The New Marshall–Olkin–Type II Exponentiated Half-Logistic–Odd Burr X-G Family of Distributions with Properties and Applications

Abstract

We develop a novel family of distributions named the Marshall–Olkin type II exponentiated half-logistic–odd Burr X-G distribution. Several mathematical properties including linear representation of the density function, Rényi entropy, probability-weighted moments, and distribution of order statistics are obtained. Different estimation methods are employed to estimate the unknown parameters of the new distribution. A simulation study is conducted to assess the effectiveness of the estimation methods. A special model of the new distribution is used to show its usefulness in various disciplines.

1. Introduction

Some recent work in statistical distribution theory includes generating new and generalized distributions by introducing additional parameter(s) to existing distributions. This results in flexible and generalized distributions. There are different generators available in the literature. These include Marshall–Olkin-G by [1], odd Burr X-G by [2], Topp–Leone-G by [3], type II half-logistic-G by [4], type II exponentiated half-logistic-G by [5], gamma-G (type 1) by [6], and gamma-G (type 2) by [7].

Some examples of recently generalized families of distributions include Marshall–Olkin–Topp–Leone-G family of distributions by [8], Marshall–Olkin extended Burr type XII distribution by [9], Marshall–Olkin generalized-G family of distributions by [10], Marshall–Olkin Fréchet distribution by [11], Marshall–Olkin generalized exponential distribution by [12], Marshall–Olkin alpha power Weibull distribution by [13], Marshall–Olkin alpha power inverse exponential distribution by [14], Marshall–Olkin logistic exponential distribution by [15], type II exponentiated half-logistic–odd Burr X-G power series class of distributions by [16], and Topp–Leone–odd Burr X-G family of distributions by [17]. Other distributions available in the literature include a comparative analysis of Poisson regression-based mean estimators by [18] as well as a paper on combining ratio and product type estimators for estimation of the finite population mean in adaptive cluster sampling design by [19].

Marshall and Olkin [1] introduced a family of distributions called the Marshall–Olkin-G (MO-G) distribution with cumulative distribution function (cdf) and probability distribution function (pdf), given by

$$\begin{array}{ccc}\hfill F_{MO-G}(x;\delta ,\mathsf{\Xi })& =& 1-{\displaystyle \frac{\delta \overline{G}(x;\mathsf{\Xi })}{1-(1-\delta )\overline{G}(x;\mathsf{\Xi })}}\hfill \end{array}$$

(1)

and

$$\begin{array}{ccc}\hfill f_{MO-G}(x;\delta ,\mathsf{\Xi })& =& \frac{\delta g(x;\mathsf{\Xi })}{{[1-(1-\delta )\overline{G}(x;\mathsf{\Xi })]}^2},\hfill \end{array}$$

(2)

respectively, for $x,\delta >0,$ where $\overline{G}(x;\mathsf{\Xi })=1-G(x;\mathsf{\Xi })$ is the survival function of the baseline distribution with the cdf $G(x;\mathsf{\Xi }).$

The type II exponentiated half-logistic-G (TIIEHL-G) was proposed by [5]. The cdf and pdf of the TIIEHL-G family of distributions are given by

$$\begin{array}{ccc}\hfill F_{TIIEHL-G}(x;\alpha ,\mathsf{\Xi })& =& 1-{\left[{\displaystyle \frac{1-G(x;\mathsf{\Xi })}{1+G(x;\mathsf{\Xi })}}\right]}^{\alpha }\hfill \end{array}$$

(3)

and

$$\begin{array}{ccc}\hfill f_{TIIEHL-G}(x;\alpha ,\mathsf{\Xi })& =& {\displaystyle \frac{2\alpha g(x;\mathsf{\Xi }){\left(1-G(x;\mathsf{\Xi })\right)}^{\alpha -1}}{{\left[1+G(x;\mathsf{\Xi })\right]}^{\alpha +1}}},\hfill \end{array}$$

(4)

respectively, where $\alpha >0$ is the shape parameter, $G(x;\mathsf{\Xi })$ is the baseline cdf, and $g(x;\mathsf{\Xi })$ is the baseline pdf, which depends on the parameter vector $\mathsf{\Xi }$.

The cdf and pdf of the odd Burr X-G (OBX-G) family of distributions (see [2]) are given by

$$\begin{array}{ccc}\hfill F_{OBX-G}(x;\beta ,\mathsf{\Xi })& =& {\left(1-exp\left[-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right]\right)}^{\beta }\hfill \end{array}$$

(5)

and

$$\begin{array}{ccc}\hfill f_{OBX-G}(x;\beta ,\mathsf{\Xi })& =& {\displaystyle \frac{2\beta g(x;\mathsf{\Xi })G(x;\mathsf{\Xi })}{{\overline{G}}^3(x;\mathsf{\Xi })}}exp\left[-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right]\hfill \\ & \times & {\left(1-exp\left[-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right]\right)}^{\beta -1},\hfill \end{array}$$

(6)

respectively, for $x,\beta >0,$ and parent cdf parameter vector $\mathsf{\Xi }.$

In this paper, we introduce a new family of generalized distributions that exhibit diverse hazard rate functions (hrfs), encompassing both monotonic and non-monotonic shapes. The new family combines the MO-G, TIIEHL-G, and OBX-G families to obtain the new and generalized MO-TIIEHL-OBX-G distribution. Another motivation for developing the new family of distributions is the construction of distributions for modeling different real datasets, as well as distributions with symmetric, left-skewed, right-skewed, and reverse-J shapes. We also seek to develop a new family of distributions which provide better fits than other generated distributions under the same transformation, and the underlying baseline cdf. Merging distributions like Marshall–Olkin and exponentiated half-logistic increases flexibility, allowing better modeling of complex data with features such as skewness and heavy tails. This helps capture a wider range of behaviors, making the model more suitable for real-world data that a standard distribution might not fit well.

This paper is organized as follows. In Section 2, we introduce the new family of distributions and its sub-families. In Section 3, the mathematical properties of the new model are explored including the expansion of the pdf, quantile function, moments, generating function, and Rényi entropy. The parameter estimates are obtained using different estimation methods in Section 4. Some special cases of the new family of distributions are given in Section 4. A Monte Carlo simulation study is conducted in Section 5 to assess the bias and mean square error of the estimates. Key risk measures and numerical studies using them are given in Section 6. Section 7 gives applications of the new model to real datasets, and conclusions are given in Section 8.

2. The New Family of Distributions

We present the new family of distributions and its sub-families in this section. The Marshall–Olkin–type II exponentiated half-logistic-G–odd Burr X-G (MO-TIIEHL-OBX-G) family of distributions obtained by combining the families in Equations (1)–(6), the cdf and pdf of which are given by

$$\begin{array}{ccc}\hfill F(x;\delta ,\alpha ,\beta ,\mathsf{\Xi })& =& 1-{\displaystyle \frac{\delta {\left(\frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}\right)}^{\alpha }}{1-\left(1-\delta \right){\left(\frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}\right)}^{\alpha }}},\hfill \end{array}$$

(7)

and

$$\begin{array}{ccc}\hfill f(x;\delta ,\alpha ,\beta ,\mathsf{\Xi })& =& {\displaystyle \frac{4\delta \alpha \beta G(x;\mathsf{\Xi })g(x;\mathsf{\Xi })}{{\overline{G}}^3(x;\mathsf{\Xi })}}exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right){\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta -1}\hfill \\ & \times & {\left(1-{\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta }\right)}^{\alpha -1}\hfill \\ & \times & {\left(1+{\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta }\right)}^{-(\alpha +1)}\hfill \\ & \times & {\left(1-\left(1-\delta \right){\left({\displaystyle \frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}}\right)}^{\alpha }\right)}^{-2},\hfill \end{array}$$

(8)

respectively, for $\delta ,\alpha ,\beta >0$, and parameter vector $\mathsf{\Xi }$, where $\delta$ is the tilt parameter, and $\alpha$ and $\beta$ are shape parameters.

The hazard rate function (hrf) is given by

$$\begin{array}{ccc}\hfill h(x;\delta ,\alpha ,\beta ,\mathsf{\Xi })& =& {\displaystyle \frac{4\delta \alpha \beta G(x;\mathsf{\Xi })g(x;\mathsf{\Xi })}{{\overline{G}}^3(x;\mathsf{\Xi })}}exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right){\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta -1}\hfill \\ & \times & {\left(1-{\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta }\right)}^{\alpha -1}\hfill \\ & \times & {\left(1+{\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta }\right)}^{-(\alpha +1)}\hfill \\ & \times & {\left(1-\left(1-\delta \right){\left({\displaystyle \frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}}\right)}^{\alpha }\right)}^{-2}\hfill \\ & \times & {\left({\displaystyle \frac{\delta {\left(\frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}\right)}^{\alpha }}{1-\left(1-\delta \right){\left(\frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}\right)}^{\alpha }}}\right)}^{-1},\hfill \end{array}$$

(9)

for $\delta ,\alpha ,\beta >0$, and parameter vector $\mathsf{\Xi }$.

Sub-Families

Several new sub-families are presented in this subsection. These new sub-families show how diverse and consequential the MO-TIIEHL-OBX-G is in the framework of new and generalized families of distributions.

• When $\alpha =1,$ we obtain the new Marshall–Olkin–type II half-logistic-G–odd Burr X-G (MO-TIIHL-OBX-G) family of distributions, with the cdf given by

  $$\begin{array}{ccc}\hfill F(x;\delta ,\beta ,\mathsf{\Xi })& =& 1-{\displaystyle \frac{\delta \left(\frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}\right)}{1-\left(1-\delta \right)\left(\frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}\right)}},\hfill \end{array}$$

  for $\delta ,\beta >0$, and parameter vector $\mathsf{\Xi }$.
• If $\delta =1$, we obtain the type II exponentiated half-logistic-G odd Burrr X-G (TIIEHL-OBX-G) with the cdf given by

  $$\begin{array}{ccc}\hfill F(x;\alpha ,\beta ,\mathsf{\Xi })& =& 1-{\left({\displaystyle \frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}}\right)}^{\alpha },\hfill \end{array}$$

  for $\alpha ,\beta >0$, and parameter vector $\mathsf{\Xi }$. This is a new family of distributions.
• When $\beta =1,$ we obtain the Marshall–Olkin–type II exponentiated half-logistic-G–odd Rayleigh-G (MO-TIIEHL-OR-G) family of distributions, with the cdf given by

  $$\begin{array}{ccc}\hfill F(x;\alpha ,\delta ,\mathsf{\Xi })& =& 1-{\displaystyle \frac{\delta {\left(\frac{exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)}{1+\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}\right)}^{\alpha }}{1-\left(1-\delta \right){\left(\frac{exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)}{1+\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}\right)}^{\alpha }}},\hfill \end{array}$$

  for $\alpha ,\delta >0$, and parameter vector $\mathsf{\Xi }$. This is a new family of distributions.
• When $\alpha =\beta =1,$ we obtain the Marshall–Olkin–type II half-logistic-G–odd Rayleigh-G (MO-TIIHL-OR-G) family of distributions, with the cdf given by

  $$\begin{array}{ccc}\hfill F(x;\delta ,\mathsf{\Xi })& =& 1-{\displaystyle \frac{\delta \left(\frac{exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)}{1+\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}\right)}{1-\left(1-\delta \right)\left(\frac{exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)}{1+\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}\right)}},\hfill \end{array}$$

  for $\delta >0$ and parameter vector $\mathsf{\Xi }$. This is a new family of distributions.
• When $\alpha =\delta =1,$ we obtain the type II half-logistic-G odd Burrr X-G (TIIHL-OBX-G) family of distributions, with the cdf given by

  $$\begin{array}{ccc}\hfill F(x;\beta ,\mathsf{\Xi })& =& 1-{\displaystyle \frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}},\hfill \end{array}$$

  for $\beta >0$ and parameter vector $\mathsf{\Xi }$. This is a new family of distributions.
• When $\beta =\delta =1,$ we obtain the type II exponentiated half-logistic-G Odd Rayleigh-G (TIIEHL-OR-G) family of distributions, with the cdf given by

  $$\begin{array}{ccc}\hfill F(x;\alpha ,\mathsf{\Xi })& =& 1-{\left({\displaystyle \frac{exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)}{1+\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}}\right)}^{\alpha },\hfill \end{array}$$

  for $\alpha >0$ and parameter vector $\mathsf{\Xi }$. This is a new family of distributions.
• When $\alpha =\beta =\delta =1,$ we obtain the type II half-logistic-G Odd Rayleigh-G (TIIHL-OR-G) family of distributions, with the cdf given by

  $$\begin{array}{ccc}\hfill F(x;\mathsf{\Xi })& =& 1-\left({\displaystyle \frac{exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)}{1+\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}}\right),\hfill \end{array}$$

  with parameter vector $\mathsf{\Xi }$. This is a new family of distributions.
  There are several generalized distributions that can be readily obtained by specifying the baseline cdf G.

3. Some Statistical Properties

The statistical properties of the MO-TIIEHL-OBX-G family of distributions are explored in this section. The statistical properties considered include expansion of the density function, quantile function, generating function, probability-weighted moments, Rényi entropy, and distribution of order statistics.

3.1. Linear Representation of the Density

Consider the generalized series expansion given by ${\left(1-z\right)}^a={\sum }_{i=0}^{\infty }{(-1)}^i\left({\displaystyle \genfrac{}{}{0pt}{}{a}{i}}\right)z^a$; for $\left|z\right|<1$, we can write

$$\begin{array}{ccc}\hfill {\left(1-\overline{\delta }{\left({\displaystyle \frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}}\right)}^{\alpha }\right)}^{-2}& =& \sum _{i=0}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{i+1}{i}}\right){\overline{\delta }}^i\hfill \\ & \times & {\left[{\displaystyle \frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}}\right]}^{\alpha i},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left(1-{\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta }\right)}^{\alpha i+\alpha -1}& =& \sum _{j=0}^{\infty }{\left(-1\right)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha i+\alpha -1}{j}}\right)\hfill \\ & \times & {\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta j},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left(1+{\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta }\right)}^{-(\alpha i+\alpha +1)}& =& \sum _{k=0}^{\infty }{\left(-1\right)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha i+\alpha +k}{k}}\right)\hfill \\ & \times & {\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta k},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta (k+j+1)-1}& =& \sum _{l=0}^{\infty }{\left(-1\right)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{\beta (k+j+1)-1}{l}}\right)\hfill \\ & \times & exp\left[-l{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right],\hfill \end{array}$$

$$\begin{array}{ccc}\hfill exp\left[-(l+1){\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right]& =& \sum _{m=0}^{\infty }{\left(-1\right)}^m{\displaystyle \frac{{\left(l+1\right)}^m{G}^{2m}(x;\mathsf{\Xi })}{m!{\overline{G}}^{2m}(x;\mathsf{\Xi })}},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left[1-G(x;\mathsf{\Xi })\right]}^{-(2m+3)}& =& \sum _{n=0}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{2m+n+2}{n}}\right)G^n(x;\mathsf{\Xi }),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left[1-\overline{G}(x;\mathsf{\Xi })\right]}^{n+2m+1}& =& \sum _{p=0}^{\infty }{\left(-1\right)}^p\left({\displaystyle \genfrac{}{}{0pt}{}{n+2m+1}{p}}\right){\overline{G}}^p(x;\mathsf{\Xi }),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left[1-G(x;\mathsf{\Xi })\right]}^p& =& \sum _{q=0}^{\infty }{\left(-1\right)}^q\left({\displaystyle \genfrac{}{}{0pt}{}{p}{q}}\right)G^q(x;\mathsf{\Xi }),\hfill \end{array}$$

and the pdf of the MO-TIIEHL-OBX-G can be written as

$$\begin{array}{ccc}\hfill f(x;\delta ,\alpha ,\beta ,\mathsf{\Xi })& =& 4\delta \alpha \beta \sum _{i,j,k,l,m,n,p,q=0}^{\infty }{\left(-1\right)}^{j+k+l+m+p+q}\left({\displaystyle \genfrac{}{}{0pt}{}{i+1}{i}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha i+\alpha -1}{j}}\right)\hfill \\ & \times & \left({\displaystyle \genfrac{}{}{0pt}{}{\alpha i+\alpha +k}{k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\beta (k+j+1)-1}{l}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{2m+n+2}{n}}\right)\hfill \\ & \times & \left({\displaystyle \genfrac{}{}{0pt}{}{n+2m+1}{p}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{p}{q}}\right){\displaystyle \frac{{\left(l+1\right)}^m}{m!}}{\overline{\delta }}^{i}g(x;\mathsf{\Xi })G^q(x;\mathsf{\Xi })\hfill \\ & =& \sum _{q=0}^{\infty }{\omega }_{q+1}{h}_{q+1}(x;\mathsf{\Xi }),\hfill \end{array}$$

(10)

where

$$\begin{array}{ccc}\hfill {\omega }_{q+1}& =& {\displaystyle \frac{4\delta \alpha \beta }{q+1}}\sum _{i,j,k,l,m,n,p=0}^{\infty }{\left(-1\right)}^{j+k+l+m+p+q}\left({\displaystyle \genfrac{}{}{0pt}{}{i+1}{i}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha i+\alpha -1}{j}}\right)\hfill \\ & \times & \left({\displaystyle \genfrac{}{}{0pt}{}{\alpha i+\alpha +k}{k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\beta k+\beta j+\beta -1}{l}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{2m+n+2}{n}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n+2m+1}{p}}\right)\hfill \\ & \times & \left({\displaystyle \genfrac{}{}{0pt}{}{p}{q}}\right){\displaystyle \frac{{\left(l+1\right)}^m}{m!}}{\overline{\delta }}^i\hfill \end{array}$$

(11)

and $h_{q+1}(x;\mathsf{\Xi })=(q+1)g(x;\mathsf{\Xi })G^q(x;\mathsf{\Xi })$ is the exponentiated-G (exp-G) density with power parameter (q + 1) and parameter vector $\mathsf{\Xi }.$ Hence, the density of MO-TIIEHL-OBX-G family of distributions can be expressed as an infinite linear combination of exp-G densities. As a result, the mathematical and statistical properties follow directly from those of the exp-G distributions.

3.2. Quantile Function

The quantile function of the MO-TIIEHL-OBX-G family of distributions is obtained by solving the non-linear equation

$$F(x;\delta ,\alpha ,\beta ,\mathsf{\Xi })=1-{\displaystyle \frac{\delta {\left(\frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}\right)}^{\alpha }}{1-\left(1-\delta \right){\left(\frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}\right)}^{\alpha }}}=u,$$

(12)

$0\le u\le 1$. Note that Equation (12) can be written as

$$\begin{array}{ccc}\hfill G(x;\mathsf{\Xi })& =& {\displaystyle \frac{{\left(-log\left[1-{\left(\frac{1-{\left(\frac{1-u}{\delta +(1-u)\overline{\delta }}\right)}^{\frac{1}{\alpha }}}{1+{\left(\frac{1-u}{\delta +(1-u)\overline{\delta }}\right)}^{\frac{1}{\alpha }}}\right)}^{\frac{1}{\beta }}\right]\right)}^{\frac{1}{2}}}{1+{\left(-log\left[1-{\left(\frac{1-{\left(\frac{1-u}{\delta +(1-u)\overline{\delta }}\right)}^{\frac{1}{\alpha }}}{1+{\left(\frac{1-u}{\delta +(1-u)\overline{\delta }}\right)}^{\frac{1}{\alpha }}}\right)}^{\frac{1}{\beta }}\right]\right)}^{\frac{1}{2}}}}.\hfill \end{array}$$

Consequently, the quantile function of the MO-TIIEHL-OBX-G family of distributions is given by

$$\begin{array}{c}\hfill Q_X(u;\delta ,\alpha ,\beta ,\mathsf{\Xi })=G^{-1}\left({\displaystyle \frac{{\left(-log\left[1-{\left(\frac{1-{\left(\frac{1-u}{\delta +(1-u)\overline{\delta }}\right)}^{\frac{1}{\alpha }}}{1+{\left(\frac{1-u}{\delta +(1-u)\overline{\delta }}\right)}^{\frac{1}{\alpha }}}\right)}^{\frac{1}{\beta }}\right]\right)}^{\frac{1}{2}}}{1+{\left(-log\left[1-{\left(\frac{1-{\left(\frac{1-u}{\delta +(1-u)\overline{\delta }}\right)}^{\frac{1}{\alpha }}}{1+{\left(\frac{1-u}{\delta +(1-u)\overline{\delta }}\right)}^{\frac{1}{\alpha }}}\right)}^{\frac{1}{\beta }}\right]\right)}^{\frac{1}{2}}}}\right).\end{array}$$

(13)

As a result, Equation (13) can be used to generate random numbers from the MO-TIIEHL-OBX-G family of distributions for the specified baseline cdf $G(x;\mathsf{\Xi }).$

3.3. Moments and Probability-Weighted Moments

The moments, moment-generating function, and probability-weighted moments (PWMs) of the MO-TIIEHL-OBX-G family of distributions are presented in this subsection. Using Equation (10), we obtain the rth moment of the MO-TIIEHL-OBX-G family of distribution as follows:

$$\begin{array}{ccc}\hfill E\left(X^r\right)& =& {\int }_{-\infty }^{\infty }{x}^r{f}_{MO-TIIEHL-OBX-G}(x;\delta ,\alpha ,\beta ,\mathsf{\Xi })dx=\sum _{q=0}^{\infty }{\omega }_{q+1}E\left(Y_{q+1}^r\right),\hfill \end{array}$$

where $E\left(Y_{q+1}^r\right)$ is the $r^{th}$ moment of $Y_{q+1}$, which follows the exp-G distribution with power parameter ($q+1$) and ${\omega }_{q+1}$ is defined as Equation (11). The moment-generating function is given by

$$\begin{array}{c}\hfill M_X\left(t\right)=E\left(e^{tX}\right)=\sum _{q=0}^{\infty }{\omega }_{q+1}E\left(e^{t{Y}_{q+1}}\right),\end{array}$$

where $E\left(e^{t{Y}_{q+1}}\right)$ is the moment-generating function of an exp-G distributed random variable $Y_{q+1}$ with power parameter ($q+1$), and ${\omega }_{q+1}$ is given by Equation (11). The PWMs of a random variable X are defined by

$$\begin{array}{c}\hfill {\omega }_{a,r}=E\left(X^a{\left[F\left(X\right)\right]}^r\right)={\int }_{-\infty }^{\infty }{x}^a{\left[F\left(x\right)\right]}^{r}f\left(x\right)dx.\end{array}$$

Using the generalized binomial expansions

$$\begin{array}{ccc}\hfill {\left[1-{\displaystyle \frac{\delta {\left(\frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}\right)}^{\alpha }}{1-\overline{\delta }{\left(\frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}\right)}^{\alpha }}}\right]}^r& =& \sum _{i=0}^{\infty }{\left(-1\right)}^i\left({\displaystyle \genfrac{}{}{0pt}{}{r}{i}}\right){\delta }^i\hfill \\ & \times & {\left({\displaystyle \frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}}\right)}^{\alpha i}\hfill \\ & \times & {\left[1-\overline{\delta }{\left({\displaystyle \frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}}\right)}^{\alpha }\right]}^{-i},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left[1-\overline{\delta }{\left({\displaystyle \frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}}\right)}^{\alpha }\right]}^{-(i+2)}& =& \sum _{j=0}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{i+j+1}{j}}\right){\overline{\delta }}^j\hfill \\ & \times & {\displaystyle \frac{{\left[1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }\right]}^{\alpha j}}{{\left[1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }\right]}^{\alpha j}}},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left(1+{\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta }\right)}^{-(\alpha j+\alpha +1)}& =& \sum _{k=0}^{\infty }{\left(-1\right)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha j+\alpha +k}{k}}\right)\hfill \\ & \times & {\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta k},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left(1-{\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta }\right)}^{\alpha j+\alpha -1}& =& \sum _{l=0}^{\infty }{\left(-1\right)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha j+\alpha -1}{l}}\right)\hfill \\ & \times & {\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta l},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta (k+l+1)-1}& =& \sum _{m=0}^{\infty }{\left(-1\right)}^m\left({\displaystyle \genfrac{}{}{0pt}{}{\beta (k+l+1)-1}{m}}\right)\hfill \\ & \times & exp\left[-m{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right],\hfill \end{array}$$

$$\begin{array}{ccc}\hfill exp\left[-(m+1){\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right]& =& \sum _{n=0}^{\infty }{\left(-1\right)}^n{\displaystyle \frac{{(m+1)}^n{G}^{2n}(x;\mathsf{\Xi })}{n!{\overline{G}}^{2n}(x;\mathsf{\Xi })}},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left[1-G(x;\mathsf{\Xi })\right]}^{-(2n+3)}& =& \sum _{p=0}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{2n+p+1}{p}}\right)G^p(x;\mathsf{\Xi }),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left[1-\overline{G}(x;\mathsf{\Xi })\right]}^{2n+p+1}& =& \sum _{q=0}^{\infty }{\left(-1\right)}^q\left({\displaystyle \genfrac{}{}{0pt}{}{2n+p+1}{q}}\right){\overline{G}}^q(x;\mathsf{\Xi }),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\left[1-G(x;\mathsf{\Xi })\right]}^q& =& \sum _{s=0}^{\infty }{\left(-1\right)}^s\left({\displaystyle \genfrac{}{}{0pt}{}{q}{s}}\right)G^s(x;\mathsf{\Xi }),\hfill \end{array}$$

the PWMs of the MO-TIIEHL-OBX-G distribution can be expressed as

$$\begin{array}{c}\hfill {\omega }_{a,r}=\sum _{s=0}^{\infty }{U}_{s+1}{\int }_{-\infty }^{\infty }{x}^a{h}_{s+1}(x;\mathsf{\Xi })dx,\end{array}$$

where

$$\begin{array}{ccc}\hfill U_{s+1}& =& {\displaystyle \frac{4\delta \alpha \beta }{(s+1)}}\sum _{i,j,k,l,m,n,p,q=0}^{\infty }{(-1)}^{i+k+l+m+n+q+s}{\delta }^i{\overline{\delta }}^j\left({\displaystyle \genfrac{}{}{0pt}{}{r}{i}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{i+j+1}{j}}\right)\hfill \\ & \times & \left({\displaystyle \genfrac{}{}{0pt}{}{\alpha j+\alpha +k}{k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha j+\alpha -1}{l}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\beta (k+l+1)-1}{m}}\right){\displaystyle \frac{{\left(m+1\right)}^n}{n!}}\hfill \\ & \times & \left({\displaystyle \genfrac{}{}{0pt}{}{2n+p+2}{p}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{2n+p+1}{q}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{q}{s}}\right),\hfill \end{array}$$

and $h_{s+1}(x;\mathsf{\Xi })=(s+1)g(x;\mathsf{\Xi })G^s(x;\mathsf{\Xi })$ is the exp-G density with power parameter $(s+1)$.

3.4. Distribution of Order Statistics

Let $X_1,X_2,\dots ,X_n$ be independent and identically distributed MO-TIIEHL-OBX-G random variables. The pdf of the $i^{th}$ order statistic $X_{i:n}$ can be written as

$$\begin{array}{c}\hfill f_{i:n}\left(x\right)={\displaystyle \frac{n!f\left(x\right)}{(i-1)!(n-i)!}}\sum _{j=0}^{n-i}{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{j}}\right){\left[F\left(x\right)\right]}^{j+i-1}.\end{array}$$

Considering the generalized binomial expansions

$$\begin{array}{ccc}\hfill {\left[1-{\displaystyle \frac{\delta {\left(\frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}\right)}^{\alpha }}{1-\overline{\delta }{\left(\frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}\right)}^{\alpha }}}\right]}^{i+j-1}& =& \sum _{k=0}^{\infty }{\left(-1\right)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{i+j-1}{k}}\right){\delta }^k\hfill \\ & \times & {\left({\displaystyle \frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}}\right)}^{\alpha k}\hfill \\ & \times & {\left[\eta \right]}^{-k},\hfill \end{array}$$

where $\eta =1-\overline{\delta }{\left({\displaystyle \frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}}\right)}^{\alpha },$

$$\begin{array}{ccc}\hfill {\left[1-\overline{\delta }{\left({\displaystyle \frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}}\right)}^{\alpha }\right]}^{-(k+2)}& =& \sum _{l=0}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{k+l+1}{l}}\right){\overline{\delta }}^l\hfill \\ & \times & {\displaystyle \frac{{\left[1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }\right]}^{\alpha l}}{{\left[1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }\right]}^{\alpha l}}},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left(1+{\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta }\right)}^{-(\alpha l+\alpha +1)}& =& \sum _{n=0}^{\infty }{\left(-1\right)}^n\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha l+\alpha +n}{n}}\right)\hfill \\ & \times & {\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta n},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left(1-{\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta }\right)}^{\alpha l+\alpha -1}& =& \sum _{m=0}^{\infty }{\left(-1\right)}^m\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha l+\alpha -1}{m}}\right)\hfill \\ & \times & {\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta m},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta (n+m+1)-1}& =& \sum _{p=0}^{\infty }{\left(-1\right)}^p\left({\displaystyle \genfrac{}{}{0pt}{}{\beta (n+m+1)-1}{p}}\right)\hfill \\ & \times & exp\left[-p{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right],\hfill \end{array}$$

$$\begin{array}{ccc}\hfill exp\left[-(p+1){\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right]& =& \sum _{q=0}^{\infty }{\left(-1\right)}^q{\displaystyle \frac{{(p+1)}^q{G}^{2q}(x;\mathsf{\Xi })}{q!{\overline{G}}^{2q}(x;\mathsf{\Xi })}},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left[1-G(x;\mathsf{\Xi })\right]}^{-(2q+3)}& =& \sum _{r=0}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{2q+r+2}{r}}\right)G^r(x;\mathsf{\Xi }),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left[1-\overline{G}(x;\mathsf{\Xi })\right]}^{r+2q+1}& =& \sum _{s=0}^{\infty }{\left(-1\right)}^s\left({\displaystyle \genfrac{}{}{0pt}{}{r+2q+1}{s}}\right){\overline{G}}^s(x;\mathsf{\Xi }),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\left[1-G(x;\mathsf{\Xi })\right]}^s& =& \sum _{t=0}^{\infty }{\left(-1\right)}^t\left({\displaystyle \genfrac{}{}{0pt}{}{s}{t}}\right)G^t(x;\mathsf{\Xi }),\hfill \end{array}$$

the pdf of the $i^{th}$ order statistic for the MO-TIIEHL-OBX-G distribution is given by

$$\begin{array}{c}\hfill f_{i:n}\left(x\right)={\displaystyle \frac{n!}{(i-1)!(n-i)!}}\sum _{j=0}^{n-i}\sum _{t=0}^{\infty }{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{j}}\right)V_{t+1}{h}_{t+1}(x;\mathsf{\Xi }),\end{array}$$

where $h_{t+1}(x;\mathsf{\Xi })=(t+1)g(x;\mathsf{\Xi })G^t(x;\mathsf{\Xi })$ is the exp-G density with power parameter $(t+1)$, and

$$\begin{array}{ccc}\hfill V_{t+1}& =& \sum _{k,l,m,n,p,q,r,s=0}^{\infty }{\displaystyle \frac{4\delta \alpha \beta }{(t+1)}}{(-1)}^{k+m+n+p+q+s+t}{\delta }^k{\overline{\delta }}^l\left({\displaystyle \genfrac{}{}{0pt}{}{j+i-1}{k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{k+l+1}{l}}\right)\hfill \\ & \times & \left({\displaystyle \genfrac{}{}{0pt}{}{\alpha l+\alpha -1}{m}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha l+\alpha +1}{n}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\beta (n+m+1)-1}{p}}\right){\displaystyle \frac{{\left(p+1\right)}^q}{q!}}\hfill \\ & \times & \left({\displaystyle \genfrac{}{}{0pt}{}{2q+r+2}{r}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{2q+r+1}{s}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{s}{t}}\right).\hfill \end{array}$$

Consequently, the distribution of the $i^{th}$ order statistic can be expressed in terms of the exp-G densities.

3.5. Rényi Entropy

The Rényi entropy of the MO-TIIEHL-OBX-G family of distributions is given in this subsection. The Rényi entropy [20] is a measure of uncertainty associated with a random variable X and is defined as

$$H_R\left(v\right)={\displaystyle \frac{1}{1-v}}log\left({\int }_0^{\infty }{f}^v\left(x\right)dx\right),$$

for $v>0$, $v\ne 1$. From Equation (8), $f^v(x;\delta ,\alpha ,\beta ,\psi )=f^v\left(x\right)$ can be written as

$$\begin{array}{ccc}\hfill f^v\left(x\right)& =& {\displaystyle \frac{{\left(4\delta \alpha \beta \right)}^v{G}^v(x;\mathsf{\Xi })g^v(x;\mathsf{\Xi })}{{\overline{G}}^{3v}(x;\mathsf{\Xi })}}exp\left(-v{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right){\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{v(\beta -1)}\hfill \\ & \times & {\left(1-{\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta }\right)}^{v(\alpha -1)}\hfill \\ & \times & {\left(1+{\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta }\right)}^{-v(\alpha +1)}\hfill \\ & \times & {\left(1-\left(1-\delta \right){\left({\displaystyle \frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}}\right)}^{\alpha }\right)}^{-2v}.\hfill \end{array}$$

Using the generalized binomial expansions

$$\begin{array}{ccc}\hfill {\left(1-\overline{\delta }{\left({\displaystyle \frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}}\right)}^{\alpha }\right)}^{-2v}& =& \sum _{i=0}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{2v+i-1}{i}}\right){\overline{\delta }}^i\hfill \\ & \times & {\left[{\displaystyle \frac{1-{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}\right)}^2\right)\right]}^{\beta }}}\right]}^{\alpha i},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left[1-{\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta }\right]}^{\alpha i+v(\alpha -1)}& =& \sum _{j=0}^{\infty }{\left(-1\right)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha i+v(\alpha -1)}{j}}\right)\hfill \\ & \times & {\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta j},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left[1+{\left(1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right)}^{\beta }\right]}^{-(\alpha i+v(\alpha +1\left)\right)}& =& \sum _{k=0}^{\infty }{\left(-1\right)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha i+v(\alpha +1)+k-1}{k}}\right)\hfill \\ & \times & {\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta k},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left[1-exp\left(-{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right)\right]}^{\beta k+\beta j+v(\beta -1)}& =& \sum _{l=0}^{\infty }{\left(-1\right)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{\beta k+\beta j+v(\beta -1)}{l}}\right)\hfill \\ & \times & exp\left[-l{\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right],\hfill \end{array}$$

$$\begin{array}{ccc}\hfill exp\left[-(l+v){\left({\displaystyle \frac{G(x;\mathsf{\Xi })}{\overline{G}(x;\mathsf{\Xi })}}\right)}^2\right]& =& \sum _{m=0}^{\infty }{\left(-1\right)}^m{\displaystyle \frac{{\left(l+v\right)}^m{G}^{2m}(x;\mathsf{\Xi })}{m!{\overline{G}}^{2m}(x;\mathsf{\Xi })}},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left[1-G(x;\mathsf{\Xi })\right]}^{-(2m+3v)}& =& \sum _{n=0}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{2m+3v+n-1}{n}}\right)G^n(x;\mathsf{\Xi }),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left[1-\overline{G}(x;\mathsf{\Xi })\right]}^{n+2m+v}& =& \sum _{p=0}^{\infty }{\left(-1\right)}^p\left({\displaystyle \genfrac{}{}{0pt}{}{n+2m+v}{p}}\right){\overline{G}}^p(x;\mathsf{\Xi }),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\left[1-G(x;\mathsf{\Xi })\right]}^p& =& \sum _{q=0}^{\infty }{\left(-1\right)}^q\left({\displaystyle \genfrac{}{}{0pt}{}{p}{q}}\right)G^q(x;\mathsf{\Xi }),\hfill \end{array}$$

we can write

$$\begin{array}{ccc}\hfill f^v\left(x\right)& =& {\left(4\delta \alpha \beta \right)}^v\sum _{i,j,k,l,m,n,p,q=0}^{\infty }{\left(-1\right)}^{j+k+l+m+p+q}\left({\displaystyle \genfrac{}{}{0pt}{}{2v+i-1}{i}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha i+v(\alpha -1)}{j}}\right)\hfill \\ & \times & \left({\displaystyle \genfrac{}{}{0pt}{}{\alpha i+v(\alpha +1)+k-1}{k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\beta k+\beta j+v(\beta -1)}{l}}\right)\hfill \\ & \times & \left({\displaystyle \genfrac{}{}{0pt}{}{2m+3v+n-1}{n}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n+2m+v}{p}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{p}{q}}\right){\overline{\delta }}^i{\displaystyle \frac{{\left(l+v\right)}^m}{m!}}{g}^v(x;\psi )G^p(x;\psi ).\hfill \end{array}$$

Thus, the Rényi entropy of the MO-TIIEHL-OBX-G family of distributions can be obtained directly from that of the exp-G distribution as follows:

$$\begin{array}{ccc}\hfill H_R\left(v\right)& =& {\displaystyle \frac{1}{1-v}}log\left[\sum _{q=0}^{\infty }{\tau }_q{e}^{(1-v)I_{REG}}\right],v>0,v\ne 1,\hfill \end{array}$$

(14)

where

$$\begin{array}{ccc}\hfill {\tau }_q& =& {\displaystyle \frac{{\left(4\delta \alpha \beta \right)}^v}{{\left(\frac{q}{v}+1\right)}^v}}\sum _{i,j,k,l,m,n,p=0}^{\infty }{\left(-1\right)}^{j+k+l+m+p+q}\left({\displaystyle \genfrac{}{}{0pt}{}{2v+i-1}{i}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha i+v(\alpha -1)}{j}}\right)\hfill \\ & \times & \left({\displaystyle \genfrac{}{}{0pt}{}{\alpha i+v(\alpha +1)+k-1}{k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\beta k+\beta j+v(\beta -1)}{l}}\right)\hfill \\ & \times & \left({\displaystyle \genfrac{}{}{0pt}{}{2m+3v+n-1}{n}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n+2m+v}{p}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{p}{q}}\right){\overline{\delta }}^i{\displaystyle \frac{{\left(l+v\right)}^m}{m!}},\hfill \end{array}$$

and $I_{REG}={\displaystyle \frac{1}{1-v}}log\left({\int }_0^{\infty }{\left[\left({\displaystyle \frac{q}{v}}+1\right)g(x;\xi ){\left(G(x;\xi )\right)}^{{\displaystyle \frac{q}{v}}}\right]}^{v}dx\right)$ is the Rényi entropy of the exp-G distribution with power parameter $({\displaystyle \frac{q}{v}}+1).$

4. Some Special Models

We present some special cases of the MO-TIIEHL-OBX-G family of distributions, particularly when the cdf $G(x;\mathsf{\Xi })$ is specified to be a Weibull, log-logistic, or standard half-logistic distribution.

4.1. Marshall–Olkin–Type II Exponentiated Half-Logistic–Odd Burr X–Weibull (MO-TIIEHL-OBX-W) Distribution

Suppose we take the baseline distribution to be the one-parameter Weibull distribution with the cdf and pdf given by $G(x;\gamma )=1-e^{{-x}^{\gamma }}$ and $g(x;\gamma )=\gamma x^{\gamma -1}{e}^{{-x}^{\gamma }}$, respectively, for $\gamma >0.$ From Equations (7) and (8), we obtain the cdf and pdf of the MO-TIIEHL-OBX-W distribution as

$$\begin{array}{ccc}\hfill F(x;\delta ,\alpha ,\beta ,\gamma )& =& 1-{\displaystyle \frac{\delta {\left(\frac{1-{\left[1-exp\left(-{\left(\frac{1-e^{{-x}^{\gamma }}}{e^{{-x}^{\gamma }}}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{1-e^{{-x}^{\gamma }}}{e^{{-x}^{\gamma }}}\right)}^2\right)\right]}^{\beta }}\right)}^{\alpha }}{1-\left(1-\delta \right){\left(\frac{1-{\left[1-exp\left(-{\left(\frac{1-e^{{-x}^{\gamma }}}{e^{{-x}^{\gamma }}}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{1-e^{{-x}^{\gamma }}}{e^{{-x}^{\gamma }}}\right)}^2\right)\right]}^{\beta }}\right)}^{\alpha }}},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill f(x;\delta ,\alpha ,\beta ,\gamma )& =& {\displaystyle \frac{4\delta \alpha \beta \left(1-e^{{-x}^{\gamma }}\right)\gamma x^{\gamma -1}{e}^{{-x}^{\gamma }}}{{\left(e^{{-x}^{\gamma }}\right)}^3}}exp\left(-{\left({\displaystyle \frac{1-e^{{-x}^{\gamma }}}{e^{{-x}^{\gamma }}}}\right)}^2\right)\hfill \\ & \times & {\left[1-exp\left(-{\left({\displaystyle \frac{1-e^{{-x}^{\gamma }}}{e^{{-x}^{\gamma }}}}\right)}^2\right)\right]}^{\beta -1}\hfill \\ & \times & {\left(1-{\left[1-exp\left(-{\left({\displaystyle \frac{1-e^{{-x}^{\gamma }}}{e^{{-x}^{\gamma }}}}\right)}^2\right)\right]}^{\beta }\right)}^{\alpha -1}\hfill \\ & \times & {\left(1+{\left[1-exp\left(-{\left({\displaystyle \frac{1-e^{{-x}^{\gamma }}}{e^{{-x}^{\gamma }}}}\right)}^2\right)\right]}^{\beta }\right)}^{-(\alpha +1)}\hfill \\ & \times & {\left(1-\left(1-\delta \right){\left({\displaystyle \frac{1-{\left[1-exp\left(-{\left(\frac{1-e^{{-x}^{\gamma }}}{e^{{-x}^{\gamma }}}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{1-e^{{-x}^{\gamma }}}{e^{{-x}^{\gamma }}}\right)}^2\right)\right]}^{\beta }}}\right)}^{\alpha }\right)}^{-2},\hfill \end{array}$$

respectively, for $x,\delta ,\alpha ,\beta$, and $\gamma >0$.

Plots of the pdf and hrf of the MO-TIIEHL-OBX-W distribution are given in Figure 1. The pdf of the MO-TIIEHL-OBX-W distribution takes several shapes including decreasing, right-skewed, and left-skewed, as well as almost symmetric, whereas the hrf displays bathtub, upside-down bathtub, upside-down bathtub followed by bathtub, decreasing, and increasing shapes.

Figure 2 presents the skewness and kurtosis plots for the MO-TIIEHL-OBX-W distribution. Skewness initially decreases and then increases for $\alpha =48.5$ and $c=1.1$, while kurtosis increases for $\alpha =1.5$ and $c=1.3$ as $\delta$ and $\beta$ increase. Additionally, skewness follows a similar decreasing–increasing pattern for $\delta =0.1$ and $\alpha =2.8$, whereas kurtosis decreases for $\delta =0.04$ and $\alpha =6.6$ as $\beta$ and $\lambda$ increase.

4.2. Marshall–Olkin–Type II Exponentiated Half-Logistic–Odd Burr X–Log-Logistic (MO-TIIEHL-OBX-LLoG) Distribution

Suppose we take the baseline distribution to be the log-logistic distribution with the cdf and pdf given by $G(x;c)=1-{(1+x^c)}^{-1}$ and $g(x;c)=c{x}^{c-1}{(1+x^c)}^{-2}$, respectively, for $c,x>0$. From Equations (7) and (8), we obtain the cdf and pdf of the MO-TIIEHL-OBX-LLoG distribution as

$$\begin{array}{ccc}\hfill F(x;\delta ,\alpha ,\beta ,c)& =& 1-{\displaystyle \frac{\delta {\left(\frac{1-{\left[1-exp\left(-{\left(\frac{1-{(1+x^c)}^{-1}}{{(1+x^c)}^{-1}}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{1-{(1+x^c)}^{-1}}{{(1+x^c)}^{-1}}\right)}^2\right)\right]}^{\beta }}\right)}^{\alpha }}{1-\left(1-\delta \right){\left(\frac{1-{\left[1-exp\left(-{\left(\frac{1-{(1+x^c)}^{-1}}{{(1+x^c)}^{-1}}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{1-{(1+x^c)}^{-1}}{{(1+x^c)}^{-1}}\right)}^2\right)\right]}^{\beta }}\right)}^{\alpha }}},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill f(x;\delta ,\alpha ,\beta ,c)& =& {\displaystyle \frac{4\delta \alpha \beta \left(1-{(1+x^c)}^{-1}\right)c{x}^{c-1}{(1+x^c)}^{-2}}{{\left({(1+x^c)}^{-1}\right)}^3}}\hfill \\ & \times & exp\left(-{\left({\displaystyle \frac{1-{(1+x^c)}^{-1}}{{(1+x^c)}^{-1}}}\right)}^2\right)\hfill \\ & \times & {\left[1-exp\left(-{\left({\displaystyle \frac{1-{(1+x^c)}^{-1}}{{(1+x^c)}^{-1}}}\right)}^2\right)\right]}^{\beta -1}\hfill \\ & \times & {\left(1-{\left[1-exp\left(-{\left({\displaystyle \frac{1-{(1+x^c)}^{-1}}{{(1+x^c)}^{-1}}}\right)}^2\right)\right]}^{\beta }\right)}^{\alpha -1}\hfill \\ & \times & {\left(1+{\left[1-exp\left(-{\left({\displaystyle \frac{1-{(1+x^c)}^{-1}}{{(1+x^c)}^{-1}}}\right)}^2\right)\right]}^{\beta }\right)}^{-(\alpha +1)}\hfill \\ & \times & {\left(1-\left(1-\delta \right){\left({\displaystyle \frac{1-{\left[1-exp\left(-{\left(\frac{1-{(1+x^c)}^{-1}}{{(1+x^c)}^{-1}}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{1-{(1+x^c)}^{-1}}{1-{(1+x^c)}^{-1}}\right)}^2\right)\right]}^{\beta }}}\right)}^{\alpha }\right)}^{-2},\hfill \end{array}$$

respectively, for $x,\delta ,\alpha ,\beta$, and $c>0.$

Plots of the pdf and hrf of the MO-TIIEHL-OBX-LLoG distribution are given in Figure 3. The pdf of the MO-TIIEHL-OBX-LLoG distribution takes several shapes including reverse-J, right-skewed, and left-skewed, as well as almost symmetric, whereas the hrf displays bathtub, upside-down bathtub, upside-down bathtub followed by bathtub, decreasing, and increasing shapes.

Figure 4 presents the skewness and kurtosis plots for the MO-TIIEHL-OBX-LLoG distribution. The skewness and kurtosis increase with higher values of $\alpha$ and c, while they both exhibit a decreasing and then increasing trend with increasing values of $\delta$ and $\beta$.

4.3. Marshall–Olkin–Type II Exponentiated Half-Logistic–Odd Burr X–Standard Half-Logistic (MO-TIIEHL-OBX-SHL) Distribution

Suppose the baseline distribution is the standard half-logistic distribution, with the cdf and pdf given by $G\left(x\right)={\displaystyle \frac{1-e^{-x}}{1+e^{-x}}}$ and $g\left(x\right)={\displaystyle \frac{2e^{-x}}{{(1+e^{-x})}^2}}$, respectively, for $x>0$. From Equations (7) and (8), we obtain the cdf and pdf of the MO-TIIEHL-OBX-SHL distribution as

$$\begin{array}{ccc}\hfill F(x;\delta ,\alpha ,\beta )& =& 1-{\displaystyle \frac{\delta {\left(\frac{1-{\left[1-exp\left(-{\left(\frac{1-e^{-x}}{2e^{-x}}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{1-e^{-x}}{2e^{-x}}\right)}^2\right)\right]}^{\beta }}\right)}^{\alpha }}{1-\left(1-\delta \right){\left(\frac{1-{\left[1-exp\left(-{\left(\frac{1-e^{-x}}{2e^{-x}}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{1-e^{-x}}{2e^{-x}}\right)}^2\right)\right]}^{\beta }}\right)}^{\alpha }}},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill f(x;\delta ,\alpha ,\beta )& =& {\displaystyle \frac{4\delta \alpha \beta \frac{2e^{-x}\left(1-e^{-x}\right)}{{\left(1+e^{-x}\right)}^3}}{{\left(1-\frac{1-e^{-x}}{1+e^{-x}}\right)}^3}}exp\left(-{\left({\displaystyle \frac{1-e^{-x}}{2e^{-x}}}\right)}^2\right)\hfill \\ & \times & {\left[1-exp\left(-{\left({\displaystyle \frac{1-e^{-x}}{2e^{-x}}}\right)}^2\right)\right]}^{\beta -1}\hfill \\ & \times & {\left(1-{\left[1-exp\left(-{\left({\displaystyle \frac{1-e^{-x}}{2e^{-x}}}\right)}^2\right)\right]}^{\beta }\right)}^{\alpha -1}\hfill \\ & \times & {\left(1+{\left[1-exp\left(-{\left({\displaystyle \frac{1-e^{-x}}{2e^{-x}}}\right)}^2\right)\right]}^{\beta }\right)}^{-(\alpha +1)}\hfill \\ & \times & {\left(1-\left(1-\delta \right){\left({\displaystyle \frac{1-{\left[1-exp\left(-{\left(\frac{1-e^{-x}}{2e^{-x}}\right)}^2\right)\right]}^{\beta }}{1+{\left[1-exp\left(-{\left(\frac{1-e^{-x}}{2e^{-x}}\right)}^2\right)\right]}^{\beta }}}\right)}^{\alpha }\right)}^{-2},\hfill \end{array}$$

respectively, for $x,\delta ,\alpha$, and $\beta >0.$

Figure 5 presents the plots of the pdf and hrf for the MO-TIIEHL-OBX-SHL distribution. The pdf of the MO-MO-TIIEHL-OBX-SHL distribution exhibits various shapes, including reverse-J, right-skewed, left-skewed, and nearly symmetric shapes. Meanwhile, the hrf demonstrates diverse patterns such as bathtub, inverted bathtub, an inverted bathtub followed by a bathtub, and an increasing trend.

Figure 6 presents the skewness and kurtosis plots for the MO-TIIEHL-OBX-SHL distribution. Both skewness and kurtosis increase with higher values of $\alpha$ and $\beta .$ However, they exhibit a decreasing-then-increasing pattern as $\delta$ and $\beta$ increase.

5. Simulation Study

The effectiveness of the parameter estimates for the MO-TIIEHL-OBX-LLoG distribution obtained through six estimation methods is examined by conducting various simulations for different sample sizes (n = 25, 50, 100, 200, 400, and 800) via the R package. The estimation methods include Maximum Likelihood Estimation (ML), Anderson–Darling (AD), Right-Tail Anderson–Darling (RAD), Ordinary Least Squares (OLS), Weighted Least Squares (WLS), and Cramér–von Mises (CVM). We simulate $N=3000$ samples for the true parameter values given in

Table 1. Simulation results.

(0.3, 1.4, 0.6, 2.2)
Sample Size (n)	Estimate	Parameter	ML	WLS	LS	RAD	AD	CVM
		$\delta$	$0.{2768}_2$	$-0.{5281}_4$	$-0.{7129}_5$	$-0.{4531}_3$	$-0.{1685}_1$	$-0.{7131}_6$
25	Abias	$\alpha$	$-0.{0801}_1$	$0.{3082}_5$	$0.{1310}_4$	$0.{6383}_6$	$0.{0842}_2$	$0.{1302}_3$
		$\beta$	$-0.{0801}_1$	$-0.{3246}_3$	$-0.{3706}_5$	$-0.{4863}_6$	$-0.{0879}_2$	$-0.{3693}_4$
		c	$0.{7989}_3$	$-0.{1072}_1$	$-0.{9644}_5$	$-1.{3811}_6$	$-0.{7171}_2$	$-0.{9069}_4$
		$\delta$	$0.{7764}_4$	$0.{7649}_3$	$0.{8797}_5$	$0.{5095}_1$	$0.{6880}_2$	$0.{8801}_6$
	RMSE	$\alpha$	$0.{5231}_3$	$0.{8418}_6$	$0.{4664}_2$	$0.{6508}_4$	$0.{7547}_5$	$0.{1659}_1$
		$\beta$	$0.{2326}_1$	$0.{3677}_2$	$0.{3984}_4$	$0.{4932}_6$	$0.{4892}_5$	$0.{3976}_3$
		c	$2.{5576}_5$	$0.{1520}_1$	$1.{7483}_4$	$1.{3924}_2$	$1.{7227}_3$	$2.{7536}_6$
		Sum of the ranks	20	25	34	33	22	33
		$\delta$	$0.{1187}_2$	$-0.{5091}_4$	$-0.{6813}_5$	$-0.{4104}_3$	$-0.{1096}_1$	$-0.{6819}_6$
50	Abias	$\alpha$	$-0.{0725}_1$	$0.{2563}_5$	$0.{1211}_4$	$0.{6853}_6$	$0.{0785}_2$	$0.{1208}_3$
		$\beta$	$-0.{0657}_1$	-$0.{2255}_3$	$-0.{3640}_5$	$-0.{4810}_6$	$-0.{0855}_2$	$-0.{3633}_4$
		c	$0.{3594}_2$	$-0.{0940}_1$	$-0.{4095}_3$	$-1.{1304}_6$	$-0.{5243}_4$	$-0.{8840}_5$
		$\delta$	$0.{3208}_1$	$0.{6475}_4$	$0.{7645}_5$	$0.{4564}_2$	$0.{5697}_3$	$0.{7903}_6$
	RMSE	$\alpha$	$0.{3641}_2$	$0.{4105}_4$	$0.{3844}_3$	$0.{6879}_6$	$0.{4184}_5$	$0.{1383}_1$
		$\beta$	$0.{1452}_1$	$0.{2910}_3$	$0.{3867}_5$	$0.{4912}_6$	$0.{2861}_2$	$0.{3862}_4$
		c	$1.{5488}_4$	$0.{1431}_1$	$1.{5427}_3$	$1.{1510}_2$	$1.{6243}_5$	$2.{0546}_6$
		Sum of the ranks	14	21	33	37	24	35
		$\delta$	$0.{0524}_1$	$-0.{4468}_5$	$-0.{4868}_6$	$-0.{3430}_3$	$-0.{0698}_2$	$-0.{3868}_4$
100	Abias	$\alpha$	$-0.{0126}_1$	$0.{0998}_3$	$0.{1166}_5$	$0.{5061}_6$	$0.{0419}_2$	$0.{1163}_4$
		$\beta$	$-0.{0306}_1$	$-0.{0714}_2$	$-0.{3387}_6$	$-0.{2962}_4$	$-0.{0837}_3$	$-0.{3383}_5$
		c	$0.{0403}_2$	$-0.{0267}_1$	$-0.{2421}_4$	$-0.{4126}_5$	$-0.{2221}_3$	$-0.{5339}_6$
		$\delta$	$0.{1659}_1$	$0.{5648}_6$	$0.{5097}_5$	$0.{3676}_3$	$0.{2699}_2$	$0.{4774}_4$
	RMSE	$\alpha$	$0.{2909}_3$	$0.{3196}_5$	$0.{2291}_2$	$0.{6607}_6$	$0.{3188}_4$	$0.{1227}_1$
		$\beta$	$0.{0904}_1$	$0.{1369}_2$	$0.{3600}_5$	$0.{3976}_6$	$0.{1839}_3$	$0.{3596}_4$
		c	$0.{3804}_2$	$0.{0766}_1$	$0.{5456}_4$	$0.{5130}_3$	$0.{6222}_5$	$1.{0356}_6$
		Sum of the ranks	12	25	37	36	22	34
		$\delta$	$0.{0348}_1$	$0.{1838}_5$	$-0.{2106}_6$	$-0.{1383}_4$	$-0.{0391}_2$	$-0.{0983}_3$
200	Abias	$\alpha$	$-0.{0299}_1$	$-0.{0512}_3$	$0.{1036}_4$	$0.{3195}_6$	$0.{0399}_2$	$0.{1099}_5$
		$\beta$	$-0.{0257}_1$	$-0.{0674}_3$	$-0.{2708}_5$	$-0.{1903}_4$	$-0.{0622}_2$	$-0.{3063}_6$
		c	$0.{0305}_2$	$-0.{0219}_1$	$0.{1265}_3$	$-0.{2912}_5$	$-0.{1780}_4$	$0.{3070}_6$
		$\delta$	$0.{1043}_1$	$0.{2517}_5$	$0.{3149}_6$	$0.{2401}_4$	$0.{1692}_3$	$0.{1568}_2$
	RMSE	$\alpha$	$0.{2063}_3$	$0.{2186}_5$	$0.{2149}_4$	$0.{4198}_6$	$0.{2002}_2$	$0.{1161}_1$
		$\beta$	$0.{0657}_2$	$0.{0326}_1$	$0.{2814}_5$	$0.{2557}_4$	$0.{0864}_3$	$0.{3228}_6$
		c	$0.{2840}_2$	$0.{0407}_1$	$0.{3196}_3$	$0.{3293}_5$	$0.{3218}_4$	$0.{6472}_6$
		Sum of the ranks	13	23	36	38	22	35
		$\delta$	$0.{0230}_2$	$0.{1134}_5$	$-0.{1207}_6$	$-0.{0642}_4$	$-0.{0217}_1$	$-0.{0406}_3$
400	Abias	$\alpha$	$-0.{0233}_2$	$-0.{0424}_4$	$0.{0372}_3$	$0.{2229}_6$	$0.{0232}_1$	$0.{1036}_5$
		$\beta$	$-0.{0168}_1$	$-0.{0211}_2$	$-0.{2371}_5$	$-0.{0767}_4$	$-0.{0312}_3$	$-0.{2708}_6$
		c	$0.{0044}_1$	$-0.{0125}_2$	$-0.{0804}_4$	$-0.{1281}_5$	$-0.{0205}_3$	$0.{2673}_6$
		$\delta$	$0.{0734}_2$	$0.{1223}_5$	$0.{1221}_4$	$0.{1376}_6$	$0.{0680}_1$	$0.{1149}_3$
	RMSE	$\alpha$	$0.{1471}_3$	$0.{1543}_5$	$0.{1447}_2$	$0.{3226}_6$	$0.{1533}_4$	$0.{1042}_1$
		$\beta$	$0.{0471}_3$	$0.{0268}_1$	$0.{2410}_5$	$0.{1775}_4$	$0.{0323}_2$	$0.{2814}_6$
		c	$0.{2134}_4$	$0.{0192}_1$	$0.{2490}_5$	$0.{2082}_3$	$0.{2055}_2$	$0.{3196}_6$
		Sum of the ranks	18	25	29	38	17	36
		$\delta$	$0.{0188}_1$	$0.{0764}_6$	$-0.{0270}_4$	$-0.{0365}_5$	$-0.{0207}_{2.5}$	$-0.{0207}_{2.5}$
800	Abias	$\alpha$	$-0.{0210}_5$	$-0.{0183}_2$	$0.{0160}_1$	$0.{0202}_{3.5}$	$0.{0202}_{3.5}$	$0.{0719}_6$
		$\beta$	$-0.{0136}_3$	$-0.{0086}_1$	$-0.{1994}_5$	$-0.{0471}_4$	$-0.{0114}_2$	$-0.{2371}_6$
		c	$-0.{0014}_1$	$-0.{0018}_2$	$-0.{0520}_4$	$-0.{0863}_5$	$-0.{0204}_3$	$-0.{1077}_6$
		$\delta$	$0.{0540}_2$	$0.{0944}_5$	$0.{0378}_1$	$0.{0870}_4$	$0.{0567}_3$	$0.{1021}_6$
	RMSE	$\alpha$	$0.{1110}_5$	$0.{1022}_2$	$0.{1051}_4$	$0.{1283}_6$	$0.{1027}_3$	$0.{0946}_1$
		$\beta$	$0.{0348}_3$	$0.{0197}_1$	$0.{2008}_5$	$0.{1082}_4$	$0.{0295}_2$	$0.{2410}_6$
		c	$0.{1421}_3$	$0.{0108}_1$	$0.{1532}_4$	$0.{1264}_2$	$0.{2037}_5$	$0.{2493}_6$
		Sum of the ranks	23	20	28	33.5	24	39.5

and

Table 2. Simulation results.

(1.4, 2.2, 1.4, 2.2)
Sample Size (n)	Estimate	Parameter	ML	WLS	LS	RAD	AD	CVM
		$\delta$	$0.{3356}_1$	$-0.{7160}_4$	$-0.{6479}_3$	$-1.{1421}_6$	$-1.{0735}_5$	$-0.{6447}_2$
25	Abias	$\alpha$	$0.{6740}_3$	$1.{1193}_5$	$0.{5130}_1$	$1.{5391}_6$	$0.{6230}_2$	$1.{1112}_4$
		$\beta$	$0.{6415}_1$	$-1.{5986}_6$	$-1.{4295}_4$	$-0.{9016}_3$	$-0.{6949}_2$	$-1.{4722}_5$
		c	$0.{9229}_2$	$-2.{9668}_6$	$-1.{1132}_3$	$-1.{6042}_5$	$-0.{6386}_1$	$-1.{3191}_4$
		$\delta$	$1.{0084}_3$	$0.{9172}_2$	$1.{6505}_6$	$1.{1422}_5$	$1.{0832}_4$	$0.{6474}_1$
	RMSE	$\alpha$	$2.{0206}_3$	$2.{1215}_4$	$1.{1442}_2$	$2.{5395}_6$	$0.{6904}_1$	$2.{1456}_5$
		$\beta$	$3.{5909}_6$	$1.{6088}_5$	$1.{4729}_2$	$1.{4904}_3$	$0.{9734}_1$	$1.{5157}_4$
		c	$2.{8593}_5$	$2.{9678}_6$	$1.{1387}_2$	$2.{6046}_4$	$0.{6417}_1$	$1.{9801}_3$
		Sum of the ranks	24	38	23	38	17	28
		$\delta$	$0.{2589}_1$	$0.{4390}_2$	$-0.{4761}_3$	$-1.{1321}_6$	$-1.{0164}_5$	$-0.{4849}_4$
50	Abias	$\alpha$	$0.{2986}_2$	$1.{0697}_6$	$0.{2319}_1$	$1.{0631}_5$	$0.{5573}_3$	$0.{8922}_4$
		$\beta$	$0.{2745}_1$	$-1.{3375}_4$	$-1.{3420}_6$	$-0.{6005}_3$	$-0.{5321}_2$	$-1.{3407}_5$
		c	$0.{3903}_1$	$-2.{0790}_6$	$-1.{0808}_4$	$-0.{6203}_3$	$-0.{6105}_2$	$-1.{2966}_5$
		$\delta$	$0.{8915}_3$	$0.{8556}_2$	$1.{5039}_6$	$1.{1322}_5$	$1.{0165}_4$	$0.{5113}_1$
	RMSE	$\alpha$	$1.{0950}_2$	$1.{4070}_5$	$1.{1206}_3$	$1.{5635}_6$	$0.{6573}_1$	$1.{3103}_4$
		$\beta$	$1.{7589}_6$	$1.{3375}_3$	$1.{3429}_5$	$1.{0055}_2$	$0.{6322}_1$	$1.{3415}_4$
		c	$1.{6086}_3$	$2.{0792}_5$	$1.{1146}_2$	$2.{2049}_6$	$0.{6105}_1$	$1.{7325}_4$
		Sum of the ranks	19	33	30	36	19	31
		$\delta$	$0.{2539}_1$	$0.{3080}_3$	$-0.{3550}_4$	$-1.{1297}_6$	$-0.{7002}_5$	$-0.{3023}_2$
100	Abias	$\alpha$	$0.{0682}_1$	$0.{4039}_3$	$0.{1922}_2$	$0.{6775}_5$	$0.{4778}_4$	$0.{7192}_6$
		$\beta$	$-0.{1117}_1$	$-1.{3357}_6$	$-1.{2765}_5$	$-0.{5068}_3$	$-0.{5000}_2$	$-0.{7654}_4$
		c	$0.{0779}_1$	$-2.{0745}_6$	$-0.{3863}_2$	$-0.{6023}_4$	$-0.{5107}_3$	$-1.{0862}_5$
		$\delta$	$0.{8356}_3$	$0.{8482}_4$	$0.{7531}_2$	$1.{1298}_6$	$1.{0016}_5$	$0.{4553}_1$
	RMSE	$\alpha$	$0.{4999}_1$	$0.{8039}_4$	$0.{9256}_6$	$0.{7972}_3$	$0.{5595}_2$	$0.{9247}_5$
		$\beta$	$0.{8215}_3$	$1.{3357}_6$	$1.{3066}_5$	$0.{6828}_2$	$0.{5200}_1$	$0.{8766}_4$
		c	$0.{6535}_2$	$2.{0745}_6$	$0.{8642}_3$	$1.{6224}_5$	$0.{5968}_1$	$1.{3863}_4$
		Sum of the ranks	13	38	29	34	23	31
		$\delta$	$0.{1821}_1$	$0.{2331}_3$	$-0.{2176}_2$	$-0.{9952}_6$	$-0.{5101}_5$	$-0.{2377}_4$
200	Abias	$\alpha$	$0.{0200}_1$	$0.{4006}_4$	$0.{1874}_2$	$0.{5652}_6$	$0.{3627}_3$	$0.{5002}_5$
		$\beta$	$-0.{0362}_1$	$-1.{0340}_5$	$-1.{0541}_6$	$-0.{4089}_3$	$-0.{3773}_2$	$-0.{5409}_4$
		c	$0.{0184}_1$	$-1.{0697}_6$	$-0.{2980}_2$	$-0.{5217}_4$	$-0.{4216}_3$	$-0.{8979}_5$
		$\delta$	$0.{7186}_3$	$0.{7331}_4$	$0.{7177}_2$	$1.{0300}_6$	$0.{9411}_5$	$0.{3718}_1$
	RMSE	$\alpha$	$0.{2678}_1$	$0.{4006}_2$	$0.{5002}_3$	$0.{5652}_5$	$0.{5056}_4$	$0.{8030}_6$
		$\beta$	$0.{7658}_4$	$1.{1340}_5$	$1.{1542}_6$	$0.{5089}_2$	$0.{4564}_1$	$0.{7543}_3$
		c	$0.{2817}_1$	$1.{6972}_6$	$0.{4989}_2$	$0.{7800}_4$	$0.{5249}_3$	$0.{8989}_5$
		Sum of the ranks	13	25	25	36	26	34
		$\delta$	$0.{1220}_2$	$0.{1355}_3$	$0.{1094}_1$	$-0.{5309}_6$	$-0.{4582}_5$	$0.{1463}_4$
400	Abias	$\alpha$	$0.{0073}_1$	$0.{3962}_4$	$-0.{0813}_2$	$0.{4608}_5$	$0.{3316}_3$	$-0.{4805}_6$
		$\beta$	$-0.{0132}_1$	$-0.{7316}_6$	-$0.{6052}_5$	$-0.{2094}_3$	$-0.{3150}_4$	$-0.{2009}_2$
		c	$-0.{0068}_1$	$-1.{0599}_6$	$-0.{1091}_2$	$-0.{3199}_3$	$-0.{3582}_4$	$-0.{4212}_5$
		$\delta$	$0.{5943}_3$	$0.{6355}_4$	$0.{5472}_2$	$0.{7309}_6$	$0.{6841}_5$	$0.{2303}_1$
	RMSE	$\alpha$	$0.{2046}_1$	$0.{3962}_3$	$0.{3558}_2$	$0.{5082}_5$	$0.{4533}_4$	$0.{7543}_6$
		$\beta$	$0.{6150}_3$	$0.{9316}_6$	$0.{7146}_5$	$0.{4094}_2$	$0.{4005}_1$	$0.{7130}_4$
		c	$0.{2099}_2$	$1.{0599}_6$	$0.{2035}_1$	$0.{4992}_4$	$0.{4805}_3$	$0.{5024}_5$
		Sum of the ranks	13	38	20	34	29	33
		$\delta$	$0.{0409}_2$	$0.{0296}_1$	$-0.{0853}_3$	$-0.{3277}_6$	$-0.{3101}_5$	$-0.{1353}_4$
800	Abias	$\alpha$	$-0.{0020}_1$	$0.{1849}_3$	$0.{0198}_2$	$0.{3562}_6$	$0.{2857}_4$	$0.{2982}_5$
		$\beta$	$-0.{0064}_1$	$-0.{3256}_5$	$-0.{3455}_6$	$-0.{1257}_2$	$-0.{2535}_4$	$-0.{1457}_3$
		c	$-0.{0019}_1$	$-0.{5027}_6$	$-0.{1022}_2$	$-0.{1315}_3$	$-0.{2669}_4$	$-0.{3425}_5$
		$\delta$	$0.{5027}_2$	$0.{5385}_4$	$0.{5212}_3$	$0.{6328}_6$	$0.{6007}_5$	$0.{2137}_1$
	RMSE	$\alpha$	$0.{1399}_1$	$0.{2852}_3$	$0.{1984}_2$	$0.{4562}_5$	$0.{3713}_4$	$0.{4982}_6$
		$\beta$	$0.{4721}_4$	$0.{6256}_6$	$0.{4355}_3$	$0.{2552}_1$	$0.{3655}_2$	$0.{5677}_5$
		c	$0.{1412}_2$	$0.{9013}_6$	$0.{1402}_1$	$0.{3203}_3$	$0.{3672}_4$	$0.{4020}_5$
		Sum of the ranks	14	34	22	32	32	34

. The tables list the average bias (Abias) and root mean square errors (RMSEs) of the model parameters as measures of the effectiveness and consistency of the parameter estimates. The Abias and RMSE for the estimated parameter, say $\widehat{\eta }$, are given by

$$Abias\left(\widehat{\eta }\right)={\displaystyle \frac{{\sum }_{i=1}^N{\widehat{\eta }}_i}{N}}-\eta ,\mathrm{and}RMSE\left(\widehat{\eta }\right)=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^N{({\widehat{\eta }}_i-\eta )}^2}{N}}},$$

respectively.

RMSE plots for different estimation methods are shown in Figure 7 and Figure 8. We note that the RMSE declines with an increase in sample size.

Table 3. Partial and overall ranks of all estimation methods.

Parameters	n	ML	WLS	LS	RAD	AD	CVM
$\delta =0.3,\alpha =1.4,\beta =0.6,c=2.2$	25	1	3	6	4.5	2	4.5
	50	1	2	4	6	3	5
	100	1	3	6	5	2	4
	200	1	3	5	6	2	4
	400	2	3	4	6	1	5
	800	2	1	4	5	3	6
$\delta =1.4,\alpha =2.2,\beta =1.4,c=2.2$	25	3	5.5	2	5.5	1	4
	50	1.5	5	3	6	1.5	4
	100	1	6	3	5	2	4
	200	1	2.5	2.5	6	4	5
	400	1	6	2	5	3	4
	800	1	5.5	2	3.5	3.5	5.5
∑ ranks		16.5	45.5	43.5	63.5	28	55
Overall rank		1	4	3	6	2	5

presents the partial and overall rankings of different estimation methods for the MO-TIIEHL-OBX-LLoG distribution across various parameter values. The findings suggest that the MO-TIIEHL-OBX-LLoG distribution exhibits stability, as the Abias and RMSE occasionally decrease with increasing sample size across all estimation methods. Furthermore, Table the results highlight that the MLE method provides the most accurate estimates for the MO-TIIEHL-OBX-LLoG parameters, followed by the AD and LS methods, whereas the RAD method exhibits the weakest performance.

6. Risk Measures

This section explores key risk measures frequently utilized by financial and actuarial professionals to evaluate market risk exposure in an investment portfolio. These measures include Value at Risk (VaR), Tail Value at Risk (TVaR), Tail Variance (TV), and Tail Variance Premium (TVP).

6.1. Value at Risk

VaR is a widely used actuarial measure for evaluating financial market risk. It is typically expressed with a specified confidence level, such as ($90\%$, $95\%$, or $99\%$). The VaR for the MO-TIIEHL-OBX-G family of distributions is defined as follows:

$$\begin{array}{cc}\hfill Va{R}_{\theta }& =G^{-1}\left({\displaystyle \frac{{\left(-log\left[1-{\left(\frac{1-{\left(\frac{1-\theta }{\delta +(1-\theta )\overline{\delta }}\right)}^{\frac{1}{\alpha }}}{1+{\left(\frac{1-\theta }{\delta +(1-\theta )\overline{\delta }}\right)}^{\frac{1}{\alpha }}}\right)}^{\frac{1}{\beta }}\right]\right)}^{\frac{1}{2}}}{1+{\left(-log\left[1-{\left(\frac{1-{\left(\frac{1-\theta }{\delta +(1-\theta )\overline{\delta }}\right)}^{\frac{1}{\alpha }}}{1+{\left(\frac{1-\theta }{\delta +(1-\theta )\overline{\delta }}\right)}^{\frac{1}{\alpha }}}\right)}^{\frac{1}{\beta }}\right]\right)}^{\frac{1}{2}}}}\right),\hfill \end{array}$$

(15)

where $\theta \in (0,1)$ is a specified level of significance.

6.2. Tail Value at Risk

The TVaR represents the expected loss given that an event exceeding the predefined probability threshold has occurred. It provides a more comprehensive assessment of risk by considering extreme losses beyond the VaR level. The TVaR for the MO-TIIEHL-OBX-G family of distributions is defined as follows:

$$\begin{array}{ccc}\hfill TVa{R}_{\theta }& =& E(X\mid X>x_{\theta })={\displaystyle \frac{1}{1-\theta }}{\int }_{Va{R}_{\theta }}^{\infty }xf\left(x\right)dx\hfill \\ & =& {\displaystyle \frac{1}{1-\theta }}\sum _{q=0}^{\infty }{\int }_{Va{R}_{\theta }}^{\infty }x{\omega }_{q+1}{h}_{q+1}(x;\mathsf{\Xi })dx,\hfill \end{array}$$

(16)

where ${\omega }_{q+1}$ is given in Equation (11) and $h_{q+1}(x;\mathsf{\Xi })=(h+1)G^q(x;\mathsf{\Xi })g(x;\mathsf{\Xi })$ is the pdf of the exp-G distribution with the power parameter $(q+1).$

6.3. Tail Variance

The TV measures the variability of losses beyond the VaR threshold, providing insight into the dispersion of extreme losses. The TV for the MO-TIIEHL-OBX-G family of distributions is defined as follows:

$$\begin{array}{ccc}\hfill T{V}_{\theta }& =& E(X^2\mid X>x_{\theta })-{\left(TVa{R}_{\theta }\right)}^2\hfill \\ & =& {\displaystyle \frac{1}{1-\theta }}{\int }_{Va{R}_{\theta }}^{\infty }{x}^{2}f\left(x\right)dx-{\left(TVa{R}_{\theta }\right)}^2\hfill \\ & =& {\displaystyle \frac{1}{1-\theta }}\sum _{q=0}^{\infty }{\omega }_{q+1}{\int }_{Va{R}_{\theta }}^{\infty }{x}^2{h}_{q+1}(x;\mathsf{\Xi })dx-{\left(TVa{R}_{\theta }\right)}^2.\hfill \end{array}$$

(17)

6.4. Tail Variance Premium

The TVP is an important risk measure widely used in insurance analysis. It quantifies the additional premium required to account for the variability of extreme losses beyond the VaR. The TVP for the MO-TIIEHL-OBX-G family of distributions is defined as follows:

$$TV{P}_{\theta }=TVa{R}_{\theta }+\lambda \left(T{V}_{\theta }\right),$$

(18)

where $0<\lambda <1$. The TVP of the MO-TIIEHL-OBX-G family of distributions can be obtained by substituting Equations (16) and (17) into Equation (18).

6.5. Numerical Study for the Risk Measures

At this juncture, we assess the effectiveness of the MO-TIIEHL-OBX-LLoG distribution in modeling heavy-tailed data by conducting a numerical simulation of key risk measures. The results are then compared with those of its sub-models as well as equi-parameter models, specifically the Alpha Power Exponentiated Log-Logistic (APExLLD) distribution proposed by [21] and the Heavy-Tailed Generalized Topp–Leone Log-Logistic (HTGenTLL) distribution introduced by [22].

Table 4. Simulation results of VaR, TVaR, TV, and TVP.

Significance Level		0.7	0.75	0.8	0.85	0.9	0.95
MO-TIIEHL-OBX-LLoG ($\delta =1.1,\alpha =0.7,\beta =5.5,c=0.2$)	VaR	2.1124	2.2503	2.4140	2.6182	2.8953	3.3454
	TVaR	1.9689	2.2814	2.7358	3.4637	4.8428	8.6341
	TV	76.5335	91.1101	112.6387	147.7041	215.1270	399.3349
	TVP	55.5424	70.6139	92.8467	129.0122	198.4572	388.0023
MO-TIIEHL-OBX-LLoG ($\delta =1.0,\alpha =0.7,\beta =5.5,c=0.2$)	VaR	2.0091	2.1085	2.2264	2.3736	2.5744	2.9037
	TVaR	1.7200	1.9960	2.3985	3.0451	4.2748	7.6771
	TV	22.2368	26.1101	31.6559	40.2540	55.2919	85.8062
	TVP	17.2858	21.5787	27.7232	37.2610	54.0375	89.1931
MO-TIIEHL-OBX-LLoG ($\delta =1.1,\alpha =1.0,\beta =5.5,c=0.2$)	VaR	1.2636	1.3617	2.0774	2.2131	2.8168	3.1372
	TVaR	0.7460	0.8606	1.0271	1.2935	1.7966	3.1723
	TV	7.1450	8.3766	10.1435	12.8951	17.7657	28.2292
	TVP	5.7475	7.1431	9.1420	12.2544	17.7858	29.9901
MO-TIIEHL-OBX-LLoG ($\delta =1.1,\alpha =0.7,\beta =5.5,c=1.0$)	VaR	1.6813	1.7574	1.8464	1.9554	2.1003	2.3285
	TVaR	1.7622	2.0546	2.4841	3.1812	4.5267	8.3453
	TV	17.1972	19.8867	23.5344	28.6380	35.4242	30.9662
	TVP	13.8003	16.9696	21.3117	27.5236	36.4085	37.7633
MO-TIIEHL-OBX-LLoG ($\delta =1.1,\alpha =1.0,\beta =5.5,c=1.0$)	VaR	1.6788	1.7337	1.7978	1.8765	1.9814	2.1486
	TVaR	0.9592	1.1081	1.3255	1.6756	2.3431	4.2007
	TV	2.8928	3.2448	3.6748	4.1516	4.2952	5.8521
	TVP	2.9842	3.5417	4.2654	5.2044	6.2088	3.3912
APExLLD ($\alpha =1.1,a=1.9,b=1.2,c=0.0$)	VaR	0.5417	0.6815	0.8652	1.1226	1.5288	2.3797
	TVaR	1.7497	1.9779	2.2802	2.7117	3.4139	4.9413
	TV	7.8451	9.0645	10.8087	13.5354	18.4869	30.8310
	TVP	7.2413	8.7764	10.9272	14.2169	20.0521	34.2308
HTGenTLL ($\lambda =1.1,\alpha =5.1,\beta =2.9,c=0.2$)	VaR	1.2015	1.3140	1.4520	1.6321	1.8938	2.3720
	TVaR	1.8703	1.9931	2.1463	2.3492	2.6472	3.1926
	TV	0.7327	0.7863	0.8622	0.9793	1.1913	1.7483
	TVP	2.3832	2.5829	2.8360	3.1816	3.7195	4.8536

presents the numerical results for the VaR, TVaR, TV, and TVP across the seven compared distributions. A distribution with higher values of these risk measures is considered to have a heavier tail. Based on the figures in Table 4, we conclude that the MO-TIIEHL-OBX-LLoG distribution exhibits a heavier tail than its sub-models and the non-nested equi-parameter APExLLD and HTGenTLL distributions. This makes it a suitable choice for modeling heavy-tailed data.

7. Applications

Examples are presented in this section to illustrate the usefulness of the MO-TIIEHL-OBX-LLoG distribution in applications using real data. Several goodness-of-fit statistics, including the -2log-likelihood (-2ln(L)), Akaike information criterion (AIC), and Bayesian information criterion (BIC) are employed to compare the MO-TIIEHL-OBX-LLoG distribution to other equi-parameter models. We also use the Cramér–von Mises ($W^{*}$), Anderson–Darling ($A^{*}$), and Kolmogorov–Smirnov (K-S) statistics as well as their associated p-values. In general, a better model fit is demonstrated by smaller values of goodness-of-fit statistics and the highest p-value for the K-S statistics. In addition, graphs including the Kaplan–Meier (K-M) survival plot, empirical cumulative distribution function (ECDF), total time on test (TTT) scaled plot, and probability plot are presented.

The MO-TIIEHL-OBX-LLoG distribution is compared to several distributions, including the type II exponentiated half-logistic Weibull (TIIEHLW) distribution by [5], the Topp–Leone–odd Burr X-log-logistic (TLOBXLLoG) distribution by [17], the Burr XII–Burr XII (BXII-BXII) distribution by [23], the logistic Burr XII (LBXII) distribution by [24], the Marshall–Olkin generalized log-logistic (MOGLL) distribution by [10], the Marshall–Olkin exponentiated Fréchet (MOEF) distribution by [25], the Alpha power exponentiated log-logistic (APExLLD) distribution by [21], and the heavy-tailed generalized Topp–Leone log-logistic (HTGenTLL) distribution by [22]. The pdf’s of these distributions are given in Appendix A.

7.1. Remission Time of Cancer Data

The dataset represents the remission times of 128 cancer patients. It was utilized in [26].

Table 5. Estimates of models and goodness-of-fit statistics for remission time data.

	Estimates				Statistics
Model	$\delta$	$\alpha$	$\beta$	c	$-2log\left(L\right)$	$AIC$	$CAIC$	$BIC$	$W^{*}$	$A^{*}$	$K-S$	p-value	SS
MO-TIIEHL-OBX-LLoG	9.6978 $\times {10}^{02}$	1.1338 $\times {10}^{01}$	3.7292	6.5701 $\times {10}^{-02}$	819.2223	827.2223	827.5475	838.6304	0.0175	0.1019	0.0307	0.9997	0.0151
	(5.4183 $\times {10}^{-05}$)	(2.0601 $\times {10}^{-02}$)	(1.1030 $\times {10}^{-01}$)	(4.2346 $\times {10}^{-03}$)
	$\alpha$	$\theta$	$\beta$
TLOBXLLoG	20.4950	0.9898	0.1528		832.2939	838.2939	838.4874	846.8500	0.1465	0.9796	0.0739	0.4862	0.1803
	(23.7196)	(0.6755)	(0.0099)
	a	$\lambda$	$\delta$	$\gamma$
TIIEHLW	9.3869 $\times {10}^{02}$	1.7964 $\times {10}^{02}$	2.9019	4.3135 $\times {10}^{-02}$	822.9009	830.9060	831.2312	842.3141	0.0685	0.4256	0.0539	0.8515	0.0673
	(2.1942 $\times {10}^{-06}$)	(1.2065 $\times {10}^{-04}$)	(2.2305 $\times {10}^{-02}$)	(2.7661 $\times {10}^{-03}$)
	a	b	$\alpha$	$\beta$
BXII-BXII	5.0574 $\times {10}^{-01}$	9.9973 $\times {10}^{-02}$	3.6341	9.9147 $\times {10}^{02}$	822.1404	830.1404	830.4656	841.5486	0.0570	0.3605	0.0509	0.8942	0.0552
	(3.6205 $\times {10}^{-01}$)	(1.2000 $\times {10}^{-01}$)	(1.6190)	(1.5951 $\times {10}^{-04}$)
	$\lambda$	d	c	s
LBXII	13.9233	0.4357	0.2974	0.0038	829.0201	837.0201	837.3453	848.4282	0.1012	0.6998	0.0483	0.9265	0.0535
	(0.0047)	(0.1491)	(0.0723)	(0.0052)
	$\delta$	a	$\alpha$	$\beta$
MOGLL	3.1630 $\times {10}^{06}$	3.7785 $\times {10}^{-02}$	3.0537 $\times {10}^{-04}$	3.9977 $\times {10}^{01}$	825.9789	833.9788	834.3040	845.3869	0.0475	0.3423	0.0550	0.8342	0.0911
	(4.6045 $\times {10}^{-12}$)	(4.2102 $\times {10}^{-03}$)	(3.4285 $\times {10}^{-04}$)	(2.2651 $\times {10}^{-06}$)
	$\alpha$	$\lambda$	$\delta$	$\beta$
MOEF	6.6142 $\times {10}^{02}$	1.5844 $\times {10}^{-01}$	1.0986 $\times {10}^{06}$	1.1521	822.1622	830.1622	830.4875	841.5704	0.0538	0.3500	0.0517	0.8832	0.0502
	(2.5092 $\times {10}^{-03}$)	(5.2148 $\times {10}^{-03}$)	(4.9159 $\times {10}^{-07}$)	(7.5640 $\times {10}^{-01}$)
	$\alpha$	a	b	c
APExLLD	3.8295 $\times {10}^{-04}$	4.8498 $\times {10}^{-01}$	3.4902 $\times {10}^{-03}$	8.7071 $\times {10}^{01}$	846.592	854.5916	854.9168	865.9997	0.2627	1.7289	0.0799	0.3873	0.1904
	(1.4896 $\times {10}^{-03}$)	(8.2971 $\times {10}^{-02}$)	(2.9078 $\times {10}^{-03}$)	(5.2280 $\times {10}^{-05}$)
	b	$\theta$	c	$\beta$
HTGenTLL	2.5660 $\times {10}^{01}$	4.3961 $\times {10}^{02}$	1.7472 $\times {10}^{-01}$	5.2477 $\times {10}^{-04}$	821.7432	829.7432	830.0684	841.1513	0.0471	0.3120	0.0480	0.9294	0.0417
	(1.6923 $\times {10}^{-04}$)	(1.3474 $\times {10}^{-08}$)	(1.7736 $\times {10}^{-02}$)	(1.3997 $\times {10}^{-04}$)

presents the maximum likelihood estimates (MLEs), standard errors (SEs) (in parentheses), and goodness-of-fit statistics for the remission time data. Based on the values in Table 5, we conclude that the MO-TIIEHL-OBX-LLoG distribution provides a better fit compared to the other equi-parameter models, since it has the lowest values of the goodness-of-fit statistics $-2log\left(L\right)$, $AIC,CAIC,BIC$, ${\mathrm{W}}^{*}$, ${\mathrm{A}}^{*}$, and K-S. The MO-TIIEHL-OBX-LLoG distribution’s p-value of the $K-S$ statistic is also the largest as compared to the other fitted distributions. Figure 9 and Figure 10 corroborate the results in Table 5.

7.2. Carbon Fiber Data

The dataset represents the breaking stress of carbon fibers with a length of 50 mm (GPa). It was recently utilized by [27] and is provided in Appendix A.

Table 6. Estimates of models and goodness-of-fit statistics for carbon fiber data.

	Estimates				Statistics
Model	$\delta$	$\alpha$	$\beta$	c	$-2log\left(L\right)$	$AIC$	$CAIC$	$BIC$	$W^{*}$	$A^{*}$	$K-S$	p-value	SS
MO-TIIEHL-OBX-LLoG	7.9960 $\times {10}^{02}$	7.9156$\times {10}^{02}$	1.3883 $\times {10}^{01}$	5.6589 $\times {10}^{-02}$	171.5707	179.5711	180.2269	188.3297	0.0812	0.4362	0.0657	0.9380	0.0616
	(2.1640 $\times {10}^{-04}$)	(7.3497 $\times {10}^{-04}$)	(2.6932 $\times {10}^{-01}$)	(5.9267 $\times {10}^{-03}$)
	$\alpha$	$\theta$	$\beta$
TLOBXLLoG	0.4149	23.3458	0.4524		184.5695	190.3248	190.7119	196.8937	0.2655	1.4483	0.1506	0.1001	0.3200
	(0.1873)	(9.6571)	(0.0266)
	a	$\lambda$	$\delta$	$\gamma$
TIIEHLW	2.6668 $\times {10}^{02}$	1.8559	4.1681 $\times {10}^{-03}$	1.8892	172.21	180.2273	180.8830	188.9859	0.0949	0.5327	0.0823	0.7629	0.0812
	(9.0535 $\times {10}^{-04}$)	(6.0362 $\times {10}^{-01}$)	(7.2830 $\times {10}^{-03}$)	(5.7951 $\times {10}^{-01}$)
	a	b	$\alpha$	$\beta$
BXII-BXII	9.9332 $\times {10}^{-02}$	8.1871 $\times {10}^{-01}$	3.6819 $\times {10}^{01}$	4.2860 $\times {10}^{02}$	172.1631	180.1631	180.8189	188.9218	0.0935	0.5281	0.0826	0.7589	0.0811
	(9.8805 $\times {10}^{-03}$)	(7.3219 $\times {10}^{-03}$)	(6.1847 $\times {10}^{-05}$)	(3.0345 $\times {10}^{-07}$)
	$\lambda$	d	c	s
LBXII	4.7440	2.4782 $\times {10}^{03}$	1.0322	5.2642 $\times {10}^{03}$	183.2949	191.2949	191.9507	200.0536	0.2592	1.3925	0.0939	0.6054	0.1537
	(4.9621 $\times {10}^{-01}$)	(1.2187 $\times {10}^{-04}$)	(5.8405 $\times {10}^{-03}$)	(5.9123 $\times {10}^{-05}$)
	$\delta$	a	$\alpha$	$\beta$
MOGLL	1.9647 $\times {10}^{04}$	9.9580 $\times {10}^{-02}$	3.5448 $\times {10}^{-01}$	4.8817 $\times {10}^{01}$	183.3055	191.3055	191.9612	200.0641	0.2682	1.4462	0.0949	0.5924	0.1570
	(1.0408 $\times {10}^{-07}$)	(7.6149 $\times {10}^{-03}$)	(5.6133 $\times {10}^{-02}$)	(4.0673 $\times {10}^{-06}$)
	$\alpha$	$\lambda$	$\delta$	$\beta$
MOEF	3.9383 $\times {10}^{05}$	2.5347 $\times {10}^{-01}$	7.3102 $\times {10}^{04}$	9.4078 $\times {10}^{-01}$	175.4444	183.4444	184.1001	192.2030	0.1436	0.7753	0.1050	0.4606	0.137
	(3.1272 $\times {10}^{-07}$)	(2.2964 $\times {10}^{-03}$)	(1.4974 $\times {10}^{-06}$)	(4.3489 $\times {10}^{-01}$)
	$\alpha$	a	b	c
APExLLD	6.5148 $\times {10}^{02}$	4.0874	2.2649	2.7501 $\times {10}^{-01}$	183.2165	191.2164	191.8722	199.9751	0.2596	1.3781	0.1147	0.3504	0.230
	(5.6474 $\times {10}^{-05}$)	(6.9509 $\times {10}^{-01}$)	(3.7404 $\times {10}^{-01}$)	(1.2474 $\times {10}^{-01}$)
	b	$\theta$	c	$\beta$
HTGenTLL	7.7041 $\times {10}^{01}$	1.8951 $\times {10}^{02}$	1.5163 $\times {10}^{-01}$	2.1003 $\times {10}^{03}$	173.4224	181.4224	182.0781	190.181	0.1136	0.6218	0.0925	0.6244	0.1027
	(1.6220)	(5.8707 $\times {10}^{-02}$)	(1.3597 $\times {10}^{-02}$)	(2.6600 $\times {10}^{-03}$)

presents the maximum likelihood estimates (MLEs), standard errors (SEs) (in parentheses), and goodness-of-fit statistics for the carbon fiber data. Based on the values in Table 6, we conclude that the MO-TIIEHL-OBX-LLoG distribution provides a better fit compared to the other equi-parameter models, since it has the lowest values of the goodness-of-fit statistics $-2log\left(L\right)$, $AIC,CAIC,BIC$, $W^{*}$, $A^{*}$, and K-S. The MO-TIIEHL-OBX-LLoG distribution’s p-value of the $K-S$ statistic is also the largest as compared to the other fitted distributions. Figure 11 and Figure 12 corroborate the results in Table 6.

7.3. Height Data

The dataset consists of height measurements (in cm) recorded for 100 Australian female athletes. It was previously utilized by [28], and the complete observations are provided in Appendix A.

Table 7. Estimates of models and goodness-of-fit statistics for height data.

	Estimates				Statistics
Model	$\delta$	$\alpha$	$\beta$	c	$-2log\left(L\right)$	$AIC$	$CAIC$	$BIC$	$W^{*}$	$A^{*}$	$K-S$	p-value	SS
MO-TIIEHL-OBX-LLoG	2.9017 $\times {10}^{03}$	2.8657 $\times {10}^{02}$	2.9446 $\times {10}^{02}$	1.3981 $\times {10}^{-01}$	696.8013	704.8026	705.2236	715.2233	0.0313	0.2033	0.0432	0.9923	0.0275
	(2.3347 $\times {10}^{-11}$)	(1.9910 $\times {10}^{-09}$)	(8.4166 $\times {10}^{-09}$)	(1.1369 $\times {10}^{-04}$)
	$\alpha$	$\theta$	$\beta$
TLOBXLLoG	3.5687 $\times {10}^{08}$	7.4386	2.4035 $\times {10}^{-01}$		772.1108	778.1111	778.3611	785.9266	0.4891	2.8872	0.2608	<0.001	1.8830
	(5.3580 $\times {10}^{-09}$)	(2.4618)	(2.7587 $\times {10}^{-03}$)
	a	$\lambda$	$\delta$	$\gamma$
TIIEHLW	2.1326 $\times {10}^{02}$	2.7418 $\times {10}^{02}$	5.0848 $\times {10}^{-02}$	8.3401 $\times {10}^{-01}$	707.9298	715.9301	716.3512	726.3508	0.0994	0.6209	0.1341	0.05483	0.3828
	(1.0112 $\times {10}^{-06}$)	(4.9965 $\times {10}^{-05}$)	(1.9763 $\times {10}^{-02}$)	(7.4975 $\times {10}^{-02}$)
	a	b	$\alpha$	$\beta$
BXII-BXII	1.3434	9.9308 $\times {10}^{-02}$	1.2136 $\times {10}^{02}$	2.1236	697.7634	705.7637	706.1848	716.1844	0.0538	0.3430	0.0545	0.9282	0.0425
	(1.8424 $\times {10}^{-03}$)	(1.3605 $\times {10}^{-04}$)	(2.7630 $\times {10}^{-08}$)	(5.1861 $\times {10}^{-06}$)
	$\lambda$	d	c	s
LBXII	8.6948 $\times {10}^{01}$	1.7098 $\times {10}^{-02}$	2.6223 $\times {10}^{01}$	1.8805 $\times {10}^{01}$	702.0292	710.0306	710.4516	720.4512	0.1093	0.7120	0.0514	0.9543	0.0574
	(5.3659 $\times {10}^{-05}$)	(1.4258 $\times {10}^{-04}$)	(1.8488 $\times {10}^{-01}$)	(5.4088 $\times {10}^{-02}$)
	$\delta$	a	$\alpha$	$\beta$
MOGLL	1.1053 $\times {10}^{06}$	3.7152 $\times {10}^{04}$	3.2956 $\times {10}^{03}$	2.6875	698.0007	706.0015	706.4225	716.4222	0.0470	0.3100	0.0635	0.8149	0.0641
	(8.8650 $\times {10}^{-11}$)	(3.7622 $\times {10}^{-08}$)	(1.1214 $\times {10}^{-06}$)	(4.1052 $\times {10}^{-03}$)
	$\alpha$	$\lambda$	$\delta$	$\beta$
MOEF	562.9600	3.7090	292.2274	1.0160	701.2151	709.2151	709.6362	719.6358	0.1033	0.6445	0.0766	0.6003	0.0971
	(0.6906)	(0.4051)	(11.9099)	(0.8383)
	$\alpha$	a	b	c
APExLLD	43.9590	30.6073	166.8402	0.9114	705.5595	713.5595	713.9806	723.9802	0.1414	0.9084	0.0779	0.5786	0.1409
	(77.6731)	(4.7926)	(5.0830)	(0.6220)
	b	$\theta$	c	$\beta$
HTGenTLL	3.1367 $\times {10}^{04}$	8.1214 $\times {10}^{01}$	7.2348 $\times {10}^{-01}$	2.4561 $\times {10}^{03}$	703.0059	711.0059	711.4270	721.4266	0.1099	0.6799	0.0824	0.5059	0.1336
	(1.9221 $\times {10}^{-09}$)	(9.0667 $\times {10}^{-08}$)	(6.1750 $\times {10}^{-04}$)	(1.4828 $\times {10}^{-09}$)

presents the maximum likelihood estimates (MLEs), standard errors (SEs) (in parentheses), and goodness-of-fit statistics for the height data. Based on the values in Table 7, we conclude that the MO-TIIEHL-OBX-LLoG distribution provide a better fit compared to the other equi-parameter models, since it has the lowest values of $-2log\left(L\right)$, $AIC,CAIC,BIC$, $W^{*}$, $A^{*}$, and K-S. The MO-TIIEHL-OBX-LLoG distribution’s p-value of the $K-S$ statistic is also the largest as compared to the other fitted distributions. Figure 13 and Figure 14 corroborate the results in Table 7.

8. Conclusions

A new generalized family of distributions called the Marshall–Olkin–type II exponentiated half-logistic–odd Burr X-G (MO-TIIEHL-OBX-G) family of distributions was developed and presented. The density of the new family of distributions can be expressed as an infinite linear combination of exponentiated-G densities. The hazard rate functions of the new family of distributions are quite flexible, with monotone and non-monotone shapes. We also obtain expressions for the moments, distribution of order statistics, probability-weighted moments, and Rényi entropy. The performance of the special case of the MO-TIIEHL-OBX-G distribution was evaluated by conducting various simulations across different sample sizes to assess the efficiency of different estimation methods. Risk measures for this distribution were also presented, and the results revealed that the Marshall–Olkin–type II exponentiated half-logistic–odd Burr X–log-logistic (MO-TIIEHL-OBX-LLoG) distribution is heavy-tailed. Finally, the special case of the MO-TIIEHL-OBX-LLoG distribution was fitted to three real datasets to illustrate the importance and applicability of the proposed family of distributions.

Future research on the MO-TIIEHL-OBX-G family of distributions could focus on extending its application to higher-dimensional data, allowing for multivariate modeling. Additionally, the proposed distribution could be applied to censored data scenarios using real-world datasets, including type-II progressive hybrid censored data, type-II double-censored data, and step-stress-accelerated data. Another promising direction is the incorporation of Bayesian estimation techniques, which could provide a more robust framework for parameter estimation, especially in the presence of small sample sizes and prior information.