A New Flexible Model for Three-Stage Phenomena: The Fragmented Kumar-Trapez Distribution

Abstract

This article proposes a solution to the problem of limiting the representation of three-stage phenomena to linear forms and addresses the stability of the second stage by introducing a novel distribution, the Fragmented Kumar-Trapez (FKT) distribution, which includes two additional parameters beyond the parameters used for an existing standard model. These parameters provide flexibility to the density function, enabling it to model a wide range of shapes. This work contributes to the understanding of distributions whose probability density functions are divided into three parts, addressing key questions such as: How to handle such distributions? How to estimate the range parameters of the trapezoidal and proposed distributions using the maximum likelihood method? How to estimate the unknown parameters of the proposed distribution using both maximum likelihood and Bayesian methods? In addition, the article explores some of the mathematical properties of the proposed distribution. Finally, a simulation study on generated data and an illustrated example are conducted to demonstrate the practical importance of the FKT distribution. WinBUGS 1.4 program is used to illustrate the application of MCMC simulation.

1. Introduction

The trapezoidal distribution has been applied to various risk analysis problems [1,2]. However, studies on trapezoidal distributions remain relatively limited. Van Dorp and Kotz [3] introduced the Generalized Trapezoidal (GT) distribution, which models phenomena involving growth, stability, and physical or mental decline. Kacker and Lawrence [4] proposed the Type-B trapezoidal distribution, analyzing key properties such as the probability density function (PDF), cumulative distribution function (CDF), inverse CDF, moment-generating function (MGF), expected value, and variance. Additionally, Lawrence et al. [5] explored cases where the marginal distribution follows a trapezoidal form in greater detail.

Trapezoidal distributions are important in describing real-world phenomena that go through three stages, such as biological growth, economic cycles, and reliability assessments. These distributions capture the general shape of a process but are constrained by their linear structure. The second stage assumes complete stability, which may not accurately reflect real scenarios where gradual increases or decreases occur.

Despite their applicability, previous studies on trapezoidal distributions are limited in number and scope. Existing research has primarily focused on defining their properties rather than developing estimation methods or expanding their flexibility. Recognizing this gap, our study aims to introduce the Fragmented Kumar-Trapez (FKT) distribution, which enhances the traditional trapezoidal model by incorporating additional parameters. These parameters allow for a more versatile representation of different three-stage phenomena, accommodating nonlinear transitions in the stability stage.

Furthermore, this study presents rigorous estimation techniques for the FKT distribution, utilizing Maximum Likelihood and Bayesian methods. By evaluating the efficiency and accuracy of these methods through simulation studies, we demonstrate the practical advantages of the proposed model. This contribution extends the applicability of trapezoidal-based distributions, offering a more refined statistical tool for analyzing complex three-stage processes.

Our interest in trapezoidal distributions and their modifications stems mainly from the idea that many physical processes in nature, the human body, and mind over time reflect the shape of the trapezoidal distribution. Therefore, in this research, we strived to develop the trapezoidal distribution into a more flexible distribution and present it as one of the trapezoidal distributions.

The trapezoidal distribution has been used to detect cancer [6,7]. It is represented by three stages: the first stage, the growth stage; the second stage, the stability stage; and the third stage, the declining stage (decay or fading). It takes the form of a quadrilateral trapezoid with two parallel sides and two non-parallel sides, which tend to be suitable for data that show a fairly rapid growth and settlement period and then somewhat rapid disappearance [8]. These stages are all linear. The first stage increases with a positive slope, the settlement period is fixed, and the last stage decreases with a negative slope, as shown in Figure 1.

The PDF of the trapezoidal distribution is defined as follows:

$$f_0\left(x\right)=\left\{\begin{array}{l}{\displaystyle \frac{h{X}_1}{c-a}};a\le x\le c\\ h;c\le x\le d\\ {\displaystyle \frac{h{\mathrm{X}}_2}{b-d}};d\le x\le b\end{array}\right.,$$

(1)

and the CDF of the trapezoidal distribution is defined as follows:

$$F_0\left(x\right)=\left\{\begin{array}{l}{\displaystyle \frac{h{X_1}^2}{2\left(c-a\right)}};a\le x\le c\\ {\displaystyle \frac{h\left(c-a\right)+2\mathrm{h}{\mathrm{X}}_3}{2}};c\le x\le d\\ 1-{\displaystyle \frac{h{X_2}^2}{2\left(b-d\right)}};d\le x\le b\end{array}\right.,$$

(2)

where $X_1=x-a$, $X_2=b-x$, $X_3=x-c$, $h={\displaystyle \frac{2}{\left(d-c+b-a\right)}}$.

In this article, the trapezoidal distribution is developed to another distribution using the Kumaraswamy [9] method, based on cumulative distribution function (CDF): $F\left(x\right)={\left[1-F_0^{\lambda }\left(x\right)\right]}^{\beta }$, where $F_0\left(x\right)$, the CDF for the trapezoidal distribution.

In the rest of this article, the Fragmented Kumar-Trapez distribution with some mathematical properties and reliability measures is addressed in Section 2. The ideas of estimating parameters for the proposed distribution and for the trapezoidal distribution are addressed in Section 3. Section 4 reports the results of an MCMC simulation study. In Section 5, an illustrated example is provided to demonstrate the applicability of the proposed distribution to real-world data. Finally, Section 6 summarizes the conclusions and suggests future research directions.

2. Fragmented Kumar-Trapezoidal Distribution

Depending on the Kumaraswamy method, using the probability density function (PDF) in Equation (1) and the cumulative distribution function (CDF) in Equation (2), the PDF of the Fragmented Kumaraswamy Trapezoidal (FKT) distribution is defined as follows:

$$\left(x\right)=\left\{\begin{array}{l}2{\lambda }_1{\lambda }_2 X_1 {\left(1-{\left({\lambda }_2{X_1}^2\right)}^{\lambda }\right)}^{\beta -1}{\left({\lambda }_2{X_1}^2\right)}^{\lambda -1}; a\le x\le c\\ {\lambda }_{1}h{\left({\lambda }_3+h X_3\right)}^{\lambda -1}{\left(1-{\left({\lambda }_3+h X_3\right)}^{\lambda }\right)}^{\beta -1}; c\le x\le d\\ 2{\lambda }_1{\lambda }_4{X}_2 {\left(1-{\lambda }_4{X_2}^2\right)}^{\lambda -1}{\left(1-{\left(1-{\lambda }_4{X_2}^2\right)}^{\lambda }\right)}^{\beta -1}; d\le x\le b\end{array}\right.,$$

(3)

where ${\lambda }_1=\lambda \beta ,{\lambda }_2={\displaystyle \frac{h}{2(c-a)}},{\lambda }_3={\displaystyle \frac{h(c-a)}{2}},{\lambda }_4={\displaystyle \frac{h}{2(b-d)}}$.

It has been proven that the function in Equation (3) is a probability density function that is as follows:

$$\begin{array}{l}\underset{a}{\overset{b}{\int }}f\left(x\right)dx=\underset{a}{\overset{c}{\int }}f\left(x\right)dx+\underset{c}{\overset{d}{\int }}f\left(x\right)dx+\underset{d}{\overset{b}{\int }}f\left(x\right)dx\\ =1-(1-{{\lambda }_3}^{\lambda }{)}^{\beta }+(1-{{\lambda }_3}^{\lambda }{)}^{\beta }-{\left(1-({\lambda }_3+h(d-c){)}^{\lambda }\right)}^{\beta }+{\left(1-(1-{\displaystyle \frac{h(b-d)}{2}}{)}^{\lambda }\right)}^{\beta }\\ =1-{\left(1-{\left({\displaystyle \frac{2d-a-c}{b-a+d-c}}\right)}^{\lambda }\right)}^{\beta }+{\left(1-{\left({\displaystyle \frac{2d-a-c}{b-a+d-c}}\right)}^{\lambda }\right)}^{\beta }=1\end{array}$$

The FKT distribution will often be abbreviated $FKT\left(\lambda ,\beta \right)$, and its behavior is studied at different values of $\beta$ with fixed $\lambda$ and with different values of $\lambda$ with fixed $\beta$. It took multiple shapes at different values of the parameters $\lambda ,\beta$ such as: left and right triangular shape at $\beta <0.5,\lambda <1$, pyramid shape at $2<\beta$, $\lambda <4$, twisted curve shape to the right at $0.5<\beta <2$, $\lambda >4$, and has taken the moderate distribution shape at $\beta >4,\lambda >0$. Based on those forms, the FKT distribution can be represented by three stages, see Figure 2 and Figure 3. These stages differed by changing the values of $\beta$ and $\lambda$. Each of these three stages can be seen in three stages (growth—growth and fading—fading) as can be seen in Figure 4.

2.1. Reliability Measures

If $\mathrm{X}$ is a random variable that follows FKT distribution, then CDF, survival function $S\left(x\right)$, hazard function $h\left(x\right)$, cumulative hazard function $H\left(x\right)$ of the random variable $X$ are defined as follows:

$$\begin{gathered} F\left(x\right)=\left\{\begin{array}{l}1-{\left[1-\theta {X_1}^{2\lambda }\right]}^{\beta };a\le x\le c\\ 1-{\left[1-{\left({\lambda }_3+h{X}_1\right)}^{\lambda }\right]}^{\beta };c\le x\le d\\ 1-\varphi +\omega -{\left[1-{\left(1-{\lambda }_4{{\mathrm{X}}_2}^2\right)}^{\lambda }\right]}^{\beta };d\le x\le b\end{array}\right., \\ S\left(x\right)=\left\{\begin{array}{l}{\left[1-\theta {X_1}^{2\lambda }\right]}^{\beta };a\le x\le c\\ {\left[1-{\left({\lambda }_3+h{X}_3\right)}^{\lambda }\right]}^{\beta };c\le x\le d\\ \varphi -\omega +{\left[1-{\left(1-{\lambda }_4{X_2}^2\right)}^{\lambda }\right]}^{\beta };d\le x\le b\end{array}\right., \\ h\left(x\right)=\left\{\begin{array}{l}{\displaystyle \frac{2{\lambda }_1{\lambda }_2{X}_1{\left(1-{\left({\lambda }_2{X_1}^2\right)}^{\lambda }\right)}^{\beta -1}{\left({\lambda }_2{X_1}^2\right)}^{\lambda -1}}{{\left(1-\theta {X_1}^{2\lambda }\right)}^{\beta }}};a\le x\le c\\ {\displaystyle \frac{{\lambda }_{1}h{\left({\lambda }_3+h{X}_3\right)}^{\lambda -1}{\left(1-{\left({\lambda }_3+h{X}_3\right)}^{\lambda }\right)}^{\beta -1}}{{\left(1-{\left({\lambda }_3+h{X}_3\right)}^{\lambda }\right)}^{\beta }}};c\le x\le d\\ {\displaystyle \frac{2{\lambda }_2{\lambda }_4{X}_2{\left(1-{\lambda }_4{X_2}^2\right)}^{\lambda -1}{\left(1-{\left(1-{\lambda }_4{X_2}^2\right)}^{\lambda }\right)}^{\beta -1}}{\varphi -\omega +{\left(1-{\left(1-{\lambda }_4{X_2}^2\right)}^{\lambda }\right)}^{\beta }}};d\le x\le b\end{array}\right., \\ H(x)=\left\{\begin{array}{l}-\mathrm{l}\mathrm{n}\left[1-{\left(1-\theta {X_1}^{2\lambda }\right)}^{\beta }\right]; a\le x\le c\\ -\mathrm{l}\mathrm{n}\left[1-{\left[1-{\left({\lambda }_3+h{X}_3\right)}^{\lambda }\right]}^{\beta }\right]; c\le x\le d\\ -\mathrm{l}\mathrm{n}\left[1-\varphi +\omega -{\left(1-{\left(1-{\lambda }_4{X_2}^2\right)}^{\lambda }\right)}^{\beta }\right]; d\le x\le b\end{array}\right. \end{gathered}$$

The random numbers of the FKT distribution are as follows:

$$x_{rv}=a+{\theta }^{{\displaystyle \frac{-1}{2\lambda }}}{\left[1-{\left(u+\varphi -\omega -3+{\left(1-{\left({\lambda }_3+h\left(x_{rv}-c\right)\right)}^{\lambda }\right)}^{\beta }+{\left(1-{\left(1-{\lambda }_4\left(b-x_{rv}\right)\right)}^{\lambda }\right)}^{\beta }\right)}^{{\displaystyle \frac{1}{\beta }}}\right]}^{{\displaystyle \frac{1}{2\lambda }}},$$

(4)

where

$$\theta ={{\lambda }_2}^{\lambda }, \varphi ={\left[1-{\left({\lambda }_3+h\left(d-c\right)\right)}^{\lambda }\right]}^{\beta }, \omega ={\left[1-{\left(1-{\displaystyle \frac{h\left(b-d\right)}{2}}\right)}^{\lambda }\right]}^{\beta }$$

Equation (4) does not have a closed form solution, so $u$ will be created as uniform random variables from $u\left(\mathrm{0,1}\right)$, and then solve for $x_{rv}$ in order to create random numbers from the FKT distribution. From Equation (4), the quantile $x_q$ of the FKT distribution can be obtained. Figure 5, Figure 6 and Figure 7 show the shape of the cumulative distribution function (CDF), the shape of the survival function, and the shape of the risk function for the FKT distribution.

2.2. Arithmetic Mean, Variance, and Random Number Generation

The mean of the FKT distribution is defined as follows:

$$\mu =\left\{\begin{array}{l}a-c{\left[1-{\left({\lambda }_2{\left(c-a\right)}^2\right)}^{\lambda }\right]}^{\beta }+{\displaystyle \frac{I1}{2\lambda \sqrt{{\lambda }_2}}}; a\le x\le c\\ {\displaystyle \sum _{j=0}^{\infty }}\left(Q_j {\Delta }_j {\left(h\lambda \left({\displaystyle \frac{1}{\lambda }}+j\right)\right)}^{-1}\right)-d{\left(1-{\left({\lambda }_3+h\left(d-c\right)\right)}^{\lambda }\right)}^{\beta }-c{\left(1-{{\lambda }^{\lambda }}_3\right)}^{\beta }; c\le x\le d\\ d{\left[1-{\left(1-{\lambda }_4{\left(b-d\right)}^2\right)}^{\lambda }\right]}^{\beta }+{\displaystyle \frac{{\sum }_{j=0}^{\infty }{Q}_j I5}{2\sqrt{{\lambda }_4}}}; d\le x\le b\end{array}\right.$$

The 2nd row moment of the FKT distribution is defined as follows:

$${\mu }_2^{\prime }=\left\{\begin{array}{l}-\left(c^2{\left(1-{\left({\lambda }_2{\left(c-a\right)}^2\right)}^{\lambda }\right)}^{\beta }-a^2-{\displaystyle \frac{1}{\lambda \sqrt{{\lambda }_2}}}\left({\displaystyle \frac{I2}{\sqrt{{\lambda }_2}}}+aI1\right)\right);a\le x\le c\\ {\displaystyle \frac{1}{\lambda h}}{\displaystyle \sum _{j=0}^{\infty }}{Q}_{\mathrm{j}}\left(\Delta 2_j{h}^{-1}{\left({\displaystyle \frac{2}{\lambda }}+j\right)}^{-1}+{\Delta }_j\left(c-{\displaystyle \frac{{\lambda }_3}{\lambda }}\right){\left({\displaystyle \frac{1}{\lambda }}+j\right)}^{-1}\right);c\le x\le d\\ {\displaystyle \frac{1}{\lambda \sqrt{{\lambda }_4}}}\left(b\lambda {\displaystyle \sum _{j=0}^{\infty }}{Q}_{\mathrm{j}}I3-{\displaystyle \frac{I4}{\sqrt{{\lambda }_4}}}\right)+d{\left[1-{\left(1-{\lambda }_4{\left(b-d\right)}^2\right)}^{\lambda }\right]}^{\beta };d\le x\le b\end{array}\right.,$$

where $Q_j={\left(-1\right)}^j\left(\begin{array}{c}\beta \\ j\end{array}\right);∆j=B^{{\displaystyle \frac{1}{\lambda }}+j}-A^{{\displaystyle \frac{1}{\lambda }}+j};∆2j=B^{{\displaystyle \frac{2}{\lambda }}+j}-A^{{\displaystyle \frac{2}{\lambda }}+j}$.

Depending on the mean and the 2nd row moment, the variance of the FKT distribution is defined as follows:

$$\left(x\right)=\left\{\begin{array}{l}\left\{-c^2{\left[1-{\left[{\lambda }_2{\left(c-a\right)}^2\right]}^{\lambda }\right]}^{\beta }+a^2+{\displaystyle \frac{1}{\lambda \sqrt{{\lambda }_2}}}\left[{\displaystyle \frac{I2}{\sqrt{{\lambda }_2}}}+aI1\right]\right\}-{\left(a-c{\left[1-{\left({\lambda }_2{\left(c-a\right)}^2\right)}^{\lambda }\right]}^{\beta }+{\displaystyle \frac{I1}{2\lambda \sqrt{{\lambda }_2}}}\right)}^2\\ {\displaystyle \frac{1}{\lambda h}}{\displaystyle \sum _{j=0}^{\infty }}{Q}_j\left\{{\displaystyle \frac{\Delta 2_j}{h\left(\frac{2}{\lambda }+j\right)}}+{\displaystyle \frac{{\Delta }_j\left(c-\frac{{\lambda }_3}{\lambda }\right)}{\left(\frac{1}{\lambda }+j\right)}}\right\}-{\left\{{\displaystyle \sum _{j=0}^{\infty }}{\displaystyle \frac{Q_j{\Delta }_j}{h\lambda \left(\frac{1}{\lambda }+j\right)}}-d{\left[1-{\left({\lambda }_3+h\left(d-c\right)\right)}^{\lambda }\right]}^{\beta }-c\left(1-{{\lambda }^{\lambda }}_3\right)\right\}}^2\\ {\displaystyle \frac{1}{\lambda \sqrt{{\lambda }_4}}}\left\{b\lambda {\displaystyle \sum _{j=0}^{\infty }}{Q}_{j}I3-{\displaystyle \frac{I4}{\sqrt{{\lambda }_4}}}\right\}+d{\left[1-{\left(1-{\lambda }_4{\left(b-d\right)}^2\right)}^{\lambda }\right]}^{\beta }-{\left\{d{\left[1-{\left(1-{\lambda }_4{\left(b-d\right)}^2\right)}^{\lambda }\right]}^{\beta }+{\displaystyle \frac{{\displaystyle {\sum }_{j=0}^{\infty }}{Q}_{j}I5}{2\sqrt{{\lambda }_4}}}\right\}}^2\end{array},\right.$$

where $A={{\lambda }^{\lambda }}_3,A_1={\left[1-{\lambda }_4{\left(b-d\right)}^2\right]}^{\lambda },B={\left[{\lambda }_3+h\left(d-c\right)\right]}^{\lambda },Q_j={\left(-1\right)}^j\left(\begin{array}{c}\beta \\ j\end{array}\right),G=A_1^{{\displaystyle \frac{1}{\lambda }}}$ and ${{\rm I}}_a\left(\alpha ,\beta \right)$ an incomplete beta function defined as follows:

$$\begin{gathered} {{\rm I}}_a\left(\alpha ,\beta \right)={\int }_0^a{x}^{\alpha -1}(1-x{)}^{\beta -1}dx,I1={{\rm I}}_A\left({\displaystyle \frac{1}{2\lambda }},\beta +1\right),I2={{\rm I}}_A\left({\displaystyle \frac{1}{\lambda }},\beta +1\right), \\ I3={{\rm I}}_{1-G}\left(j\lambda +1,{\displaystyle \frac{1}{2}}\right),I4={{\rm I}}_{1-A_1}\left({\displaystyle \frac{1}{\lambda }},\beta +1\right),I5={{\rm I}}_{1-G}\left({\lambda }_{j+1,{\displaystyle \frac{1}{2}}}\right). \end{gathered}$$

The quantile function $Q_x\left(p\right)$ of the FKT distribution is defined as follows:

$$Q_x(p)=\left\{\begin{array}{l}a+{\left({\displaystyle \frac{1-{\left(1-p\right)}^{\frac{1}{\beta }}}{\theta }}\right)}^{{\displaystyle \frac{1}{2\lambda }}}; a\le x\le c\\ c+{\displaystyle \frac{{\left(1-{\left(1-p\right)}^{\frac{1}{\beta }}\right)}^{\frac{1}{\lambda }}-{\lambda }_3}{h}}; c\le x\le d\\ b-\sqrt{{\displaystyle \frac{1-{\left(1-{\left(1-\varphi +\omega -p\right)}^{\frac{1}{\beta }}\right)}^{\frac{1}{\lambda }}}{{\lambda }_4}}}; d\le x\le b\end{array}\right.$$

2.3. Inferences of Fragmented Kumaraswamy Trapezoid Parameters

The parameters of the FKT distribution are related to three different analytical expressions appearing in the definition of the structure of the FKT distribution. The derivation of the ML estimators for the parameters seems to be similar but more complicated compared to the one presented in Johnson and Kotz [10]. They dealt with two different analytical expressions of the structure of the standard triangular distribution. For more details, see [11]. Let $X_1,X_2,\dots ,X_n$ an independently and identically distributed (iid) random sample of size $n$ and the order statistics be $x_{\left(1\right)}<x_{\left(2\right)}<\dots <x_{\left(n\right)}$, then the likelihood and the log likelihood function of the STK PDF with parameters $\varpi =(a,c,d,b)$ are as follows:

$$L(\underset{\_}{x},\varpi )=\left\{\begin{array}{l}{L}_1(\underset{\_}{x};a,c)={\left(2{\lambda }_1{\lambda }_2\right)}^r{\displaystyle \prod _{i=1}^r}\left\{\left(x_{\left(i\right)}-a\right){\left[1-{\left({\theta }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}{\left[{\theta }_i\left(x\right)\right]}^{\lambda -1}\right\}\\ L_2(\underset{\_}{x};c,d)={\left({\lambda }_{1}h\right)}^{s-r}{\displaystyle \prod _{i=r+1}^s}\left\{{\left[B_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left(B_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}\\ L_3(\underset{\_}{x};d,b)={\left(2{\lambda }_1{\lambda }_4\right)}^{n-s}{\displaystyle \prod _{i=s+1}^n}\left\{\left(b-x_{\left(i\right)}\right){\left[{\phi }_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left({\phi }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}\end{array}\right.$$

(5)

$$\mathrm{l}\mathrm{n}L\left(\underset{\_}{x},\varpi \right)=\left\{\begin{array}{l}r\mathrm{l}\mathrm{n}\left(2\lambda \beta \right)+r\lambda \mathrm{l}\mathrm{n}{\lambda }_2+\left(\beta -1\right){\displaystyle \sum _{i=1}^r}\mathrm{l}\mathrm{n}\left(1-{\left({\theta }_i\left(x\right)\right)}^{\lambda }\right)+\left(2\lambda -1\right){\displaystyle \sum _{i=1}^r}\mathrm{l}\mathrm{n}\left(x_{\left(i\right)}-a\right)\\ \left(s-r\right)\mathrm{l}\mathrm{n}\left(h\lambda \beta \right)+\left(\lambda -1\right){\displaystyle \sum _{i=r+1}^s}\mathrm{l}\mathrm{n}\left[B_i\left(x\right)\right]+\left(\beta -1\right){\displaystyle \sum _{i=r+1}^s}\mathrm{l}\mathrm{n}\left[1-{\left(B_i\left(x\right)\right)}^{\lambda }\right]\\ \left(n-s\right)\mathrm{l}\mathrm{n}\left(2{\lambda }_4\lambda \beta \right)+{\displaystyle \sum _{i=s+1}^n}\mathrm{l}\mathrm{n}\left(b-x_{\left(i\right)}\right)+\left(\lambda -1\right){\displaystyle \sum _{i=s+1}^n}\mathrm{l}\mathrm{n}\left[{\phi }_i\left(x\right)\right]+\left(\beta -1\right){\displaystyle \sum _{i=s+1}^n}\mathrm{l}\mathrm{n}\left[1-{\left({\phi }_i\left(x\right)\right)}^{\lambda }\right]\end{array}\right.$$

(6)

Above $r,s$ are implicitly defined by $x_{\left(r\right)}<\widehat{c}<x_{\left(r+1\right)}$, $x_{\left(s\right)}<\widehat{d}<x_{\left(s+1\right)}$ and $x_{\left(1\right)}\equiv \widehat{a},x_{\left(n\right)}\equiv \widehat{b}$.

The ML estimator of $c$ and $d$ maximizing the likelihood in Equation (6) is $x_{\left(\widehat{r}\right)}\equiv \widehat{c},x_{\left(\widehat{s}\right)}\equiv \widehat{d}$, where

$$\widehat{r}=\underset{r\in \left\{1,...,s\right\}}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}{M}_{\left(r\right)};\widehat{s}=\underset{s\in \left\{r,...,n\right\}}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}{M}_{\left(s\right)},$$

(7)

such that,

$$M{R}_{\left(r\right)}={\displaystyle \prod _{i=1}^{r-1}}\left\{2{\lambda }_1{\lambda }_2\left(x_{\left(i\right)}-a\right){\left[1-{\left({\theta }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}{\left[{\theta }_i\left(x\right)\right]}^{\lambda -1}\right\}.{\displaystyle \prod _{i=r+1}^s}\left\{h{\lambda }_1{\left[B_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left(B_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}$$

(8)

$$S_{\left(s\right)}={\displaystyle \prod _{i=r+1}^{s-1}}\left\{h{\lambda }_1{\left[B_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left(B_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}.{\displaystyle \prod _{i=s+1}^n}\left\{2{\lambda }_1{\lambda }_4\left(b-x_{\left(i\right)}\right){\left[{\phi }_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left({\phi }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\},$$

(9)

with $h=2{\left(x_{\left(s\right)}-x_{\left(r\right)}+x_{\left(n\right)}-x_{\left(1\right)}\right)}^{-1}{\lambda }_1=\lambda \beta$ ${\lambda }_2={\displaystyle \frac{h}{2\left(x_{\left(r\right)}-x_{\left(1\right)}\right)}},{\lambda }_3={\displaystyle \frac{h\left(x_{\left(r\right)}-x_{\left(1\right)}\right)}{2}},$ ${\lambda }_4={\displaystyle \frac{h}{2\left(x_{\left(n\right)}-x_{\left(s\right)}\right)}},{\theta }_i\left(x\right)=\left({\lambda }_2{\left(x_{\left(i\right)}-x_{\left(1\right)}\right)}^2\right),$ $B_i\left(x\right)=\left({\lambda }_3+h\left(x_{\left(i\right)}-x_{\left(r\right)}\right)\right),{\phi }_i\left(x\right)=\left(1-{\lambda }_4{\left(x_{\left(n\right)}-x_{\left(i\right)}\right)}^2\right)$.

By taking the partial derivative of $\mathrm{l}\mathrm{n}L(\underset{\_}{x},\varpi )$ with respect to $\beta ,\lambda$, the closed form of $\widehat{\beta }$ is obtained as follows:

$$\widehat{\beta }=-3n{\left\{{\sum }_{i=1}^r\mathrm{l}\mathrm{n}\left(1-{\left({\theta }_i\left(x\right)\right)}^{\lambda }\right)+{\sum }_{i=r+1}^s\mathrm{l}\mathrm{n}\left(1-{\left(B_i\left(x\right)\right)}^{\lambda }\right)+{\sum }_{i=s+1}^n\mathrm{l}\mathrm{n}\left(1-{\left({\phi }_i\left(x\right)\right)}^{\lambda }\right)\right\}}^{-1}.$$

Unfortunately, to obtain $\widehat{\lambda }$, the computation of $\mathrm{l}\mathrm{n}L(\underset{\_}{x},\varpi )$ cannot be performed by solving the normal equations for $\lambda ,$ then it will be solving numerically, such that:

$$\begin{array}{l}{\displaystyle \frac{\partial \mathrm{l}\mathrm{n}L(\underset{\_}{x},\varpi )}{\partial \lambda }}={\displaystyle \frac{n}{\lambda }}+r\mathrm{l}\mathrm{n}{\lambda }_2+2{\displaystyle \sum _{i=1}^r}\mathrm{l}\mathrm{n}\left(x_{\left(i\right)}-a\right)+{\displaystyle \sum _{i=r+1}^s}\mathrm{l}\mathrm{n}\left\{B_i\left(x\right)\right\}+{\displaystyle \sum _{i=s+1}^n}\mathrm{l}\mathrm{n}\left\{{\phi }_i\left(x\right)\right\}\\ -\left(\beta -1\right)\left({\displaystyle \sum _{i=1}^r}{\displaystyle \frac{{\left({\theta }_i\left(x\right)\right)}^{\lambda }\mathrm{l}\mathrm{n}\left({\theta }_i\left(x\right)\right)}{1-{\left({\theta }_i\left(x\right)\right)}^{\lambda }}}+{\displaystyle \sum _{i=r+1}^s}{\displaystyle \frac{{\left(B_i\left(x\right)\right)}^{\lambda }\mathrm{l}\mathrm{n}\left(B_i\left(x\right)\right)}{1-{\left(B_i\left(x\right)\right)}^{\lambda }}}+{\displaystyle \sum _{i=s+1}^n}{\displaystyle \frac{{\left({\phi }_i\left(x\right)\right)}^{\lambda }\mathrm{l}\mathrm{n}\left({\phi }_i\left(x\right)\right)}{1-{\left({\phi }_i\left(x\right)\right)}^{\lambda }}}\right).\end{array}$$

Let $FI=F{I}_{ij}, i,j=\mathrm{1,2}$, the Fisher information (FI) matrix of $\lambda$ and $\beta$, then the observed FI matrix is as follows:

$$\widehat{F}I=\left[\begin{array}{ll}{\widehat{F}}_{11}& {\widehat{F}}_{12}\\ {\widehat{F}}_{21}& {\widehat{F}}_{22}\end{array}\right],$$

where

$$\begin{gathered} {\widehat{F}}_{11}=-\left[\left({\displaystyle \frac{{\partial }^2\mathrm{l}\mathrm{n}L(\underset{\_}{x},\varpi )}{\partial {\beta }^2}}\right)\downarrow \widehat{\beta }\right]={\displaystyle \frac{n}{{\widehat{\beta }}^2}}{\widehat{F}}_{22}=-\left[\left({\displaystyle \frac{{\partial }^2\mathrm{l}\mathrm{n}l}{\partial {\lambda }^2}}\right)\downarrow \widehat{\lambda }\right] \\ ={\displaystyle \frac{n}{{\lambda }^2}}+\left(\beta -1\right)\left\{{\displaystyle \sum _{i=1}^r}{\displaystyle \frac{{\left({\theta }_i\left(x\right)\right)}^{-\lambda }{\left(\mathrm{l}\mathrm{n}\left({\theta }_i\left(x\right)\right)\right)}^2}{{\left({\left({\theta }_i\left(x\right)\right)}^{-\lambda }-1\right)}^2}}+{\displaystyle \sum _{i=r+1}^s}{\displaystyle \frac{{\left(B_i\left(x\right)\right)}^{-\lambda }{\left(\mathrm{l}\mathrm{n}\left(B_i\left(x\right)\right)\right)}^2}{{\left({\left(B_i\left(x\right)\right)}^{-\lambda }-1\right)}^2}}+{\displaystyle \sum _{i=s+1}^n}{\displaystyle \frac{{\left({\phi }_i\left(x\right)\right)}^{-\lambda }{\left(\mathrm{l}\mathrm{n}\left({\phi }_i\left(x\right)\right)\right)}^2}{{\left({\left({\phi }_i\left(x\right)\right)}^{-\lambda }-1\right)}^2}}\right\}, \\ {\widehat{F}}_{12}={\widehat{F}}_{21}=-\left[\left({\displaystyle \frac{{\partial }^2\mathrm{l}\mathrm{n}l}{\partial \beta \partial \lambda }}\right)\downarrow \widehat{\beta },\widehat{\lambda }\right]={\displaystyle \sum _{i=1}^r}{\displaystyle \frac{{\left({\theta }_i\left(x\right)\right)}^{\lambda }\mathrm{l}\mathrm{n}\left({\theta }_i\left(x\right)\right)}{1-{\left({\theta }_i\left(x\right)\right)}^{\lambda }}}+{\displaystyle \sum _{i=r+1}^s}{\displaystyle \frac{{\left(B_i\left(x\right)\right)}^{\lambda }\mathrm{l}\mathrm{n}\left(B_i\left(x\right)\right)}{1-{\left(B_i\left(x\right)\right)}^{\lambda }}}+{\displaystyle \sum _{i=s+1}^n}{\displaystyle \frac{{\left({\phi }_i\left(x\right)\right)}^{\lambda }\mathrm{l}\mathrm{n}\left({\phi }_i\left(x\right)\right)}{1-{\left({\phi }_i\left(x\right)\right)}^{\lambda }}} \end{gathered}$$

Therefore, the estimated variance–covariance matrix of $\widehat{\lambda }$ and $\widehat{\beta }$ is as follows:

$$\mathsf{\Sigma }={\widehat{F}}^{-1}=\left[\begin{array}{ll}{\mathsf{\Sigma }}_{11}& {\mathsf{\Sigma }}_{12}\\ {\mathsf{\Sigma }}_{21}& {\mathsf{\Sigma }}_{22}\end{array}\right].$$

The asymptotic distributions of ${\displaystyle \frac{\widehat{\beta }-E\left(\widehat{\beta }\right)}{\sqrt{{\mathsf{\Sigma }}_{11}}}}$ and ${\displaystyle \frac{\widehat{\lambda }-E\left(\widehat{\lambda }\right)}{\sqrt{{\mathsf{\Sigma }}_{22}}}}$ are used to construct the asymptotic CIs of $\lambda$ and $\beta$. Thus, a $\left(1-\alpha \right)100\%$ confidence intervals of $\widehat{\lambda }$ and $\widehat{\beta }$ are as follows: $\widehat{\beta }\pm Z_{{\displaystyle \raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\sqrt{{\mathsf{\Sigma }}_{11}}$ and $\widehat{\lambda }\pm Z_{{\displaystyle \raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\sqrt{{\mathsf{\Sigma }}_{22}}$.

3. Bayesian Estimates

The Bayesian method is based on the assumption that the parameter $\theta$ is a random variable with a probability distribution $\pi \left(\theta \right)$ that expresses our previous information about the parameter $\theta$. This distribution is called the prior distribution because it depends on the researcher’s beliefs and previous experience with the parameter before the data were collected or before the sample was obtained. There are prior distributions and non-prior distributions. To apply the Bayesian method, we must have the posterior distribution, which is represented by the function that contains all the information contained in the likelihood function, in addition to the information about the parameter $\theta$. That is, we are looking for a function that includes all the information about the parameter $\theta$ after obtaining the sample. The posterior distribution $\pi \left(\theta /x\right)$ of the parameter $\theta$ is a conditional probability distribution of the parameter $\theta$ on the condition of obtaining the sample. That is, it describes the degree of our belief in the different values of the parameter $\theta$ after obtaining the sample.

The posterior distribution in Bayesian analysis replaces the likelihood function, in that it includes all the information about the parameter $\theta$ before obtaining the sample and the information contained in the sample about the parameter $\theta$. Therefore, if we want to estimate the parameter $\theta$, we can use the same method that we followed with the likelihood function. That is, we can estimate the parameter $\theta$ with the value that maximizes the posterior distribution. Since the posterior distribution is a probability distribution, the distribution can be summarized by one of the measures of central tendency, that is, we can estimate the parameter by its expected value or the median of the posterior distribution [8]. Therefore, the parameters of the Kumaraswamy trapezoid distribution will be estimated assuming that the prior distribution accompanying the parameter $\left(\beta \right)$ is the exponential distribution with the parameter $\left(\theta \right)$, the gamma distribution with two parameters $\left(\alpha ,\theta \right)$, and the uniform distribution.

3.1. Bayesian Estimates of the Shape Parameter $\beta$

The likelihood function in Equation (5) can be re-written as follows:

$$\begin{gathered} L\left(\underset{\_}{x},\lambda ,\beta \right)={\left(2{\lambda }_1{\lambda }_2\right)}^r{\displaystyle \prod _{i=1}^r}\left\{\left(x_{\left(i\right)}-a\right){\left[1-{\left({\theta }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}{\left[{\theta }_i\left(x\right)\right]}^{\lambda -1}\right\} \\ \begin{array}{l}+{\left({\lambda }_{1}h\right)}^{s-r}.{\displaystyle \prod _{i=r+1}^s}\left\{{\left[B_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left(B_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}\\ +{\left(2{\lambda }_1{\lambda }_4\right)}^{n-s}{\displaystyle \prod _{i=s+1}^n}\left\{\left(b-x_{\left(i\right)}\right){\left[{\phi }_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left({\phi }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}\end{array} \end{gathered}$$

(10)

Case 1:

Assume that the prior of $\left(\beta \right)$ is exponential $\left(\theta \right)$, i.e., $\pi \left(\beta \right)=\theta e^{-\theta \beta },\beta ,\theta >0$. Then, the posterior density function of $\left(\beta \right)$ is as follows:

$$\begin{array}{l}\pi \left(\beta /\underset{\_}{x}\right)\\ ={\displaystyle \frac{\theta e^{-\theta \beta }}{{\displaystyle {\int }_0^{\infty }}\pi \left(\beta \right)L(\underset{\_}{x},\lambda ,\beta )\mathrm{d}\beta }}\left\{{\left(2{\lambda }_1{\lambda }_2\right)}^r{\displaystyle \prod _{i=1}^r}\left\{\left(x_{\left(i\right)}-a\right){\left[1-{\left({\theta }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}{\left[{\theta }_i\left(x\right)\right]}^{\lambda -1}\right\}+{\left({\lambda }_{1}h\right)}^{s-r}\right.\\ .\left.{\displaystyle \prod _{i=r+1}^s}\left\{{\left[B_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left(B_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}+{\left(2{\lambda }_1{\lambda }_4\right)}^{n-s}{\displaystyle \prod _{i=s+1}^n}\left\{\left(b-x_{\left(i\right)}\right){\left[{\phi }_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left({\phi }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}\right\}\end{array}$$

Therefore, the Bayesian estimator $\tilde{\beta }$ of the shape parameter $\beta$ is as follows:

$$\tilde{\beta }={\displaystyle \frac{\theta }{A1}}\left\{{\displaystyle \prod _{i=1}^r}{\displaystyle \frac{M1\left(x_i\right)\mathsf{\Gamma }\left(r+2\right)}{{\left(\theta -\mathrm{l}\mathrm{n}M2\left(x_i\right)\right)}^{r+2}}}+{\displaystyle \prod _{i=r+1}^s}{\displaystyle \frac{M3\left(x_i\right)\mathsf{\Gamma }\left(s-r+2\right)}{{\left(\theta -\mathrm{l}\mathrm{n}M4\left(x_i\right)\right)}^{s-r+2}}}+{\displaystyle \prod _{i=s+1}^n}{\displaystyle \frac{M5\left(x_i\right)\mathsf{\Gamma }\left(n-s+2\right)}{{\left(\theta -\mathrm{l}\mathrm{n}M6\left(x_i\right)\right)}^{n-s+2}}}\right\},$$

(11)

where

$$\begin{gathered} \begin{array}{l}M1\left(x_i\right)={\left(2\lambda {\lambda }_2\right)}^r{\displaystyle \prod _{i=1}^r}\left\{\left(x_{\left(i\right)}-a\right){\left[{\theta }_i\left(x\right)\right]}^{\lambda -1}{\left(M2\left(x_i\right)\right)}^{-1}\right\};M2\left(x_i\right)=\left[1-{\left({\theta }_i\left(x\right)\right)}^{\lambda }\right]\\ M3\left(x_i\right)={\left(\lambda h\right)}^{s-r}{\displaystyle \prod _{i=r+1}^s}\left\{{\left[B_i\left(x\right)\right]}^{\lambda -1}{\left(M4\left(x_i\right)\right)}^{-1}\right\};M4\left(x_i\right)=\left[1-{\left(B_i\left(x\right)\right)}^{\lambda }\right]\\ M5\left(x_i\right)=\left.{\left(2\lambda {\lambda }_4\right)}^{n-s}{\displaystyle \prod _{i=s+1}^n}\left\{\left(b-x_{\left(i\right)}\right){\left[{\phi }_i\left(x\right)\right]}^{\lambda -1}{\left(M6\left(x_i\right)\right)}^{-1}\right\}\right\};M6\left(x_i\right)=\left[1-{\left({\phi }_i\left(x\right)\right)}^{\lambda }\right]\end{array} \\ A1=\underset{0}{\overset{\infty }{\int }}\pi \left(\beta \right)L(\underset{\_}{x},\lambda ,\beta )\mathrm{d}\beta \\ =\theta \left\{{\displaystyle \prod _{i=1}^r}{\displaystyle \frac{M1\left(x_i\right)\mathsf{\Gamma }\left(r+1\right)}{{\left(\theta -\mathrm{l}\mathrm{n}M2\left(x_i\right)\right)}^{r+1}}}+{\displaystyle \prod _{i=r+1}^s}{\displaystyle \frac{M3\left(x_i\right)\mathsf{\Gamma }\left(s-r+1\right)}{{\left(\theta -\mathrm{l}\mathrm{n}M4\left(x_i\right)\right)}^{s-r+1}}}+{\displaystyle \prod _{i=s+1}^n}{\displaystyle \frac{M5\left(x_i\right)\mathsf{\Gamma }\left(n-s+1\right)}{{\left(\theta -\mathrm{l}\mathrm{n}M6\left(x_i\right)\right)}^{n-s+1}}}\right\} \end{gathered}$$

Case 2:

Assume that the prior of $\left(\beta \right)$ is gamma $\left(\alpha ,\theta \right)$, i.e., $\pi \left(\beta \right)={\displaystyle \frac{{\theta }^{\alpha }}{\mathsf{\Gamma }\left(\alpha \right)}}{\beta }^{\alpha -1}{e}^{-\theta \beta },\beta ,\theta ,\alpha >0$. Then, the posterior density function of $\left(\beta \right)$ is as follows:

$$\begin{array}{l}\pi \left(\beta /\underset{\_}{x}\right)\\ ={\displaystyle \frac{{\theta }^{\alpha }{\beta }^{\alpha -1}{e}^{-\theta \beta }}{\mathsf{\Gamma }\left(\alpha \right){\int }_0^{\infty }\pi \left(\beta \right)L(\underset{\_}{x},\lambda ,\beta )\mathrm{d}\beta }}\left\{{\left(2{\lambda }_1{\lambda }_2\right)}^r{\displaystyle \prod _{i=1}^r}\left\{\left(x_{\left(i\right)}-a\right){\left[1-{\left({\theta }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}{\left[{\theta }_i\left(x\right)\right]}^{\lambda -1}\right\}+{\left({\lambda }_{1}h\right)}^{s-r}\right.\\ .\left.{\displaystyle \prod _{i=r+1}^s}\left\{{\left[B_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left(B_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}+{\left(2{\lambda }_1{\lambda }_4\right)}^{n-s}{\displaystyle \prod _{i=s+1}^n}\left\{\left(b-x_{\left(i\right)}\right){\left[{\phi }_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left({\phi }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}\right\}\end{array}$$

Therefore, the Bayesian estimator $\tilde{\beta }$ of the shape parameter $\beta$ is as follows:

$$\tilde{\beta }={\displaystyle \frac{{\theta }^{\alpha }}{\mathsf{\Gamma }\left(\alpha \right)A2}}.\left\{{\displaystyle \prod _{i=1}^r}{\displaystyle \frac{M1\left(x_i\right)\mathsf{\Gamma }\left(r+\alpha +1\right)}{{\left(\theta -\mathrm{l}\mathrm{n}M2\left(x_i\right)\right)}^{r+\alpha +1}}}+{\displaystyle \prod _{i=r+1}^s}{\displaystyle \frac{M3\left(x_i\right)\mathsf{\Gamma }\left(s-r+\alpha +1\right)}{{\left(\theta -\mathrm{l}\mathrm{n}M4\left(x_i\right)\right)}^{s-r+\alpha +1}}}+{\displaystyle \prod _{i=s+1}^n}{\displaystyle \frac{M5\left(x_i\right)\mathsf{\Gamma }\left(n-s+\alpha +1\right)}{{\left(\theta -\mathrm{l}\mathrm{n}M6\left(x_i\right)\right)}^{n-s+\alpha +1}}}\right\},$$

(12)

where

$$A2={\displaystyle \frac{{\theta }^{\alpha }}{\mathsf{\Gamma }\left(\alpha \right)}}\left\{{\displaystyle \prod _{i=1}^r}{\displaystyle \frac{M1\left(x_i\right)\mathsf{\Gamma }\left(r+\alpha \right)}{{\left(\theta -\mathrm{l}\mathrm{n}M2\left(x_i\right)\right)}^{r+\alpha }}}+{\displaystyle \prod _{i=r+1}^s}{\displaystyle \frac{M3\left(x_i\right)\mathsf{\Gamma }\left(s-r+\alpha \right)}{{\left(\theta -\mathrm{l}\mathrm{n}M4\left(x_i\right)\right)}^{s-r+\alpha }}}+{\displaystyle \prod _{i=s+1}^n}{\displaystyle \frac{M5\left(x_i\right)\mathsf{\Gamma }\left(n-s+\alpha \right)}{{\left(\theta -\mathrm{l}\mathrm{n}M6\left(x_i\right)\right)}^{n-s+\alpha }}}\right\}$$

Case 3:

Assume that the prior of $\left(\beta \right)$ is uniform $\left(\gamma ,\alpha \right)$, i.e., $\pi \left(\beta \right)={\displaystyle \frac{1}{\gamma -\alpha }},\gamma <\beta <\alpha ,\gamma ,\alpha >0$. Then, the posterior density function of $\left(\beta \right)$ is as follows:

$$\begin{array}{l}\pi (\beta /\underset{\_}{x})={\displaystyle \frac{1}{\left({\displaystyle {\int }_0^{\infty }}\pi \left(\beta \right)L(\underset{\_}{x},\lambda ,\beta )\mathrm{d}\beta \right)\left(\gamma -\alpha \right)}}\left\{{\left(2{\lambda }_1{\lambda }_2\right)}^r{\displaystyle \prod _{i=1}^r}\left\{\left(x_{\left(i\right)}-a\right){\left[1-{\left({\theta }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right.\right.\\ .\left.{\left[{\theta }_i\left(x\right)\right]}^{\lambda -1}\right\}+{\left({\lambda }_{1}h\right)}^{s-r}{\displaystyle \prod _{i=r+1}^s}\left\{{\left[B_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left(B_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}\\ \left.+{\left(2{\lambda }_1{\lambda }_4\right)}^{n-s}{\displaystyle \prod _{i=s+1}^n}\left\{\left(b-x_{\left(i\right)}\right){\left[{\phi }_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left({\phi }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}\right\}\end{array}$$

Therefore, the Bayesian estimator $\tilde{\beta }$ of the shape parameter $\beta$ is as follows:

$$\tilde{\beta }={\displaystyle \frac{1}{A3\left(\gamma -\alpha \right)}}.\left\{{\displaystyle \prod _{i=1}^r}{\displaystyle \frac{M1\left(x_i\right)\mathsf{\Gamma }\left(r+2\right)}{{\left(-\mathrm{l}\mathrm{n}M2\left(x_i\right)\right)}^{r+2}}}+{\displaystyle \prod _{i=r+1}^s}{\displaystyle \frac{M3\left(x_i\right)\mathsf{\Gamma }\left(s-r+2\right)}{{\left(-\mathrm{l}\mathrm{n}M4\left(x_i\right)\right)}^{s-r+2}}}+{\displaystyle \prod _{i=s+1}^n}{\displaystyle \frac{M5\left(x_i\right)\mathsf{\Gamma }\left(n-s+2\right)}{{\left(-\mathrm{l}\mathrm{n}M6\left(x_i\right)\right)}^{n-s+2}}}\right\},$$

(13)

where

$$A3={\displaystyle \frac{1}{\left(\gamma -\alpha \right)}}.\left\{{\displaystyle \prod _{i=1}^r}{\displaystyle \frac{M1\left(x_i\right)\mathsf{\Gamma }\left(r+1\right)}{{\left(-\mathrm{l}\mathrm{n}M2\left(x_i\right)\right)}^{r+1}}}+{\displaystyle \prod _{i=r+1}^s}{\displaystyle \frac{M3\left(x_i\right)\mathsf{\Gamma }\left(s-r+1\right)}{{\left(-\mathrm{l}\mathrm{n}M4\left(x_i\right)\right)}^{s-r+1}}}+{\displaystyle \prod _{i=s+1}^n}{\displaystyle \frac{M5\left(x_i\right)\mathsf{\Gamma }\left(n-s+1\right)}{{\left(-\mathrm{l}\mathrm{n}M6\left(x_i\right)\right)}^{n-s+1}}}\right\}$$

3.2. Bayesian Estimates of the Shape Parameter $\lambda$

Case 1:

Assume that the prior of $\left(\lambda \right)$ is exponential $\left(\theta \right)$, i.e., $\pi \left(\lambda \right)=\theta e^{-\theta \lambda },\lambda ,\theta >0$. Then, the posterior density function of $\left(\lambda \right)$ is as follows:

$$\begin{array}{l}\pi \left(\lambda /\underset{\_}{x}\right)\\ ={\displaystyle \frac{\theta e^{-\theta \lambda }}{{\displaystyle {\int }_0^{\infty }}\pi \left(\lambda \right)L(\underset{\_}{x},\lambda ,\beta )\mathrm{d}\lambda }}\left\{{\left(2{\lambda }_1{\lambda }_2\right)}^r{\displaystyle \prod _{i=1}^r}\left\{\left(x_{\left(i\right)}-a\right){\left[1-{\left({\theta }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}{\left[{\theta }_i\left(x\right)\right]}^{\lambda -1}\right\}+{\left({\lambda }_{1}h\right)}^{s-r}\right.\\ .\left.{\displaystyle \prod _{i=r+1}^s}\left\{{\left[B_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left(B_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}+{\left(2{\lambda }_1{\lambda }_4\right)}^{n-s}{\displaystyle \prod _{i=s+1}^n}\left\{\left(b-x_{\left(i\right)}\right){\left[{\phi }_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left({\phi }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}\right\}\end{array}$$

Therefore, the Bayesian estimator $\tilde{\lambda }$ of the shape parameter $\lambda$ is as follows:

$$\begin{array}{l}\tilde{\lambda }=E\left(\lambda /\underset{\_}{x}\right)\\ =\underset{0}{\overset{\infty }{\int }}\lambda \pi \left(\lambda /\underset{\_}{x}\right)\mathrm{d}\lambda \\ ={\displaystyle \frac{\theta }{{\displaystyle {\int }_0^{\infty }}\pi \left(\lambda \right)L(\underset{\_}{x},\lambda ,\beta )\mathrm{d}\lambda }}\underset{0}{\overset{\infty }{\int }}\lambda e^{-\theta \lambda }\left\{{\left(2{\lambda }_1{\lambda }_2\right)}^r{\displaystyle \prod _{i=1}^r}\left\{\left(x_{\left(i\right)}-a\right)\right.\right.\\ \left.{\left[1-{\left({\theta }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}{\left[{\theta }_i\left(x\right)\right]}^{\lambda -1}\right\}+{\left({\lambda }_{1}h\right)}^{s-r}{\displaystyle \prod _{i=r+1}^s}\left\{{\left[B_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left(B_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}\\ \left.+{\left(2{\lambda }_1{\lambda }_4\right)}^{n-s}{\displaystyle \prod _{i=s+1}^n}\left\{\left(b-x_{\left(i\right)}\right){\left[{\phi }_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left({\phi }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}\right\}\mathrm{d}\lambda \end{array}$$

(14)

Case 2:

Assume that the prior of $\left(\lambda \right)$ is Weibull $\left(k,\theta \right)$, i.e., $\pi \left(\lambda \right)=k\theta {\lambda }^{k-1}{e}^{-\theta {\lambda }^k},\lambda >0, k,\theta >0$. Then, the posterior density function of $\left(\lambda \right)$ is as follows:

$$\begin{array}{l}\pi \left(\lambda /\underset{\_}{x}\right)\\ ={\displaystyle \frac{k\theta {\lambda }^{k-1}{e}^{-\theta {\lambda }^k}}{{\displaystyle {\int }_0^{\infty }}\pi \left(\lambda \right)L(\underset{\_}{x},\lambda ,\beta )\mathrm{d}\lambda }}\left\{{\left(2{\lambda }_1{\lambda }_2\right)}^r{\displaystyle \prod _{i=1}^r}\left\{\left(x_{\left(i\right)}-a\right){\left[1-{\left({\theta }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}{\left[{\theta }_i\left(x\right)\right]}^{\lambda -1}\right\}+{\left({\lambda }_{1}h\right)}^{s-r}\right.\\ .\left.{\displaystyle \prod _{i=r+1}^s}\left\{{\left[B_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left(B_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}+{\left(2{\lambda }_1{\lambda }_4\right)}^{n-s}{\displaystyle \prod _{i=s+1}^n}\left\{\left(b-x_{\left(i\right)}\right){\left[{\phi }_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left({\phi }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}\right\}\end{array}$$

Therefore, the Bayesian estimator $\tilde{\lambda }$ of the shape parameter $\lambda$ is as follows:

$$\begin{array}{l}\tilde{\lambda }=E\left(\lambda /\underset{\_}{x}\right)\\ =\underset{0}{\overset{\infty }{\int }}\lambda \pi \left(\lambda /\underset{\_}{x}\right)\mathrm{d}\lambda \\ ={\displaystyle \frac{k\theta }{{\displaystyle {\int }_0^{\infty }}\pi \left(\lambda \right)L(\underset{\_}{x},\lambda ,\beta )\mathrm{d}\lambda }}\underset{0}{\overset{\infty }{\int }}\lambda {\lambda }^{k-1}{e}^{-\theta {\lambda }^k}\left\{{\left(2{\lambda }_1{\lambda }_2\right)}^r{\displaystyle \prod _{i=1}^r}\left\{\left(x_{\left(i\right)}-a\right)\right.\right.\\ .\left.{\left[1-{\left({\theta }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}{\left[{\theta }_i\left(x\right)\right]}^{\lambda -1}\right\}+{\left({\lambda }_{1}h\right)}^{s-r}{\displaystyle \prod _{i=r+1}^s}\left\{{\left[B_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left(B_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}\\ \left.+{\left(2{\lambda }_1{\lambda }_4\right)}^{n-s}{\displaystyle \prod _{i=s+1}^n}\left\{\left(b-x_{\left(i\right)}\right){\left[{\phi }_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left({\phi }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}\right\}d\lambda \end{array}$$

(15)

Case 3:

Assume that the prior of $\left(\lambda \right)$ is uniform $\left(\gamma ,\alpha \right)$, i.e., $\pi \left(\lambda \right)={\displaystyle \frac{1}{\gamma -\alpha }},\gamma <\lambda <\alpha ,\gamma ,\alpha >0$. Then, the posterior density function of $\left(\lambda \right)$ is as follows:

$$\begin{array}{l}\pi \left(\lambda /\underset{\_}{x}\right)\\ ={\displaystyle \frac{1}{\left({\displaystyle {\int }_0^{\infty }}\pi \left(\lambda \right)L(\underset{\_}{x},\lambda ,\beta )\mathrm{d}\lambda \right)\left(\gamma -\alpha \right)}}\left\{{\left(2{\lambda }_1{\lambda }_2\right)}^r{\displaystyle \prod _{i=1}^r}\left\{\left(x_{\left(i\right)}-a\right){\left[1-{\left({\theta }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}{\left[{\theta }_i\left(x\right)\right]}^{\lambda -1}\right\}\right.\\ +{\left({\lambda }_{1}h\right)}^{s-r}{\displaystyle \prod _{i=r+1}^s}\left\{{\left[B_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left(B_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}+{\left(2{\lambda }_1{\lambda }_4\right)}^{n-s}{\displaystyle \prod _{i=s+1}^n}\left\{\left(b-x_{\left(i\right)}\right){\left[{\phi }_i\left(x\right)\right]}^{\lambda -1}\right.\\ .\left.\left.{\left[1-{\left({\phi }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}\right\}\end{array}$$

Therefore, the Bayesian estimator $\tilde{\lambda }$ of the shape parameter $\lambda$ is as follows:

$$\begin{array}{l}\tilde{\lambda }=E\left(\lambda /\underset{\_}{x}\right)\\ =\underset{0}{\overset{\infty }{\int }}\lambda \pi \left(\lambda /\underset{\_}{x}\right)\mathrm{d}\lambda \\ ={\displaystyle \frac{1}{\left({\displaystyle {\int }_0^{\infty }}\pi \left(\lambda \right)L(\underset{\_}{x},\lambda ,\beta )\mathrm{d}\lambda \right)\left(\gamma -\alpha \right)}}\underset{0}{\overset{\infty }{\int }}\lambda \left\{{\left(2{\lambda }_1{\lambda }_2\right)}^r{\displaystyle \prod _{i=1}^r}\left\{\left(x_{\left(i\right)}-a\right)\right.\right..\\ \left.{\left[1-{\left({\theta }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}{\left[{\theta }_i\left(x\right)\right]}^{\lambda -1}\right\}+{\left({\lambda }_{1}h\right)}^{s-r}{\displaystyle \prod _{i=r+1}^s}\left\{{\left[B_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left(B_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}\\ +\left.{\left(2{\lambda }_1{\lambda }_4\right)}^{n-s}{\displaystyle \prod _{i=s+1}^n}\left\{\left(b-x_{\left(i\right)}\right){\left[{\phi }_i\left(x\right)\right]}^{\lambda -1}{\left[1-{\left({\phi }_i\left(x\right)\right)}^{\lambda }\right]}^{\beta -1}\right\}\right\}d\lambda \end{array}$$

(16)

Given the complexity of evaluating the integrals in Equations (14) and (15) for Bayesian estimation of the unknown parameters, we employ numerical simulations via Markov Chain Monte Carlo (MCMC) integration. Specifically, we utilize the WinBUGS 1.4 software to estimate the expected values of the unknown parameters. The MCMC algorithm is particularly effective for addressing challenges associated with posterior distributions, especially in cases where explicit solutions to posterior integrals are difficult to obtain [12]. This algorithm constructs a Markov chain that converges to a stationary distribution, allowing for the generation of samples from the posterior distribution $\pi (\theta │t)$. A key technique within MCMC is Gibbs sampling, which iteratively draws conditional samples, ultimately enabling statistical inference about the parameters based on the generated samples [13].

The Gibbs sampling method is one of the simplest and most widely used techniques within the MCMC framework. The Gibbs sampling procedure follows these steps:

Let $\pi (\theta │t)$ represent the posterior distribution of the unknown parameter vector $\theta =(\theta ₁,\theta ₂,\dots ,\theta \mathrm{ₖ})$. Given an initial set of parameter values $\theta ⁰=(\theta ₁⁰,\theta ₂⁰,\dots ,\theta \mathrm{ₖ}⁰)$, the sampling process proceeds as follows:

• Step 1: The parameter $\left({\theta }_1^1\right)$ is sampled from the conditional distribution $\pi \left({\theta }_1\right|{\theta }_2^0,{\theta }_3^0,\cdots ,{\theta }_k^0,\underset{\_}{t})$.
• Step 2: The parameter $\left({\theta }_2^1\right)$ is sampled from the conditional distribution $\pi \left({\theta }_2\right|{\theta }_1^1,{\theta }_3^0,\cdots ,{\theta }_k^0,\underset{\_}{t})$.
• Step k: The parameter ${\theta }_k^1$ is sampled from the conditional distribution $\pi \left({\theta }_k\right|{\theta }_1^1,{\theta }_2^1,\cdots ,{\theta }_{k-1}^1,\underset{\_}{t})$.

These steps (from Step 1 to Step $k$) are repeated $m$ times, generating a Markov chain:

$${\theta }^1=\left({\theta }_1^1,{\theta }_2^1,\cdots ,{\theta }_k^1\right),{\theta }^2=\left({\theta }_1^2,{\theta }_2^2,\cdots ,{\theta }_k^2\right),\dots ,{\theta }^m=\left({\theta }_1^m,{\theta }_2^m,\cdots ,{\theta }_k^m\right)$$

When $m$ is sufficiently large, the sampled values are considered simulated observations from the marginal posterior distribution of the parameter. The sample mean of these values serves as the Bayesian estimate of the parameter [14].

4. Simulation Study

This section illustrates the application of the various theoretical results developed in the previous sections on basis of generated data from FKT distribution as well as to assess the performance of SKT-parameters estimation for eleven different simulated random samples of sizes ranging from 10 to 20,000 data units by ML and Bayesian methods. The computations are performed using WinBUGS 1.4, MatheCad15, and R programs. MLEs for the unknown parameter $\beta ,\lambda$ were obtained for five different random samples, and WinBUGS 1.4 program was used to perform a Monte Carlo. The iterations continue until the Monte Carlo standard error value is less than 5% [15].

Table 1. Calculations of $M_{\left(r\right)}$ and $M_{\left(s\right)}$.

x(n)	${\mathit{x}}_{\left(\widehat{\mathit{s}}\right)}$	${\mathit{x}}_{\left(\widehat{\mathit{r}}\right)}$	x(1)	$\widehat{\mathit{s}}$	$\widehat{\mathit{r}}$	n
0.50	0.483	0.473	0.460	7	2	10
3.20	3.146	3.144	3.000	16	2	20
11.0	10.133	9.857	7.000	27	2	40
0.90	0.800	0.76	0.200	65	9	80
8.00	6.956	5.8208	2.000	95	19	100

summarizes the calculations of the ML estimates (MLEs) of the shape parameters with some measurements. The estimations of $\widehat{a}=x_{\left(1\right)},\widehat{b}=x_{\left(n\right)},\widehat{r},\widehat{\mathrm{s}},\widehat{c}=x_{\left(\widehat{r}\right)} \mathrm{and} \widehat{d}=x_{\left(\widehat{s}\right)}$ are estimated using Equations (7)–(9) at the initial values c₀, d₀, and for more explained, graphs of $M{R}_{\left(r\right)}$ and $M{S}_{\left(s\right)}$ defined by Equations (6)–(8) as shown in Figure 8. It shows that, for example at $n=10$, the maximum value of $M_{\left(r\right)}$ is the value corresponding to $x_{\left(2\right)}$, that is, $\widehat{r}=2$, and this leads us to that $x_{\left(\widehat{r}\right)}=x_{\left(2\right)}=0.473$ and accordingly $\widehat{c}=x_{\left(2\right)}=0.473$. The maximum value of $M_{\left(s\right)}$ is the value corresponding to $x_{\left(7\right)}$, that is, $\widehat{s}=7$, and this leads us to that $x_{\left(\widehat{s}\right)}=x_{\left(7\right)}=0.483$ and accordingly $\widehat{d}=x_{\left(7\right)}=0.483$, see Figure 8 and so on for the remaining samples.

As can be seen from

Table 2. ML estimates for the shape parameter beta.

UB	LB	$\mathbf{SD} (\widehat{\mathit{\beta }})$	$\widehat{\mathit{\beta }}$	$\mathit{n}$
1.0290	1.4669	0.3533	1.248	10
0.8858	1.0801	0.2216	0.983	20
0.9695	1.0684	0.1596	1.019	40
0.2716	0.6473	0.0835	0.559	80
0.6124	0.6435	0.0792	0.628	100

and

Table 3. ML estimates for the shape parameter lambda.

Iteration	UB	LB	$\mathbf{SD} (\widehat{\mathit{\lambda }})$	MC Error	Median	$\widehat{\mathit{\lambda }}$	$\mathit{n}$
1000	0.9733	0.02949	0.2899	0.009137	0.5015	0.5001	10
5000	0.9759	0.02713	0.2872	0.004419	0.5016	0.5026	20
4000	0.9733	0.02694	0.2858	0.004279	0.5028	0.5028	40
6000	0.9733	0.02726	0.2844	0.003688	0.5020	0.5011	80
6000	0.9742	0.02779	0.2843	0.003537	0.5020	0.5017	100

that the estimates of both parameters are reasonable for all considered random samples. The results show the stability of these estimated around the actual parameter values by which the data have been simulated, especially for the large sample sizes. It is clear from the results that the parameters’ estimates are reasonable for all the random samples studied. The results show the stability of the values of these estimators, especially for large sample sizes. Note that the value of the Monte Carlo standard error is small and decreases as the sample size increases.

Monte Carlo simulation was conducted to simulate Bayesian estimates of the unknown parameter ($\lambda$) using the WinBUGS 1.4 program at different sample sizes. The generated data and the estimated values of ($r,s$), and the estimated values of the parameters of the range of the random variable for the KT distribution ($c,d$) and three prior distributions (exponential–gamma–uniform) were used at each sample size to obtain Bayesian estimates of the unknown parameter ($\beta$). The Markov chain began by relying on different initial values for the unknown parameter ($\lambda$) and three prior distributions, as shown in

Table 4. Bayesian estimates of the shape parameters.

Prior distribution: $\beta ~Exp(\theta =7),\lambda ~\mathrm{E}\mathrm{x}\mathrm{p}\left(5\right)$, $\left({\lambda }_0=0.25\right)$
Iteration	UB	LB	MC error	D	Median	$\widehat{\lambda }$	$\widehat{\beta }$	$n$
1000	0.7922	0.006089	0.006415	0.1852	0.1374	0.1987	0.4445921	10
2000	0.7931	0.004873	0.004696	0.1844	0.1376	0.1934	0.5060638	20
3000	0.7433	0.005131	0.003898	0.1839	0.1376	0.1946	0.1708156	40
4000	0.7229	0.005508	0.003776	0.1833	0.1376	0.1959	0.4884839	80
5000	0.7214	0.004892	0.003043	0.1827	0.1381	0.1969	0.3119205	100
Prior distribution: $\beta ~Unif(0,\alpha =2),\lambda ~Unif\left(\mathrm{0.5,5}\right)$, $\left({\lambda }_0=3.5\right)$
Iteration	UB	LB	MC error	SD	Median	$\widehat{\lambda }$	$\widehat{\beta }$	$n$
1000	4.865	0.6114	0.04258	1.334	2.794	2.750	1.105512	10
2000	4.892	0.5857	0.02918	1.275	2.768	2.756	5.486938	20
3000	4.895	0.6349	0.02584	1.273	2.762	2.757	5.567503	40
3000	4.895	0.6349	0.02373	1.272	2.762	2.757	4.378639	80
5830	4.892	0.6146	0.02243	1.265	2.855	2.796	2.839402	100
Prior distribution: $\beta ~\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}\left(\alpha =2,\theta =7\right),\lambda ~\mathrm{W}\mathrm{e}\mathrm{i}\mathrm{b}\left(\mathrm{2,4}\right)$, $\left({\lambda }_0=0.25\right)$
Iteration	UB	LB	MC error	SD	Median	$\widehat{\lambda }$	$\widehat{\beta }$	$n$
1000	0.9951	0.08724	0.007543	0.2436	0.4113	0.4430	0.9609388	10
2000	0.9299	0.07321	0.005562	0.2230	0.4147	0.4381	4.8025200	20
2650	0.9326	0.07612	0.004862	0.2227	0.4118	0.4375	0.3978001	40
3100	0.9185	0.06871	0.003663	0.2217	0.4087	0.4324	1.0746990	80
4000	0.9501	0.08033	0.002153	0.2216	0.4086	0.4321	1.812131	100

. The Markov chain continues for a number $\left(m\right)$ of iterations at different sample sizes. Table 4 shows the number of iterations that were run for each Markov chain. The iterations continue until the Monte Carlo standard error value is less than 5% [15].

It is clear from the results of Table 4 that the parameter estimates are good for the random samples studied. The results show the stability of the parameter estimates through the simulated data, especially for large sample sizes, as we note that the value of the Monte Carlo standard error is small and decreases as the sample size increases. Noted also that Bayesian estimates of the parameters in the case of the prior exponential distribution were the best compared to the estimates in the cases of Weibull and uniform prior distributions. Also, Bayesian estimates in the cases of Weibull and exponential prior distributions were the best as they provided the least error compared to the maximum likelihood method. While in the case of a uniform prior distribution, the results showed that the maximum likelihood method is the best as it provided the least error compared to the Bayesian method.

The following plots are examples of the Weibull prior at $n=100$:

(i)

The trace plots for 20,000 iterations are presented in Figure 9.

(ii)

Chains for which convergence looks reasonable. Gelman–Rubin convergence statistic, R, is introduced. Figure 10 shows that the Gelman–Rubin statistic has converged, i.e., R is close to one [16].

(iii)

The shape of the posterior density of the parameter $\lambda$ is shown in Figure 11.

(iv)

The accuracy of the posterior estimate is calculated in terms of Monte Carlo standard error (MC error) of the mean according to Gelman et al. [8]. The simulation is run until the MC error for each node is less than 0.05 of the sample standard deviation. Therefore, the rule of MC error has been achieved. Table 3 and Table 4 show that the MC error for each node is less than 0.05 of the sample standard deviation.

(v)

Quantiles: The running mean with 95% confidence intervals plotted against the iteration number is shown in Figure 12.

It is clear from the results of

Table 5. Comparison between estimation methods.

Bayesian Estimates						ML Estimates
Uniform Prior Distribution		Weibull Prior Distribution		Exponential Prior Distribution
MC Error	$\widehat{\mathit{\lambda }}$	MC Error	$\widehat{\mathit{\lambda }}$	MC Error	$\widehat{\mathit{\lambda }}$	MC Error	$\widehat{\mathit{\lambda }}$	$\mathit{n}$
0.04258	2.750	0.007543	0.4430	0.006415	0.1987	0.009137	0.5001	10
0.02918	2.756	0.005562	0.4381	0.004696	0.1934	0.004419	0.5026	20
0.02584	2.757	0.004862	0.4375	0.003898	0.1946	0.004279	0.5028	40
0.02373	2.757	0.003663	0.4324	0.003776	0.1959	0.003688	0.5011	80
0.02243	2.796	0.002153	0.4321	0.003043	0.1969	0.003537	0.5017	100

that the parameter estimates are good for the random samples studied, and the results show the stability of the parameter estimates, especially for large sample sizes. As we note that the value of the MC error is small and decreases as the sample size increases. Also, we found that the Bayesian estimates of the parameter in the case of the prior exponential distribution were the best compared to the estimates in the cases of the prior Weibull and the prior uniform distributions. In addition, the Bayes estimates in the cases of the prior Weibull and the prior exponential distributions were the best as they provided the minimum MC error compared to the maximum likelihood method. While in the case of a prior uniform distribution, the results showed that the maximum likelihood method is the best as it provided the minimum MC error compared to the Bayes method in the case of a prior uniform distribution, see Figure 13.

The source code used in this study is available at this link (https://github.com/TahaRadwan2025/MCMC_Model_Code Accessed date: 10 March 2025).

5. Illustrated Example

The data were taken from Lawless [17]. These data represent the ages of a random sample of 10 pieces of industrial equipment, measured in months. The data are as follows: 81, 72, 70, 60, 41, 31, 31, 30, 29, 21. The results showed that the data fit the Kumaraswamy trapezoid distribution using the Kolmogorov–Smirnov test, where the test value was (0.27434), and it is less than the critical value (0.54179) at a significance level (α = 0.01). The ML of the random variable range parameters of the Kumaraswamy trapezoidal distribution was estimated using the initial values $\left(a=16,\begin{array}{l}b=89,\begin{array}{l}\lambda =0.9,\beta =3\end{array}\end{array}\right)$, and the calculations for $M_{\left(r\right)}$ and $M_{\left(s\right)}$ are summarized in

Table 6. Parameter estimates of the range of the random variable for the Kumaraswamy trapezoidal distribution of the industrial equipment data.

${\mathit{M}}_{\left(\mathit{s}\right)}$	${\mathit{M}}_{\left(\mathit{r}\right)}$	The Data
1.645 × 10−10	6.203 × 10−14	21	$x_{\left(1\right)}$
0.332 × 10−10	1.171 × 10−13	29	$x_{\left(2\right)}$
2.311 × 10−10	1.645 × 10−15	30	$x_{\left(3\right)}$
2.112 × 10−9	0.0002 × 10−15	31	$x_{\left(4\right)}$
1.073 × 10−10	0.0011 × 10−16	31	$x_{\left(5\right)}$
0.671 × 10−10	0.0002 × 10−16	41	$x_{\left(6\right)}$
2.876 × 10−10	0.0003 × 10−17	60	$x_{\left(7\right)}$
0.088 × 10−9	0.0031 × 10−18	70	$x_{\left(8\right)}$
4.14910 × 10−9	0.0023 × 10−18	72	$x_{\left(9\right)}$
5.28710 × 10−8	0.0033 × 10−17	81	$x_{\left(10\right)}$

.

We noted from the results of Table 6 that the maximum value of $M_{\left(r\right)}$ correspond to the value $x_{\left(2\right)}$, that is, $\widehat{r}=2$, and this leads us to that $x_{\left(\widehat{r}\right)}=x_{\left(2\right)}=29$, and therefore, $\widehat{c}=x_{\left(2\right)}=29$. Also, the maximum value of $x_{\left(10\right)}$ correspond to the value $M_{\left(s\right)}$, that is, $\widehat{s}=10$, and this leads us to that $x_{\left(\widehat{s}\right)}=x_{\left(10\right)}=81$, and therefore, $\widehat{d}=x_{\left(10\right)}=81$, and this is clear from Figure 14.

Figure 15 shows that the Kumaraswamy trapezoidal function is considered more flexible than the trapezoidal distribution. As the Kumaraswamy trapezoidal function took the form of a model that goes through three stages (implementation and propagation stage—slow decline stage—fading stage) and is not limited to linear shapes as is the case in the trapezoidal function in Figure 1.

It is clear from the results of

Table 7. Bayesian estimates of the shape parameters.

$\mathbf{P}\mathbf{r}\mathbf{i}\mathbf{o}\mathbf{r} \mathbf{D}\mathbf{i}\mathbf{s}\mathbf{t}\mathbf{r}\mathbf{i}\mathbf{b}\mathbf{u}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}:\mathit{\beta }~\mathbf{G}\mathbf{a}\mathbf{m}\mathbf{m}\mathbf{a}\left(\mathit{\alpha }=2,\mathit{\theta }=7\right),\mathit{\lambda }~\mathbf{W}\mathbf{e}\mathbf{i}\mathbf{b}\left(\mathrm{2,4}\right)$$,\left({\mathit{\lambda }}_0=0.90\right)$
Iteration	UB	LB	MC Error	SD	Median	$\widehat{\mathit{\lambda }}$	$\widehat{\mathit{\beta }}$	$\mathit{n}$
50,000	0.9539	0.07951	0.0009758	0.2305	0.4165	0.4425	3.000	10

that the parameter estimates are good for the real data, as we note that the value of the Monte Carlo standard error is small. The trace plots for 50,000 iterations are presented in Figure 16. Figure 17 shows that the Gelman–Rubin statistics are believed to have converged, and Figure 18 shows the shape of the posterior density of the parameter $\lambda$. Also, plotting the running mean with running 95% confidence intervals against iteration number is shown in Figure 19.

6. Conclusions

In this article, a new form of trapezoidal distribution is discussed, which apparently did not find any interest of previous studies on such distributions. This article is the first that discusses and deals with this distribution. Trapezoidal distribution was developed to the Fragmented Kumar-Trapez (FKT) distribution, its parameters are related to three different analytical expressions appearing in the definition of the structure of the KT distribution. Some mathematical properties, reliability measures, and maximum likelihood and Bayesian FKT parameters estimation are presented.

It is noted that trapezoidal distribution takes the form of a quadrilateral trapezoid with two parallel sides and two non-parallel sides and is represented by three stages, all are linear: the first stage increases with a positive slope, the settlement period is fixed, and the last stage decreases with a negative slope.

On the other hand, the new FKT distribution appears to be more flexible than the trapezoidal distribution; it took multiple shapes at different values of the parameters $\lambda ,\beta$ (left and right triangular shape—pyramid shape—twisted curve shape to the right—moderate distribution shape). Based on those forms, the KT distribution can be represented by three stages, differing by changing the values of $\beta ,\lambda$. So, it is reflected in the shape of a phenomenon that can be represented by three stages that are not limited to linear shapes but rather showed growth and dissolution, convex, nonlinear, or concave. To assess the performance of FKT parameters estimation, generated data were created from FKT distribution for different simulated random samples. The results show that the estimates of FKT parameters are reasonable for all considered random samples, with stability of these estimated around the actual parameter values as the sample sizeincreases.

The results also showed that the parameter estimates are good, and that the value of the MC error is small and decreases as the sample size increases. It also turns out that the Bayesian estimates in the case of the prior exponential distribution are better compared to the estimates in the case of the prior Weibull distribution and the prior uniform distribution. Also, the Bayesian estimates in the case of the prior Weibull distribution and the prior exponential distribution were the best, as they provided minimum MC error compared to the maximum likelihood method. While in the case of the prior uniform distribution, the results showed that the maximum likelihood method is the best as it provided minimum MC error compared to the Bayes method in the case of the prior uniform distribution.

Through the results of this article, it can be recommended to expand the use of different estimation methods and make comparisons between them; conduct further research to develop similar distributions whose probability density function is partitioned into two parts and three or more; also, complete the derivation of other properties of the Kumaraswamy trapezoidal distribution. In addition, we recommend using the general idea of Johnson and Kotz [10] to estimate the parameters of the range of the random variable for segmented distributions. Finally, we recommend conducting further research and developing similar distributions, especially those whose probability density function is divided into two or more parts.