Modeling Healthcare Data with a Novel Flexible Three-Parameter Distribution

Abstract

Developing flexible lifetime distributions is essential for accurately modeling reliability and lifetime data across various scientific and engineering contexts. In this work, we introduce a new three-parameter lifetime distribution, which extends the well-known two-parameter Sarhan–Tadj–Hamilton model. We derive and discuss several of its important theoretical properties, including the reliability characteristics and moments. The parameter estimation is carried out using both maximum likelihood and Bayesian approaches, providing a comprehensive comparison of inferential techniques. To further examine the efficiency and robustness of the proposed estimators, a detailed Monte Carlo simulation study is conducted under different sample sizes and parameter settings. The practical usefulness of the distribution is illustrated through its application to three real-world datasets, namely cancer and COVID-19 data, where it demonstrates superior fit and flexibility compared to existing and nested lifetime models. These findings highlight the potential of the proposed model as a valuable addition to the toolbox of applied statisticians and reliability practitioners.

1. Introduction

Sarhan et al. [1] proposed two new statistical distributions. The first is a single-parameter model that shares similarities with the Lindley distribution, and the second is obtained by applying a power transformation to the first. These are Sarhan–Tadj–Hamilton-I and Sarhan–Tadj–Hamilton-II and are referred to as STH-I$\left(\alpha \right)$ and STH-II$(\alpha ,\beta )$, respectively. They demonstrated that the hazard rate function (HRF) of STH-I$\left(\alpha \right)$ can only take a unimodal shape, while STH-II$(\alpha ,\beta )$ is far more flexible. Depending on the chosen parameters, its HRF may decrease, increase, exhibit a single peak, or even exhibit more complex patterns such as decreasing–increasing–decreasing or increasing–decreasing–increasing.

The survival function of the STH-I distribution is defined as follows:

$$S_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-}\mathrm{I}}(x;\alpha )={\displaystyle \frac{1}{1+\alpha }}\left[\alpha +\left(1+\alpha x\right)e^{-\alpha x}\right]e^{-\alpha x},x\ge 0;\alpha >0.$$

(1)

Similarly, the survival function of the STH-II distribution is given by

$$S_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x;\alpha ,\beta )={\displaystyle \frac{1}{1+\alpha }}\left[\alpha +\left(1+\alpha x^{\beta }\right)e^{-\alpha x^{\beta }}\right]e^{-\alpha x^{\beta }},x\ge 0;\alpha ,\beta >0.$$

(2)

Sarhan et al. [1] also demonstrated that the parameter $\alpha$ governs the shape of the probability density function (PDF) of the STH-I$\left(\alpha \right)$. Specifically, when $\alpha =3.535$, the PDF is symmetric; for $\alpha <3.535$, it is left-skewed; and for $\alpha >3.535$, it is right-skewed. Moreover, they observed that a random variable $X\sim \mathrm{S}\mathrm{T}\mathrm{H}\text{-II}(\alpha ,\beta )$ can be expressed as a mixture of the following three components: $\mathrm{Weibull}(\alpha ,\beta )$, $\mathrm{Weibull}(\alpha ,2\beta )$, and the $\mathrm{power}\mathrm{gamma}(\alpha ,2,2\beta )$, combined with mixture weights $a_1=\beta /(1+\beta )$, $a_2=1/\left[2(1+\beta )\right]$, and $a_3=1/\left[2(1+\beta )\right]$.

Generalizing a distribution is an important strategy in statistical modeling because it allows researchers to capture a wider variety of data patterns with greater accuracy. Real-world data often exhibit irregular shapes, varying spreads, or extreme values that standard distributions cannot adequately model. By introducing additional parameters, modifying the functional form, or combining distributions, statisticians can adjust skewness, tail behavior, and other characteristics, thereby improving the flexibility of the model. Such generalizations enable better data fitting, more reliable predictions, and a deeper understanding of the underlying phenomena.

A widely used method for creating generalized models involves the exponentiation of existing distributions (see Gupta and Kundu [2]). This strategy became particularly prominent in the mid-1990s and has since inspired the development of many exponentiated distribution families. Among these are the exponentiated Lindley (EL) distribution by Nadarajah et al. [3], the exponentiated quasi Lindley (EQL) distribution by Elbatal et al. [4], the exponentiated XLindley (EXL) distribution by Alomair et al. [5], and the exponentiated power Lindley (EPL) distribution by Ashour and Eltehiwy [6].

In this study, we introduce a novel three-parameter probability model, termed the exponentiated Sarhan–Tadj–Hamilton (ESTH-II) distribution, and examine its main statistical properties. The ESTH-II model contains, as a nested case, the new exponentiated Sarhan–Tadj–Hamilton-I (ESTH-I) distribution. The PDF of ESTH-II can display either a decreasing trend or a unimodal shape, while its HRF may be unimodal, decreasing–increasing–decreasing–increasing, or purely increasing. The parameter estimation is performed using both maximum likelihood (MLE) and Bayesian methods. The practical utility of the proposed distribution is illustrated through its applications to real datasets, demonstrating a superior fit compared to competing models from the same family, including EL, EQL, EXL, and EPL.

The primary goal of introducing this distribution is to leverage the flexibility afforded by its power and exponentiated parameters, allowing it to model a broad spectrum of lifetime data accurately.

The organization of this work unfolds in a clear sequence. First, Section 2 introduces the ESTH-II distribution and explores its asymptotic behavior. Section 3 examines its key statistical characteristics. Section 4 then describes the estimation of its parameters using both maximum likelihood and Bayesian methods. Section 5 presents the findings from simulation studies, and Section 6 applies the model to three real datasets, offering comparisons with both nested and alternative models. Finally, Section 7 provides concluding insights and outlines potential avenues for future investigation.

2. Exponentiated STH-II Distribution

The cumulative distribution function (CDF) of the ESTH-II distribution, defined by the three parameters $\theta =(\alpha ,\beta ,\gamma )$, is formulated as follows:

$$F_{\mathrm{E}\mathrm{S}\mathrm{T}\mathrm{H}\text{-}\mathrm{I}\mathrm{I}}(x;\alpha ,\beta ,\gamma )={\left[1-S_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-}\mathrm{I}\mathrm{I}}(x;\alpha ,\beta )\right]}^{\gamma },\alpha ,\beta ,\gamma >0.$$

Substituting from (2) into the above equation, we get

$$\begin{array}{c}\hfill F_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-}\mathrm{I}\mathrm{I}}(x;\alpha ,\beta ,\gamma )={\left\{1-{\displaystyle \frac{1}{1+\beta }}\left[\beta +\left(1+\beta x^{\alpha }\right)e^{-\beta x^{\alpha }}\right]e^{-\beta x^{\alpha }}\right\}}^{\gamma },x\ge 0;\alpha ,\beta ,\gamma >0.\end{array}$$

(3)

The PDF of the ESTH-II distribution is

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

(4)

$$\begin{array}{ccc}& & {\left\{1-{\displaystyle \frac{1}{1+\beta }}\left[\beta +\left(1+\beta x^{\alpha }\right)e^{-\beta x^{\alpha }}\right]e^{-\beta x^{\alpha }}\right\}}^{\gamma -1}.\hfill \end{array}$$

(5)

The survival function of the ESTH-II is given by

$$\begin{array}{c}\hfill S_{\mathrm{E}\mathrm{S}\mathrm{T}\mathrm{H}\text{-}\mathrm{I}\mathrm{I}}(x;\alpha ,\beta ,\gamma )=1-{\left\{1-{\displaystyle \frac{1}{1+\beta }}\left[\beta +\left(1+\beta x^{\alpha }\right)e^{-\beta x^{\alpha }}\right]e^{-\beta x^{\alpha }}\right\}}^{\gamma },\end{array}$$

(6)

and the HRF is defined as

$$\begin{array}{ccc}\hfill h_{\mathrm{E}\mathrm{S}\mathrm{T}\mathrm{H}\text{-}\mathrm{I}\mathrm{I}}(x;\alpha ,\beta ,\gamma )& =& {\displaystyle \frac{\alpha \beta \gamma x^{\alpha -1}\left[\beta +\left(1+2\beta x^{\alpha }\right)e^{-\beta x^{\alpha }}\right]e^{-\beta x^{\alpha }}}{(1+\beta )\left\{1-{\left\{1-\frac{1}{1+\beta }\left[\beta +\left(1+\beta x^{\alpha }\right)e^{-\beta x^{\alpha }}\right]e^{-\beta x^{\alpha }}\right\}}^{\gamma }\right\}}}\times \hfill \\ & & {\left\{1-{\displaystyle \frac{1}{1+\beta }}\left[\beta +\left(1+\beta x^{\alpha }\right)e^{-\beta x^{\alpha }}\right]e^{-\beta x^{\alpha }}\right\}}^{\gamma -1}.\hfill \end{array}$$

(7)

Finally, the reversed hazard rate function (RHRF) of ESTH-II $(\alpha ,\beta ,\gamma )$ distribution is

$$\begin{array}{ccc}\hfill r_{\mathrm{E}\mathrm{S}\mathrm{T}\mathrm{H}\text{-}\mathrm{I}\mathrm{I}}(x;\alpha ,\beta ,\gamma )& =& {\displaystyle \frac{\alpha \beta \gamma x^{\alpha -1}\left[\beta +\left(1+2\beta x^{\alpha }\right)e^{-\beta x^{\alpha }}\right]e^{-\beta x^{\alpha }}}{(1+\beta )\left\{1-\frac{1}{1+\beta }\left[\beta +\left(1+\beta x^{\alpha }\right)e^{-\beta x^{\alpha }}\right]e^{-\beta x^{\alpha }}\right\}}}.\hfill \end{array}$$

(8)

The ESTH-II $(\alpha ,\beta ,\gamma )$ distribution is a flexible probability model that encompasses a wide range of sub-models. In particular, by applying specific restrictions to its parameters, several nested models naturally arise as follows:

• When $\gamma =1$, the ESTH-II distribution reduces to the STH-II distribution with parameters $\alpha$ and $\beta$.
• Setting $\beta =1$, the ESTH-II yields the ESTH-I model, representing a new model.

Asymptotic Behavior

In the following lemma, we present the asymptotic behavior of the PDF of the ESTH-II distribution.

Lemma 1.

The asymptotic behavior of $f_{ESTH\text{-}II}(x;\alpha ,\beta ,\gamma )$ at $x\to 0$ is given as

$$\underset{x\to 0}{lim}{f}_{ESTH\text{-}II}(x;\alpha ,\beta ,\gamma )=\left\{\begin{array}{c}0,\alpha \gamma >1;\hfill \\ {\beta }^{\gamma },\alpha \gamma =1;\hfill \\ \infty ,\alpha \gamma <1.\hfill \end{array}\right.$$

(9)

Moreover, the asymptotic behavior of $f_{ESTH\text{-}II}$ as $x\to \infty$ is

$$\underset{x\to \infty }{lim}{f}_{ESTH\text{-}II}(x;\alpha ,\beta ,\gamma )=0.$$

(10)

Proof. 

Set $t=\beta x^{\alpha }$, so that $t\to 0$ as $x\to 0$. We may rewrite the PDF in (4) as

$$f_{ESTH\text{-}II}(x;\alpha ,\beta ,\gamma )={\displaystyle \frac{\alpha \beta \gamma x^{\alpha -1}}{1+\beta }}\mathrm{\Psi }\left(t\right){\mathrm{\Phi }}^{\gamma -1}\left(t\right),$$

where

$$\mathrm{\Psi }\left(t\right)=\left[\beta +\left(1+2t\right)e^{-t}\right]e^{-t},\mathrm{\Phi }\left(t\right)=1-{\displaystyle \frac{1}{1+\beta }}\left[\beta +\left(1+t\right)e^{-t}\right]e^{-t}.$$

Using the Taylor expansion $e^{-t}=1-t+{\displaystyle \frac{1}{2}}{t}^2+O\left(t^3\right)$, we obtain

$$(1+2t)e^{-t}=1+t-{\displaystyle \frac{3}{2}}{t}^2+O\left(t^3\right),$$

and therefore

$$\mathrm{\Psi }\left(t\right)=(\beta +1+t+{\displaystyle \frac{1}{2}}{t}^2+O\left(t^3\right))(1-t+{\displaystyle \frac{1}{2}}{t}^2+O\left(t^3\right))\sim 1+\beta +O\left(t\right).$$

(11)

Similarly, $(1+t)e^{-t}=1-{\displaystyle \frac{1}{2}}{t}^2+O\left(t^3\right),$ so $[\beta +(1+t)e^{-t}]e^{-t}=(\beta +1)-t+O\left(t^2\right)$. It follows that $\mathrm{\Phi }\left(t\right)=1-{\displaystyle \frac{1}{\beta +1}}[\beta +(1+t)e^{-t}]e^{-t}\sim t+O\left(t^2\right)$. From the above, we have

$${\mathrm{\Phi }}^{\gamma -1}\left(t\right)\sim t^{\gamma -1}(1+o\left(1\right)).$$

(12)

Combining the expansions (11) and (12), we obtain

$$f_{ESTH\text{-}II}(x;\alpha ,\beta ,\gamma )\sim {\displaystyle \frac{\alpha \beta \gamma x^{\alpha -1}}{1+\beta }}(1+\beta +O\left(t\right))t^{\gamma -1}(1+o\left(1\right)).$$

By simplifying constants and substituting $t=\beta x^{\alpha }$, we obtain

$$f_{ESTH\text{-}II}(x;\alpha ,\beta ,\gamma )\sim \alpha \beta \gamma x^{\alpha -1}{\beta }^{\gamma -1}{x}^{\alpha (\gamma -1)}(1+o\left(1\right))=\alpha \gamma {\beta }^{\gamma }{x}^{\alpha \gamma -1}(1+o\left(1\right)).$$

(13)

This implies that the limit of $f_{ESTH}(x;\alpha ,\beta ,\gamma )$ as $x\to 0$ follows from the known limits of $x^{\alpha \gamma -1}$, as follows:

• If $\alpha \gamma >1,$ then $f_{ESTH\text{-}II}(x;\alpha ,\beta ,\gamma )\to 0$.
• If $\alpha \gamma =1,$ then $f_{ESTH\text{-}II}(x;\alpha ,\beta ,\gamma )\to {\beta }^{\gamma }$.
• If $\alpha \gamma <1,$ then $f_{ESTH\text{-}II}(x;\alpha ,\beta ,\gamma )\to \infty$.

   This completes the proof of the limit relation (9). Since $\mathrm{\Psi }\left(t\right)\to 0$ and ${\mathrm{\Phi }}^{\gamma -1}\left(t\right)\to 1$ as $t\to \infty$, the limit relation (10) follows, completing the proof of the lemma.    □

Since the survival function of the ESTH-II distribution satisfies $S_{ESTH\text{-}II}(x;\alpha ,\beta ,\gamma )\to 1\mathrm{as} x\to 0\mathrm{and}{S}_{EST\mathrm{H}\text{-}II}(x;\alpha ,\beta ,\gamma )\to 0\mathrm{as}x\to \infty ,$ Lemma 1 implies that the limiting behavior of the HRF is analogous to that of the PDF.

Figure 1 shows the PDF and HRF of the ESTH-II distribution for different parameter values. The results indicate that the HRF of the ESTH-II distribution is highly versatile, taking forms such as unimodal, decreasing–increasing–decreasing–increasing, or purely increasing, depending on the parameter settings. Likewise, the PDF can appear as unimodal or monotonically decreasing. This flexibility in the shapes of both the PDF and HRF makes the ESTH-II distribution a powerful tool for modeling diverse types of real-world data across various applications.

3. Statistical Properties

In this section, we examine the key statistical properties of the ESTH-II distribution, including the quantile function, k-th moments, the moment-generating function, skewness, and kurtosis.

3.1. Quantiles and Random Samples’ Generation

Let X∼ESTH-II$(\alpha ,\beta ,\gamma )$. The $100q$-th quantile of X, denoted by $x_q$ for $q\in (0,1)$, is the value that satisfies

$$1-q^{1/\gamma }={\displaystyle \frac{1}{1+\beta }}\left[\beta +(1+\beta x_q^{\alpha })e^{-\beta x_q^{\alpha }}\right]e^{-\beta x_q^{\alpha }}.$$

(14)

As a closed-form expression for $x_q$ is unavailable, we determine its value numerically using Algorithm 1.

Algorithm 1 Quantile computation for ESTH-II$(\alpha ,\beta ,\gamma )$

Input: Parameters $\alpha >0$, $\beta >0$, $\gamma >0$; quantile level $q\in (0,1)$

1: Solve for $v\in (0,1)$: $(1-q^{1/\gamma })(1+\beta )-\left[\beta +(1-lnv)v\right]v=0$ 2: Compute: $x_q\leftarrow {\left(-{\displaystyle \frac{lnv}{\beta }}\right)}^{1/\alpha }$ 3: return  $x_q$

3.2. Random Sample Generation

Although the proposed distribution has a closed-form CDF, which allows the use of the inverse CDF method for generating random samples, its structure does not permit an explicit closed-form expression for the quantile function. As a result, quantile values must be computed numerically. Algorithm 2 describes the procedure for generating random samples from an $\mathrm{E}\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}(\alpha ,\beta ,\gamma )$ distribution.

Algorithm 2 Random number generation from $\mathrm{E}\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}(\alpha ,\beta ,\gamma )$

Input: Parameters $\alpha >0$, $\beta >0$, $\gamma >0$; sample size $n\in \mathbb{N}$

1: for $i=1$ to n do 2:       Generate $u_i\sim \mathcal{U}(0,1)$ 3:       Solve for $v\in (0,1)$: $u_i^{1/\gamma }(1+\beta )-\left[\beta +(1-lnv)v\right]v=0$ 4:       Compute: $T\leftarrow -{\displaystyle \frac{lnv}{\beta }}$ 5:       Set: $X_i\leftarrow T^{1/\alpha }$ 6: end for 7: return$\{X_1,X_2,\dots ,X_n\}$

3.3. Effect of Model Parameters on Quartiles

Using Algorithm 1, we calculate the first quartile ($Q_1$), median, and third quartile ($Q_3$) for various values of the parameters $\alpha$, $\beta$, and $\gamma$. This enables us to investigate how these parameters influence the distribution. Figure 2 illustrates the quantiles produced by Algorithm 1 across a set of chosen parameter values. The chosen model parameter values were selected to reflect different distributional shapes. The panels highlight how variations in $\alpha$, $\beta$, and $\gamma$ shift the distribution, alter its scale, and modify the balance between its lower, central, and upper portions. From the patterns observed in Figure 2, we can draw the following conclusions.

1.

Quartiles as a Function of $\mathit{\alpha }$: When $\beta$ and $\gamma$ are held constant, increasing $\alpha$ generally raises the lower quartile and the median. For small $\beta$ and $\gamma$, the upper quartile decreases as $\alpha$ increases, indicating a tighter distribution with a reduced upper tail. This effect is strongest when both $\beta$ and $\gamma$ are small. For larger values of $\beta$ and $\gamma$, the upper quartile remains nearly constant, and the three quartiles converge, yielding a more symmetric, compact distribution.

2.

Quartiles as a Function of $\mathit{\beta }$: As $\beta$ increases, all three quartiles decrease, regardless of the fixed values of $\alpha$ and $\gamma$. The decline is steeper when $\alpha$ and $\gamma$ are small, and more gradual when they are large. As $\beta$ grows, the gap between quartiles narrows, reducing spread and skewness, and pushing the distribution toward symmetry.

3.

Quartiles as a Function of $\mathit{\gamma }$: When $\alpha$ and $\beta$ are small, increasing $\gamma$ raises all quartiles, especially the upper one, indicating a heavier right tail and greater spread. For larger $\alpha$ and $\beta$, the influence of $\gamma$ diminishes, and the distribution remains narrower and more symmetric.

3.4. Moments

The next theorem presents the closed-form expressions for the moments of the ESTH-II distribution. These formulae are particularly valuable because they apply not only to the full model but also specialize to several well-known sub-distributions. By choosing the appropriate parameter settings, one can immediately recover the moments of the STH-I, STH-II, and ESTH-I families. This unified approach highlights the versatility of the ESTH-II framework, illustrating how the general model seamlessly incorporates a variety of related distributions within a single coherent structure.

Theorem 1.

The k-th moment of the ESTH-II distribution can be expressed as follows:

$$\mathbb{E}\left(X^k\right)={\displaystyle \frac{k\Gamma \left(\frac{k}{\alpha }\right)}{\alpha {\beta }^{k/\alpha }}}\sum _{m=1}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{\gamma }{m}}\right){\displaystyle \frac{{(-1)}^{m+1}}{{(1+\beta )}^m}}\sum _{n=0}^m\left({\displaystyle \genfrac{}{}{0pt}{}{m}{n}}\right){\beta }^{m-n}U\left({\displaystyle \frac{k}{\alpha }},{\displaystyle \frac{k}{\alpha }}+n+1,n+m\right).$$

(15)

Proof. 

The k-th moment of the ESTH-II distribution is

$$\mathbb{E}\left(X^k\right)=k{\int }_0^{\infty }{x}^{k-1}{S}_{\mathrm{E}\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}\left(x\right)dx.$$

Let $u=\beta x^{\alpha }$, with $dx={\displaystyle \frac{1}{\alpha {\beta }^{1/\alpha }}}{u}^{1/\alpha -1}du$, as follows:

$$\mathbb{E}\left(X^k\right)={\displaystyle \frac{k}{\alpha {\beta }^{k/\alpha }}}{\int }_0^{\infty }{u}^{k/\alpha -1}\left[1-{\left(1-{\displaystyle \frac{\beta +(1+u)e^{-u}}{1+\beta }}{e}^{-u}\right)}^{\gamma }\right]du.$$

Using the binomial expansion ${\left(1-z\right)}^{\gamma }={\sum }_{m=0}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{\gamma }{m}}\right){(-1)}^m{z}^m$, where $z={\displaystyle \frac{\beta +(1+u)e^{-u}}{1+\beta }}{e}^{-u}$, the $k\text{-}$th moment can be written as follows:

$$\mathbb{E}\left(X^k\right)={\displaystyle \frac{k}{\alpha {\beta }^{k/\alpha }}}\sum _{m=1}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{\gamma }{m}}\right){\displaystyle \frac{{(-1)}^{m+1}}{{(1+\beta )}^m}}\sum _{n=0}^m\left({\displaystyle \genfrac{}{}{0pt}{}{m}{n}}\right){\beta }^{m-n}{\int }_0^{\infty }{u}^{k/\alpha -1}{(1+u)}^n{e}^{-(n+m)u}du.$$

(16)

Using the Kummer function of the second kind (see, Abramowitz and Stegun [7]), which is defined as

$$U(a,b,z)={\displaystyle \frac{1}{\Gamma \left(a\right)}}{\int }_0^{\infty }{t}^{a-1}{(1+t)}^{b-a-1}{e}^{-zt}dt.$$

By assigning $a={\displaystyle \frac{k}{\alpha }}$, $b={\displaystyle \frac{k}{\alpha }}+n+1$, and $z=n+m$, we obtain

$${\int }_0^{\infty }{u}^{k/\alpha -1}{(1+u)}^n{e}^{-(n+m)u}du=\Gamma \left({\displaystyle \frac{k}{\alpha }}\right)U\left({\displaystyle \frac{k}{\alpha }},{\displaystyle \frac{k}{\alpha }}+n+1,n+m\right).$$

(17)

By substituting Equation (17) into Equation (16), we arrive at the final moment expression shown in (15), thereby completing the proof.    □

From the result established in the preceding theorem, we observe that for $\gamma =1$ (corresponding to the STH-II distribution), the binomial coefficients in Equation (15) take on a particularly simple form, as follows:

$$\left({\displaystyle \genfrac{}{}{0pt}{}{1}{m}}\right)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}m=1,\hfill \\ 0\hfill & \mathrm{if}m>1.\hfill \end{array}\right.$$

Consequently, only the term with $m=1$ contributes, so that the $k\text{-}$th moment reduces to

$$\mathbb{E}\left(X^k\right){|}_{\gamma =1}={\displaystyle \frac{k\Gamma \left(\frac{k}{\alpha }\right)}{\alpha {\beta }^{k/\alpha }}}·{\displaystyle \frac{1}{1+\beta }}\sum _{n=0}^1\left({\displaystyle \genfrac{}{}{0pt}{}{1}{n}}\right){\beta }^{1-n}U\left({\displaystyle \frac{k}{\alpha }},{\displaystyle \frac{k}{\alpha }}+n+1,n+1\right).$$

(18)

Here, the summation in Equation (18) contains only two terms:

• Term 1: For $n=0$,

  $$\left({\displaystyle \genfrac{}{}{0pt}{}{1}{0}}\right){\beta }^{1-0}U\left({\displaystyle \frac{k}{\alpha }},{\displaystyle \frac{k}{\alpha }}+1,1\right)=\beta (\mathrm{using}U(a,a+1,1)=1).$$
• Term 2: For $n=1$,

  $$\left({\displaystyle \genfrac{}{}{0pt}{}{1}{1}}\right){\beta }^{0}U\left({\displaystyle \frac{k}{\alpha }},{\displaystyle \frac{k}{\alpha }}+2,2\right)=U\left({\displaystyle \frac{k}{\alpha }},{\displaystyle \frac{k}{\alpha }}+2,2\right)=2^{-k/\alpha }\left(1+{\displaystyle \frac{k}{\alpha }}\right).$$

Combining these two contributions gives the final expression for the $k\text{-}$th moment of STH-II distribution in the form

$$\mathbb{E}\left(X^k\right){|}_{\gamma =1}={\displaystyle \frac{k\Gamma \left(\frac{k}{\alpha }\right)}{\alpha {\beta }^{k/\alpha }(1+\beta )}}\left[\beta +2^{-k/\alpha }\left(1+{\displaystyle \frac{k}{\alpha }}\right)\right].$$

(19)

3.5. Moment-Generating Function

Based on the result presented in Equation (15), Theorem 1, the moment-generating function of the random variable X∼ESTH-II$(\alpha ,\beta ,\gamma )$ can be represented as follows:

$${\mathbb{M}}_X\left(t\right)=\sum _{i=0}^{\infty }\sum _{m=1}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{\gamma }{m}}\right){\displaystyle \frac{{(-1)}^{m+1}i\Gamma \left(\frac{i}{\alpha }\right)t^i}{{(1+\beta )}^m\alpha {\beta }^{i/\alpha }i!}}\sum _{n=0}^m\left({\displaystyle \genfrac{}{}{0pt}{}{m}{n}}\right){\beta }^{m-n}U\left({\displaystyle \frac{i}{\alpha }},{\displaystyle \frac{i}{\alpha }}+n+1,n+m\right).$$

(20)

3.6. Skewness and Kurtosis

The skewness ($\xi$) and kurtosis ($\eta$), which measure the degree of asymmetry and the heaviness of the tails relative to the normal distribution, can be expressed in terms of these moments. The k-th moment is given by the following:

$$\xi ={\displaystyle \frac{\mathbb{E}\left(X^3\right)-3\mathbb{E}\left(X\right)\mathbb{E}\left(X^2\right)+2{\left(\mathbb{E}\left(X\right)\right)}^3}{{\left[\mathbb{E}\left(X^2\right)-{\left(\mathbb{E}\left(X\right)\right)}^2\right]}^{3/2}}}$$

(21)

and

$$\eta ={\displaystyle \frac{\mathbb{E}\left(X^4\right)-4\mathbb{E}\left(X\right)\mathbb{E}\left(X^3\right)+6{\left(\mathbb{E}\left(X\right)\right)}^2\mathbb{E}\left(X^2\right)-3{\left(\mathbb{E}\left(X\right)\right)}^4}{{\left[\mathbb{E}\left(X^2\right)-{\left(\mathbb{E}\left(X\right)\right)}^2\right]}^2}}.$$

(22)

Table 1. Descriptive statistics for the ESTH-II distribution. The top panel varies $\alpha$, the middle panel varies $\beta$, and the bottom panel varies $\gamma$, each with three sets of fixed parameter pairs. These chosen parameter values reflect different behaviors of the distributions.

	$\mathit{E}\left(\mathit{X}\right)$	$\mathit{E}\left({\mathit{X}}^2\right)$	$\mathit{E}\left({\mathit{X}}^3\right)$	$\mathit{E}\left({\mathit{X}}^4\right)$	${\mathit{\sigma }}_{\mathit{x}}^2$	$\mathit{\xi }$	$\mathit{\eta }$	$\mathit{E}\left(\mathit{X}\right)$	$\mathit{E}\left({\mathit{X}}^2\right)$	$\mathit{E}\left({\mathit{X}}^3\right)$	$\mathit{E}\left({\mathit{X}}^4\right)$	${\mathit{\sigma }}_{\mathit{x}}^2$	$\mathit{\xi }$	$\mathit{\eta }$	$\mathit{E}\left(\mathit{X}\right)$	$\mathit{E}\left({\mathit{X}}^2\right)$	$\mathit{E}\left({\mathit{X}}^3\right)$	$\mathit{E}\left({\mathit{X}}^4\right)$	${\mathit{\sigma }}_{\mathit{x}}^2$	$\mathit{\xi }$	$\mathit{\eta }$
$\alpha$	$(\beta ,\gamma )=(0.5,0.5)$							$(\beta ,\gamma )=(1,1)$							$(\beta ,\gamma )=(2,2)$
1.0	1.035	2.917	13.5	89.5	1.845	2.662	14.3	0.875	1.500	3.938	14.3	0.734	2.129	10.4	0.684	0.728	1.086	2.124	0.260	1.742	7.989
1.5	0.865	1.363	2.917	7.76	0.614	1.404	5.6	0.831	0.989	1.500	2.743	0.299	1.107	4.757	0.734	0.670	0.728	0.920	0.131	0.943	4.362
2.0	0.829	1.035	1.610	2.92	0.348	0.850	3.57	0.835	0.875	1.076	1.500	0.178	0.639	3.437	0.775	0.684	0.673	0.728	0.083	0.571	3.461
2.5	0.824	0.917	1.209	1.79	0.238	0.516	2.86	0.847	0.840	0.935	1.139	0.123	0.354	2.999	0.806	0.709	0.672	0.679	0.059	0.347	3.152
3.0	0.828	0.865	1.035	1.36	0.179	0.282	2.58	0.860	0.831	0.875	0.989	0.091	0.156	2.856	0.831	0.734	0.684	0.670	0.044	0.195	3.042
3.5	0.836	0.841	0.946	1.15	0.142	0.104	2.48	0.872	0.831	0.848	0.915	0.071	0.008	2.836	0.850	0.756	0.701	0.674	0.034	0.083	3.013
4.0	0.844	0.829	0.896	1.04	0.117	-0.037	2.48	0.882	0.835	0.835	0.875	0.057	-0.108	2.872	0.865	0.775	0.718	0.684	0.027	-0.002	3.021
4.5	0.852	0.825	0.865	0.964	0.098	-0.153	2.52	0.891	0.841	0.831	0.853	0.047	-0.201	2.937	0.877	0.792	0.734	0.696	0.022	-0.070	3.046
5.0	0.860	0.824	0.847	0.917	0.084	-0.251	2.59	0.899	0.847	0.830	0.840	0.039	-0.279	3.014	0.888	0.806	0.749	0.709	0.019	-0.125	3.079
$\beta$	$(\alpha ,\gamma )=(0.5,0.5)$							$(\alpha ,\gamma )=(1,1)$							$(\alpha ,\gamma )=(2,2)$
1.0	0.818	7.241	191.9	10283.5	6.572	10.4	224.2	0.875	1.500	3.938	14.25	0.734	2.129	10.4	1.071	1.301	1.760	2.616	0.155	0.582	3.556
1.5	0.387	1.626	19.82	478.5	1.476	10.1	206.1	0.600	0.711	1.289	3.200	0.351	2.119	10.2	0.887	0.894	1.005	1.242	0.108	0.578	3.502
2.0	0.227	0.556	3.880	53.1	0.504	9.8	195.3	0.458	0.417	0.578	1.094	0.207	2.106	10.0	0.775	0.684	0.673	0.728	0.083	0.571	3.461
2.5	0.149	0.240	1.083	9.518	0.218	9.7	188.1	0.371	0.274	0.309	0.472	0.136	2.094	9.86	0.698	0.555	0.492	0.480	0.068	0.564	3.431
3.0	0.106	0.120	0.379	2.321	0.109	9.6	183.1	0.313	0.194	0.184	0.236	0.097	2.085	9.76	0.640	0.467	0.380	0.340	0.057	0.559	3.408
3.5	0.079	0.067	0.156	0.701	0.060	9.5	179.3	0.270	0.145	0.119	0.131	0.072	2.076	9.67	0.595	0.404	0.305	0.254	0.050	0.554	3.390
4.0	0.061	0.040	0.072	0.248	0.036	9.42	176.4	0.237	0.113	0.081	0.079	0.056	2.070	9.61	0.558	0.355	0.252	0.197	0.044	0.550	3.375
4.5	0.049	0.025	0.036	0.099	0.023	9.36	174.1	0.212	0.090	0.058	0.050	0.045	2.064	9.55	0.527	0.317	0.213	0.157	0.039	0.547	3.363
5.0	0.040	0.017	0.020	0.043	0.015	9.32	172.2	0.192	0.073	0.042	0.033	0.037	2.059	9.51	0.501	0.287	0.183	0.128	0.036	0.544	3.353
$\gamma$	$(\alpha ,\beta )=(0.5,0.5)$							$(\alpha ,\beta )=(1,1)$							$(\alpha ,\beta )=(2,2)$
1.0	5.330	176.000	17280.000	3575040.100	147.600	8.300	148.500	0.875	1.500	3.940	14.251	0.734	2.129	10.401	0.602	0.458	0.410	0.417	0.095	0.650	3.401
1.5	7.421	259.901	25844.800	5359651.511	204.912	7.120	111.220	1.115	2.090	5.700	21.100	0.849	1.911	9.100	0.704	0.585	0.553	0.582	0.089	0.586	3.420
2.0	9.261	341.611	34362.600	7142371.300	255.910	6.471	92.000	1.301	2.616	7.389	27.680	0.922	1.79	8.45	0.775	0.684	0.673	0.728	0.083	0.571	3.462
2.5	10.920	421.200	42835.711	8923232.500	302.000	6.031	80.211	1.454	3.090	8.983	34.142	0.975	1.716	8.074	0.829	0.766	0.778	0.861	0.079	0.570	3.499
3.0	12.441	499.000	51265.700	10702266.300	344.311	5.710	72.100	1.584	3.520	10.505	40.458	1.016	1.667	7.82	0.871	0.835	0.871	0.983	0.076	0.576	3.530
3.5	13.800	575.100	59654.600	12479502.300	383.600	5.470	66.110	1.696	3.930	12.000	46.600	1.050	1.630	7.630	0.906	0.895	0.955	1.095	0.073	0.582	3.555
4.0	15.100	649.700	68003.800	14254968.500	420.400	5.270	61.500	1.796	4.300	13.400	52.700	1.080	1.602	7.490	0.936	0.948	1.031	1.200	0.071	0.589	3.576
4.5	16.400	722.800	76314.600	16028691.700	455.000	5.110	57.900	1.890	4.650	14.700	58.600	1.098	1.580	7.370	0.963	0.995	1.100	1.300	0.069	0.596	3.594
5.0	17.500	794.600	84588.600	17800697.400	487.800	4.970	54.900	1.970	4.980	16.100	64.500	1.120	1.560	7.300	0.986	1.038	1.166	1.390	0.067	0.603	3.609

summarizes key descriptive statistics of the ESTH-II distribution, encompassing the first four moments, the variance (${\sigma }_x^2$), and the parameters $\xi$ and $\eta$. Additionally, Figure 3 and Figure 4 depict the behavior of $\xi$, $\eta$, and ${\sigma }_x^2$ as each parameter ($\alpha$, $\beta$, and $\gamma$) varies individually, while the remaining two parameters are kept constant.

The numerical results presented in Table 1, along with the visual representations in Figure 3 and Figure 4, reveal several noteworthy patterns in the behavior of the ESTH-II distribution across different parameter settings. These findings illustrate how variations in $\alpha$, $\beta$, and $\gamma$ affect the shape, dispersion, and higher-order characteristics of the distribution. Below, we highlight the main insights, focusing on the observed trends and the sensitivity of the distribution to changes in its parameters.

• Effect of $\mathit{\alpha }$: When $\beta$ and $\gamma$ are held fixed, $\alpha$ tends to elevate the central tendency of the ESTH-II distribution while simultaneously decreasing both skewness and kurtosis. This reflects a movement toward a more symmetric distribution with less influence from extreme values. The effect is particularly noticeable when $\beta$ and $\gamma$ are small, whereas at higher $\alpha$ values, the distribution curves begin to flatten. In essence, larger $\alpha$ produces distributions that are more balanced, with lighter tails and reduced variability.
• Effect of $\mathit{\beta }$: Holding $\alpha$ and $\gamma$ constant, variations in $\beta$ primarily affect the tail structure and concentration of the distribution. Increasing $\beta$ raises the mean and higher-order moments, shifting the mass toward larger values. The skewness and kurtosis are initially high for small $\beta$, but stabilize as $\beta$ grows. Variance, however, decreases steadily with increasing $\beta$. The sensitivity to $\beta$ is strongest when both $\alpha$ and $\gamma$ are small, while for larger values, the curves flatten and their influence diminishes.
• Effect of $\mathit{\gamma }$: With $(\alpha ,\beta )$ fixed, increasing $\gamma$ produces systematic changes in the moments. For small $\gamma$, the distribution is highly skewed and heavy-tailed, especially when $(\alpha ,\beta )$ are small (e.g., 0.5). As $\gamma$ increases, both the mean and variance rise steadily, while the skewness and kurtosis decrease, reflecting a move toward symmetry and lighter tails. This effect is more pronounced when $(\alpha ,\beta )$ are low, whereas for higher values (e.g., 2), the distribution stabilizes quickly and changes occur at a slower rate. Overall, $\gamma$ acts as a scale and tail control factor, strongly influencing the distribution’s shape.

In summary, $\alpha$ primarily governs symmetry and tail behavior, $\beta$ regulates overall spread and tail decay, and $\gamma$ serves as a shape-modifying exponent, with its impact most evident when $\alpha$ and $\beta$ are small.

4. Parameter Estimation

In this section, we employ two fundamental approaches to statistical inference: maximum likelihood estimation and Bayesian estimation, to determine the parameters of the ESTH-II model. We also derive asymptotic confidence intervals for the maximum likelihood estimates (MLEs) of $\alpha$, $\beta$, and $\gamma$ based on complete sample data. All inferential procedures in this study are developed under the assumption of complete (uncensored) data.

4.1. Maximum Likelihood Estimation

Let $X_1,X_2,\dots ,X_n$ be a random sample from the ESTH-II$(\alpha ,\beta ,\gamma )$ distribution. Using the relationship between the PDF of the ESTH-II distribution and the PDF and CDF of the STH-II distribution, we write the following:

$$f_{\mathrm{E}\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x;\alpha ,\beta ,\gamma )=\gamma f_{\text{STH-II}}(x;\alpha ,\beta ){\left[F_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x;\alpha ,\beta )\right]}^{\gamma -1}.$$

From this, the log-likelihood function for the sample becomes

$$\mathcal{L}(\alpha ,\beta ,\gamma )=nlog\gamma +{\mathcal{L}}_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(\alpha ,\beta )+(\gamma -1)\sum _{i=1}^{n}log{F}_{\text{STH-II}}(x_i;\alpha ,\beta ),$$

(23)

where ${\mathcal{L}}_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(\alpha ,\beta )$ denotes the log-likelihood function corresponding to the STH-II distribution, which can be expressed as follows:

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\text{STH-II}}(\alpha ,\beta )& =nln\alpha +nln\beta -nln(1+\beta )+(\alpha -1)\sum _{i=1}^{n}ln{x}_i-\beta \sum _{i=1}^n{x}_i^{\alpha }\hfill \\ \hfill & +\sum _{i=1}^{n}ln\left(\beta +(1+2\beta x_i^{\alpha })e^{-\beta x_i^{\alpha }}\right).\hfill \end{array}$$

Note that ${\mathcal{L}}_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(\alpha ,\beta )$ is independent of $\gamma$, which simplifies the estimation process.

To find the maximum likelihood estimates, we take the derivative of the $\mathcal{L}(\alpha ,\beta ,\gamma )$ with respect to each parameter. The resulting expressions for the partial derivatives are as follows:

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \mathcal{L}}{\partial \alpha }}& ={\displaystyle \frac{\partial {\mathcal{L}}_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}}{\partial \alpha }}+(\gamma -1)\sum _{i=1}^n{\displaystyle \frac{\frac{\partial }{\partial \alpha }{F}_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x_i;\alpha ,\beta )}{F_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x_i;\alpha ,\beta )}},\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \mathcal{L}}{\partial \beta }}& ={\displaystyle \frac{\partial {\mathcal{L}}_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}}{\partial \beta }}+(\gamma -1)\sum _{i=1}^n{\displaystyle \frac{\frac{\partial }{\partial \beta }{F}_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x_i;\alpha ,\beta )}{F_{\text{STH-II}}(x_i;\alpha ,\beta )}},\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \mathcal{L}}{\partial \gamma }}& ={\displaystyle \frac{n}{\gamma }}+\sum _{i=1}^{n}log{F}_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x_i;\alpha ,\beta ).\hfill \end{array}$$

(26)

By setting the derivatives equal to zero, we obtain the system of likelihood equations. Solving this system leads to the MLEs of the model parameters. In particular, starting from (26), the estimator of $\gamma$ can be written explicitly in terms of $\alpha$ and $\beta$, as follows:

$$\widehat{\gamma }=-{\displaystyle \frac{n}{{\sum }_{i=1}^{n}log{F}_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x_i;\alpha ,\beta )}}.$$

(27)

Substituting (27) into (24) and (25) yields a system of two nonlinear equations involving $\alpha$ and $\beta$. Since these equations typically cannot be solved analytically, their solutions must be obtained using numerical techniques. Once numerical estimates for $\alpha$ and $\beta$ are available, the corresponding value of $\gamma$ can be directly computed from (27).

Confidence Intervals

To construct confidence intervals for the parameters $(\alpha ,\beta ,\gamma )$, we rely on the asymptotic distribution of the MLEs. Under the usual regularity conditions presented in Casella and Berger [8], the MLE $\widehat{\mathit{\theta }}=(\widehat{\alpha },\widehat{\beta },\widehat{\gamma })$ follows an approximately normal distribution, i.e.,

$$\widehat{\mathit{\theta }}\stackrel{·}{\sim }\mathcal{N}(\mathit{\theta },I_n^{-1}\left(\mathit{\theta }\right))$$

where $\mathit{\theta }=(\alpha ,\beta ,\gamma )$ denotes the true parameter vector and $I_n\left(\mathit{\theta }\right)$ represents the observed Fisher information matrix. This matrix is computed as the negative Hessian of the log-likelihood function evaluated at $\widehat{\mathit{\theta }}$, as follows:

$$I_n\left(\mathit{\theta }\right)=-{\left.\left(\begin{array}{ccc}{\displaystyle \frac{{\partial }^2\mathcal{L}}{\partial {\alpha }^2}}& {\displaystyle \frac{{\partial }^2\mathcal{L}}{\partial \alpha \partial \beta }}& {\displaystyle \frac{{\partial }^2\mathcal{L}}{\partial \alpha \partial \gamma }}\\ {\displaystyle \frac{{\partial }^2\mathcal{L}}{\partial \beta \partial \alpha }}& {\displaystyle \frac{{\partial }^2\mathcal{L}}{\partial {\beta }^2}}& {\displaystyle \frac{{\partial }^2\mathcal{L}}{\partial \beta \partial \gamma }}\\ {\displaystyle \frac{{\partial }^2\mathcal{L}}{\partial \gamma \partial \alpha }}& {\displaystyle \frac{{\partial }^2\mathcal{L}}{\partial \gamma \partial \beta }}& {\displaystyle \frac{{\partial }^2\mathcal{L}}{\partial {\gamma }^2}}\end{array}\right)\right|}_{\mathit{\theta }=\widehat{\mathit{\theta }}}$$

The practical implementation involves the following three steps:

1.

Calculate the standard errors $\mathrm{SE}\left({\widehat{\theta }}_j\right)=\sqrt{{\left[I_n^{-1}\left(\widehat{\mathit{\theta }}\right)\right]}_{jj}}$, $j=1,2,3.$

2.

Determine the critical value $z_{1-\vartheta /2}$ from the standard normal distribution.

3.

Construct the $(1-\vartheta )\times 100\%$ confidence intervals as follows:

$${\widehat{\theta }}_j\pm z_{1-\vartheta /2}·\mathrm{SE}\left({\widehat{\theta }}_j\right),j=1,2,3.$$

(28)

The second partial derivatives of $\mathcal{L}(\alpha ,\beta ,\gamma )$ can be derived as in the following.

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^2\mathcal{L}}{\partial {\alpha }^2}}& ={\displaystyle \frac{{\partial }^2{\mathcal{L}}_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}}{\partial {\alpha }^2}}+(\gamma -1)\sum _{i=1}^n{\displaystyle \frac{F_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x_i;\alpha ,\beta )·\frac{{\partial }^2{F}_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x_i;\alpha ,\beta )}{\partial {\alpha }^2}-{\left(\frac{\partial F_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x_i;\alpha ,\beta )}{\partial \alpha }\right)}^2}{{\left[F_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x_i;\alpha ,\beta )\right]}^2}},\hfill \\ \hfill {\displaystyle \frac{{\partial }^2\mathcal{L}}{\partial {\beta }^2}}& ={\displaystyle \frac{{\partial }^2{\mathcal{L}}_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}}{\partial {\beta }^2}}+(\gamma -1)\sum _{i=1}^n{\displaystyle \frac{F_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x_i;\alpha ,\beta )·\frac{{\partial }^2{F}_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x_i;\alpha ,\beta )}{\partial {\beta }^2}-{\left(\frac{\partial F_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x_i;\alpha ,\beta )}{\partial \beta }\right)}^2}{{\left[F_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x_i;\alpha ,\beta )\right]}^2}},\hfill \\ \hfill {\displaystyle \frac{{\partial }^2\mathcal{L}}{\partial {\gamma }^2}}& =-{\displaystyle \frac{n}{{\gamma }^2}},\hfill \\ \hfill {\displaystyle \frac{{\partial }^2\mathcal{L}}{\partial \alpha \partial \beta }}& ={\displaystyle \frac{{\partial }^2{\mathcal{L}}_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}}{\partial \alpha \partial \beta }}+(\gamma -1)\sum _{i=1}^n{\displaystyle \frac{F_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x_i;\alpha ,\beta )·\frac{{\partial }^2{F}_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x_i;\alpha ,\beta )}{\partial \alpha \partial \beta }-\frac{\partial F_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x_i;\alpha ,\beta )}{\partial \alpha }·\frac{\partial F_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x_i;\alpha ,\beta )}{\partial \beta }}{{\left[F_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x_i;\alpha ,\beta )\right]}^2}},\hfill \\ \hfill {\displaystyle \frac{{\partial }^2\mathcal{L}}{\partial \alpha \partial \gamma }}& =\sum _{i=1}^n{\displaystyle \frac{1}{F_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x_i;\alpha ,\beta )}}·{\displaystyle \frac{\partial F_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x_i;\alpha ,\beta )}{\partial \alpha }},\hfill \\ \hfill {\displaystyle \frac{{\partial }^2\mathcal{L}}{\partial \beta \partial \gamma }}& =\sum _{i=1}^n{\displaystyle \frac{1}{F_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x_i;\alpha ,\beta )}}·{\displaystyle \frac{\partial F_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}(x_i;\alpha ,\beta )}{\partial \beta }},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial F_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}}{\partial \alpha }}=& {\displaystyle \frac{e^{-\alpha x^{\beta }}}{{(1+\alpha )}^2}}\left[\alpha +(1+\alpha x^{\beta })e^{-\alpha x^{\beta }}\right]-{\displaystyle \frac{e^{-\alpha x^{\beta }}}{1+\alpha }}\left[1-x^{\beta }{e}^{-\alpha x^{\beta }}\left(\alpha x^{\beta }+\alpha x^{\beta }(1+\alpha x^{\beta }-{\left(\alpha x^{\beta }\right)}^2)\right)\right],\hfill \\ \\ \hfill {\displaystyle \frac{\partial F_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}}{\partial \beta }}=& {\displaystyle \frac{\alpha x^{\beta }logx{e}^{-\alpha x^{\beta }}}{1+\alpha }}\left[(1+\alpha x^{\beta })e^{-\alpha x^{\beta }}+\alpha x^{\beta }{e}^{-\alpha x^{\beta }}(1-\alpha x^{\beta })\right],\hfill \\ \\ \hfill {\displaystyle \frac{{\partial }^2{F}_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}}{\partial {\alpha }^2}}=& -\left\{{\displaystyle \frac{2e^{-\alpha x^{\beta }}}{{(1+\alpha )}^3}}\left[\alpha +(1+\alpha x^{\beta })e^{-\alpha x^{\beta }}\right]\right.\hfill \\ \hfill & -{\displaystyle \frac{2e^{-\alpha x^{\beta }}}{{(1+\alpha )}^2}}\left[1-x^{\beta }{e}^{-\alpha x^{\beta }}\left(\alpha x^{\beta }+\alpha x^{\beta }(1+\alpha x^{\beta }-{\left(\alpha x^{\beta }\right)}^2)\right)\right],\hfill \\ \hfill & +\left.{\displaystyle \frac{e^{-\alpha x^{\beta }}}{1+\alpha }}\left[x^{\beta }{e}^{-\alpha x^{\beta }}\left(2x^{\beta }(1+\alpha x^{\beta }-{\left(\alpha x^{\beta }\right)}^2)+\alpha x^{\beta }(1+3\alpha x^{\beta }-3{\left(\alpha x^{\beta }\right)}^2-{\left(\alpha x^{\beta }\right)}^3)\right)\right]\right\},\hfill \\ \\ \hfill {\displaystyle \frac{{\partial }^2{F}_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}}{\partial {\beta }^2}}=& -\left\{-{\displaystyle \frac{\alpha x^{\beta }{(logx)}^2{e}^{-\alpha x^{\beta }}}{1+\alpha }}\left[(1+\alpha x^{\beta })e^{-\alpha x^{\beta }}+\alpha x^{\beta }{e}^{-\alpha x^{\beta }}(1-\alpha x^{\beta })\right]\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{\alpha x^{\beta }logx{e}^{-\alpha x^{\beta }}}{1+\alpha }}\left[x^{\beta }logx{e}^{-\alpha x^{\beta }}\left((1-\alpha x^{\beta })(1-2\alpha x^{\beta })-{\left(\alpha x^{\beta }\right)}^2\right)\right]\right\},\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^2{F}_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}}{\partial \alpha \partial \beta }}=& -\left\{{\displaystyle \frac{x^{\beta }logx{e}^{-\alpha x^{\beta }}}{{(1+\alpha )}^2}}\left[(1+\alpha x^{\beta })e^{-\alpha x^{\beta }}+\alpha x^{\beta }{e}^{-\alpha x^{\beta }}(1-\alpha x^{\beta })\right]\right.\hfill \\ \hfill & -\left.{\displaystyle \frac{\alpha x^{\beta }logx{e}^{-\alpha x^{\beta }}}{1+\alpha }}\left[x^{\beta }{e}^{-\alpha x^{\beta }}\left((1-\alpha x^{\beta })(1-2\alpha x^{\beta })-{\left(\alpha x^{\beta }\right)}^2\right)\right]\right\}.\hfill \end{array}$$

4.2. Likelihood Intervals

When the normality assumption of MLEs is doubtful or when the Fisher information matrix is analytically intractable, likelihood-based intervals provide a robust alternative for the parameters $\alpha$, $\beta$, and $\gamma$. Unlike intervals derived from asymptotic normality, this approach can achieve more accurate coverage in finite samples. Following Kalbfleisch [9], we define the $100p\%$, $0<p<1$ likelihood interval for a parameter $\theta \in \{\alpha ,\beta ,\gamma \}$ as the set of values satisfying the following:

$$r_{max}\left(\theta \right)=\mathcal{L}\left(\theta \right)-\mathcal{L}\left(\widehat{\theta }\right)\ge logp,$$

(29)

where $\mathcal{L}\left(\theta \right)$ is the log-likelihood function and $\widehat{\theta }$ is the maximum likelihood estimate of $\theta$. For approximate $95\%$ confidence intervals, we adopt $p=0.147$ (the $14.7\%$ likelihood interval), since $-2log\left(0.147\right)\approx 3.92\approx {\chi }_{1,0.95}^2$.

4.3. Bayesian Inference via Markov Chain Monte Carlo (MCMC)

• Model and priors.

Let $x=(x_1,\dots ,x_n)$ be i.i.d. from the ESTH-II distribution with parameters $(\alpha ,\beta ,\gamma )$ and density in (4). Assume independent gamma priors, as follows:

$$\alpha \sim \mathrm{Gamma}(a_{\alpha },b_{\alpha }),\beta \sim \mathrm{Gamma}(a_{\beta },b_{\beta }),\gamma \sim \mathrm{Gamma}(a_{\gamma },b_{\gamma }).$$

Up to a normalizing constant, the joint posterior density is

$$\begin{array}{cc}\hfill \pi (\alpha ,\beta ,\gamma \mid \mathbf{x})\propto & {\gamma }^{n+a_{\gamma }-1}{\alpha }^{n+a_{\alpha }-1}{\beta }^{n+a_{\beta }-1}{(1+\beta )}^{-n}\hfill \\ \hfill \times & exp\left(-b_{\alpha }\alpha -b_{\beta }\beta -b_{\gamma }\gamma -\beta \sum _{i=1}^n{x}_i^{\alpha }\right)\hfill \\ \hfill \times & \left(\prod _{i=1}^n{x}_i^{\alpha -1}\right)·\left(\prod _{i=1}^n\left[\beta +\left(1+2\beta x_i^{\alpha }\right)e^{-\beta x_i^{\alpha }}\right]\right)\hfill \\ \hfill \times & {\left(\prod _{i=1}^n\left\{1-{\displaystyle \frac{1}{1+\beta }}\left[\beta +\left(1+\beta x_i^{\alpha }\right)e^{-\beta x_i^{\alpha }}\right]e^{-\beta x_i^{\alpha }}\right\}\right)}^{\gamma -1}.\hfill \end{array}$$

(30)

Bayesian estimation allows us to derive estimators for model parameters using their posterior distributions, both as point estimates and as interval estimates:

(i)

Under the quadratic loss function (QLF), the Bayes estimator is the posterior mean, and the corresponding Bayes risk is the posterior variance.

(ii)

Under the absolute error loss function (ALF), the Bayes estimator is the posterior median, and the corresponding Bayes risk is the posterior expected absolute deviation from the posterior median.

(iii)

A $(1-\vartheta )100\%$ credible interval for ${\theta }_j$ is obtained from the $\vartheta /2$ and $(1-\vartheta /2)$ quantiles of its marginal posterior distribution.

The posterior density in (30) is both nonconjugate and highly nonlinear with respect to all three parameters. Specifically, the parameter $\alpha$ appears within powers and exponential functions such as $x^{\alpha }$, $\beta$ influences the model both polynomially and through exponential terms, and $\gamma$ enters multiplicatively via products involving $F_{\mathrm{S}\mathrm{T}\mathrm{H}\text{-II}}$. As a result, obtaining closed-form Bayes estimators, including posterior means or analytical credible intervals, is not feasible.

To overcome this difficulty, we employ MCMC techniques (see, Gelman et al. [10]), which allow for the practical approximation of the posterior distribution and facilitate computation of quantities like posterior means, medians, and credible intervals. For a concise and accessible explanation of the Metropolis–Hastings algorithm, including its implementation within a Gibbs sampling framework, the reader is referred to Chib and Greenberg [11]. Algorithm 3 outlines the steps of the Metropolis-within-Gibbs sampler that is applied to the ESTH-II posterior.

Algorithm 3 Metropolis-within-Gibbs sampler for the ESTH-II posterior

Input: Data $x=(x_1,\dots ,x_n)$; prior hyperparameters $(a_{\alpha },b_{\alpha })$, $(a_{\beta },b_{\beta })$, and $(a_{\gamma },b_{\gamma })$; iterations N; burn-in B; and proposal kernels $q_{\alpha }$, $q_{\beta }$, $q_{\gamma }$

Output: Posterior draws ${\left\{({\alpha }^{\left(t\right)},{\beta }^{\left(t\right)},{\gamma }^{\left(t\right)})\right\}}_{t=B+1}^N$

1: Initialize $({\alpha }^{\left(0\right)},{\beta }^{\left(0\right)},{\gamma }^{\left(0\right)})$    2: for $t=1$ to N do    3:       Update $\alpha$ (Metropolis step):    4:       Propose ${\alpha }^{★}\sim q_{\alpha }(·\mid {\alpha }^{(t-1)})$    5:       Compute $$r_{\alpha }={\displaystyle \frac{\pi ({\alpha }^{★},{\beta }^{(t-1)},{\gamma }^{(t-1)}\mid x)q_{\alpha }({\alpha }^{(t-1)}\mid {\alpha }^{★})}{\pi ({\alpha }^{(t-1)},{\beta }^{(t-1)},{\gamma }^{(t-1)}\mid x)q_{\alpha }({\alpha }^{★}\mid {\alpha }^{(t-1)})}},P_{\alpha }=min\{1,r_{\alpha }\}$$    6:         With probability $P_{\alpha }$, set ${\alpha }^{\left(t\right)}={\alpha }^{★}$; else ${\alpha }^{\left(t\right)}={\alpha }^{(t-1)}$    7:         Update $\beta$ (Metropolis step):    8:         Propose ${\beta }^{★}\sim q_{\beta }(·\mid {\beta }^{(t-1)})$    9:         Compute $r_{\beta }$ analogously; accept with probability $P_{\beta }$  10:       Update $\gamma$ (Metropolis step):  11:       Propose ${\gamma }^{★}\sim q_{\gamma }(·\mid {\gamma }^{(t-1)})$  12:       Compute $r_{\gamma }$ analogously; accept with probability $P_{\gamma }$  13: end for  14: Discard the first B iterations)

In MCMC, a Markov chain is constructed so that its stationary distribution coincides with the target posterior $\pi (\alpha ,\beta ,\gamma \mid x)$. After discarding a burn-in period, averages over the generated samples yield consistent estimates of posterior expectations. These can be used to obtain point estimators such as means or medians, as well as interval estimates through credible sets.

The Metropolis-within-Gibbs sampler converges under fairly mild and familiar conditions on the proposal distributions used in the Metropolis updates. When the transition mechanism is irreducible with respect to the posterior density $\pi (\theta \mid x)$ being aperiodic and satisfying detailed balance, the resulting Markov chain is ergodic. As a result, the sequence of draws ${\left\{{\theta }^{\left(t\right)}\right\}}_{t\ge 1}$ approaches the posterior distribution $\pi (\theta \mid x)$ as the number of iterations increases. In addition, for any integrable function h, the averages

$${\displaystyle \frac{1}{T}}\sum _{t=1}^{T}h\left({\theta }^{\left(t\right)}\right)$$

converge almost surely to the corresponding posterior expectation $\mathbb{E}\left[h\right(\theta )\mid x]$. Further discussion of these conditions and their implications can be found in Chib and Greenberg [11].

Assessing the reliability of the MCMC output requires careful consideration of both convergence and mixing. In practice, analysts often begin by inspecting trace plots to check whether the simulated values have settled into a stable pattern. Autocorrelation plots are also useful for understanding the dependence between successive samples and for judging the overall efficiency of the sampler. Another common strategy is to initiate the algorithm from several well-separated starting points and then apply convergence diagnostics, such as the Gelman–Rubin statistic (see Gelman and Rubin [12]), which helps determine whether the simulation has reached the target posterior distribution.

5. Simulation Study

To evaluate how the proposed estimators perform in realistic, finite samples, we conducted a comprehensive Monte Carlo simulation study. While asymptotic theory offers valuable guidance, it often fails to capture the true behavior of estimators when sample sizes are small or moderate. Simulation studies overcome this gap by providing an empirical framework under controlled settings, allowing for a clear comparison of bias, efficiency, and the accuracy of uncertainty measures across different estimation approaches.

The specific aims of the simulation study were as follows:

(1)

To examine the accuracy of maximum likelihood (MLE) and Bayesian estimators (under quadratic and absolute loss functions) in terms of average point estimates (APE) and mean squared error (MSE);

(2)

To evaluate the reliability of their interval estimates through coverage probabilities (CP);

(3)

To investigate how sample size influences the overall quality of parameter estimation.

5.1. Simulation Design and Algorithm

The simulation was designed to be comprehensive and reproducible. For a set of true parameter vectors ${\mathit{\theta }}_0\in \{(1,0.1,1),(1,0.1,1.5),(1,1.2,1.5)$, and $(1.2,1,1.5)\}$, we generated datasets across a range of sample sizes ($n=30,50,70,100,150$). For each combination of parameters and sample size, $M=2000$ Monte Carlo replicates were performed. The chosen parameter values were selected to represent a variety of distributional behaviors. Algorithm 4 details the exact procedure implemented in R4.x.

Algorithm 4 Monte Carlo Simulation Procedure

Input: True parameter vectors ${\mathit{\theta }}_0$; sample sizes $\{30,50,70,100,150\}$; and number of replicates $M=2000$.

Output: Arrays of estimates $\widehat{\mathbf{\Theta }}$, MSE values, and coverage indicators.

1: Initialize arrays to store results for APE, MSE, and CP.    2: for each true parameter vector ${\mathit{\theta }}_0$ do    3:       for each sample size n do    4:             for $m=1$ to M do    5:                   Generate a complete dataset ${\mathbf{x}}_m$ of size n from $F(·|{\theta }_0)$.    6:                   Apply the MLE method to ${\mathbf{x}}_m$ to obtain estimates ${\widehat{\theta }}_{\mathrm{M}\mathrm{L}\mathrm{E}}$ and 95% confidence intervals.    7:                   Apply the Bayesian method (QLF) to ${\mathbf{x}}_m$ to obtain estimates ${\widehat{\theta }}_{BQ}$ and 95% credible intervals.    8:                   Apply the Bayesian method (ALF) to ${\mathbf{x}}_m$ to obtain estimates ${\widehat{\theta }}_{BA}$ and 95% credible intervals.    9:                   Store all point estimates and interval check results.  10:         end for  11:         Calculate APE: $\mathrm{APE}\left(\widehat{\theta }\right)={\displaystyle \frac{1}{M}}{\sum }_{m=1}^M{\widehat{\theta }}_m$, where ${\widehat{\theta }}_m$= ${\widehat{\theta }}_{\mathrm{M}\mathrm{L}\mathrm{E}}$, ${\widehat{\theta }}_{BQ}$, ${\widehat{\theta }}_{BA}$ for sample ${\mathbf{x}}_m$.  12:         Calculate MSE: $\mathrm{MSE}\left(\widehat{\theta }\right)={\displaystyle \frac{1}{M}}{\sum }_{m=1}^M{({\widehat{\theta }}_m-{\theta }_0)}^2$.  13:         Calculate CP: Proportion of intervals containing ${\theta }_0$.  14:     end for  15: end for  16: Return: All aggregated results (APE, MSE, and CP).

This rigorous design ensures that the reported metrics are stable and that the comparative analysis is based on a substantial number of samples, minimizing the impact of random variation. For the Bayesian analysis, we used $M=\mathrm{12,000}$ MCMC iterations with a burn-in period of $B=2000$, and the initial values of the chains were chosen close to the maximum likelihood estimates of the parameters.

5.2. Simulation Results

The simulation results are summarized in

Table 2. Simulation results for APE, MSE, and CP for different true parameter sets $\mathit{\theta }=(\alpha ,\beta ,\gamma )$ and estimation methods.

		$\mathit{\theta }=(1,0.1,1)$									$\mathit{\theta }=(1,0.1,1.5)$
		MLE			QLF			ALF			MLE			QLF			ALF
$\mathit{n}$	Par.	APE	MSE	CP	APE	MSE	CP	APE	MSE	CP	APE	MSE	CP	APE	MSE	CP	APE	MSE	CP
30	$\alpha$	1.008	0.0040	94.6	1.0013	0.0038	94.3	1.0027	0.0038	94.3	1.0038	0.0028	94.8	0.9987	0.0027	94.2	0.9994	0.0027	94.2
	$\beta$	0.102	0.0003	95.1	0.1024	0.0003	93.9	0.1013	0.00034	93.9	0.1008	0.0002	94.4	0.1012	0.0002	94.3	0.1004	0.0002	94.3
	$\gamma$	1.038	0.0362	95.6	1.0386	0.0364	95.1	1.0273	0.0351	95.1	1.5691	0.0995	95.3	1.57	0.0996	93.2	1.5521	0.0953	93.2
50	$\alpha$	1.0058	0.0025	94.0	1.0017	0.0025	92.3	1.0024	0.0025	92.3	1.0036	0.0016	95.7	1.0007	0.0016	94.4	1.001	0.0016	94.4
	$\beta$	0.1014	0.0002	94.7	0.1016	0.0002	93.5	0.101	0.0002	93.5	0.101	0.0001	95.3	0.1012	0.0001	94.5	0.1008	0.0001	94.5
	$\gamma$	1.0287	0.0227	95.2	1.0289	0.0228	93.3	1.0221	0.0223	93.3	1.5313	0.0491	95.7	1.531	0.0495	94.7	1.5206	0.0490	94.7
70	$\alpha$	1.0051	0.0016	95.2	1.0022	0.0017	94.3	1.0026	0.0017	94.3	1.0045	0.0012	94.2	1.0026	0.0012	93.2	1.0028	0.0012	93.2
	$\beta$	0.1014	0.00014	96.6	0.1016	0.0001	93.8	0.1011	0.0001	93.8	0.1011	0.0001	95.4	0.1012	0.0001	94.9	0.1009	0.0001	94.9
	$\gamma$	1.0102	0.0155	94.7	1.011	0.0157	93.0	1.0071	0.0157	93.0	1.5249	0.0350	96.0	1.5262	0.0353	93.4	1.5189	0.0349	93.4
100	$\alpha$	1.003	0.0012	94.4	1.001	0.0012	92.7	1.0012	0.0012	92.7	1.0032	0.0008	95.2	1.0016	0.0008	93.0	1.0015	0.0008	93.0
	$\beta$	0.1008	0.0001	94.1	0.1009	0.0001	93.2	0.1006	0.0001	93.2	0.1008	0.0001	95.9	0.1009	0.0001	93.8	0.1007	0.0001	93.8
	$\gamma$	1.0111	0.0110	95.2	1.0115	0.0110	93.3	1.0081	0.0110	93.3	1.5122	0.0221	95.8	1.5125	0.0220	93.9	1.5066	0.0217	93.9
150	$\alpha$	1.003	0.0008	95.1	1.0017	0.0008	91.7	1.0019	0.0008	91.7	1.0008	0.0005	96.1	1.0000	0.0005	93.7	1.0002	0.0005	93.7
	$\beta$	0.1008	0.0001	94.7	0.1008	0.0001	93.8	0.1005	0.0001	93.8	0.1003	0	96.1	0.1003	0	94.5	0.1001	0	94.5
	$\gamma$	1.0079	0.0070	95.5	1.0091	0.0069	93.4	1.0068	0.0069	93.4	1.5075	0.0150	95.0	1.5083	0.0150	93.8	1.5046	0.0152	93.8
		$\mathit{\theta }$ = (1, 1.2, 1.5)									$\mathit{\theta }\mathbf{=}\mathbf{(}\mathbf{1.2}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{1.5}\mathbf{)}$
		MLE			QLF			ALF			MLE			QLF			ALF
		APE	MSE	CP	APE	MSE	CP	APE	MSE	CP	APE	MSE	CP	APE	MSE	CP	APE	MSE	CP
30	$\alpha$	1.0239	0.0224	95.0	1.0166	0.0218	94.7	1.0125	0.0216	94.7	1.2303	0.0326	94.8	1.2201	0.0317	93.3	1.2157	0.0314	93.3
	$\beta$	1.2164	0.0381	94.5	1.2222	0.0385	95.7	1.2101	0.0376	95.7	1.0242	0.0294	93.0	1.0296	0.0297	93.7	1.0188	0.0285	93.7
	$\gamma$	1.5522	0.0817	96.1	1.5512	0.0812	95.4	1.5339	0.0784	95.4	1.5451	0.0873	96.0	1.5450	0.0872	95.7	1.5270	0.0845	95.7
50	$\alpha$	1.0209	0.0140	94.8	1.0164	0.0136	94.1	1.0140	0.0134	94.1	1.2186	0.0170	95.4	1.2122	0.0168	94.5	1.2100	0.0169	94.5
	$\beta$	1.2088	0.0230	94.4	1.2118	0.0231	94.4	1.2045	0.0229	94.4	1.0093	0.0154	94.7	1.0126	0.0155	95.1	1.0061	0.0153	95.1
	$\gamma$	1.5433	0.0521	94.8	1.5419	0.0515	93.3	1.5325	0.0507	93.3	1.5367	0.0516	95.1	1.5373	0.0515	94.7	1.5274	0.0508	94.7
70	$\alpha$	1.0081	0.0087	95.0	1.0050	0.0087	94.5	1.0029	0.0086	94.5	1.2197	0.0124	94.4	1.2148	0.0122	94.0	1.2130	0.0121	94.0
	$\beta$	1.2093	0.0178	94.6	1.2123	0.0180	94.0	1.2072	0.0179	94.0	1.0128	0.0117	94.4	1.0152	0.0118	94.4	1.0107	0.0117	94.4
	$\gamma$	1.5224	0.0359	95.2	1.5227	0.0359	93.5	1.5156	0.0354	93.5	1.5179	0.0305	95.6	1.5177	0.0306	95.2	1.5109	0.0307	95.2
100	$\alpha$	1.0018	0.0059	94.9	0.9993	0.0059	94.0	0.9982	0.0059	94.0	1.2080	0.0086	95.0	1.2045	0.0084	94.9	1.2033	0.0085	94.9
	$\beta$	1.2048	0.0126	94.5	1.2067	0.0125	94.3	1.2030	0.0124	94.3	1.0055	0.0080	95.0	1.0068	0.0080	93.8	1.0034	0.0079	93.8
	$\gamma$	1.5120	0.0229	95.3	1.5118	0.0228	94.0	1.5068	0.0228	94.0	1.5138	0.0221	95.6	1.5138	0.0221	94.1	1.5086	0.0223	94.1
150	$\alpha$	1.0071	0.0047	93.5	1.0054	0.0046	92.8	1.0045	0.0047	92.8	1.2063	0.0058	93.8	1.2041	0.0058	93.1	1.2033	0.0058	93.1
	$\beta$	1.2058	0.0077	95.5	1.2070	0.0077	94.4	1.2048	0.0077	94.4	1.0064	0.0053	94.9	1.0073	0.0054	93.7	1.0054	0.0054	93.7
	$\gamma$	1.5076	0.0148	94.8	1.5070	0.0147	93.5	1.5046	0.0150	93.5	1.5069	0.0157	94.7	1.5062	0.0157	93.2	1.5017	0.0155	93.2

, with MSEs illustrated in Figure 5. Table 2 reports the APE, MSE, and CP for the nominal 95% confidence and credible intervals across all parameters, methods, and sample sizes.

5.3. Discussion of Simulation Results

The simulation study yields several important insights into the performance of the estimators. Across all sample sizes, the three approaches (MLE, Bayesian with QLF, and Bayesian with ALF) produce point estimates with very low bias, as is indicated by the fact that APE values in Table 2 are consistently close to the true parameter values. Among these, the Bayesian method with ALF demonstrates the highest accuracy in estimating the parameter $\gamma$.

Regarding precision, measured by MSE, Table 2 shows that all three estimators perform similarly for each parameter and sample size. This suggests that the choice of estimation method has little effect on the variability of the estimates. As expected, MSE declines for all methods as the sample size grows.

The most striking differences emerge in the assessment of uncertainty. Coverage probabilities indicate that MLE-based confidence intervals are consistently closer to the nominal 95% level. In contrast, Bayesian credible intervals whether using QLF or ALF tend to slightly under cover, especially for smaller samples. This implies that the Bayesian intervals are somewhat too narrow, giving an overly optimistic impression of certainty.

In conclusion, the Bayesian ALF estimator offers slightly more accurate point estimates for $\gamma$, whereas MLE provides better-calibrated measures of uncertainty. Therefore, the choice of method should be guided by the primary objective of the analysis: Bayesian ALF when precision of point estimates is paramount and MLE when reliable interval estimation is the priority.

6. Applications

In this section, we demonstrate the application of the proposed models on three real-world datasets, which are described below and presented in Appendix C.

6.1. Objectives

The primary objectives of this section are threefold. First, we assess the empirical performance of the baseline distributions, STH-I and STH-II, together with their exponentiated extensions, ESTH-I and ESTH-II, when fitted to real data. Second, we evaluate whether the additional shape flexibility introduced by the exponentiation mechanism leads to a statistically and practically meaningful improvement in model fit. To this end, model comparison is carried out using the Akaike Information Criterion (AIC), which balances goodness of fit against model complexity.

Since ESTH-II nests the remaining three models as special cases, we further employ a likelihood ratio test to formally examine whether the more general ESTH-II model provides a significant improvement over its sub-models. The likelihood ratio test statistic, denoted by $\Lambda$, is defined as follows:

$$\Lambda =-2\left({\mathcal{L}}_0-{\mathcal{L}}_1\right),$$

(31)

where ${\mathcal{L}}_0$ and ${\mathcal{L}}_1$ represent the log-likelihood values evaluated at the MLEs under the null hypothesis ($H_0$) and the alternative hypothesis ($H_1$), respectively. Under $H_0$, the statistic $\Lambda$ follows a chi-square distribution with k degrees of freedom, denoted by ${\chi }_k^2$, where k is the number of restricted parameters.

Finally, to place the proposed models within a broader modeling context, we compare their performance with four alternative distributions that possess a similar CDF structure: the EL distribution (Nadarajah et al. [3]), the EQL distribution (Elbatal et al. [4]), the EXL distribution (Alomair et al. [5]), and the EPL distribution (Ashour and Eltehiwy [6]). The CDFs of these four distributions are provided in Appendix A.

6.2. Data Description

• Data I: This dataset consists of mortality rates for 76 independent COVID-19 patients from the United Kingdom. Each observation corresponds to a single patient-level outcome, and no patient contributes more than one observation to the dataset. The data were collected during the period from April 15 to June 30, 2020; this time span specifies the data collection window only and does not imply temporal ordering, serial dependence, or repeated measurements. Consequently, the observations are treated as independent and identically distributed (i.i.d.), which is consistent with the analysis in Kilai et al. [13].
• Data II: This dataset contains remission times, measured in months, for 128 bladder cancer patients. Each observation represents the remission time of an individual patient and is therefore assumed to be independent. The data were analyzed under the i.i.d. framework in Arshad et al. [14] and were originally reported by Lee and Wang [15].
• Data III: The dataset reported by Efron [16] represents the survival times of patients diagnosed with head and neck cancer, and treated using radiotherapy (RT). Each survival time corresponds to a distinct patient, and the observations are conventionally modeled as independent lifetime data.

6.3. Results and Analysis

Table 3. The MLE(SE), $\mathcal{L}$, AIC, $A^{*}$, $W^{*}$, K-S, and its corresponding p-value for all models using the three datasets.

Model	$\widehat{\mathit{\alpha }}$ (SE)	$\widehat{\mathit{\beta }}$ (SE)	$\widehat{\mathit{\gamma }}$ (SE)	$\mathcal{L}$	AIC	${\mathit{A}}^{*}$	${\mathit{W}}^{*}$	K-S	p-Value
Data I
STH-I	0.3387 (0.0418)	-	-	−146.6420	295.2841	1.0881	0.1657	0.1694	0.0255
ESTH-I	0.7128 (0.1053)	-	0.2607 (0.0451)	−143.9384	291.8769	1.1438	0.1749	0.1050	0.3716
STH-II	0.7855 (0.0693)	0.4562 (0.0677)	-	−142.4348	288.8697	0.8164	0.1213	0.0797	0.7203
EL	0.4998 (0.0674)	-	0.6172 (0.0929)	−145.3657	294.7314	1.4137	0.2192	0.1111	0.3045
EQL	0.3681 (0.0792)	21.6017 (81.7220)	0.7996 (0.1171)	−142.5148	291.0296	0.9318	0.1400	0.0986	0.4505
EXL	0.4778 (0.0626)	-	0.7092 (0.1076)	−144.0222	292.0444	1.2099	0.1856	0.1055	0.3663
EPL	0.7326 (1.2454)	0.576 (0.1378)	1.8578 (8.3797)	−139.9435	285.887	0.3535	0.0501	0.0569	0.9663
ESTH-II	0.3579 $\left(0.0293\right)$	2.0223 $\left(0.1169\right)$	7.0935 $\left(0.8137\right)$	−139.934	285.869	0.3555	0.0509	0.0559	0.9714
Data II
STH-I	0.0925 (0.0078)	-	-	−404.938	811.876	0.5143	0.0742	0.0756	0.4575
ESTH-I	0.0813 (0.0949)	-	0.8243 (0.0098)	−403.484	810.968	0.5839	0.0856	0.0609	0.7287
STH-II	0.8492 (0.0571)	0.1371 (0.0231)	-	−401.726	807.452	0.3748	0.0529	0.0511	0.8915
EL	0.1555 (0.0165)	-	0.5794 (0.0684)	−406.1204	816.2407	0.8033	0.1156	0.0779	0.4177
EQL	0.1148 (0.0167)	25.6046 (65.5004)	0.9208 (0.1064)	−402.5419	811.0837	0.4730	0.0696	0.0552	0.8303
EXL	0.1571 (0.0162)	-	0.6595 (0.0794)	−405.8700	815.7401	0.8249	0.1215	0.0782	0.4141
EPL	0.7326 (0.2914)	0.5759 (0.1078)	1.8578 (1.8581)	−400.8387	807.6773	0.3916	0.0591	0.0492	0.9165
ESTH-II	0.6801 $\left(0.0254\right)$	0.2759 $\left(0.0197\right)$	1.5913 $\left(0.1406\right)$	−400.583	807.166	0.3396	0.0499	0.0461	0.9483
Data III
STH-I	0.003 (0.0004)	-	-	−284.0535	570.1071	1.1542	0.2005	0.1809	0.0986
ESTH-I	0.8752 (0.1663)	-	0.0031 (0.0005)	−283.8067	571.6135	1.1829	0.2057	0.1673	0.1516
STH-II	0.8204 (0.0777)	0.0098 (0.0046)	-	−282.3783	568.7567	0.8925	0.1537	0.1256	0.4549
EL	0.0058 (0.0011)	-	0.4998 (0.1008)	−283.0355	570.0711	1.0423	0.1799	0.1626	0.1746
EQL	0.0052 (0.0010)	9.9409 (13.4971)	1.0789 (0.2229)	−282.0996	570.1991	0.8996	0.1549	0.1512	0.2415
EXL	0.0059 (0.0011)	-	0.5144 (0.1029)	−283.1906	570.3811	1.0599	0.1828	0.1659	0.1581
EPL	1.2948 (1.2365)	0.2698 (0.1182)	25.7942 (49.5767)	−277.5244	561.0487	0.1228	0.0191	0.0593	0.9953
ESTH-II	0.3481 $\left(0.0111\right)$	0.4684 $\left(0.0279\right)$	15.8227 $\left(2.3854\right)$	−277.3516	560.7032	0.1159	0.0175	0.0559	0.9979

and

Table 4. The hypotheses testing results using the three datasets.

Data	${\mathit{H}}_0$	vs.	${\mathit{H}}_{\mathit{A}}$	df	$\mathbf{\Lambda }$	p-Value
I	$\beta =1,\gamma =1$ (STH-I)	vs.	$\beta =1,\gamma >0$ (ESTH-I)	1	5.407	0.0201
	$\beta =1,\gamma =1$ (STH-I)	vs.	$\beta >0,\gamma >0$ (ESTH-II)	2	13.416	0.0012
	$\beta =1$ (ESTH-I)	vs.	$\beta >0$ (ESTH-II)	1	8.008	0.0047
	$\gamma =1$ (STH-II)	vs.	$\gamma >0$ (ESTH-II)	1	5.001	0.0253
II	$\beta =1,\gamma =1$ (STH-I)	vs.	$\beta =1,\gamma >0$ (ESTH-I)	1	2.908	0.0881
	$\beta =1,\gamma =1$ (STH-I)	vs.	$\beta >0,\gamma >0$ (ESTH-II)	2	8.71	0.0128
	$\beta =1$ (ESTH-I)	vs.	$\beta >0$ (ESTH-II)	1	5.801	0.0160
	$\gamma =1$ (STH-II)	vs.	$\gamma >0$ (ESTH-II)	1	2.286	0.1306
III	$\beta =1,\gamma =1$ (STH-I)	vs.	$\beta =1,\gamma >0$ (ESTH-I)	1	0.497	0.4807
	$\beta =1,\gamma =1$ (STH-I)	vs.	$\beta >0,\gamma >0$ (ESTH-II)	2	13.407	0.0012
	$\beta =1$ (ESTH-I)	vs.	$\beta >0$ (ESTH-II)	1	12.909	0.0003
	$\gamma =1$ (STH-II)	vs.	$\gamma >0$ (ESTH-II)	1	10.051	0.0015

, together with Figure 6, Figure 7 and Figure 8, provide a comprehensive assessment of the performance of the ESTH-II distribution relative to its competing models across the three datasets.

Table 3 reports the maximum likelihood estimates along with several goodness-of-fit and model selection criteria, while the corresponding likelihood-based inference is visually supported by Figure 6, which displays the maximum logarithmic relative likelihood functions for each parameter. The well-defined and unimodal profiles in Figure 6 confirm the uniqueness and stability of the maximum likelihood estimates reported in Table 3 for all datasets.

Across all three datasets, the ESTH-II model consistently achieves the lowest AIC values and the smallest Anderson–Darling and Cramér–von Mises statistics (Table 3), indicating a superior global fit. This strong numerical performance is further reflected in Figure 7, where the fitted PDF and CDF of the ESTH-II distribution closely track the empirical counterparts, outperforming the nested sub-models. In particular, for Data III, the K–S P-value of 0.9979 (Table 3) corresponds to an almost perfect overlap between the fitted and empirical distributions in Figure 7, a level of agreement not achieved by the competing models.

The role of the additional shape parameters is further clarified by the likelihood ratio tests reported in Table 4. For Data I, the rejection of the restricted sub-models demonstrates that both $\beta$ and $\gamma$ contribute significantly to model adequacy, a finding that is consistent with the flexibility observed in the fitted hazard rate functions shown in Figure 8. In Data II, the primary improvement arises from the inclusion of $\beta$, although the full ESTH-II model remains preferable overall. For Data III, both parameters again prove essential, yielding a highly significant improvement ($p=0.0012$), which is visually corroborated by the accurate hazard shape captured in Figure 8.

Finally, Figure 8 complements the inferential results by illustrating that the ESTH-II distribution is capable of accommodating complex hazard structures across datasets, as evidenced by the fitted, scaled TTT-transforms and hazard rate functions. When interpreted jointly with the numerical results in Table 3 and Table 4, these graphical diagnostics confirm that the ESTH-II model is not only statistically superior but also practically flexible. These conclusions are further reinforced by the Bayesian analysis, where posterior estimates under both weakly informative and informative priors demonstrate the robustness and efficiency of the ESTH-II framework.

6.4. BayesianInference

We employed a Bayesian framework to estimate the three parameters of the ESTH-II distribution for each dataset, implemented through Algorithm 3 in the R environment. Two alternative sets of hyperparameters were considered for the prior distributions: a weakly informative, and an informative specification calibrated from preliminary estimates. In the weakly informative prior, the values of the hyperparameters are chosen to be equal and sufficiently small to represent a weakly informative prior, reflecting minimal prior knowledge about the parameters. While in the informative prior case, the hyperparameters are specified using an empirical Bayes approach, where their values are determined from the posterior means and posterior variances obtained under the weakly informative prior.

• Prior I (the weakly informative prior): $a_{\alpha }=a_{\beta }=a_{\gamma }=b_{\alpha }=b_{\beta }=b_{\gamma }=0.001$ for all datasets.
• Prior II (the informative prior):

  (1)

  For Data I: $a_{\alpha }=158.687$, $b_{\alpha }=445.375$, $a_{\beta }=320.077$, $b_{\beta }=158.133$, $a_{\gamma }=76.858$, and $b_{\gamma }=10.894$;

  (2)

  For Data II: $a_{\alpha }=766.818$, $b_{\alpha }=1130.5$, $a_{\beta }=190.854$, $b_{\beta }=690.75$, $a_{\gamma }=119.453$, and $b_{\gamma }=74.887$;

  (3)

  For Data III: $a_{\alpha }=1204.784$, $b_{\alpha }=3471$, $a_{\beta }=275.538$, $b_{\beta }=586.875$, $a_{\gamma }=44.411$, and $b_{\gamma }=2.797$.

Table 5. Comprehensive results for the ESTH-II model. The table shows Bayesian point estimates (with exact estimates in parentheses) under two priors, alongside 95% asymptotic confidence intervals, 14.7% Likelihood intervals, and 95% Bayesian credible intervals (MCMC and exact) using weakly informative and informative priors for all datasets.

		Bayesian Estimates		Confidence Intervals
Data	Par.	Weakly Informative Prior	Informative	Asymptotic	Likelihood
I	$\alpha$	0.3563 (0.3554)	0.3573 (0.3565)	$(0.3005,0.4152)$	$(0.3022,0.4163)$
	$\beta$	2.0241 (2.0233)	2.0224 (2.0189)	$(1.7932,2.2514)$	$(1.8038,2.2611)$
	$\gamma$	7.055 (7.0025)	7.086 (7.0515)	$(5.4987,8.6883)$	$(5.6122,8.7958)$
II	$\alpha$	0.6783 (0.6781)	0.6776 (0.6770)	$(0.6332,0.7282)$	$(0.6309,0.7299)$
	$\beta$	0.2763 (0.2759)	0.2757 (0.2754)	$(0.2391,0.3188)$	$(0.239,0.3157)$
	$\gamma$	1.5951 (1.5813)	1.5900 (1.5862)	$(1.3449,1.9039)$	$(1.3295,1.878)$
III	$\alpha$	0.3471 (0.3473)	0.3471 (0.3474)	$(0.3263,0.3700)$	$(0.3283,0.372)$
	$\beta$	0.4695 (0.4693)	0.4691 (0.4679)	$(0.4137,0.5231)$	$(0.4086,0.5161)$
	$\gamma$	15.8795 (15.8048)	15.7988 (15.7699)	$(11.148,20.498)$	$(11.335,20.438)$
		Credible Intervals (MCMC)		Exact Credible Intervals
		Weakly Informative Prior	Informative	Weakly Informative Prior	Informative
I	$\alpha$	$(0.3029,0.4126)$	$(0.3225,0.3965)$	$(0.301,0.4155)$	$(0.3181,0.3976)$
	$\beta$	$(1.8033,2.2537)$	$(1.8724,2.1821)$	$(1.8059,2.265)$	$(1.8706,2.1881)$
	$\gamma$	$(5.5687,8.7483)$	$(6.0191,8.1624)$	$(5.6038,8.7817)$	$(6.0153,8.2568)$
II	$\alpha$	$(0.6303,0.7299)$	$(0.6472,0.7105)$	$(0.6298,0.7291)$	$(0.6454,0.7141)$
	$\beta$	$(0.2373,0.3145)$	$(0.249,0.3056)$	$(0.2395,0.3168)$	$(0.2497,0.3046)$
	$\gamma$	$(1.3156,1.8669)$	$(1.3835,1.8225)$	$(1.3303,1.8809)$	$(1.4035,1.8013)$
III	$\alpha$	$(0.3244,0.3699)$	$(0.3319,0.3619)$	$(0.3266,0.3700)$	$(0.3335,0.3624)$
	$\beta$	$(0.4178,0.5265)$	$(0.4329,0.5067)$	$(0.4188,0.5283)$	$(0.4323,0.5100)$
	$\gamma$	$(11.281,20.719)$	$(12.654,19.463)$	$(11.533,20.843)$	$(12.741,19.350)$

reports the Bayes point estimates under quadratic error loss and absolute error loss (in brackets), along with the exact and approximate 95% credible intervals obtained using two sets of hyperparameters. For comparison, the table also presents the 95% confidence intervals, both approximate and likelihood-based, for each parameter of the ESTH-II model across all three datasets. The results in Table 5 show close agreement between Bayesian point estimates and their exact counterparts, indicating that MCMC sampling provides stable estimates under both weakly informative and informative priors. The choice of prior mainly affects interval width: informative priors yield noticeably tighter credible intervals, especially for parameters with higher variance, such as $\gamma$ in Data I and III, while central estimates remain essentially unchanged.

Comparing frequentist and Bayesian approaches, the asymptotic and likelihood confidence intervals are almost indistinguishable, though their likelihood intervals tend to be slightly narrower, reflecting their stronger finite sample performance. Bayesian credible intervals, whether exact or MCMC-based, align closely with these ranges, reinforcing the robustness of the inference across methods.

To better understand these differences, we assessed the MCMC sampling behavior under both prior settings, as is shown in Figure 9, Figure 10 and Figure 11. With weakly informative priors, the chains exhibited greater variability and slower autocorrelation decay, requiring many iterations for approximate independence. In contrast, the informative priors, calibrated from the weakly informative case, produced more stable trace plots and faster mixing, yielding effectively independent samples with fewer iterations. Importantly, the posterior means remained consistent across both prior choices, reinforcing the robustness of the estimates while underscoring the computational advantages of informative priors. Hence, in the subsequent analysis, we rely on the results from the informative priors to ensure more stable and efficient inference without compromising robustness.

We present in Figure 12 the marginal posterior densities obtained via MCMC together with the exact densities, with the limits of the exact 95% credible intervals marked. The results show that the posteriors become more concentrated under informative hyperparameters, indicating improved efficiency. Furthermore, the close agreement between the MCMC and exact densities confirms the accuracy of the sampling approach.

Finally, we investigated the convergence of the MCMC chains using the Gelman–Rubin diagnostic. The Gelman–Rubin statistic was computed for each dataset under the informative prior setting. For this purpose, we ran ten independent MCMC chains initialized at overdispersed starting values. Each chain consisted of 12,000 iterations; the first 2000 iterations were discarded as burn-in, and the remaining 10,000 draws from each chain were used to compute the Gelman–Rubin statistic, $\widehat{R}$.

The following steps were employed to calculate $\widehat{R}$:

1.

Run m-independent MCMC chains, each of length k (after burn-in), for the model parameters $\alpha$, $\beta$, and $\gamma$. Let ${\alpha }_{ij}$ denote the jth draw from chain i for parameter $\alpha$, where $i=1,2,\dots ,m$ and $j=1,2,\dots ,k$.

2.

Compute the chain means and the overall mean, as follows:

(a)

    The mean of chain i is

$${\overline{\alpha }}_i={\displaystyle \frac{1}{k}}\sum _{j=1}^k{\alpha }_{ij}.$$

(b)

    The overall mean is

$${\overline{\alpha }}_{·}={\displaystyle \frac{1}{m}}\sum _{i=1}^m{\overline{\alpha }}_i.$$

3.

Compute the average within-chain variance W as follows:

$$W={\displaystyle \frac{1}{m}}\sum _{i=1}^m{s}_i^2,\mathrm{where}{s}_i^2={\displaystyle \frac{1}{k-1}}\sum _{j=1}^k{\left({\alpha }_{ij}-{\overline{\alpha }}_i\right)}^2.$$

4.

Compute the between-chain variance B as follows:

$$B={\displaystyle \frac{k}{m-1}}\sum _{i=1}^m{\left({\overline{\alpha }}_i-{\overline{\alpha }}_{·}\right)}^2.$$

5.

Estimate the marginal posterior variance $\widehat{V}$ as follows:

$$\widehat{V}={\displaystyle \frac{k-1}{k}}W+{\displaystyle \frac{1}{k}}B.$$

6.

Compute the Gelman–Rubin statistic as follows:

$$\widehat{R}=\sqrt{{\displaystyle \frac{\widehat{V}}{W}}}.$$

7.

Repeat Steps 2–6 for the remaining parameters $\beta$ and $\gamma$.

Table 6. The Gelman-Rubin diagnostic results for the three datasets.

	$\mathit{\alpha }$		$\mathit{\beta }$		$\mathit{\gamma }$
Chain	Mean	Variance	Mean	Variance	Mean	Variance
Data I
1	0.3564989	0.0004269295	2.02334	0.007152241	7.132806	0.3271977
2	0.3563541	0.000417599	2.024036	0.006679194	7.120189	0.3339793
3	0.3577726	0.0004414557	2.023744	0.006983468	7.070339	0.3552804
4	0.3583323	0.0004080709	2.016696	0.007076394	7.086573	0.3478988
5	0.3562034	0.0004278291	2.025041	0.007150677	7.100076	0.3207258
6	0.3570126	0.0004112886	2.026344	0.00667269	7.085031	0.2933487
7	0.3564653	0.0003857774	2.017757	0.006998961	7.141821	0.3232892
8	0.3555877	0.0004231039	2.016525	0.007146827	7.110605	0.309977
9	0.3570127	0.0004160165	2.025359	0.007346811	7.128938	0.3478143
10	0.3577661	0.0004010358	2.017941	0.007464367	7.075059	0.300759
Overall Mean	0.3569006	0.0004159106	2.021678	0.007067163	7.105144	0.326027
B	0.007159882		0.1554755		6.471491
$\widehat{V}$	0.000416585		0.007082004		0.3266415
$\widehat{R}$	1.00081		1.001049		1.000942
Data II
1	0.6784543	0.0003306328	0.2768565	0.000208984	1.593156	0.009165
2	0.6796594	0.0003116209	0.2756564	0.0002080026	1.599705	0.010285
3	0.6804415	0.0002757593	0.277311	0.0002102146	1.590641	0.009200
4	0.6790418	0.0003251517	0.2760122	0.0001940104	1.585264	0.010553
5	0.6786966	0.0003789456	0.2767015	0.0001899971	1.589106	0.008785
6	0.6800975	0.0003474061	0.2771244	0.000200715	1.589256	0.009304
7	0.6788874	0.0003249929	0.2757779	0.0002125022	1.591236	0.008897
8	0.6784231	0.000306496	0.2766208	0.0002044866	1.593038	0.009305
9	0.6789308	0.0003413619	0.2761224	0.0001986317	1.585157	0.010066
10	0.6793144	0.0003657088	0.2755594	0.0001962817	1.585929	0.008804
Overall Mean	0.6791947	0.0003308076	0.2763743	0.0002023826	1.590249	0.009436
B	0.004643392		0.003966625		0.1983402
$\widehat{V}$	0.0003312389		0.000202759		0.009455271
$\widehat{R}$	1.000652		1.00093		1.001
Data III
1	0.3477564	0.00005954546	0.4696218	0.0003901486	15.7587	2.913201
2	0.3483826	0.00004775203	0.4696503	0.00037734	15.82331	2.727301
3	0.3478268	0.00004.881005	0.4691026	0.0003719368	15.72394	2.875778
4	0.3482534	0.00005.169352	0.4701666	0.0004067735	15.80332	2.911068
5	0.3476546	0.00004.815847	0.470234	0.0003833537	15.75937	2.843547
6	0.347817	0.00005.547186	0.4692357	0.0003968226	15.79249	2.793001
7	0.3476789	0.00005130219	0.4693357	0.0003678061	15.81117	2.788853
8	0.3483103	0.00005414653	0.46988	0.000368036	15.77399	2.872296
9	0.3478863	0.00005513234	0.4692499	0.0004054088	15.87618	2.736006
10	0.3480249	0.00005040109	0.4702613	0.0003989066	15.87353	2.961118
Overall Mean	0.3479591	0.00005224135	0.4696738	0.0003866533	15.7996	2.842217
B	0.0007210634		0.001955259		24.18081
$\widehat{V}$	0.00005230824		0.0003868101		2.844351
$\widehat{R}$	1.00064		1.000203		1.000375

reports all quantities required to compute $\widehat{R}$ for each parameter across the three real datasets. As shown in the table, all values of $\widehat{R}$ are at most $1.001$, indicating the excellent convergence of the MCMC chains for all model parameters.

7. Conclusions and Future Work

This paper introduces a novel three-parameter lifetime distribution that generalizes the Sarhan–Tadj–Hamilton model, offering enhanced flexibility for capturing a wide variety of hazard rate shapes observed in real-world data. This makes it a versatile tool for modeling complex lifetime and reliability phenomena. We thoroughly investigated its theoretical properties, including moments, generating functions, and reliability measures, providing deeper insight into its behavior. Parameter estimation was performed using both maximum likelihood and Bayesian approaches, ensuring robust inference across different data scenarios.

The performance and applicability of the model were assessed through comprehensive simulation studies, which demonstrated accurate parameter recovery and strong efficiency. Furthermore, three real datasets from diverse domains were analyzed to illustrate the practical utility of the distribution. In all cases, it outperformed existing nested and competing models in terms of goodness-of-fit and predictive accuracy, highlighting its practical superiority.

Looking ahead, several avenues for future research are envisaged. These include extending the distribution to regression models for analyzing covariate-dependent lifetimes, employing advanced Bayesian computation techniques such as Markov chain Monte Carlo for more flexible inference, and developing multivariate or mixture formulations to handle dependent or heterogeneous data. The potential applications are extensive, spanning reliability engineering, biomedical survival analysis, actuarial risk assessment, and environmental hazard modeling, where flexible lifetime distributions are critical for accurate risk quantification and decision making. Overall, the proposed distribution provides both a theoretically sound and practically effective framework for modern reliability and survival studies, with considerable scope for further methodological and applied advancements.