frac{1}{-\rho +1}}log\left[1-(1-\rho )I_{\rho }\left(X\right)\right].$$

(23)

3.8. Mean Time to Failure

Suppose $X\sim OFW-W(\alpha ,\beta ,\gamma )$, then the mean time to failure of the variable X, say $MTTF$, can be found by:

$$MTTF=\underset{0}{\overset{\infty }{\int }}xf(x; \alpha , \beta , \gamma )dx={\mu }^{\left(1\right)}.$$

From Equation (14), as $r=1$, then the $MTTF$ will be:

$$\begin{array}{ccc}\hfill MTTF& =& \sum _{i=0}^{\infty }\sum _{j=0}^{\infty }\sum _{n=0}^j\left({\displaystyle \genfrac{}{}{0pt}{}{j}{n}}\right){\displaystyle \frac{{(-1)}^{i+m+n}{(i+1)}^j{\alpha }^{j-n}{\beta }^n\Gamma \left(1+\frac{1}{\gamma }\right)}{i!j!}}\hfill \\ & & \times \left\{\alpha \sum _{m=0}^{j-2n}\left({\displaystyle \genfrac{}{}{0pt}{}{j-2n}{m}}\right){\left({\displaystyle \frac{1}{2n+m-(j+1)}}\right)}^{1+{\displaystyle \frac{1}{\gamma }}}+\beta \sum _{m=0}^{j-2(n+1)}\left({\displaystyle \genfrac{}{}{0pt}{}{j-2(n+1)}{m}}\right){\left({\displaystyle \frac{1}{2n+m-(j-1)}}\right)}^{1+{\displaystyle \frac{1}{\gamma }}}\right\}.\hfill \end{array}$$

(24)

3.9. The Mean Inactive (Waiting) and Residual Lifetimes

Assume $T\sim OFW-W(\alpha , \beta , \gamma )$, then the mean inactive lifetime of T, say $M_w\left(t\right)$, is given as:

$$\begin{array}{ccc}\hfill M_w\left(t\right)& =& E\left(t-T|T\le t\right)={\displaystyle \frac{1}{F\left(t\right)}}\underset{0}{\overset{t}{\int }}F\left(x\right)dx\hfill \\ & =& {\displaystyle \frac{1}{F\left(t\right)}}\left[t-\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }\sum _{n=0}^j\sum _{m=0}^{j-2n}\sum _{\ell =0}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{j}{n}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{-2n+j}{m}}\right){\displaystyle \frac{{(-1)}^{i+m+n}{i}^j{\alpha }^{j-n}{\beta }^n{\left(-2n+j-m\right)}^{\ell }{t}^{1+\gamma \ell }}{i!j!\ell !(1+\gamma \ell )}}\right].\hfill \end{array}$$

(25)

Also, the mean residual lifetime of $T\sim OFW-W(\alpha , \beta , \gamma )$, say $M_r\left(t\right)$, is given as:

$$\begin{array}{ccc}\hfill M_r\left(t\right)& =& E\left(T-t|T>t\right)={\displaystyle \frac{1}{R\left(t\right)}}\underset{t}{\overset{\infty }{\int }}R\left(x\right)dx\hfill \\ & =& {\displaystyle \frac{1}{1-F\left(t\right)}}\left[{\mu }^{\left(1\right)}-\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }\sum _{n=0}^j\left({\displaystyle \genfrac{}{}{0pt}{}{j}{n}}\right){\displaystyle \frac{{(-1)}^{i+n}{i}^j{\alpha }^{j-n}{\beta }^n}{i!j!}}\underset{0}{\overset{t}{\int }}{\left(e^{x^{\gamma }}-1\right)}^{j-2n}dx\right]\hfill \\ & =& {\displaystyle \frac{1}{1-F\left(t\right)}}\left[{\mu }^{\left(1\right)}-\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }\sum _{n=0}^j\sum _{m=0}^{j-2n}\sum _{\ell =0}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{j}{n}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{-2n+j}{m}}\right){\displaystyle \frac{{(-1)}^{i+m+n}{i}^j{\alpha }^{j-n}{\beta }^n{\left(-2n+j-m\right)}^{\ell }{t}^{1+\gamma \ell }}{i!j!\ell !(1+\gamma \ell )}}\right].\hfill \end{array}$$

(26)

4. Parameter Estimation

4.1. Maximum Likelihood (ML) Procedure

Assume $x_1,x_2,...,x_n$ is a randomly selected sample with size n from the OFW-W. Then, the log-likelihood function $L(\alpha , \beta , \gamma )$ of this sample is expressed as:

$$\begin{array}{ccc}\hfill L(\alpha , \beta , \gamma )& =& nln\left(\gamma \right)+\sum _{i=1}^n{x}_i^{\gamma }+(\gamma -1)\sum _{i=1}^{n}ln\left(x_i\right)+\sum _{i=1}^{n}ln\left[\beta {(e^{x_i^{\gamma }}-1)}^{-2}+\alpha \right]\hfill \\ & & +\alpha \sum _{i=1}^n(e^{x_i^{\gamma }}-1)-\beta \sum _{i=1}^n{(e^{x_i^{\gamma }}-1)}^{-1}-\sum _{i=1}^n{e}^{\alpha (e^{x_i^{\gamma }}-1)-\beta {(e^{x_i^{\gamma }}-1)}^{-1}}.\hfill \end{array}$$

(27)

The ML estimators of the OFW-W model, say ${\widehat{\alpha }}_{ML}, {\widehat{\beta }}_{ML},$ and ${\widehat{\gamma }}_{ML},$ are found by maximizing Equation (27). The corresponding normal equations of $L(\alpha , \beta , \gamma )$ will take the forms:

$$\sum _{i=1}^n{\displaystyle \frac{1}{\widehat{\alpha }+\widehat{\beta }{(e^{x_i^{\widehat{\gamma }}}-1)}^{-2}}}+\sum _{i=1}^n(e^{x_i^{\widehat{\gamma }}}-1)-\sum _{i=1}^n(e^{x_i^{\widehat{\gamma }}}-1)e^{\widehat{\alpha }(e^{x_i^{\widehat{\gamma }}}-1)-\widehat{\beta }{(e^{x_i^{\widehat{\gamma }}}-1)}^{-1}}=0,$$

(28)

$$\sum _{i=1}^n{\displaystyle \frac{1}{\widehat{\alpha }{(e^{x_i^{\widehat{\gamma }}}-1)}^2+\widehat{\beta }}}-\sum _{i=1}^n{(e^{x_i^{\widehat{\gamma }}}-1)}^{-1}+\sum _{i=1}^n{(e^{x_i^{\widehat{\gamma }}}-1)}^{-1}{e}^{\widehat{\alpha }(e^{x_i^{\widehat{\gamma }}}-1)-\widehat{\beta }{(e^{x_i^{\widehat{\gamma }}}-1)}^{-1}}=0$$

(29)

and

$$\begin{array}{ccc}& & {\displaystyle \frac{n}{\widehat{\gamma }}}+\sum _{i=1}^{n}ln\left(x_i\right)+\sum _{i=1}^n{x}_i^{\widehat{\gamma }}ln\left(x_i\right)-2\widehat{\beta }+\sum _{i=1}^n{\displaystyle \frac{x_i^{\widehat{\gamma }}{e}^{x_i^{\widehat{\gamma }}}ln\left(x_i\right){(e^{x_i^{\widehat{\gamma }}}-1)}^{-3}}{\widehat{\alpha }+\widehat{\beta }{(e^{x_i^{\widehat{\gamma }}}-1)}^{-2}}}+\widehat{\alpha }\sum _{i=1}^n{x}_i^{\widehat{\gamma }}{e}^{x_i^{\widehat{\gamma }}}ln\left(x_i\right)\hfill \\ & & +\widehat{\beta }\sum _{i=1}^n{x}_i^{\widehat{\gamma }}{e}^{x_i^{\widehat{\gamma }}}ln\left(x_i\right){(e^{x_i^{\widehat{\gamma }}}-1)}^{-2}-\sum _{i=1}^n{x}_i^{\widehat{\gamma }}{e}^{x_i^{\widehat{\gamma }}}ln\left(x_i\right)\left(\widehat{\alpha }+\widehat{\beta }{(e^{x_i^{\widehat{\gamma }}}-1)}^{-2}\right)e^{\widehat{\alpha }(e^{x_i^{\widehat{\gamma }}}-1)-\widehat{\beta }{(e^{x_i^{\widehat{\gamma }}}-1)}^{-1}}=0.\hfill \end{array}$$

(30)

Equations (28)–(30) can be solved numerically, using R software 4.2.2., to obtain ${\widehat{\alpha }}_{ML}, {\widehat{\beta }}_{ML},$ and ${\widehat{\gamma }}_{ML}$.

4.2. Least Squares (LS) Procedure

Assume $x_1,x_2,....,x_n$ is a random sample with size n taken from the OFW-W and the order statistics for this sample is $x_{\left(1\right)},x_{\left(2\right)},....,x_{\left(n\right)}$. Then, the LS estimators of the OFW-W model, say ${\widehat{\alpha }}_{LS}, {\widehat{\beta }}_{LS},$ and ${\widehat{\gamma }}_{LS},$ can be found by solving the system of non-linear equations defined as:

$$\sum _{i=1}^n\left[F\left(x_{\left(i\right)}|\alpha , \beta , \gamma \right)-{\displaystyle \frac{i}{n+1}}\right]{\mathrm{\Delta }}_{\omega }\left(x_{\left(i\right)}|\alpha , \beta , \gamma \right)=0; \omega =1, 2, 3,$$

(31)

with respect to the parameters $\alpha , \beta ,$ and $\gamma ,$ where

$${\mathrm{\Delta }}_1\left(x_{\left(i\right)}|\alpha , \beta , \gamma \right)={\displaystyle \frac{\partial }{\partial \alpha }}F\left(x_{\left(i\right)}|\alpha , \beta , \gamma \right), {\mathrm{\Delta }}_2\left(x_{\left(i\right)}|\alpha , \beta , \gamma \right)={\displaystyle \frac{\partial }{\partial \beta }}F\left(x_{\left(i\right)}|\alpha , \beta , \gamma \right), {\mathrm{\Delta }}_3\left(x_{\left(i\right)}|\alpha , \beta , \gamma \right)={\displaystyle \frac{\partial }{\partial \gamma }}F\left(x_{\left(i\right)}|\alpha , \beta , \gamma \right).$$

(32)

Note that the solution of ${\mathrm{\Delta }}_{\omega }\left(x_{\left(i\right)}|\alpha , \beta , \gamma \right)$ can be obtained numerically.

4.3. Weighted Least Squares (WLS) Procedure

Consider a random sample of size n taken from the OFW-W model, where the order statistics for this sample is $x_{\left(1\right)}, x_{\left(2\right)}, ...., x_{\left(n\right)}$. Then, the WLS estimators of the OFW-W model, say ${\widehat{\alpha }}_{WLS}, {\widehat{\beta }}_{WLS},$ and ${\widehat{\gamma }}_{WLS},$ can be obtained by solving the non-linear system of equations defined by:

$$\sum _{i=1}^n{\displaystyle \frac{(n+2){(n+1)}^2}{i(n-i+1)}}\left[F\left(x_{\left(i\right)}|\alpha , \beta , \gamma \right)-{\displaystyle \frac{i}{n+1}}\right]{\mathrm{\Delta }}_{\omega }\left(x_{\left(i\right)}|\alpha , \beta , \gamma \right)=0; \omega =1, 2, 3,$$

(33)

for the parameters $\alpha , \beta ,$ and $\gamma$, where ${\mathrm{\Delta }}_{\omega }\left(x_{\left(i\right)}|\alpha , \beta , \gamma \right)$ is defined in Equation (32).

4.4. Cramér–Von Mises (CVM) Procedure

Let $x_{\left(1\right)}, x_{\left(2\right)}, ...., x_{\left(n\right)}$ be the order statistics of a random sample of size n taken from the OFW-W model. Then, the CVM estimators of the OFW-W model, say ${\widehat{\alpha }}_{CVM}, {\widehat{\beta }}_{CVM},$ and ${\widehat{\gamma }}_{CVM},$ can be obtained by solving the non-linear system of equations defined by:

$$\sum _{i=1}^n\left[F\left(x_{\left(i\right)}|\alpha , \beta , \gamma \right)+{\displaystyle \frac{1-2i}{2n}}\right]{\mathrm{\Delta }}_{\omega }\left(x_{\left(i\right)}|\alpha , \beta , \gamma \right)=0; \omega =1, 2, 3,$$

(34)

for the parameters $\alpha , \beta ,$ and $\gamma$, where ${\mathrm{\Delta }}_{\omega }\left(x_{\left(i\right)}|\alpha , \beta , \gamma \right)$ is defined in Equation (32).

4.5. Percentile Estimator (PC) Procedure

Let $u_i=i/(n+1)$ be the unbiased estimator of $F\left(x_{\left(i\right)}|\alpha , \beta , \gamma \right)$. Then, the PC estimators of the OFW-W model, say ${\widehat{\alpha }}_{PC}, {\widehat{\beta }}_{PC},$ and ${\widehat{\gamma }}_{PC},$ can be obtained by minimizing the function:

$$P(\alpha , \beta , \gamma )=\sum _{i=1}^n{\left[x_{\left(i\right)}-Q\left(u_i\right)\right]}^2,$$

(35)

with respect to $\alpha , \beta ,$ and $\gamma$, where $Q\left(u_i\right)=F^{-1}\left(x_{\left(i\right)}|\alpha , \beta , \gamma \right)$ is the quantile function of the OFW-W model.

4.6. Maximum Product of Spacings (MPS) Procedure

Assume $x_1, x_2, ...., x_n$ is a random sample of size n taken from the OFW-W model. For $i=1, 2, ...,n+1,$ let:

$$J_i(\alpha , \beta , \gamma )=F\left(x_{\left(i\right)}|\alpha , \beta , \gamma \right)-F\left(x_{(i-1)}|\alpha , \beta , \gamma \right)$$

be the uniform spacings of a random sample from the OFW-W, where $F\left(x_{\left(0\right)}|\alpha , \beta , \gamma \right)=0, F\left(x_{(n+1)}|\alpha , \beta , \gamma \right)=1,$ and ${\sum }_{i=1}^{n+1}{J}_i(\alpha , \beta , \gamma )=1.$ The MPS estimators of the OFW-W model, say ${\widehat{\alpha }}_{MPS}, {\widehat{\beta }}_{MPS},$ and ${\widehat{\gamma }}_{MPS},$ can be obtained by maximizing the geometric mean of the spacings defined as:

$$W(\alpha , \beta , \gamma )={\left[\prod _{i=1}^{n+1}{J}_i(\alpha , \beta , \gamma )\right]}^{{\displaystyle \frac{1}{n+1}}},$$

(36)

with respect to the parameters $\alpha , \beta ,$ and $\gamma$.

4.7. Anderson–Darling (AD) and Right-Tail Anderson–Darling (RAD) Procedures

The ADE and RADE are another types of minimum distance estimators. Let $x_1, x_2, ...., x_n$ be a random sample with size n taken from the OFW-W model. Then, the AD estimators of the OFW-W model, say ${\widehat{\alpha }}_{AD}, {\widehat{\beta }}_{AD},$ and ${\widehat{\gamma }}_{AD},$ can be derived by minimizing:

$$AD(\alpha , \beta , \gamma )=-n-{\displaystyle \frac{1}{n}}\sum _{i=1}^n(2i-1)\left[logF\left(x_{\left(i\right)}|\alpha , \beta , \gamma \right)+log\left(1-F\left(x_{\left(i\right)}|\alpha , \beta , \gamma \right)\right)\right],$$

(37)

for the parameters $\alpha , \beta ,$ and $\gamma$. On the other hand, the RAD estimators of the OFW-W model, say ${\widehat{\alpha }}_{RAD}, {\widehat{\beta }}_{RAD},$ and ${\widehat{\gamma }}_{RAD},$ can be found by minimizing the function:

$$RAD(\alpha , \beta , \gamma )={\displaystyle \frac{n}{2}}-2\sum _{i=1}^{n}F\left(x_{(i:n)}|\alpha , \beta , \gamma \right)-{\displaystyle \frac{1}{n}}\sum _{i=1}^n(2i-1)\left[log\left(1-F\left(x_{(n+1-i:n)}|\alpha , \beta , \gamma \right)\right)\right],$$

(38)

for the parameters $\alpha , \beta ,$ and $\gamma .$

5. Simulation Results

The efficiency and behavior of the MLEs, CVMEs, LSEs, PCEs, WLSEs, MPSEs, ADEs, and RADEs for the OFW-W parameters will be tested and examined, in this section, by conducting a simulation study for some choices of $\alpha , \beta , \gamma ,$ and n depending on three various criteria: mean squared errors (MSE), mean relative errors (MRE), and biases. Random samples of sizes $n=20, 50, 150,$ and 300 are generated from the OFW-W distribution with four various sets of parametric values $\left(\alpha , \beta , \gamma \right)\in \left\{\left(0.8, 2.9, 3.1\right), \left(0.5, 1.9, 1.1\right), \left(1.8, 1.3, 1.5\right), \left(0.3, 0.9, 0.6\right)\right\}$. Each sample is repeated $N=1000$ times to calculate the bias, MSE, and MRE of the unknown parameters using 1000 replications of random samples by the formulas:

$$\left|Bias\left(\mathrm{\Omega }\right)\right|={\displaystyle \frac{1}{1000}}\sum _{k=1}^{1000}\left|\widehat{\mathrm{\Omega }}-\mathrm{\Omega }\right|, MSE\left(\mathrm{\Omega }\right)={\displaystyle \frac{1}{1000}}\sum _{k=1}^{1000}{\left(\widehat{\mathrm{\Omega }}-\mathrm{\Omega }\right)}^2, MRE\left(\mathrm{\Omega }\right)={\displaystyle \frac{1}{1000}}\sum _{k=1}^{1000}{\displaystyle \frac{\left|\widehat{\mathrm{\Omega }}-\mathrm{\Omega }\right|}{\mathrm{\Omega }}},$$

where $\widehat{\mathrm{\Omega }}$ is an estimator of the vector parameter $\mathrm{\Omega }={\left(\alpha , \beta , \gamma \right)}^{\prime }.$ According to the empirical results of the simulation offered in

Table 1. Simulation results of the eight estimation methods for $\left(\alpha , \beta , \gamma \right)=\left(0.8, 2.9, 3.1\right)$.

n	Est	Par	LSE	CVME	MLE	WLSE	RADE	MPSE	PCE	ADE
20	$\left|Bias\right|$	$\widehat{\alpha }$	0.${5613}_1$	0.${6255}_5$	0.${7060}_7$	0.${6974}_6$	0.${6105}_4$	0.${7404}_8$	0.${5629}_2$	0.${5868}_3$
		$\widehat{\beta }$	0.${0940}_3$	0.${0939}_2$	0.${1036}_5$	0.${1064}_7$	0.${0924}_1$	0.${1747}_8$	0.${0951}_4$	0.${1054}_6$
		$\widehat{\gamma }$	0.${9375}_1$	1.${0724}_4$	1.${1747}_6$	1.${1953}_7$	1.${1037}_5$	1.${2710}_8$	1.${0159}_3$	0.${9807}_2$
	$MSE$	$\widehat{\alpha }$	0.${3106}_2$	0.${3857}_6$	0.${0317}_1$	0.${4795}_7$	0.${3674}_5$	0.${5404}_8$	0.${3123}_3$	0.${3395}_4$
		$\widehat{\beta }$	0.${0087}_3$	0.${0088}_4$	0.${0008}_1$	0.${0112}_7$	0.${0085}_2$	0.${0301}_8$	0.${0089}_5$	0.${0110}_6$
		$\widehat{\gamma }$	0.${9375}_2$	1.${1337}_5$	1.${3603}_7$	1.${4085}_8$	1.${2008}_6$	0.${5782}_1$	1.${0174}_4$	0.${9481}_3$
	$MRE$	$\widehat{\alpha }$	0.${3742}_1$	0.${4171}_5$	0.${4707}_7$	0.${4650}_6$	0.${4070}_4$	0.${4936}_8$	0.${3753}_2$	0.${3912}_3$
		$\widehat{\beta }$	0.${1880}_3$	0.${1877}_2$	0.${2071}_5$	0.${2128}_7$	0.${1848}_1$	0.${3494}_8$	0.${1902}_4$	0.${2109}_6$
		$\widehat{\gamma }$	0.${3125}_1$	0.${3575}_4$	0.${3916}_6$	0.${3984}_7$	0.${3679}_5$	0.${4237}_8$	0.${3386}_3$	0.${3269}_2$
Sum of Ranks			${17}_1$	${37}_5$	${45}_6$	${62}_7$	${33}_3$	${65}_8$	${30}_2$	${35}_4$
50	$\left|Bias\right|$	$\widehat{\alpha }$	0.${3518}_1$	0.${3844}_4$	0.${4220}_6$	0.${4283}_7$	0.${3980}_5$	0.${5040}_8$	0.${3663}_2$	0.${3704}_3$
		$\widehat{\beta }$	0.${0571}_1$	0.${0605}_3$	0.${0655}_6$	0.${0631}_5$	0.${0609}_4$	0.${1329}_8$	0.${0582}_2$	0.${0680}_7$
		$\widehat{\gamma }$	0.${6022}_1$	0.${6365}_2$	0.${7404}_6$	0.${7621}_7$	0.${6962}_5$	0.${8668}_8$	0.${6560}_4$	0.${6460}_3$
	$MSE$	$\widehat{\alpha }$	0.${1220}_2$	0.${1457}_5$	0.${0679}_1$	0.${1809}_7$	0.${1562}_6$	0.${2504}_8$	0.${1323}_3$	0.${1353}_4$
		$\widehat{\beta }$	0.${0032}_2$	0.${0036}_4$	0.${0014}_1$	0.${0039}_6$	0.${0037}_5$	0.${0174}_8$	0.${0033}_3$	0.${0046}_7$
		$\widehat{\gamma }$	0.${4022}_1$	0.${4994}_5$	0.${5405}_6$	0.${5725}_7$	0.${4778}_4$	0.${7406}_8$	0.${4242}_3$	0.${4114}_2$
	$MRE$	$\widehat{\alpha }$	0.${2345}_1$	0.${2563}_4$	0.${2813}_6$	0.${2856}_7$	0.${2653}_5$	0.${3360}_8$	0.${2442}_2$	0.${2470}_3$
		$\widehat{\beta }$	0.${1136}_1$	0.${1211}_3$	0.${1310}_6$	0.${1261}_5$	0.${1218}_4$	0.${2659}_8$	0.${1163}_2$	0.${1359}_7$
		$\widehat{\gamma }$	0.${2007}_1$	0.${2122}_2$	0.${2468}_6$	0.${2540}_7$	0.${2321}_5$	0.${2889}_8$	0.${2187}_4$	0.${2153}_3$
Sum of Ranks			${11}_1$	${32}_3$	${44}_6$	${58}_7$	${43}_5$	${72}_8$	${25}_2$	${39}_4$
150	$\left|Bias\right|$	$\widehat{\alpha }$	0.${1970}_1$	0.${2101}_3$	0.${2624}_7$	0.${2437}_6$	0.${2260}_5$	0.${3170}_8$	0.${2159}_4$	0.${2034}_1$
		$\widehat{\beta }$	0.${0317}_1$	0.${0326}_2$	0.${0383}_7$	0.${0353}_{4.5}$	0.${0343}_3$	0.${0858}_8$	0.${0353}_{4.5}$	0.${0366}_6$
		$\widehat{\gamma }$	0.${3304}_1$	0.${3632}_3$	0.${4266}_7$	0.${4189}_6$	0.${3875}_5$	0.${5118}_8$	0.${3593}_2$	0.${3840}_4$
	$MSE$	$\widehat{\alpha }$	0.${0382}_1$	0.${0435}_3$	0.${1756}_8$	0.${0585}_6$	0.${0503}_5$	0.${0990}_7$	0.${0460}_4$	0.${0408}_2$
		$\widehat{\beta }$	0.${0010}_{1.5}$	0.${0010}_{1.5}$	0.${0042}_7$	0.${0012}_4$	0.${0012}_4$	0.${0073}_8$	0.${0012}_4$	0.${0013}_6$
		$\widehat{\gamma }$	0.${1074}_1$	0.${1400}_3$	0.${1794}_7$	0.${1730}_6$	0.${1481}_5$	0.${2582}_8$	0.${1273}_2$	0.${1454}_4$
	$MRE$	$\widehat{\alpha }$	0.${1313}_1$	0.${1401}_3$	0.${1749}_7$	0.${1625}_6$	0.${1506}_5$	0.${2113}_8$	0.${1439}_4$	0.${1356}_2$
		$\widehat{\beta }$	0.${0634}_1$	0.${0653}_2$	0.${0765}_7$	0.${0706}_4$	0.${0687}_3$	0.${1716}_8$	0.${0707}_5$	0.${0732}_6$
		$\widehat{\gamma }$	0.${1101}_1$	0.${1211}_3$	0.${1422}_7$	0.${1396}_6$	0.${1292}_5$	0.${1706}_8$	0.${1198}_2$	0.${1280}_4$
Sum of Ranks			9.$5_1$	23.$5_2$	${64}_7$	48.$5_6$	${40}_5$	${71}_8$	31.$5_3$	${35}_4$
300	$\left|Bias\right|$	$\widehat{\alpha }$	0.${1369}_1$	0.${1472}_3$	0.${1793}_6$	0.${1836}_7$	0.${1503}_4$	0.${2232}_8$	0.${1534}_5$	0.${1462}_2$
		$\widehat{\beta }$	0.${0229}_1$	0.${0244}_4$	0.${0285}_7$	0.${0266}_5$	0.${0234}_2$	0.${0643}_8$	0.${0240}_3$	0.${0267}_6$
		$\widehat{\gamma }$	0.${2325}_1$	0.${2546}_2$	0.${3034}_7$	0.${3025}_6$	0.${2601}_3$	0.${3884}_8$	0.${2631}_4$	0.${2747}_5$
	$MSE$	$\widehat{\alpha }$	0.${0185}_1$	0.${0214}_3$	0.${4914}_8$	0.${0332}_6$	0.${0223}_4$	0.${0491}_7$	0.${0232}_5$	0.${0211}_2$
		$\widehat{\beta }$	0.${0005}_{1.5}$	0.${0006}_{3.5}$	0.${0106}_8$	0.${0007}_5$	0.${0005}_{1.5}$	0.${0041}_7$	0.${0006}_{3.5}$	0.${0008}_6$
		$\widehat{\gamma }$	0.${0325}_1$	0.${0639}_2$	0.${0908}_7$	0.${0902}_6$	0.${0667}_3$	0.${1488}_8$	0.${0638}_4$	0.${0744}_5$
	$MRE$	$\widehat{\alpha }$	0.${0913}_1$	0.${0981}_3$	0.${1195}_6$	0.${1224}_7$	0.${1002}_4$	0.${1489}_8$	0.${1022}_5$	0.${0975}_2$
		$\widehat{\beta }$	0.${0458}_1$	0.${0488}_4$	0.${0570}_7$	0.${0532}_5$	0.${0468}_2$	0.${1285}_8$	0.${0480}_3$	0.${0535}_6$
		$\widehat{\gamma }$	0.${0775}_1$	0.${0849}_2$	0.${1011}_7$	0.${1008}_6$	0.${0867}_3$	0.${1295}_8$	0.${0877}_4$	0.${0916}_5$
Sum of Ranks			9.$5_1$	26.$5_{2.5}$	${63}_7$	${53}_6$	26.$5_{2.5}$	${70}_8$	36.$5_4$	${39}_5$

,

Table 2. Simulation results of the eight estimation methods for $\left(\alpha , \beta , \gamma \right)=\left(0.5, 1.9, 1.1\right)$.

n	Est	Par	LSE	CVME	MLE	WLSE	RADE	MPSE	PCE	ADE
20	$\left|Bias\right|$	$\widehat{\alpha }$	0.${5423}_1$	0.${6043}_5$	0.${6821}_7$	0.${6738}_6$	0.${5898}_4$	0.${7153}_8$	0.${5438}_2$	0.${5669}_3$
		$\widehat{\beta }$	0.${0945}_3$	0.${0944}_2$	0.${1042}_5$	0.${1070}_7$	0.${0929}_1$	0.${1757}_8$	0.${0956}_4$	0.${1060}_6$
		$\widehat{\gamma }$	0.${5921}_1$	0.${8938}_3$	0.${9924}_7$	0.${9765}_6$	0.${9116}_4$	1.${0804}_8$	0.${8305}_2$	0.${9712}_5$
	$MSE$	$\widehat{\alpha }$	0.${3000}_2$	0.${3726}_6$	0.${0306}_1$	0.${4633}_7$	0.${3549}_5$	0.${5221}_8$	0.${3018}_3$	0.${3279}_4$
		$\widehat{\beta }$	0.${0088}_4$	0.${0087}_3$	0.${0008}_1$	0.${0112}_7$	0.${0085}_2$	0.${0303}_8$	0.${0090}_5$	0.${0111}_6$
		$\widehat{\gamma }$	0.${8107}_4$	0.${7198}_2$	0.${8873}_7$	0.${8590}_6$	0.${7486}_3$	1.${0516}_8$	0.${6214}_1$	0.${8497}_5$
	$MRE$	$\widehat{\alpha }$	0.${3615}_1$	0.${4030}_5$	0.${4547}_7$	0.${4492}_6$	0.${3932}_4$	0.${4769}_8$	0.${3625}_2$	0.${3779}_3$
		$\widehat{\beta }$	0.${1891}_3$	0.${1887}_2$	0.${2083}_5$	0.${2140}_7$	0.${1858}_1$	0.${3513}_8$	0.${1913}_4$	0.${2120}_6$
		$\widehat{\gamma }$	0.${2702}_1$	0.${2979}_3$	0.${3308}_7$	0.${3255}_6$	0.${3039}_4$	0.${3601}_8$	0.${2768}_2$	0.${3237}_5$
Sum of Ranks			${20}_1$	${31}_4$	${47}_6$	${58}_7$	${28}_3$	${72}_8$	${25}_2$	${43}_5$
50	$\left|Bias\right|$	$\widehat{\alpha }$	0.${3398}_1$	0.${3714}_4$	0.${4077}_6$	0.${4138}_7$	0.${3845}_5$	0.${4869}_8$	0.${3539}_2$	0.${3579}_3$
		$\widehat{\beta }$	0.${0574}_1$	0.${0609}_3$	0.${0659}_6$	0.${0634}_5$	0.${0612}_4$	0.${1337}_8$	0.${0585}_2$	0.${0683}_7$
		$\widehat{\gamma }$	0.${2297}_1$	0.${5647}_4$	0.${6135}_5$	0.${6424}_7$	0.${5472}_3$	0.${7437}_8$	0.${5379}_2$	0.${6239}_6$
	$MSE$	$\widehat{\alpha }$	0.${1178}_2$	0.${1407}_5$	0.${0656}_1$	0.${1748}_7$	0.${1509}_6$	0.${2419}_8$	0.${1278}_3$	0.${1307}_4$
		$\widehat{\beta }$	0.${0032}_2$	0.${0036}_4$	0.${0014}_1$	0.${0039}_6$	0.${0037}_5$	0.${0175}_8$	0.${0033}_3$	0.${0046}_7$
		$\widehat{\gamma }$	0.${5050}_8$	0.${2873}_3$	0.${3391}_4$	0.${3718}_6$	0.${2698}_2$	0.${4983}_7$	0.${2607}_1$	0.${3507}_5$
	$MRE$	$\widehat{\alpha }$	0.${2266}_1$	0.${2476}_4$	0.${2718}_6$	0.${2759}_7$	0.${2563}_5$	0.${3246}_8$	0.${2359}_2$	0.${2386}_3$
		$\widehat{\beta }$	0.${1142}_1$	0.${1218}_3$	0.${1317}_6$	0.${1268}_5$	0.${1224}_4$	0.${2674}_8$	0.${1169}_2$	0.${1367}_7$
		$\widehat{\gamma }$	0.${1683}_1$	0.${1882}_4$	0.${2045}_5$	0.${2142}_7$	0.${1824}_3$	0.${2479}_8$	0.${1793}_2$	0.${2080}_6$
Sum of Ranks			${18}_1$	${34}_3$	${40}_5$	${57}_7$	${37}_4$	${71}_8$	${19}_2$	${48}_6$
150	$\left|Bias\right|$	$\widehat{\alpha }$	0.${1903}_1$	0.${2030}_3$	0.${2535}_7$	0.${2354}_6$	0.${2183}_5$	0.${3062}_8$	0.${2086}_4$	0.${1965}_2$
		$\widehat{\beta }$	0.${0319}_1$	0.${0328}_2$	0.${0385}_7$	0.${0355}_{4.5}$	0.${0345}_3$	0.${0863}_8$	0.${0355}_{4.5}$	0.${0368}_6$
		$\widehat{\gamma }$	0.${0806}_1$	0.${3037}_2$	0.${3576}_6$	0.${3535}_5$	0.${3091}_3$	0.${4613}_8$	0.${3106}_4$	0.${3645}_7$
	$MSE$	$\widehat{\alpha }$	0.${0369}_1$	0.${0421}_3$	0.${1696}_8$	0.${0566}_6$	0.${0486}_5$	0.${0957}_7$	0.${0445}_4$	0.${0394}_2$
		$\widehat{\beta }$	0.${0010}_1$	0.${0011}_2$	0.${0043}_7$	0.${0012}_4$	0.${0012}_4$	0.${0073}_8$	0.${0012}_4$	0.${0013}_6$
		$\widehat{\gamma }$	0.${2989}_8$	0.${0831}_1$	0.${1152}_5$	0.${1126}_4$	0.${0861}_2$	0.${1917}_7$	0.${0869}_3$	0.${1197}_6$
	$MRE$	$\widehat{\alpha }$	0.${1268}_1$	0.${1354}_3$	0.${1690}_7$	0.${1570}_6$	0.${1455}_5$	0.${2041}_8$	0.${1391}_4$	0.${1310}_2$
		$\widehat{\beta }$	0.${0638}_1$	0.${0656}_2$	0.${0769}_7$	0.${0709}_4$	0.${0691}_3$	0.${1725}_8$	0.${0710}_5$	0.${0737}_6$
		$\widehat{\gamma }$	0.${0996}_1$	0.${1012}_2$	0.${1192}_6$	0.${1178}_5$	0.${1030}_3$	0.${1538}_8$	0.${1035}_4$	0.${1215}_7$
Sum of Ranks			${16}_1$	${20}_2$	${60}_7$	44.$5_6$	${33}_3$	${70}_8$	36.$5_4$	${44}_5$
300	$\left|Bias\right|$	$\widehat{\alpha }$	0.${1323}_1$	0.${1422}_3$	0.${1732}_6$	0.${1773}_7$	0.${1452}_4$	0.${2156}_8$	0.${1482}_5$	0.${1413}_2$
		$\widehat{\beta }$	0.${0230}_1$	0.${0245}_4$	0.${0286}_7$	0.${0268}_5$	0.${0235}_2$	0.${0646}_8$	0.${0241}_3$	0.${0269}_6$
		$\widehat{\gamma }$	0.${0356}_1$	0.${2235}_4$	0.${2541}_6$	0.${2496}_5$	0.${2141}_2$	0.${3392}_8$	0.${2194}_3$	0.${2621}_7$
	$MSE$	$\widehat{\alpha }$	0.${0179}_1$	0.${0206}_3$	0.${4747}_8$	0.${0321}_6$	0.${0215}_4$	0.${0475}_7$	0.${0224}_5$	0.${0204}_2$
		$\widehat{\beta }$	0.${0005}_{1.5}$	0.${0006}_{3.5}$	0.${0106}_8$	0.${0007}_5$	0.${0005}_{1.5}$	0.${0041}_7$	0.${0006}_{3.5}$	0.${0008}_6$
		$\widehat{\gamma }$	0.${1989}_8$	0.${0450}_3$	0.${0582}_5$	0.${0561}_4$	0.${0413}_1$	0.${1037}_7$	0.${0434}_2$	0.${0619}_6$
	$MRE$	$\widehat{\alpha }$	0.${0882}_1$	0.${0948}_3$	0.${1155}_6$	0.${1182}_7$	0.${0968}_4$	0.${1438}_8$	0.${0987}_5$	0.${0942}_2$
		$\widehat{\beta }$	0.${0460}_1$	0.${0490}_4$	0.${0573}_7$	0.${0535}_5$	0.${0471}_2$	0.${1292}_8$	0.${0482}_3$	0.${0538}_6$
		$\widehat{\gamma }$	0.${0663}_1$	0.${0746}_4$	0.${0847}_6$	0.${0832}_5$	0.${0714}_2$	0.${1131}_8$	0.${0731}_3$	0.${0874}_7$
Sum of Ranks			16.$5_1$	31.$5_3$	${59}_7$	${49}_6$	22.$5_2$	${69}_8$	32.$5_4$	${44}_5$

,

Table 3. Simulation results of the eight estimation methods for $\left(\alpha , \beta , \gamma \right)=\left(1.8, 1.3, 1.5\right)$.

n	Est	Par	LSE	CVME	MLE	WLSE	RADE	MPSE	PCE	ADE
20	$\left|Bias\right|$	$\widehat{\alpha }$	0.${3787}_1$	0.${4178}_5$	0.${4384}_7$	0.${4353}_6$	0.${4117}_4$	0.${4998}_8$	0.${4103}_3$	0.${4027}_2$
		$\widehat{\beta }$	0.${3327}_2$	0.${3395}_4$	0.${3576}_5$	0.${3816}_6$	0.${3359}_3$	0.${4863}_8$	0.${3294}_1$	0.${3899}_7$
		$\widehat{\gamma }$	0.${8350}_3$	0.${8347}_2$	0.${9577}_5$	0.${9690}_7$	0.${8423}_4$	1.${0781}_8$	0.${8319}_1$	0.${9677}_6$
	$MSE$	$\widehat{\alpha }$	0.${1463}_1$	0.${1781}_5$	0.${1961}_7$	0.${1933}_6$	0.${1730}_4$	0.${2549}_8$	0.${1718}_3$	0.${1655}_2$
		$\widehat{\beta }$	0.${1085}_2$	0.${1130}_4$	0.${1254}_5$	0.${1427}_6$	0.${1106}_3$	0.${2319}_8$	0.${1064}_1$	0.${1491}_7$
		$\widehat{\gamma }$	0.${6277}_3$	0.${6275}_2$	0.${8261}_5$	0.${8459}_7$	0.${6391}_4$	1.${0470}_8$	0.${6234}_1$	0.${8437}_6$
	$MRE$	$\widehat{\alpha }$	0.${2525}_1$	0.${2785}_5$	0.${2923}_7$	0.${2901}_6$	0.${2745}_4$	0.${3332}_8$	0.${2736}_3$	0.${2685}_2$
		$\widehat{\beta }$	0.${2218}_2$	0.${2263}_4$	0.${2384}_5$	0.${2544}_6$	0.${2240}_3$	0.${3242}_8$	0.${2196}_1$	0.${2599}_7$
		$\widehat{\gamma }$	0.${2782}_{2.5}$	0.${2782}_{2.5}$	0.${3192}_5$	0.${3230}_7$	0.${2808}_4$	0.${3594}_8$	0.${2773}_1$	0.${3226}_6$
Sum of Ranks			17.$5_2$	33.$5_4$	${51}_6$	${57}_7$	${33}_3$	${72}_8$	${15}_1$	${45}_5$
50	$\left|Bias\right|$	$\widehat{\alpha }$	0.${2412}_1$	0.${2591}_4$	0.${2778}_6$	0.${2971}_7$	0.${2517}_3$	0.${3372}_8$	0.${2515}_2$	0.${2613}_5$
		$\widehat{\beta }$	0.${1998}_1$	0.${2070}_4$	0.${2267}_5$	0.${2497}_7$	0.${2050}_2$	0.${3383}_8$	0.${2069}_3$	0.${2480}_6$
		$\widehat{\gamma }$	0.${4887}_1$	0.${5177}_2$	0.${5856}_5$	0.${5949}_6$	0.${5413}_4$	0.${7613}_8$	0.${5190}_3$	0.${5955}_7$
	$MSE$	$\widehat{\alpha }$	0.${0594}_1$	0.${0686}_4$	0.${0787}_6$	0.${0900}_7$	0.${0647}_3$	0.${1160}_8$	0.${0646}_2$	0.${0693}_5$
		$\widehat{\beta }$	0.${0391}_1$	0.${0420}_{3.5}$	0.${0504}_5$	0.${0611}_7$	0.${0412}_2$	0.${1122}_8$	0.${0420}_{3.5}$	0.${0603}_6$
		$\widehat{\gamma }$	0.${2152}_1$	0.${2415}_2$	0.${3089}_5$	0.${3188}_6$	0.${2639}_4$	0.${5221}_8$	0.${2427}_3$	0.${3194}_7$
	$MRE$	$\widehat{\alpha }$	0.${1609}_1$	0.${1727}_4$	0.${1852}_6$	0.${1980}_7$	0.${1678}_3$	0.${2248}_8$	0.${1677}_2$	0.${1738}_5$
		$\widehat{\beta }$	0.${1331}_1$	0.${1380}_{3.5}$	0.${1511}_5$	0.${1665}_7$	0.${1366}_2$	0.${2256}_8$	0.${1380}_{3.5}$	0.${1653}_6$
		$\widehat{\gamma }$	0.${1629}_1$	0.${1726}_2$	0.${1952}_5$	0.${1983}_6$	0.${1804}_4$	0.${2538}_8$	0.${1730}_3$	0.${1985}_7$
Sum of Ranks			$9_1$	${29}_4$	${48}_5$	${60}_7$	${27}_3$	${72}_8$	${25}_2$	${54}_6$
150	$\left|Bias\right|$	$\widehat{\alpha }$	0.${1290}_1$	0.${1430}_4$	0.${1661}_6$	0.${1694}_7$	0.${1370}_2$	0.${1974}_8$	0.${1435}_5$	0.${1422}_3$
		$\widehat{\beta }$	0.${1122}_1$	0.${1197}_3$	0.${1381}_6$	0.${1303}_5$	0.${1201}_4$	0.${2115}_8$	0.${1149}_2$	0.${1401}_7$
		$\widehat{\gamma }$	0.${2752}_1$	0.${2937}_2$	0.${3409}_6$	0.${3306}_5$	0.${3024}_3$	0.${4899}_8$	0.${3060}_4$	0.${3451}_7$
	$MSE$	$\widehat{\alpha }$	0.${0170}_1$	0.${0209}_4$	0.${0281}_6$	0.${0293}_7$	0.${0192}_2$	0.${0398}_8$	0.${0210}_5$	0.${0206}_3$
		$\widehat{\beta }$	0.${0123}_1$	0.${0140}_3$	0.${0187}_6$	0.${0166}_5$	0.${0141}_4$	0.${0445}_8$	0.${0129}_2$	0.${0192}_7$
		$\widehat{\gamma }$	0.${0682}_1$	0.${0777}_2$	0.${1047}_6$	0.${0985}_5$	0.${0824}_3$	0.${2162}_8$	0.${0844}_4$	0.${1073}_7$
	$MRE$	$\widehat{\alpha }$	0.${0860}_1$	0.${0953}_4$	0.${1107}_6$	0.${1129}_7$	0.${0914}_2$	0.${1317}_8$	0.${0957}_5$	0.${0948}_3$
		$\widehat{\beta }$	0.${0748}_1$	0.${0798}_3$	0.${0922}_6$	0.${0869}_5$	0.${0800}_4$	0.${1410}_8$	0.${0767}_2$	0.${0934}_7$
		$\widehat{\gamma }$	0.${0917}_1$	0.${0979}_2$	0.${1136}_6$	0.${1102}_5$	0.${1008}_3$	0.${1633}_8$	0.${1020}_4$	0.${1151}_7$
Sum of Ranks			$9_1$	${27}_{2.5}$	${54}_7$	${51}_{5.5}$	${27}_{2.5}$	${72}_8$	${33}_4$	${51}_{5.5}$
300	$\left|Bias\right|$	$\widehat{\alpha }$	0.${1002}_3$	0.${0974}_1$	0.${1117}_7$	0.${1109}_6$	0.${0995}_2$	0.${1417}_8$	0.${1045}_4$	0.${1085}_5$
		$\widehat{\beta }$	0.${0825}_1$	0.${0852}_3$	0.${0958}_6$	0.${0954}_5$	0.${0831}_2$	0.${1574}_8$	0.${0878}_4$	0.${0988}_7$
		$\widehat{\gamma }$	0.${1961}_1$	0.${2067}_3$	0.${2365}_5$	0.${2442}_6$	0.${2155}_4$	0.${3486}_8$	0.${2058}_2$	0.${2528}_7$
	$MSE$	$\widehat{\alpha }$	0.${0102}_3$	0.${0097}_1$	0.${0127}_7$	0.${0126}_6$	0.${0101}_2$	0.${0205}_8$	0.${0112}_4$	0.${0120}_5$
		$\widehat{\beta }$	0.${0067}_1$	0.${0071}_3$	0.${0090}_6$	0.${0089}_5$	0.${0068}_2$	0.${0243}_8$	0.${0076}_4$	0.${0096}_7$
		$\widehat{\gamma }$	0.${0346}_1$	0.${0385}_3$	0.${0504}_5$	0.${0537}_6$	0.${0419}_4$	0.${2162}_8$	0.${0381}_2$	0.${0576}_7$
	$MRE$	$\widehat{\alpha }$	0.${0668}_3$	0.${0649}_1$	0.${0744}_7$	0.${0739}_6$	0.${0664}_2$	0.${0945}_8$	0.${0697}_4$	0.${0723}_5$
		$\widehat{\beta }$	0.${0550}_1$	0.${0568}_3$	0.${0639}_6$	0.${0636}_5$	0.${0555}_2$	0.${1049}_8$	0.${0586}_4$	0.${0659}_7$
		$\widehat{\gamma }$	0.${0654}_1$	0.${0689}_3$	0.${0788}_5$	0.${0814}_6$	0.${0718}_4$	0.${1162}_8$	0.${0686}_2$	0.${0843}_7$
Sum of Ranks			${15}_1$	${21}_2$	${54}_6$	${51}_5$	${24}_3$	${72}_8$	${30}_4$	${57}_7$

,

Table 4. Simulation results of the eight estimation methods for $\left(\alpha , \beta , \gamma \right)=\left(0.3, 0.9, 0.6\right)$.

n	Est	Par	LSE	CVME	MLE	WLSE	RADE	MPSE	PCE	ADE
20	$\left|Bias\right|$	$\widehat{\alpha }$	0.${1358}_1$	0.${1501}_5$	0.${1590}_7$	0.${1585}_6$	0.${1446}_4$	0.${1748}_8$	0.${1367}_2$	0.${1392}_3$
		$\widehat{\beta }$	0.${6860}_2$	0.${6881}_3$	0.${7596}_6$	0.${7402}_5$	0.${7136}_4$	0.${9543}_8$	0.${6730}_1$	0.${7934}_7$
		$\widehat{\gamma }$	0.${1550}_1$	0.${1713}_5$	0.${1814}_7$	0.${1809}_6$	0.${1650}_4$	0.${1995}_8$	0.${1561}_2$	0.${1589}_3$
	$MSE$	$\widehat{\alpha }$	0.${6594}_2$	0.${6613}_3$	0.${7301}_6$	0.${7115}_5$	0.${6859}_4$	0.${9172}_8$	0.${6468}_1$	0.${7626}_7$
		$\widehat{\beta }$	0.${1087}_1$	0.${1492}_5$	0.${1838}_7$	0.${1637}_6$	0.${1387}_3$	0.${1842}_8$	0.${1175}_2$	0.${1464}_4$
		$\widehat{\gamma }$	0.${0213}_1$	0.${0260}_5$	0.${0291}_7$	0.${0290}_6$	0.${0241}_4$	0.${0352}_8$	0.${0215}_2$	0.${0224}_3$
	$MRE$	$\widehat{\alpha }$	0.${4392}_2$	0.${4418}_3$	0.${5385}_6$	0.${5113}_5$	0.${4751}_4$	0.${8498}_8$	0.${4226}_1$	0.${5875}_7$
		$\widehat{\beta }$	0.${2231}_1$	0.${2614}_5$	0.${2901}_7$	0.${2738}_6$	0.${2520}_3$	0.${2904}_8$	0.${2319}_2$	0.${2589}_4$
		$\widehat{\gamma }$	0.${3098}_1$	0.${3427}_5$	0.${3629}_7$	0.${3618}_6$	0.${3301}_4$	0.${3991}_8$	0.${3121}_2$	0.${3178}_3$
Sum of Ranks			${12}_1$	${39}_4$	${60}_7$	${51}_6$	${34}_3$	${72}_8$	${15}_2$	${41}_5$
50	$\left|Bias\right|$	$\widehat{\alpha }$	0.${2198}_2$	0.${2204}_3$	0.${2434}_6$	0.${2372}_5$	0.${2286}_4$	0.${3057}_8$	0.${2156}_1$	0.${2542}_7$
		$\widehat{\beta }$	0.${2120}_1$	0.${2178}_3$	0.${2494}_7$	0.${2485}_6$	0.${2217}_4$	0.${2937}_8$	0.${2166}_2$	0.${2373}_5$
		$\widehat{\gamma }$	0.${0913}_1$	0.${1056}_4$	0.${1112}_6$	0.${1146}_7$	0.${1004}_2$	0.${1377}_8$	0.${1015}_3$	0.${1076}_5$
	$MSE$	$\widehat{\alpha }$	0.${3932}_1$	0.${4260}_4$	0.${4687}_6$	0.${4583}_5$	0.${4149}_2$	0.${6413}_8$	0.${4178}_3$	0.${4772}_7$
		$\widehat{\beta }$	0.${0436}_1$	0.${0461}_3$	0.${0604}_7$	0.${0599}_6$	0.${0477}_4$	0.${0837}_8$	0.${0455}_2$	0.${0547}_5$
		$\widehat{\gamma }$	0.${0074}_1$	0.${0099}_4$	0.${0109}_6$	0.${0116}_7$	0.${0089}_2$	0.${0168}_8$	0.${0091}_3$	0.${0102}_5$
	$MRE$	$\widehat{\alpha }$	0.${1562}_1$	0.${1832}_4$	0.${2219}_6$	0.${2121}_5$	0.${1739}_2$	0.${4154}_8$	0.${1763}_3$	0.${2300}_7$
		$\widehat{\beta }$	0.${1413}_1$	0.${1452}_3$	0.${1663}_7$	0.${1656}_6$	0.${1478}_4$	0.${1958}_8$	0.${1444}_2$	0.${1582}_5$
		$\widehat{\gamma }$	0.${1826}_1$	0.${2111}_4$	0.${2223}_6$	0.${2291}_7$	0.${2008}_2$	0.${2753}_8$	0.${2030}_3$	0.${2152}_5$
Sum of Ranks			${10}_1$	${32}_4$	${57}_7$	${54}_6$	${26}_3$	${72}_8$	${22}_2$	${51}_5$
150	$\left|Bias\right|$	$\widehat{\alpha }$	0.${1311}_1$	0.${1420}_4$	0.${1563}_6$	0.${1528}_5$	0.${1383}_2$	0.${2138}_8$	0.${1393}_3$	0.${1591}_7$
		$\widehat{\beta }$	0.${1186}_1$	0.${1272}_2$	0.${1347}_6$	0.${1397}_7$	0.${1287}_3$	0.${1743}_8$	0.${1303}_4$	0.${1316}_5$
		$\widehat{\gamma }$	0.${0561}_1$	0.${0562}_2$	0.${0654}_7$	0.${0648}_6$	0.${0591}_3$	0.${0797}_8$	0.${0598}_4$	0.${0626}_5$
	$MSE$	$\widehat{\alpha }$	0.${2231}_1$	0.${2307}_2$	0.${2634}_5$	0.${2689}_6$	0.${2481}_4$	0.${4181}_8$	0.${2461}_3$	0.${2706}_7$
		$\widehat{\beta }$	0.${0137}_1$	0.${0157}_2$	0.${0176}_6$	0.${0190}_7$	0.${0162}_3$	0.${0294}_8$	0.${0165}_4$	0.${0168}_5$
		$\widehat{\gamma }$	0.${0028}_{1.5}$	0.${0028}_{1.5}$	0.${0038}_7$	0.${0037}_6$	0.${0031}_3$	0.${0056}_8$	0.${0032}_4$	0.${0035}_5$
	$MRE$	$\widehat{\alpha }$	0.${0503}_1$	0.${0538}_2$	0.${0701}_5$	0.${0731}_6$	0.${0622}_4$	0.${1766}_8$	0.${0612}_3$	0.${0740}_7$
		$\widehat{\beta }$	0.${0791}_1$	0.${0847}_2$	0.${0898}_6$	0.${0932}_7$	0.${0858}_3$	0.${1160}_8$	0.${0869}_4$	0.${0877}_5$
		$\widehat{\gamma }$	0.${1122}_1$	0.${1124}_2$	0.${1309}_7$	0.${1295}_6$	0.${1182}_3$	0.${1594}_8$	0.${1197}_4$	0.${1252}_5$
Sum of Ranks			9.$5_1$	19.$5_2$	${55}_6$	${56}_7$	${28}_3$	${72}_8$	${33}_4$	${51}_5$
300	$\left|Bias\right|$	$\widehat{\alpha }$	0.${0744}_1$	0.${0769}_2$	0.${0878}_5$	0.${0896}_6$	0.${0827}_4$	0.${1394}_8$	0.${0820}_3$	0.${0902}_7$
		$\widehat{\beta }$	0.${0862}_2$	0.${0849}_1$	0.${1001}_6$	0.${1024}_7$	0.${0907}_4$	0.${1264}_8$	0.${0868}_3$	0.${0959}_5$
		$\widehat{\gamma }$	0.${0379}_1$	0.${0403}_3$	0.${0459}_6$	0.${0466}_7$	0.${0414}_4$	0.${0598}_8$	0.${0392}_2$	0.${0445}_5$
	$MSE$	$\widehat{\alpha }$	0.${1604}_1$	0.${1652}_3$	0.${1886}_6$	0.${1837}_5$	0.${1662}_4$	0.${3122}_8$	0.${1647}_2$	0.${2028}_7$
		$\widehat{\beta }$	0.${0072}_2$	0.${0070}_1$	0.${0097}_6$	0.${0102}_7$	0.${0080}_4$	0.${0294}_8$	0.${0074}_3$	0.${0089}_5$
		$\widehat{\gamma }$	0.${0013}_1$	0.${0014}_{2.5}$	0.${0019}_6$	0.${0020}_7$	0.${0015}_4$	0.${0032}_8$	0.${0014}_{2.5}$	0.${0018}_5$
	$MRE$	$\widehat{\alpha }$	0.${0260}_1$	0.${0276}_3$	0.${0359}_6$	0.${0341}_5$	0.${0279}_4$	0.${0984}_8$	0.${0274}_2$	0.${0415}_7$
		$\widehat{\beta }$	0.${0575}_2$	0.${0566}_1$	0.${0667}_6$	0.${0683}_7$	0.${0605}_4$	0.${0843}_8$	0.${0579}_3$	0.${0640}_5$
		$\widehat{\gamma }$	0.${0759}_1$	0.${0806}_3$	0.${0919}_6$	0.${0931}_7$	0.${0828}_4$	0.${1198}_8$	0.${0785}_2$	0.${0889}_5$
Sum of Ranks			${12}_1$	19.$5_2$	${53}_6$	${58}_7$	${36}_4$	${72}_8$	22.$5_3$	${51}_5$

and

Table 5. Ranking of estimation methods based on simulation results.

	n	LSE	CVME	MLE	WLSE	RADE	MPSE	PCE	ADE
Schema I	20	1	5	6	7	3	8	2	4
	50	1	3	6	7	5	8	2	4
	150	1	2	7	6	5	8	3	4
	300	1	2.5	7	6	2.5	8	4	5
Schema II	20	1	4	6	7	3	8	2	5
	50	1	3	5	7	4	8	2	6
	150	1	2	7	6	3	8	4	5
	300	1	3	7	6	2	8	4	5
Schema III	20	2	4	6	7	3	8	1	5
	50	1	4	5	7	3	8	2	6
	150	1	2.5	7	5.5	2.5	8	4	5.5
	300	1	2	6	5	3	8	4	7
Schema IV	20	1	4	7	6	3	8	2	5
	50	1	4	7	6	3	8	2	5
	150	1	2	6	7	3	8	4	5
	300	1	2	6	7	4	8	3	5
Sum of Ranks		17	49	101	102.5	52	128	45	81.5
Mean of Ranks		1.06	3.06	6.31	6.41	3.25	8.00	2.81	5.09
SD of Ranks		0.25	0.981	0.704	0.664	0.837	0.0	1.047	0.779
Overall Rank		1	3	6	7	4	8	2	5

, we observe that the consistency characteristic for the ML, CVM, LS, PC, WLS, MPS, AD, and RAD estimators is satisfied as n grows. This means that all studied procedures can be utilized effectively to assess the OFW-W parameters. Figure 4 presents the mean and standard deviation for the ranking of the various estimation methods based on the simulation results.

6. Data Analysis

To clarify the importance and fitting capability of the OFW-W distribution in practice, we will analyze four applications to complete real, upper record and type-II right censored data. Our model will be compared with five fitted models; called, Weibull (W), exponentiated Weibull (EW), Topp-Leaone Weibull (ToLW), Odd log-logistic Weibull (OLogLW), and Type-I generalized exponentiated Weibull (TIGEW) distributions. The various criteria used here to make the required comparison with the other competitive models are the Kolmogorov–Smirnov (K-S) statistic, the p-values, and the measures for goodness of fit (GOF); especially the log-likelihood values (say $-L$), Cramér–von Mises (say $W^{*}$), and Anderson–Darling (say $A^{*}$) statistics. Finally, the WLS, LS, ML, MPS, CVM, AD, RAD, and PC estimators will be computed for the OFW-W model, with the corresponding K-S and p-values, to compare between the eight various estimation procedures.

6.1. Data I: Solidity of Glass Fibers

The data studied here were initially collected by the researchers at the UK’s National Physical Laboratory (NPL) and was introduced by Smith and Naylor [30] to represent 63 readings of the solidity of 1.5 cm glass fibers.

Table 6. The MLEs, p-values, K-S values, and GOF measures for data I.

Statistics	Models
	W	EW	ToLW	OLogLW	TIGEW	OFW-W
$\widehat{\alpha }$	−	−	$26.465$	$1.780$	$0.041$	$2.715$
$\widehat{\beta }$	−	−	$0.236$	−	$8.348$	$12.767$
$\widehat{\gamma }$	$1.718$	$2.051$	$1.857$	$0.486$	$0.562$	$0.292$
$\widehat{\theta }$	−	$5.828$	−	−	−	−
K-S	$0.622$	$0.231$	$0.221$	$0.641$	$0.255$	$0.116$
p-values	$0.00$	$0.002$	$0.004$	$0.00$	$0.001$	$0.485$
$-L$	$79.780$	$23.879$	$21.058$	$78.226$	$52.664$	$14.990$
$W^{*}$	$0.469$	$0.565$	$0.469$	$0.677$	$1.359$	$0.228$
$A^{*}$	$2.573$	$3.103$	$2.571$	$3.701$	$7.126$	$1.256$

specifies the maximum likelihood (ML) estimators, Kolmogorov–Smirnov (K-S) values, and the corresponding p-values for all compared distributions. It is evident that the new model has the largest p-value and the smallest K-S value. Moreover, the goodness-of-fit (GOF) measures confirm that the OFW-W model has the smallest values of $-L$, $A^{*}$, and $W^{*}$. Based on these results, the OFW-W model is the best for describing the first dataset compared to the other five competitive models.

$$\begin{array}{ccccccccccccc}\hfill 0.55& 0.93& 1.25\hfill & 1.36& 1.49& 1.52& 1.58& 1.61& 1.64& 1.68& 1.73& 1.81& 2.00\\ \hfill 0.74& 1.04& 1.27\hfill & 1.39& 1.49& 1.53& 1.59& 1.61& 1.66& 1.68& 1.76& 1.82& 2.01\\ \hfill 0.77& 1.11& 1.28\hfill & 1.42& 1.50& 1.54& 1.60& 1.62& 1.66& 1.69& 1.76& 1.84& 2.24\\ \multicolumn{1}{c}{0.81}& \multicolumn{1}{c}{1.13}& \multicolumn{1}{c}{1.29}& \multicolumn{1}{c}{1.48}& \multicolumn{1}{c}{1.51}& \multicolumn{1}{c}{1.55}& \multicolumn{1}{c}{1.61}& 1.62& 1.66& 1.70& 1.77& 1.84& 0.84\\ \multicolumn{1}{c}{1.24}& \multicolumn{1}{c}{1.30}& \multicolumn{1}{c}{1.48}& \multicolumn{1}{c}{1.51}& \multicolumn{1}{c}{1.55}& \multicolumn{1}{c}{1.61}& \multicolumn{1}{c}{1.63}& 1.67& 1.70& 1.78& 1.89& \end{array}$$

Figure 5 gives the quantile-quantile (QQ), box, total time test (TTT), Kernel density (KD), histogram, and the Violin plots. These non-parametric plots show that data I are asymmetric left skewed and has some extreme values. In Figure 6, the appreciated PDF, CDF, probability-probability (P-P), and SF plots of data I are shown. Figure 7 gives the contour and profile of log-likelihood functions plots for $\alpha , \beta ,$ and $\gamma$ of data set I, which reveals that the parameters of the OFW-W distribution are unimodal functions. The WLS, LS, ML, MPS, CVM, AD, RAD, and PC estimators are offered in

Table 7. The various estimators, p-values, and K-S values of data I.

Method	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\beta }}$	$\widehat{\mathit{\gamma }}$	K-S	p-Values
WLSE	$1.553$	$10.413$	$0.495$	$0.125$	$0.399$
LSE	$1.923$	$10.205$	$0.366$	$0.080$	$0.884$
MLE	$1.492$	$8.366$	$0.408$	$0.116$	$0.485$
MPSE	$2.843$	$14.658$	$0.343$	$0.089$	$0.800$
CVME	$2.588$	$13.841$	$0.363$	$0.080$	$0.884$
ADE	$1.629$	$8.968$	$0.392$	$0.101$	$0.661$
RADE	$2.714$	$14.377$	$0.358$	$0.084$	$0.847$
PCE	$1.538$	$8.283$	$0.379$	$0.112$	$0.497$

with the K-S values and p-values of the OFW-W model. The results obtained in Table 7 show that all estimation methods perform well for fitting data I and the LSE and CVME procedures are the most effective among them. On the other hand, Figure 8 offers the appreciated PDFs, CDFs, and SFs of data I, in accordance with the estimators given in Table 7.

6.2. Data II: Milk Production of SINDI Cows

This dataset comprises the total milk production from the first calving of 107 SINDI cows. The data were analyzed by Cordeiro and Birto [31]. Specifically, the dataset includes:

$$\begin{array}{ccccccccccccc}\hfill 0.4365& 0.4260& 0.5140\hfill & 0.6907& 0.7471& 0.2605& 0.6196& 0.8781& 0.4990& 0.605& 0.6891& 0.5770& 0.5394\\ \hfill 0.1479& 0.2356& 0.6012\hfill & 0.1525& 0.5483& 0.6927& 0.7261& 0.3323& 0.0671& 0.2361& 0.4800& 0.5707& 0.7131\\ \hfill 0.5853& 0.6768& 0.5350\hfill & 0.4151& 0.6789& 0.4576& 0.3259& 0.2303& 0.7687& 0.4371& 0.3383& 0.6114& 0.3480\\ \multicolumn{1}{c}{0.4564}& \multicolumn{1}{c}{0.7804}& \multicolumn{1}{c}{0.3406}& \multicolumn{1}{c}{0.4823}& \multicolumn{1}{c}{0.5912}& \multicolumn{1}{c}{0.5744}& \multicolumn{1}{c}{0.5481}& 0.1131& 0.7290& 0.0168& 0.5529& 0.4530& 0.3891\\ \multicolumn{1}{c}{0.4752}& \multicolumn{1}{c}{0.3134}& \multicolumn{1}{c}{0.3175}& \multicolumn{1}{c}{0.1167}& \multicolumn{1}{c}{0.6750}& \multicolumn{1}{c}{0.5113}& \multicolumn{1}{c}{0.5447}& 0.4143& 0.5627& 0.5150& 0.0776& 0.3945& 0.4553\\ \hfill 0.4470& 0.5285& 0.5232\hfill & 0.6465& 0.0650& 0.8492& 0.8147& 0.3627& 0.3906& 0.4438& 0.4612& 0.3188& 0.2160\\ \hfill 0.6707& 0.6220& 0.5629\hfill & 0.4675& 0.6844& 0.3413& 0.4332& 0.0854& 0.3821& 0.4694& 0.3635& 0.4111& 0.5349\\ \hfill 0.3751& 0.1546& 0.4517\hfill & 0.2681& 0.4049& 0.5553& 0.5878& 0.4741& 0.3598& 0.7629& 0.5941& 0.6174& 0.6860\\ \hfill 0.0609& 0.6488& 0.2747\hfill & & & & & & & & & \end{array}$$

The ML estimators, p-values, and K-S values for the compared distributions are presented in

Table 8. The MLS, p-values, K-S values, and GOF measures for data II.

Statistics	Models
	W	EW	ToLW	OLogLW	TIGEW	OFW-W
$\widehat{\alpha }$	−	−	$0.114$	$5.350$	$1.899$	$4.313$
$\widehat{\beta }$	−	−	$2.564$	−	$0.054$	$9.775$
$\widehat{\gamma }$	$1.457$	$7.755$	$7.613$	$0.465$	$7.743$	$0.132$
$\widehat{\theta }$	−	$0.144$	−	−	−	−
K-S	$0.470$	$0.245$	$0.099$	$0.097$	$0.179$	$0.082$
p-values	0	$5.3\times {10}^{-6}$	$0.242$	$0.262$	$0.002$	$0.470$
$-L$	$-41.279$	$-37.847$	$-24.545$	$-24.215$	$-29.34$	$-22.059$
$W^{*}$	$0.571$	$0.439$	$0.258$	$0.234$	$0.366$	$0.215$
$A^{*}$	$3.509$	$2.726$	$1.642$	$1.526$	$2.365$	$1.414$

. The measures for goodness of fit (GOF) show that the OFW-W model has the smallest values of $-L$, $A^{*}$, and $W^{*}$. Moreover, the OFW-W has the smallest K-S value and the largest p-value. These results confirm that the OFW-W model works quite well in modeling data II compared to the other five distributions.

Figure 9 represents the QQ, box, TTT, KD, histogram, and violin plots, showing that dataset II appears to be nearly symmetric with no extreme values found. Figure 10 displays the estimated PDF, CDF, P-P, and SF plots for this data. Figure 11 presents the contour and profile plots of the log-likelihood functions for $\alpha , \beta ,$ and $\gamma$ of dataset II, revealing that the OFW-W parameters are unimodal.

The WLS, LS, ML, MPS, CVM, AD, RAD, and PC estimators are listed in

Table 9. The various estimators, p-values, and K-S values of data II.

Method	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\beta }}$	$\widehat{\mathit{\gamma }}$	K-S	p-Values
WLSE	$3.237$	$6.393$	$0.212$	$0.074$	$0.593$
LSE	$3.233$	$6.394$	$0.213$	$0.075$	$0.578$
MLE	$4.318$	$9.790$	$0.1315$	$0.082$	$0.470$
MPSE	$2.054$	$2.702$	$0.501$	$0.082$	$0.465$
CVME	$4.315$	$9.78$	$0.132$	$0.082$	$0.469$
ADE	$4.119$	$9.212$	$0.144$	$0.067$	$0.727$
RADE	$2.067$	$2.628$	$0.459$	$0.082$	$0.462$
PCE	$2.704$	$4.923$	$0.251$	$0.067$	$0.727$

along with the p-values and K-S values of the OFW-W model. We can observe in Table 9 that the different estimation methods perform best for modeling data set II, with ADE and PCE being the most effective procedures. The estimated CDFs, PDFs, and SFs are illustrated in Figure 12.

6.3. Data III: Type-II Right Censored Data

In some areas of study, like reliability, medicine, engineering studies, and others, the experiment under study can be ended when the kth failure $x_{\left(k\right)}$ is observed. In this situation, the data are named a type-II right censored which saves cost and time but part of the information about the parameters is lost in the censored data, see Zheng and Park [32]. Let $x_1, x_2, ....., x_n$ be a randomly selected sample from the OFW-W, then the $L(\alpha , \beta , \gamma )$ of $x_{\left(1\right)}, x_{\left(2\right)}, ....., x_{\left(n\right)}, k\le n$ is determined using:

$$L(\alpha , \beta , \gamma )=ln\left({\ell }_{cen.}\right)=ln\left\{{\displaystyle \frac{n!}{(n-k)!}}{\left(R\left(x_k\right)\right)}^{n-k}\prod _{i=1}^{k}f\left(x_i\right)\right\},$$

(39)

Substituting Equations (6) and (7) into Equation (39), we get:

$$\begin{array}{ccc}\hfill L(\alpha , \beta , \gamma )& =& ln\left({\displaystyle \frac{n!}{(n-k)!}}\right)-(n-k)e^{\alpha (e^{x_k^{\gamma }}-1)-\beta {(e^{x_k^{\gamma }}-1)}^{-1}}+kln\left(\gamma \right)+\sum _{i=1}^k\left(x_i^{\gamma }\right)+(\gamma -1)\sum _{i=1}^{k}ln\left(x_i\right)\hfill \\ & & +\alpha \sum _{i=1}^k(e^{x_i^{\gamma }}-1)-\beta \sum _{i=1}^k{(e^{x_i^{\gamma }}-1)}^{-1}+\sum _{i=1}^{k}ln\left[\alpha +\beta {(e^{x_i^{\gamma }}-1)}^{-2}\right]-\sum _{i=1}^k{e}^{\alpha (e^{x_i^{\gamma }}-1)-\beta {(e^{x_i^{\gamma }}-1)}^{-1}}.\hfill \end{array}$$

(40)

By differentiating Equation (40) for $\alpha , \beta$, and $\gamma$, we will obtain:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial L}{\partial \alpha }}& =& -(n-k)(e^{x_k^{\gamma }}-1)e^{\alpha (e^{x_k^{\gamma }}-1)-\beta {(e^{x_k^{\gamma }}-1)}^{-1}}+\sum _{i=1}^k(e^{x_i^{\gamma }}-1)\hfill \\ & & +\sum _{i=1}^k{\displaystyle \frac{1}{\beta {(e^{x_i^{\gamma }}-1)}^{-2}+\alpha }}-\sum _{i=1}^k(e^{x_i^{\gamma }}-1)e^{\alpha (e^{x_i^{\gamma }}-1)-\beta {(e^{x_i^{\gamma }}-1)}^{-1}}.\hfill \end{array}$$

(41)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial L}{\partial \beta }}& =& (n-k){(e^{x_k^{\gamma }}-1)}^{-1}{e}^{\alpha (e^{x_k^{\gamma }}-1)-\beta {(e^{x_k^{\gamma }}-1)}^{-1}}-\sum _{i=1}^k{(e^{x_i^{\gamma }}-1)}^{-1}\hfill \\ & & +\sum _{i=1}^k{\displaystyle \frac{1}{\alpha {(e^{x_i^{\gamma }}-1)}^2+\beta }}+\sum _{i=1}^k{(e^{x_i^{\gamma }}-1)}^{-1}{e}^{\alpha (e^{x_i^{\gamma }}-1)-\beta {(e^{x_i^{\gamma }}-1)}^{-1}}\hfill \end{array}$$

(42)

and

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial L}{\partial \gamma }}& =& {\displaystyle \frac{k}{\gamma }}-(n-k)\left(x_k^{\gamma }\right)ln\left(x_k\right)\left(e^{x_k^{\gamma }}\right)\left[\alpha +\beta {(e^{x_k^{\gamma }}-1)}^{-2}\right]+\sum _{i=1}^k\left(1+x_i^{\gamma }\right)ln\left(x_i\right)+\sum _{i=1}^k\left(x_i^{\gamma }\right)ln\left(x_i\right)\left(e^{x_i^{\gamma }}\right)\hfill \\ & & \times \left\{{\displaystyle \frac{-2\beta {(e^{x_i^{\gamma }}-1)}^{-3}}{\alpha +\beta {(e^{x_i^{\gamma }}-1)}^{-2}}}+\left(\alpha +\beta {(e^{x_i^{\gamma }}-1)}^{-2}\right)\left(1-e^{\alpha (e^{x_i^{\gamma }}-1)-\beta {(e^{x_i^{\gamma }}-1)}^{-1}}\right)\right\}.\hfill \end{array}$$

(43)

By solving the resulting system of non-linear equations of ${\displaystyle \frac{\partial L}{\partial \alpha }},$${\displaystyle \frac{\partial L}{\partial \beta }}$, and ${\displaystyle \frac{\partial L}{\partial \gamma }}$, we can compute the ML estimators $({\widehat{\alpha }}_{ML}, {\widehat{\beta }}_{ML}, {\widehat{\gamma }}_{ML})$. McCool [33] studied the censored data, which gives the life of tiredness, in hours, for 10.0 bearings of a particular type. The data studied are 152.7, 172, 172.5, 173.5, 193, 204.7, 216.5, 234.9, 262.6, and 422.6. Eight values have been selected and analyzed from these data.

Table 10. The MLEs, –L values, K-S values, and the p-values of data III.

Models	MLEs	$-\mathbf{L}$	K-S	p-Values
W	$\widehat{\gamma }=2.476$	$25.635$	$0.376$	$0.159$
OFW-W	$\widehat{\alpha }=0.947, \widehat{\beta }=219.746, \widehat{\gamma }=0.395$	$20.945$	$0.125$	$0.987$

presents the MLEs, –L values, p-values, and K-S values for the W and OFW-W models. The OFW-W has the largest p-value and the smallest –L and K-S values. This result emphasizes that the OFW-W distribution describes the studied data perfectly, as opposed to the W distribution.

6.4. Data IV: Upper Recorded Data

The upper recorded values have a great and wide interest in a lot of fields, like sportive events, seismological and meteorological studies, weather forecasting, and others. Assume $x_1, x_2, ....., x_n$ is a random sample chosen from $OFW-W(\alpha , \beta , \gamma )$ and $X=\left\{X_{U\left(1\right)}, X_{U\left(2\right)}, ....., X_{U\left(n\right)}\right\}$ represent the upper recorded values chosen from it, then the $L(\alpha , \beta , \gamma )$ of X will be:

$$L(\alpha , \beta , \gamma )=ln\left\{f(x_{U\left(n\right)};\alpha , \beta , \gamma )\prod _{i=1}^{n-1}{\displaystyle \frac{f(x_{U\left(i\right)};\alpha , \beta , \gamma )}{R(x_{U\left(i\right)};\alpha , \beta , \gamma )}}\right\},$$

(44)

Substituting Equations (6) and (7) into Equation (44), we get:

$$\begin{array}{ccc}\hfill L(\alpha , \beta , \gamma )& =& nln\left(\gamma \right)+x_{U\left(n\right)}^{\gamma }+(\gamma -1)ln\left(x_{U\left(n\right)}\right)+ln\left[\alpha +\beta {(e^{x_{U\left(n\right)}^{\gamma }}-1)}^{-2}\right]-e^{\alpha \left(e^{x_{U\left(n\right)}^{\gamma }}-1\right)-\beta {\left(e^{x_{U\left(n\right)}^{\gamma }}-1\right)}^{-1}}\hfill \\ & & +\alpha \left(e^{x_{U\left(n\right)}^{\gamma }}-1\right)-\beta {\left(e^{x_{U\left(n\right)}^{\gamma }}-1\right)}^{-1}+\sum _{i=1}^{n-1}\left(x_{U\left(i\right)}^{\gamma }\right)+(\gamma -1)\sum _{i=1}^{n-1}ln\left(x_{U\left(i\right)}\right)\hfill \\ & & +\sum _{i=1}^{n-1}ln\left[\alpha +\beta {(e^{x_{U\left(i\right)}^{\gamma }}-1)}^{-2}\right]+\alpha \sum _{i=1}^{n-1}\left(e^{x_{U\left(i\right)}^{\gamma }}-1\right)-\beta \sum _{i=1}^{n-1}{\left(e^{x_{U\left(i\right)}^{\gamma }}-1\right)}^{-1}.\hfill \end{array}$$

(45)

If we differentiate Equation (45), for $\alpha , \beta$, and $\gamma$, we will obtain:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial L}{\partial \alpha }}& =& {\displaystyle \frac{1}{\alpha +\beta {(e^{x_{U\left(n\right)}^{\gamma }}-1)}^{-2}}}+\sum _{i=1}^{n-1}\left(e^{x_{U\left(i\right)}^{\gamma }}-1\right)+\sum _{i=1}^{n-1}{\displaystyle \frac{1}{\alpha +\beta {(e^{x_{U\left(i\right)}^{\gamma }}-1)}^{-2}}}\hfill \\ & & +\left(e^{x_{U\left(n\right)}^{\gamma }}-1\right)\left\{1-e^{\alpha \left(e^{x_{U\left(n\right)}^{\gamma }}-1\right)-\beta {\left(e^{x_{U\left(n\right)}^{\gamma }}-1\right)}^{-1}}\right\}.\hfill \end{array}$$

(46)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial L}{\partial \beta }}& =& {\displaystyle \frac{1}{\alpha {(e^{x_{U\left(n\right)}^{\gamma }}-1)}^2+\beta }}-\sum _{i=1}^{n-1}{\left(e^{x_{U\left(i\right)}^{\gamma }}-1\right)}^{-1}+\sum _{i=1}^{n-1}{\displaystyle \frac{1}{\alpha {(e^{x_{U\left(i\right)}^{\gamma }}-1)}^2+\beta }}\hfill \\ & & -{\left(e^{x_{U\left(n\right)}^{\gamma }}-1\right)}^{-1}\left\{1-e^{\alpha \left(e^{x_{U\left(n\right)}^{\gamma }}-1\right)-\beta {\left(e^{x_{U\left(n\right)}^{\gamma }}-1\right)}^{-1}}\right\}.\hfill \end{array}$$

(47)

and

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial L}{\partial \gamma }}& =& {\displaystyle \frac{n}{\gamma }}+(1+x_{U\left(n\right)}^{\gamma })ln\left(x_{U\left(n\right)}\right)+\sum _{i=1}^{n-1}(1-x_{U\left(i\right)}^{\gamma })ln\left(x_{U\left(i\right)}\right)+\left\{\left(x_{U\left(n\right)}^{\gamma }\right)ln\left(x_{U\left(n\right)}\right)\left(e^{x_{U\left(n\right)}^{\gamma }}\right)\right\}\hfill \\ & & \times \left\{\left(\alpha +\beta {(e^{x_{U\left(n\right)}^{\gamma }}-1)}^{-2}\right)\left(1-e^{\alpha \left(e^{x_{U\left(n\right)}^{\gamma }}-1\right)-\beta {\left(e^{x_{U\left(n\right)}^{\gamma }}-1\right)}^{-1}}\right)+{\displaystyle \frac{-2\beta {(e^{x_{U\left(n\right)}^{\gamma }}-1)}^{-3}}{\alpha +\beta {(e^{x_{U\left(n\right)}^{\gamma }}-1)}^{-2}}}\right\}\hfill \\ & & +\sum _{i=1}^{n-1}\left(x_{U\left(i\right)}^{\gamma }\right)ln\left(x_{U\left(i\right)}\right)\left(e^{x_{U\left(i\right)}^{\gamma }}\right)\left\{{\displaystyle \frac{-2\beta {(e^{x_{U\left(i\right)}^{\gamma }}-1)}^{-3}}{\alpha +\beta {(e^{x_{U\left(i\right)}^{\gamma }}-1)}^{-2}}}+\left(\alpha +\beta {(e^{x_{U\left(i\right)}^{\gamma }}-1)}^{-2}\right)\right\}.\hfill \end{array}$$

(48)

Equating Equations (46)–(48) to zero and solving the resulting system of ${\displaystyle \frac{\partial L}{\partial \alpha }},$${\displaystyle \frac{\partial L}{\partial \beta }}$, and ${\displaystyle \frac{\partial L}{\partial \gamma }}$ for $\alpha , \beta$, and $\gamma$, we will get the ML estimators $({\widehat{\alpha }}_{ML},{\widehat{\beta }}_{ML},{\widehat{\gamma }}_{ML})$. Lawless [34] introduced the record data analyzed here, which gives the breakdown lifetimes for a random sample of $n=11$ electrically isolated fluids and exposed to 30 kilovolts. The data studied are 2.84, 3.12, 3.05, 5.17, 4.93, 4.97, 3.02, 3.77, 5.27, 3.86, and 2.05. The upper recorded values are 2.84, 3.12, 5.17, and 5.27.

Table 11. The MLEs, –L values, K-S values, and the p-values of data IV.

Models	MLEs	–L	K-S	p-Values
W	$\widehat{\gamma }=5.836$	$5.398$	$0.627$	$0.103$
OFW-W	$\widehat{\alpha }=0.648, \widehat{\beta }=2.945, \widehat{\gamma }=1.846$	$2.657$	$0.433$	$0.429$

shows the MLEs, $-L$ values, p-values, and K-S values for the W and OFW-W distributions. The OFW-W has the largest p-value and the smallest $-L$ and K-S values. This result emphasizes that OFW-W fits data IV better than the W distribution.

7. Conclusions

The odd flexible Weibull-Weibull (OFW-W) distribution, a new continuous lifetime model with three parameters, has been proposed. It is considered a new flexible extension of the Weibull distribution. The OFW-W model is appropriate for describing symmetric and asymmetric positively and negatively skewed datasets with extreme observations and is characterized by an increasing, decreasing, bathtub-shaped, and unimodal hazard rate function (HRF). Some essential features and characteristics of the OFW-W model are discussed in detail. The unknown OFW-W parameters are obtained via eight different estimation methods, namely: WLSE, LSE, MLE, MPSE, CVME, ADE, RADE, and PCE. A detailed simulation study is performed to evaluate the behavior of the WLSEs, LSEs, MLEs, MPSEs, CVMEs, ADEs, RADEs, and PCEs based on the estimated mean squared errors (MSEs), biases, and mean relative errors (MREs). The practical importance of the OFW-W distribution is illustrated using four real datasets from engineering and medical fields, including complete, upper record, and Type-II right censored data, and it is compared with five other lifetime distributions. It was found that the new OFW-W model is more capable and flexible in fitting various forms of data than the other models compared. Some practical examples of areas where the proposed distribution might be applied include: (i) Human Health: It can model patient survival times in clinical trials, aiding in the assessment of treatment effectiveness for various diseases; (ii) Epidemiology: It can analyze the time until the onset of diseases in a population, offering insights into risk factors and health interventions; (iii) Reliability Engineering: It can be utilized to model the lifespan of machinery components, helping manufacturers understand failure rates and improve quality control; (iv) Actuarial Sciences: It can be employed to model the duration until claims are filed, assisting insurers in setting premiums and effectively managing risk; (v) Environmental Studies: It can assist in modeling the duration until a specific pollution level is attained in a particular area, facilitating environmental protection strategies; and (vi) Network Reliability: It can be used to model the time until network failures occur, assisting in the design of more robust communication systems.

Future research directions may include:

• Exploring further extensions of the OFW-W model to accommodate additional parameters for enhanced flexibility;
• Investigating the application of the OFW-W distribution in other fields, such as finance and social sciences, to evaluate its versatility;
• Developing robust estimation techniques to improve the accuracy of parameter estimation in challenging data scenarios;
• Conducting real-world case studies to validate the model’s effectiveness in various applications beyond those presented.

By addressing these areas, we aim to strengthen the understanding and applicability of the OFW-W distribution in diverse research contexts.