Exploring the New Exponentiated Harris-G Family of Distributions and Its Applications

Abstract

This paper introduces a new family of distributions called exponentiated Harris-G. This new distribution is a weighted distribution of the well established exponentiated-G distributions. The model allows for easy derivation of statistical properties based on the exponentiated-G distribution. Several statistical properties for the new model were derived. The paper considered different parameter estimation techniques and the maximum likelihood estimation technique emerged as the best technique. This was evaluated via Monte Carlo simulation studies of the proposed family. Estimation techniques were ranked based on the lowest values of the root mean square error and average bias. The proposed model showed enhanced flexibility in data modeling when compared to some selected competing models. This was demonstrated through application of the special case to two real-world datasets.

1. Introduction

The theory of probability distributions dates back to some novel work by Pearson [1] where he discussed the development of statistical models from differential equations. Several authors, thereafter, developed some techniques that sought to produce models that can address skewness, kurtosis, or both in data modeling. Some notable techniques prior to 1980 include the method of quantiles Hastings et al. [2] Tukey [3] and method of translation or transformation Johnson [4]. These methods gave birth to classical distributions which have wider applications in lifetime and reliability analysis, hydrology, and decision science. An increase in the availability of data exposed some weaknesses in the modeling capabilities of classical models; for instance, classical distributions could model data with monotonic failure rates except for a few distributions like the Topp–Leone distribution Topp and Leone [5], which could model data with a bathtub failure rate but had a limitation because it was a unit distribution. To overcome these shortcomings, researchers post 1980 proposed several methods to generalize classical distributions that include methods of generating skew distributions Azzalini [6], Azzalini [7] and adding parameters (exponentiated family) Mudholkar and Srivastava [8], and the beta-generated family Eugene et al. [9] extends any parent distribution by applying the beta distribution’s logit as a generator, enabling flexible modeling of skewness, bimodality, and heavy tails. The transformed-transformer (T-X) approach Alzaatreh et al. [10] provides a unifying framework that generates new distributions by transforming a random variable T using another variable X as a transformer, subsuming many existing families (including the beta-generated) as special cases. The composite method Cooray and Ananda [11] combines two distinct distributions, such as a lognormal and a Pareto, into a single model to capture both the central body and the extreme tail of data, which is especially valuable in actuarial science. Finally, the Marshall–Olkin family Marshall and Olkin [12] introduces a tilt parameter into any survival function, producing a new family with a hazard rate that can be uniformly raised or lowered relative to the baseline, thereby enhancing flexibility in reliability and survival analysis.

There are some generators which do not involve generalization of classical distributions, for example, the Marshall–Olkin and the exponentiated-G (Exp-G) Gupta et al. [13]. The advantage of these generators is that they combine well with other generators that generalize classical distributions.

The cumulative distribution function (cdf) and probability density function (pdf) of the Exp-G family of distributions (FoD) are

$$\begin{array}{ccc}\hfill F_{Exp-G}(z;\lambda ,\underline{\mathrm{{\rm Y}}})& =& {\left[G(z;\underline{\mathrm{{\rm Y}}})\right]}^{\lambda }\hfill \end{array}$$

(1)

and

$$\begin{array}{ccc}\hfill f_{Exp-G}(z;\lambda ,\underline{\mathrm{{\rm Y}}})& =& \lambda g(z;\underline{\mathrm{{\rm Y}}}){\left[G(z;\underline{\mathrm{{\rm Y}}})\right]}^{\lambda -1},\hfill \end{array}$$

(2)

for $\lambda >0$, and parent parameter vector $\underline{\mathrm{{\rm Y}}}$. Aly and Benkherouf Aly and Benkherouf [14] pioneered the Harris-G (H-G) FoD with cdf

$$\begin{array}{ccc}\hfill F_{H-G}(z;v,\delta ,\underline{\mathrm{{\rm Y}}})& =& 1-{\displaystyle \frac{{\delta }^{\frac{1}{v}}\overline{G}(z;\underline{\mathrm{{\rm Y}}})}{{[1-\overline{\delta }{\overline{G}}^v(z;\underline{\mathrm{{\rm Y}}})]}^{\frac{1}{v}}}}\hfill \end{array}$$

(3)

and pdf

$$\begin{array}{ccc}\hfill f_{H-G}(z;v,\delta ,\underline{\mathrm{{\rm Y}}})& =& {\delta }^{{\displaystyle \frac{1}{v}}}{[1-\overline{\delta }{\overline{G}}^v(z;\underline{\mathrm{{\rm Y}}})]}^{-(1+{\displaystyle \frac{1}{v}})}g(z;\underline{\mathrm{{\rm Y}}}),\hfill \end{array}$$

(4)

for $\delta ,v>0$, and parent parameter vector $\underline{\mathrm{{\rm Y}}}$. For $v=1$, Equation (4) becomes the pdf of the Marshall–Olkin-G FoD.

The development of the New Exponentiated Harris-G family of distributions is primarily motivated by the need for enhanced flexibility in modeling failure and survival data across diverse fields. While the baseline Harris-G family already accommodates non-monotonic hazard rates including bathtub and unimodal shapes through its tilt and shape parameters, the exponentiated version introduces an additional shape parameter that significantly enriches the distributional tails, skewness, and kurtosis. This extra parameter allows the new family to capture a broader spectrum of aging behaviors, ranging from early failure (infant mortality) to wear-out phases, which simpler generalizations often miss. Moreover, the exponentiated Harris-G generator preserves closed-form expressions for the cdf, quantile function, and hazard rate, facilitating straightforward maximum likelihood estimation. Practical applications demonstrate that the Exponentiated Harris-G family consistently outperforms both its nested and non-nested models. Thus, the primary motivation is to provide researchers with a mathematically coherent, computationally feasible, and empirically superior distributional framework for analyzing complex real-world data.

Structure of the paper: The proposed model is presented in Section 2 and the statistical properties are discussed in Section 3. Two particular cases are examined in Section 4, while additional properties are covered in Section 5. Various estimation techniques and simulation study outcomes are detailed in Section 6 and Section 7, respectively. Real-data applications are provided in Section 8, and Section 9 offers concluding remarks.

2. The Proposed Family

In this section, we derive the new FoD by taking Equation (3) as the parent distribution in Equation (1). The new FoD called the exponentiated Harris-G (ExpH-G) has cdf

$$\begin{array}{ccc}\hfill F_{ExpH-G}(z;\lambda ,\delta ,v,\underline{\mathrm{{\rm Y}}})& =& {\left[1-{\displaystyle \frac{{\delta }^{\frac{1}{v}}\overline{G}(z;\underline{\mathrm{{\rm Y}}})}{{[1-\overline{\delta }{\overline{G}}^v(z;\underline{\mathrm{{\rm Y}}})]}^{\frac{1}{v}}}}\right]}^{\lambda }\hfill \end{array}$$

(5)

and pdf

$$\begin{array}{ccc}\hfill f_{ExpH-G}(z;\lambda ,\delta ,v,\underline{\mathrm{{\rm Y}}})& =& \lambda {\delta }^{{\displaystyle \frac{1}{v}}}{[1-\overline{\delta }{\overline{G}}^v(z;\underline{\mathrm{{\rm Y}}})]}^{-(1+{\displaystyle \frac{1}{v}})}g(z;\underline{\mathrm{{\rm Y}}})\hfill \\ & \times & {\left[1-{\displaystyle \frac{{\delta }^{\frac{1}{v}}\overline{G}(z;\underline{\mathrm{{\rm Y}}})}{{[1-\overline{\delta }{\overline{G}}^v(z;\underline{\mathrm{{\rm Y}}})]}^{\frac{1}{v}}}}\right]}^{\lambda -1},\hfill \end{array}$$

(6)

for $\lambda ,\delta ,v>0$ and parent parameter vector $\underline{\mathrm{{\rm Y}}}$. The parameter $\lambda$ controls the overall shape and the lower tail, while $\delta$ controls the upper tail heaviness and the hazard rate’s long-term behavior. A complete proof demonstrating that the cdf of the ExpH-G is a valid cdf is provided in the Supplementary Materials.

Tail Behavior Analysis

We examine the survival function $\overline{F}\left(z\right)=1-F\left(z\right)$ for a large z; that is, $\overline{G}\left(z\right)\to 0$. Expanding the inner term:

$${\displaystyle \frac{{\delta }^{1/v}\overline{G}}{{(1-\overline{\delta }{\overline{G}}^v)}^{1/v}}}={\delta }^{1/v}\overline{G}\left(1+{\displaystyle \frac{1}{v}}\overline{\delta }{\overline{G}}^v+O({\overline{G}}^{2v})\right).$$

Thus the cdf becomes

$$F\left(z\right)={\left[1-{\delta }^{1/v}\overline{G}+o\left(\overline{G}\right)\right]}^{\lambda }=1-\lambda {\delta }^{1/v}\overline{G}+o\left(\overline{G}\right).$$

Hence

$$\overline{F}\left(z\right)\sim \lambda {\delta }^{1/v}\overline{G}\left(z\right)\mathrm{as}z\to \infty .$$

Comparing with the baseline survival $\overline{G}\left(z\right)$:

$${\displaystyle \frac{\overline{F}\left(z\right)}{\overline{G}\left(z\right)}}\to \lambda {\delta }^{1/v}.$$

Therefore:

• If $\lambda >1$, the tail of the proposed model is heavier than that of the $\lambda =1$ subfamily (and heavier than the baseline if $\lambda {\delta }^{1/v}>1$).
• If $\lambda <1$, the tail is lighter.

This explicit control of tail weight via $\lambda$ is a distinctive feature not shared by the beta-G or Kumaraswamy-G generators.

3. Some Statistical Properties

This section presents some statistical properties of the ExpH-G FoD.

3.1. Quantile Function

The quantile function (QF) of the ExpH-G FoD is found by inverting Equation (5). The QF of ExpH-G FoD is

$$\begin{array}{ccc}\hfill Q_Z\left(u\right)& =& G^{-1}\left(1-{\left[{\displaystyle \frac{{(1-u^{\frac{1}{\lambda }})}^v}{{(1-u^{\frac{1}{\lambda }})}^v\overline{\delta }+\delta }}\right]}^{{\displaystyle \frac{1}{v}}}\right),\hfill \end{array}$$

(7)

for $\lambda ,\delta ,v>0$, where $u\in [0,1]$ and G is the parent distribution.

3.2. Series Expansion

This subsection provides a series expansion representation of the ExpH-G family.

The pdf of the ExpH-G FoD can be expressed as

$$\begin{array}{ccc}\hfill f(z;\lambda ,\delta ,v,\underline{\mathrm{{\rm Y}}})& =& \sum _{l=0}^{\infty }{c}_{l+1}{g}_{l+1}(z;\underline{\mathrm{{\rm Y}}}),\hfill \end{array}$$

(8)

where $g_{l+1}(z;\underline{\mathrm{{\rm Y}}})=(l+1)g(z;\underline{\mathrm{{\rm Y}}})G^l(z;\underline{\mathrm{{\rm Y}}})$ is the exponentiated-G (Exp-G) distribution with parameter $(l+1)$ and

$$\begin{array}{ccc}\hfill c_{l+1}& =& \sum _{i,k=0}^{\infty }{(-1)}^{i+k}\left({\displaystyle \genfrac{}{}{0pt}{}{\lambda -1}{i}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\frac{-1}{v}(i+1)-1}{k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{i+vk}{l}}\right)\hfill \\ & \times & \lambda {\delta }^{{\displaystyle \frac{1}{v}}(i+1)}{\overline{\delta }}^k\left({\displaystyle \frac{1}{l+1}}\right).\hfill \end{array}$$

(9)

For all derivations, see the Supplementary Materials.

4. Particular Cases

This section examines two particular cases of the ExpH-G family. The first arises when the baseline distribution is taken as the log-logistic (LLoG) distribution, while the second corresponds to the Weibull (W) distribution.

4.1. ExpH-LLoG Distribution

If we take the log-logistic distribution as the baseline distribution with cdf $G\left(z\right)=1-{(1+z^{\beta })}^{-1}$ and pdf $g\left(z\right)=\beta z^{\beta -1}{(1+z^{\beta })}^{-2}$, we get the ExpH-LLoG distribution with cdf and pdf

$$\begin{array}{ccc}\hfill F_{ExpH-LLoG}(z;\lambda ,\delta ,v,\beta )& =& {\left[1-{\displaystyle \frac{{\delta }^{\frac{1}{v}}{(1+z^{\beta })}^{-1}}{{[1-\overline{\delta }{(1+z^{\beta })}^{-v}]}^{\frac{1}{v}}}}\right]}^{\lambda }\hfill \end{array}$$

(10)

and

$$\begin{array}{ccc}\hfill f_{ExpH-LLoG}(z;\lambda ,\delta ,v,\beta )& =& \lambda {\delta }^{{\displaystyle \frac{1}{v}}}{[1-\overline{\delta }{(1+z^{\beta })}^{-v}]}^{-(1+{\displaystyle \frac{1}{v}})}\beta z^{\beta -1}{(1+z^{\beta })}^{-2}\hfill \\ & \times & {\left[1-{\displaystyle \frac{{\delta }^{\frac{1}{v}}{(1+z^{\beta })}^{-1}}{{[1-\overline{\delta }{(1+z^{\beta })}^{-v}]}^{\frac{1}{v}}}}\right]}^{\lambda -1},\hfill \end{array}$$

(11)

for $\lambda ,\delta ,v,\beta >0$. Graphs of the pdfs (Figure 1) show several shapes including left-skewed and reverse-J shapes. Graphs of hazard rate functions (hrfs) display monotonic and non-monotonic shapes.

4.2. ExpH-W Distribution

If we take the Weibull distribution as the baseline distribution with cdf $G\left(z\right)=1-e^{-z^{\theta }}$ and pdf $g\left(z\right)=\theta z^{\theta -1}{e}^{-z^{\theta }}$, we get the ExpH-W distribution with cdf and pdf

$$\begin{array}{ccc}\hfill F_{ExpH-W}(z;\lambda ,\delta ,v,\theta )& =& {\left[1-{\displaystyle \frac{{\delta }^{\frac{1}{v}}{e}^{-z^{\theta }}}{{[1-\overline{\delta }{e}^{-v{z}^{\theta }}]}^{\frac{1}{v}}}}\right]}^{\lambda }\hfill \end{array}$$

(12)

and

$$\begin{array}{ccc}\hfill f_{ExpH-W}(z;\lambda ,\delta ,v,\theta )& =& \lambda {\delta }^{{\displaystyle \frac{1}{v}}}{[1-\overline{\delta }{e}^{-v{z}^{\theta }}]}^{-(1+{\displaystyle \frac{1}{v}})}\theta z^{\theta -1}{e}^{-z^{\theta }}\hfill \\ & \times & {\left[1-{\displaystyle \frac{{\delta }^{\frac{1}{v}}{e}^{-z^{\theta }}}{{[1-\overline{\delta }{e}^{-v{z}^{\theta }}]}^{\frac{1}{v}}}}\right]}^{\lambda -1},\hfill \end{array}$$

(13)

for $\lambda ,\delta ,v,\theta >0$. Graphs of the pdfs (Figure 2) show several shapes including right-skewed, J, and reverse-J. Graphs of hazard rate functions (hrfs) display monotonic and non-monotonic shapes.

5. Additional Statistical Properties

5.1. Distribution of Order Statistics

Let $Z_1,Z_2,\dots ,Z_n$ be a random sample from ExpH-G FoD. The pdf of the j-th order statistic is defined as

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

where $F\left(z\right)$ and f(z) are the cdf and pdf of the ExpH-G FoD. Therefore, the distribution of the j-th-order statistics from the ExpH-G FoD is

$$\begin{array}{c}\hfill f_{j:n}\left(z\right)={\displaystyle \frac{n!}{(j-1)!(n-j)!}}\sum _{q=0}^{n-j}\sum _{l=0}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{n-j}{q}}\right){(-1)}^q{c}_{l+1}^{\ast }{g}_{l+1}(z;\underline{\mathrm{{\rm Y}}}),\end{array}$$

where $g_{l+1}(z;\underline{\mathrm{{\rm Y}}})=(l+1)g(z;\underline{\mathrm{{\rm Y}}})G^l(z;\underline{\mathrm{{\rm Y}}})$ is the Exp-G distribution with power parameter $(l+1)$ and

$$\begin{array}{ccc}\hfill c_{l+1}^{\ast }& =& \sum _{i,k=0}^{\infty }{(-1)}^{i+k+l}\left({\displaystyle \genfrac{}{}{0pt}{}{\lambda (j+q)-1}{i}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\frac{-1}{v}(i+1)-1}{k}}\right)\hfill \\ & \times & \left({\displaystyle \genfrac{}{}{0pt}{}{i+vk}{l}}\right)\left({\displaystyle \frac{\lambda {\delta }^{\frac{1}{v}(i+1)}{\overline{\delta }}^k}{l+1}}\right).\hfill \end{array}$$

(14)

For all derivations, see the Supplementary Materials.

5.2. Uncertainty Measure

Uncertainty measures play a crucial role in various fields of statistics, particularly in information theory and machine learning. Notable examples include Shannon entropy Shannon [15] and Rényi entropy Rényi [16]. Rényi entropy quantifies the randomness or uncertainty within a system and generalizes the Shannon entropy. The Rényi entropy of the ExpH-G FoD is

$$\begin{array}{c}\hfill I_R\left(ϵ\right)={\displaystyle \frac{1}{1-ϵ}}log\left(\sum _{l=0}^{\infty }{w}_l{e}^{(1-ϵ)I_{REG}}\right),\end{array}$$

(15)

where $I_{REG}={\displaystyle \frac{1}{1-ϵ}}log({\int }_0^{\infty }{\left[(1+{\displaystyle \frac{l}{ϵ}})g(z;\underline{\mathrm{{\rm Y}}})G^{{\displaystyle \frac{l}{ϵ}}}(z;\underline{\mathrm{{\rm Y}}})\right]}^{ϵ}dz)$ is the Rényi entropy for an Exp-G distribution with power parameter $({\displaystyle \frac{1}{ϵ}}+1)$ and

$$\begin{array}{ccc}\hfill w_l& =& \sum _{i,k=0}^{\infty }{(-1)}^{i+k+l}\left({\displaystyle \genfrac{}{}{0pt}{}{ϵ(\lambda -1)}{i}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\frac{-1}{v}(i+ϵ)-ϵ}{k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{i+vk}{l}}\right)\hfill \\ & \times & {\lambda }^{ϵ}{\delta }^{{\displaystyle \frac{1}{v}}(ϵ+i)}{\overline{\delta }}^k{\displaystyle \frac{1}{{\left(1+\frac{l}{ϵ}\right)}^{ϵ}}}.\hfill \end{array}$$

For all derivations, see the Supplementary Materials.

5.3. Moments

In this subsection, we derive the raw moments, moment generating function (mgf), and conditional moments of the ExpH-G FoD.

5.3.1. Moments and Moment Generating Functions

Let $X\sim$Exp$(l+1)$; then, the j-th raw moment, ${\mu }_j^{\prime }$, of ExpH-G FoD is

$$\begin{array}{c}\hfill {\mu }_j^{\prime }=\mathbb{E}\left[Z^j\right]={\int }_{-\infty }^{\infty }{z}^{j}f\left(z\right)dz=\sum _{l=0}^{\infty }{c}_{l+1}\mathbb{E}\left[X^j\right],\end{array}$$

(16)

where $\mathbb{E}\left[X^j\right]$ is the j-th moment of the Exp-G distribution with parameter $(l+1)$ and $c_{l+1}$ is given in Equation (9). The mgf of ExpH-G FoD is

$$\begin{array}{c}\hfill {\mathcal{M}}_Z\left(s\right)=\mathbb{E}\left[e^{sZ}\right]=\sum _{l=0}^{\infty }{c}_{l+1}{\mathcal{M}}_{l+1}\left(s\right),\end{array}$$

(17)

for $\left|s\right|<1$, where ${\mathcal{M}}_{l+1}\left(s\right)$ is the mgf of X and $c_{l+1}$ is given by Equation (9).

5.3.2. Conditional Moments

The j-th conditional moment of the ExpH-G FoD is obtained as

$$\begin{array}{c}\hfill \mathbb{E}\left[Z^j\right|Z>s]={\displaystyle \frac{1}{\overline{F}\left(s\right)}}{\int }_s^{\infty }{z}^{j}f\left(z\right)dz=\sum _{l=0}^{\infty }{c}_{l+1}\mathbb{E}\left[X^j{I}_{\{X^j>s\}}\right],\end{array}$$

(18)

where $\mathbb{E}\left[X^j{I}_{\{X^j>s\}}\right]={\int }_s^{\infty }{x}^j{g}_{l+1}(x;\underline{\mathrm{{\rm Y}}})dz$ and $c_{l+1}$ is as given in Equation (9).

6. Parameter Estimation

This section presents several estimation techniques for the parameters of the ExpH-G family. The methods considered are maximum likelihood (ML), least squares (LS), ordinary least squares (OLS), Anderson–Darling (AD), and Cramér–von Mises (CVM).

ML Estimation

Consider independent random variables Z_i $\sim \text{ExpH-G}(\lambda ,\delta ,v,\underline{\mathrm{{\rm Y}}})$, $i=1,\dots ,m$, with parameter vector $\underline{\theta }={(\lambda ,\delta ,v,\underline{\mathrm{{\rm Y}}})}^{\top }$. The total log-likelihood is

$$\begin{array}{ccc}\hfill \ell \left(\underline{\theta }\right)& =& mln\left(\lambda \right)+{\displaystyle \frac{m}{v}}ln\left(\delta \right)-\left(1+{\displaystyle \frac{1}{v}}\right)\sum _{i=1}^{m}ln[1-\overline{\delta }{\overline{G}}^v(z_i;\underline{\mathrm{{\rm Y}}})]\hfill \\ & +& \sum _{i=1}^{m}ln\left[g(z_i;\underline{\mathrm{{\rm Y}}})\right]+(\lambda -1)\sum _{i=1}^{m}ln\left[1-{\displaystyle \frac{{\delta }^{\frac{1}{v}}\overline{G}(z_i;\underline{\mathrm{{\rm Y}}})}{{[1-\overline{\delta }{\overline{G}}^v(z_i;\underline{\mathrm{{\rm Y}}})]}^{\frac{1}{v}}}}\right].\hfill \end{array}$$

(19)

The score vector elements and the expressions for estimating the LS, OLS, AD, and CVM are in the Supplementary Materials.

7. Simulation Design and Results

In this section, we present simulation results for the ML, AD, OLS, WLS, and CVM estimation techniques. Simulation experiments were carried out under different sample sizes to find the best estimation technique for estimating the parameters of the ExpH-W model, a special case within the ExpH-G FoD framework. Two metrics were utilized to evaluate the performance of the estimators: the root mean square error (RMSE) and the average bias (ABias). The RMSE and ABias expressions are

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}\left(\widehat{\mathrm{\Omega }}\right)=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^M{({\widehat{\mathrm{\Omega }}}_i-\mathrm{\Omega })}^2}{M}}},\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{A}\mathrm{B}\mathrm{i}\mathrm{a}\mathrm{s}(\widehat{\mathrm{\Omega }})={\displaystyle \frac{{\sum }_{i=1}^M{\widehat{\mathrm{\Omega }}}_i}{M}}-\mathrm{\Omega },$$

where $\widehat{\mathrm{\Omega }}$ denotes the estimated parameter.

Table 1. Simulation results for $(\lambda ,\delta ,v,\theta )=(0.2,0.2,0.7,0.9)$.

Parameter	m	ML	AD	RMSE OLS	WLS	CVM	ML	AD	Abias OLS	WLS	CVM
$\lambda$	30	0.1468 (1)	0.3958 (2)	0.7406 (5)	0.5383 (4)	0.5049 (3)	0.0803 (1)	0.2222 (2)	0.3588 (5)	0.2366 (3)	0.3004 (4)
$\delta$	30	0.8536 (5)	0.1492 (3)	0.1330 (1)	0.1885 (4)	0.1484 (2)	0.4949 (5)	−0.0485 (2)	−0.0613 (3)	0.0290 (1)	−0.0914 (4)
v	30	1.8633 (2)	2.1753 (3)	3.1832 (4)	0.1331 (1)	4.2749 (5)	0.7945 (2)	0.8512 (3)	1.7417 (4)	0.0869 (1)	2.1626 (5)
$\theta$	30	1.5071 (5)	0.3767 (3)	0.3455 (2)	0.1307 (1)	0.3882 (4)	0.3701 (4)	−0.3730 (5)	−0.2690 (3)	−0.0242 (1)	−0.2616 (2)
Sum of Ranks		13	11	12	10	14	12	12	15	6	15
$\lambda$	50	0.0915 (1)	0.2869 (3)	0.2962 (4)	0.2542 (2)	0.3027 (5)	0.0610 (1)	0.2063 (3)	0.1973 (2)	0.2272 (5)	0.2184 (4)
$\delta$	50	0.4006 (5)	0.1350 (4)	0.1203 (3)	0.1172 (2)	0.1171 (1)	0.4747 (5)	0.0403 (3)	−0.0298 (2)	0.0282 (1)	−0.0573 (4)
v	50	1.2714 (2)	1.2825 (3)	2.5839 (5)	0.0759 (1)	2.0697 (4)	0.4588 (2)	0.4797 (3)	1.1085 (5)	0.0734 (1)	1.0397 (4)
$\theta$	50	0.3013 (2)	0.3657 (5)	0.3120 (4)	0.0561 (1)	0.3047 (3)	0.2305 (2)	0.3620 (5)	−0.2484 (4)	−0.0233 (1)	−0.2318 (3)
Sum of Ranks		14	13	15	6	12	10	14	13	8	15
$\lambda$	100	0.0798 (1)	0.2238 (4)	0.2185 (3)	0.2470 (5)	0.1858 (2)	0.0576 (1)	0.1955 (4)	0.1640 (3)	0.2126 (5)	0.1562 (2)
$\delta$	100	0.2455 (5)	0.0861 (1)	0.0892 (2)	0.0902 (3)	0.0916 (4)	0.4441 (5)	0.0333 (3)	0.0161 (1)	0.0238 (2)	−0.0596 (4)
v	100	0.5649 (2)	0.5715 (3)	0.8145 (4)	0.0736 (1)	1.0229 (5)	0.1156 (2)	−0.5582 (5)	0.1260 (3)	−0.0672 (1)	0.2277 (4)
$\theta$	100	0.2335 (2)	0.3614 (5)	0.3070 (4)	0.0491 (1)	0.2992 (3)	0.1007 (2)	−0.3485 (5)	−0.2699 (3)	−0.0356 (1)	−0.2604 (4)
Sum of Ranks		10	13	13	10	14	10	17	10	9	14
$\lambda$	200	0.0617 (1)	0.2033 (4)	0.1738 (2)	0.2284 (5)	0.1800 (3)	0.0481 (1)	0.1826 (4)	0.1461 (2)	0.2107 (5)	0.1526 (3)
$\delta$	200	0.1270 (5)	0.0854 (4)	0.0793 (2)	0.0811 (3)	0.0744 (1)	0.3825 (5)	0.0332 (3)	0.0173 (1)	0.0244 (2)	0.0533 (4)
v	200	0.4547 (2)	0.5511 (4)	0.4988 (3)	0.0594 (1)	0.5734 (5)	0.1056 (2)	−0.5413 (5)	−0.1212 (3)	−0.0589 (1)	−0.1850 (4)
$\theta$	200	0.2067 (2)	0.3576 (5)	0.3065 (4)	0.0483 (1)	0.2852 (3)	0.0219 (1)	−0.3333 (5)	−0.2812 (4)	−0.0371 (2)	−0.2554 (3)
Sum of Ranks		10	17	11	10	12	9	17	10	10	14
$\lambda$	400	0.0476 (1)	0.1970 (4)	0.1539 (2)	0.2257 (5)	0.1578 (3)	0.0364 (1)	0.1856 (4)	0.1369 (2)	0.2068 (5)	0.1396 (3)
$\delta$	400	0.1022 (5)	0.0778 (3)	0.0753 (1)	0.0809 (4)	0.0763 (2)	0.0432 (4)	0.0324 (3)	0.0140 (1)	0.0233 (2)	0.0530 (5)
v	400	0.4051 (2)	0.5437 (5)	0.4678 (3)	0.0523 (1)	0.4799 (4)	−0.0858 (2)	−0.3589 (5)	−0.1167 (3)	−0.0504 (1)	−0.3279 (4)
$\theta$	400	0.1272 (2)	0.3001 (4)	0.3060 (5)	0.0424 (1)	0.2162 (3)	0.0216 (1)	−0.2433 (4)	−0.2992 (5)	−0.0399 (2)	−0.1973 (3)
Sum of Ranks		10	16	11	11	12	8	16	11	10	15
$\lambda$	800	0.0401 (1)	0.1785 (4)	0.1438 (3)	0.2155 (5)	0.1399 (2)	0.0361 (1)	0.1727 (4)	0.1329 (3)	0.2093 (5)	0.1320 (2)
$\delta$	800	0.0918 (5)	0.0627 (1)	0.0707 (2)	0.0803 (4)	0.0760 (3)	0.0230 (4)	0.0271 (3)	0.0119 (1)	0.0175 (2)	0.0534 (5)
v	800	0.3842 (2)	0.5152 (5)	0.4643 (3)	0.0466 (1)	0.4745 (4)	0.0335 (1)	−0.3899 (4)	−0.1087 (3)	−0.0500 (2)	−0.4375 (5)
$\theta$	800	0.1220 (2)	0.2910 (4)	0.3045 (5)	0.0416 (1)	0.2127 (3)	0.0200 (1)	−0.2417 (4)	−0.3063 (5)	−0.0412 (2)	−0.2043 (3)
Sum of Ranks		10	14	13	11	12	7	15	12	11	15
$\lambda$	1000	0.0373 (1)	0.1745 (4)	0.1389 (3)	0.2098 (5)	0.1373 (2)	0.0273 (1)	0.1707 (5)	0.1322 (3)	0.1362 (4)	0.1293 (2)
$\delta$	1000	0.0387 (1)	0.0574 (2)	0.0639 (3)	0.0801 (5)	0.0751 (4)	0.0127 (2)	0.0224 (4)	0.0045 (1)	0.0132 (3)	0.0529 (5)
v	1000	0.2030 (2)	0.5012 (5)	0.4594 (3)	0.0389 (1)	0.4741 (4)	−0.0030 (1)	−0.3511 (4)	−0.1065 (3)	−0.0503 (2)	−0.4440 (5)
$\theta$	1000	0.0366 (1)	0.2901 (4)	0.3031 (5)	0.0405 (2)	0.2114 (3)	0.0197 (1)	0.2014 (4)	−0.3140 (5)	−0.0423 (2)	−0.2024 (3)
Sum of Ranks		5	15	14	13	13	5	17	12	11	15

and

Table 2. Simulation results for $(\lambda ,\delta ,v,\theta )=(0.7,0.2,0.7,1.1)$.

Parameter	m	ML	AD	RMSE OLS	WLS	CVM	ML	AD	Abias OLS	WLS	CVM
$\lambda$	30	0.5988 (1)	3.5330 (2)	5.0166 (3)	5.2952 (5)	5.0907 (4)	−0.3870 (1)	0.6461 (2)	2.0737 (4)	2.7005 (3)	2.3113 (5)
$\delta$	30	2.6190 (5)	0.1428 (1)	0.1886 (3)	0.1912 (4)	0.1599 (2)	0.7468 (5)	−0.0208 (2)	−0.0340 (3)	0.0183 (1)	−0.0584 (4)
v	30	1.9582 (1)	2.0534 (2)	2.9499 (4)	2.3728 (3)	3.7246 (5)	1.0438 (3)	0.8529 (1)	1.7460 (4)	0.8609 (2)	2.1656 (5)
$\theta$	30	0.3320 (1)	0.5388 (2)	0.8533 (4)	0.6874 (3)	1.0024 (5)	0.2114 (3)	0.1538 (2)	0.5694 (5)	0.1261 (1)	0.4884 (4)
Sum of Ranks		8	7	14	15	16	12	7	16	7	18
$\lambda$	50	0.4441 (1)	1.4505 (4)	3.2583 (5)	0.9318 (3)	0.7657 (2)	−0.3360 (2)	0.3364 (3)	0.8622 (5)	0.3719 (4)	0.0739 (1)
$\delta$	50	0.6299 (5)	0.1167 (2)	0.1320 (4)	0.1044 (1)	0.1263 (3)	0.4163 (5)	−0.0111 (1)	−0.0297 (3)	−0.0118 (2)	−0.0584 (4)
v	50	1.2797 (1)	1.5626 (2)	2.9412 (5)	2.3106 (3)	2.8286 (4)	1.0133 (3)	0.5326 (2)	1.7347 (5)	0.4816 (1)	1.2098 (4)
$\theta$	50	0.2557 (1)	0.4711 (3)	0.7871 (5)	0.4710 (2)	0.7143 (4)	0.1698 (3)	0.0970 (2)	0.4738 (5)	0.0772 (1)	0.3009 (4)
Sum of Ranks		8	11	19	9	13	13	8	18	8	13
$\lambda$	100	0.4436 (1)	0.6051 (2)	0.8122 (5)	0.6433 (3)	0.7131 (4)	−0.2004 (2)	0.2430 (5)	0.2143 (4)	0.2098 (3)	0.0396 (1)
$\delta$	100	0.5436 (5)	0.0976 (2)	0.1053 (3)	0.0956 (1)	0.1148 (4)	0.4496 (5)	−0.0051 (1)	−0.0255 (3)	0.0061 (2)	−0.0517 (4)
v	100	1.2634 (2)	1.0762 (1)	2.3179 (5)	1.1842 (3)	1.7482 (4)	0.2501 (1)	0.3681 (3)	0.8583 (4)	0.3523 (2)	1.0001 (5)
$\theta$	100	0.1054 (1)	0.4452 (2)	0.5813 (4)	0.4706 (3)	0.6890 (5)	0.0616 (2)	0.0173 (1)	0.1784 (4)	0.0644 (3)	0.3028 (5)
Sum of Ranks		9	7	17	10	17	10	10	15	10	15
$\lambda$	200	0.4369 (1)	0.4826 (2)	0.6204 (4)	0.5322 (3)	0.6878 (5)	−0.1206 (2)	0.1238 (3)	0.2131 (5)	0.1986 (4)	0.0209 (1)
$\delta$	200	0.5385 (5)	0.0829 (1)	0.0965 (4)	0.0885 (2)	0.0915 (3)	0.4821 (5)	−0.0044 (1)	0.0151 (3)	0.0056 (2)	−0.0460 (4)
v	200	0.7909 (1)	0.9209 (3)	1.0134 (4)	0.8917 (2)	1.3076 (5)	0.1958 (1)	0.2438 (3)	0.3037 (4)	0.2048 (2)	0.6181 (5)
$\theta$	200	0.1038 (1)	0.4187 (3)	0.5423 (4)	0.4064 (2)	0.6586 (5)	0.0109 (1)	0.0150 (2)	0.1457 (4)	−0.0192 (3)	0.2754 (5)
Sum of Ranks		8	9	16	9	18	9	9	16	11	15
$\lambda$	400	0.3558 (1)	0.3975 (3)	0.4130 (4)	0.3583 (2)	0.5865 (5)	−0.0546 (2)	0.1233 (4)	0.0974 (3)	0.1708 (5)	0.0207 (1)
$\delta$	400	0.5236 (5)	0.0761 (2)	0.0828 (3)	0.0639 (1)	0.0863 (4)	0.5011 (5)	0.0005 (1)	0.0075 (3)	0.0020 (2)	−0.0238 (4)
v	400	0.6038 (2)	0.7090 (3)	0.9378 (4)	0.5844 (1)	1.2151 (5)	−0.1191 (2)	0.1252 (3)	0.2169 (4)	0.0915 (1)	0.5436 (5)
$\theta$	400	0.0901 (1)	0.3813 (3)	0.4996 (4)	0.3269 (2)	0.6118 (5)	0.0073 (1)	0.0173 (3)	0.0751 (4)	0.0165 (2)	0.2644 (5)
Sum of Ranks		9	11	15	6	19	10	11	14	10	15
$\lambda$	800	0.2650 (1)	0.3450 (2)	0.4077 (4)	0.3559 (3)	0.4990 (5)	−0.0347 (2)	0.0911 (4)	0.0451 (3)	0.1247 (5)	0.0196 (1)
$\delta$	800	0.4192 (5)	0.0658 (2)	0.0751 (3)	0.0638 (1)	0.0794 (4)	0.4172 (5)	−0.0004 (1)	−0.0140 (4)	0.0014 (2)	−0.0115 (3)
v	800	0.5210 (1)	0.5417 (2)	0.8795 (4)	0.5677 (3)	1.0105 (5)	0.1056 (3)	0.0795 (1)	0.2434 (4)	0.0512 (2)	0.3884 (5)
$\theta$	800	0.0875 (1)	0.2932 (2)	0.4798 (4)	0.3239 (3)	0.5229 (5)	0.0057 (1)	0.0134 (4)	0.0131 (3)	−0.0112 (2)	0.1593 (5)
Sum of Ranks		8	8	15	10	19	11	10	14	11	14
$\lambda$	1000	0.1556 (1)	0.3409 (3)	0.3957 (4)	0.3121 (2)	0.4092 (5)	−0.0356 (2)	0.0396 (3)	0.0435 (4)	0.1186 (5)	0.0164 (1)
$\delta$	1000	0.3862 (5)	0.0551 (1)	0.0666 (3)	0.0613 (2)	0.0696 (4)	0.4862 (5)	−0.0003 (1)	−0.0053 (3)	0.0013 (2)	−0.0067 (4)
v	1000	0.3626 (1)	0.4385 (2)	0.7884 (4)	0.4497 (3)	0.8383 (5)	−0.4626 (5)	0.0413 (2)	0.2010 (3)	−0.0198 (1)	0.2736 (4)
$\theta$	1000	0.0809 (1)	0.2651 (2)	0.4276 (4)	0.2790 (3)	0.4752 (5)	0.0039 (1)	−0.0132 (4)	0.0130 (3)	−0.0118 (2)	0.1448 (5)
Sum of Ranks		8	8	15	10	19	13	10	13	10	14

present the RMSE and Abias outcomes from the simulation study for selected parameter values using different estimation approaches. The superscript attached to each value indicates its rank among the estimators. For instance, in Table 1, the ML estimate of $\theta$ for $n=30$ ranks 5th in the RMSE while its corresponding Abias ranks 4th compared to the other methods. The aggregated ranking across estimators for each sample size is presented in the row labeled “SUM of RANKS”.

Table 3. Partial rank, total rank, and overall rank.

Parameters	m	ML	AD	OLS	WLS	CVM
	30	25 (3)	23 (2)	27 (4)	16 (1)	29 (5)
	50	24 (2)	27 (3.5)	28 (5)	14 (1)	27 (3.5)
$\lambda =0.2,\delta =0.2,v=0.7,\theta =0.9$	100	20 (2)	30 (5)	23 (3)	19 (1)	28 (4)
	200	19 (1)	34 (5)	21 (3)	20 (2)	26 (4)
	400	18 (1)	32 (5)	22 (3)	21 (2)	27 (4)
	800	17 (1)	29 (5)	25 (3)	22 (2)	27 (4)
	1000	10 (1)	32 (5)	26 (3)	24 (2)	28 (4)
	30	20 (2)	14 (1)	30 (4)	22 (3)	34 (5)
	50	21 (3)	19 (1)	37 (5)	17 (2)	26 (4)
$\lambda =0.7,\delta =0.2,v=0.7,\theta =1.1$	100	19 (2)	17 (1)	32 (4.5)	20 (3)	32 (4.5)
	200	17 (1)	18 (2)	32 (4)	20 (3)	33 (5)
	400	19 (2)	22 (3)	29 (4)	16 (1)	34 (5)
	800	19 (2)	18 (1)	29 (4)	21 (3)	33 (5)
	1000	21 (3)	18 (1)	28 (4)	20 (2)	33 (5)
Sum of Ranks		26	40.5	53.5	28	62
Overall rank		1	3	4	2	5

provides the overall rankings that combine both RMSE and Abias. A superscript in this table indicates the partial rank assigned to each estimation approach, while the cumulative totals are displayed in the row designated as SUM of RANKS. The overall rank presented shows the best method.

Table 1 and Table 2 reveal that the RMSE typically decreases with increasing sample size, whereas the Abias exhibits less consistent behavior. Table 3 indicates that the ML estimator outperforms the competing methods for parameter estimation in the ExpH-W distribution, particularly for moderate to large sample sizes. However, it is important to note that ML estimation does not perform well for small samples, where alternative methods may be more reliable.

Figure 3 and Figure 4 present the RMSE plots for each parameter, demonstrating the variation in the RMSE values of the ExpH-W parameters as n increases for different estimation techniques. The graphs indicate that, for all five estimation methods analyzed, the RMSE consistently decreases with increasing n.

8. Applications

To demonstrate the dominance of the ExpH-W distribution, we utilize two real-world datasets in this section. The ML estimation method is employed to estimate the model parameters. To determine which distribution fits these datasets better, we utilize the following goodness-of-fit (GoF) statistics: the −2loglikelihood statistic (2log(L)), Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Anderson–Darling ($A^{\ast }$) value, Cramér von Mises ($W^{\ast }$) value, and Kolmogorov–Smirnov (K-S) value along with its p-value. The optimal model is the one that has the lowest values for all GoF statistics and the biggest p-value for the K-S statistic.

We compare our proposed model with the following non-nested models: the Harris–Weibull (HW) distribution Aly and Benkherouf [14], gamma type II exponentiated half logistic-Topp–Leone–Weibull (RBTIIEHLTLW) distribution  Charumbira et al. [17], Weibull–Lomax (WLx) distribution [18], odd log-logistic exponentiated Weibull (OLLEW) distribution Afify et al. [19], exponentiated half logistic generalized Weibull–Poisson (EHLWP) distribution Chipepa et al. [20], exponentiated Lindley odd log-logistic Weibull (ELOLLW) distribution Korkmaz et al. [21], Harris–Weibull Poisson (HWP) distribution Charumbira et al. [22], and Kumaraswamy–Weibull (KumW) distribution Cordeiro and De Castro [23].

8.1. Growth Hormone Data

The dataset consists of the time from when the growth hormone medication was administered to children until they reach the target age. The dataset was analyzed by Alizadeh et al. [24].

The asymptotic 95% confidence intervals (estimate ± margin of error) for the parameters are: $\lambda \in [1.7840\pm 1.5071$, $\delta \in [3.1373\times {10}^{10}\pm 5.5826\times {10}^{-11}]$, $v\in [15.3200\pm 2.7921]$, $\theta \in [0.6743\pm 0.1271]$. Standard errors for the ML estimates were obtained from the inverse of the observed Fisher information matrix, that is, the inverse of the Hessian of the negative log-likelihood evaluated at the ML estimation. Numerical Hessian computation was performed using the mle2 function (package bbmle) with the nlminb optimizer in R Software Version 4.5.2.

Table 4. Growth hormone data: parameter estimates and GoF statistics.

			Estimates						GoF Statistics
Model	$\mathit{\lambda }$	$\mathit{\delta }$	$\mathit{v}$	$\mathit{\theta }$	−2log(L)	AIC	CAIC	BIC	${\mathit{W}}^{\ast }$	${\mathit{A}}^{\ast }$	K-S	p-Value
ExpH-W	1.7840	3.1373 $\times {10}^{10}$	15.3200	0.6743	153.0958	161.0958	162.4291	167.3172	0.0305	0.1986	0.0867	0.9549
	(0.7689)	(2.8483 $\times {10}^{-11}$)	(1.4245)	(0.06487)
	$\delta$	v	$\theta$
HW	$3.1373\times {10}^{10}$	$1.4393\times {10}^1$	$6.0698\times {10}^{-1}$		$154.6682$	$160.6682$	$161.4424$	$165.3343$	$0.0480$	$0.2920$	$0.1302$	$0.5933$
	$(2.0241\times {10}^{-11})$	$(9.8584\times {10}^{-1})$	$(5.4142\times {10}^{-2})$
	$\delta$	a	b	$\lambda$
RBTIIEHLTLW	1.9022	4.9008 $\times {10}^5$	134.3900	0.06787	163.6136	171.6136	172.947	177.8350	0.1477	0.933	0.1463	0.4424
	(0.37494)	(3.9165 $\times {10}^{-7}$)	(0.02882)	(4.9684 $\times {10}^{-3}$)
	a	b	$\alpha$	$\beta$
WLx	2.2065 $\times {10}^3$	4.4219	7.7393 $\times {10}^{-2}$	0.8239	161.0881	169.0881	170.4215	175.3095	0.1103	0.7102	0.1232	0.6631
	(5.0757 $\times {10}^{-5}$)	(1.8868)	(1.3194 $\times {10}^{-2}$)	(1.3124)
	$\alpha$	$\beta$	$\gamma$	$\theta$
OLLEW	0.0115	0.0908	3.5189	14.6611	158.6735	166.6735	168.0068	172.8949	0.0573	0.4133	0.0990	0.8829
	(0.0193)	(0.0124)	(0.7393)	(0.0966)
	$\alpha$	$\beta$	$\delta$	$\theta$
EHLWP	7.0598 $\times {10}^{-2}$	5.0729	8.0153 $\times {10}^2$	1.7194 $\times {10}^2$	160.3848	168.3848	169.7182	174.6062	0.1026	0.663	0.12098	0.6849
	(8.4735 $\times {10}^{-3}$)	(8.3097 $\times {10}^{-2}$)	(1.0306 $\times {10}^{-4}$)	(3.3213 $\times {10}^{-5}$)
	$\beta$	$\lambda$	$\theta$	$\gamma$
ELOLLW	2.6629 $\times {10}^{-5}$	6.4083 $\times {10}^{-2}$	6.6610	1.9932	164.9772	172.9772	174.3105	179.1986	0.1639	1.0262	0.1454	0.4500
	(2.1143)	(1.2802 $\times {10}^{-2}$)	(1.8971 $\times {10}^{-4}$)	(0.2437)
	$\theta$	$\delta$	$\beta$	v
HWP	1.3697 $\times {10}^{-7}$	7.9607 $\times {10}^4$	0.6445	6.0891	155.5646	163.5646	164.8979	169.786	0.0382	0.2585	0.1098	0.7930
	(5.7727 $\times {10}^{-2}$)	(4.5399 $\times {10}^{-6}$)	(5.5434 $\times {10}^{-2}$)	(0.5655)
	a	b	$\alpha$	$\beta$
KumW	48.7672	34.5299	16.7869	0.2135	160.2115	168.2115	169.5449	174.4329	0.1002	0.6496	0.1213	0.6819
	(4.4684)	(0.6746)	(4.2412)	(0.0183)

illustrates that the ExpH-W distribution provides a superior fit for the growth hormone data as evidenced by its lowest AIC, BIC, CAIC, AD, and CVM K-S, and highest p value of the K-S statistic compared to non-nested models. These findings underscore the effectiveness of the ExpH-W distribution in capturing the underlying patterns in the data.

The profile log-likelihood plots in Figure 5 show that the parameters of the ExpH-W distribution are identifiable for growth hormone data. Figure 6 show the fitted density and the corresponding probability plots, which suggest a strong fit for the ExpH-W distribution compared to the selected competing models. As shown in Figure 7, the strong alignment of the fitted K-M survival curve with its empirical line, and of the ECDF with its empirical line, indicates that the model effectively captures the growth hormone data. The scaled TTT transform plot in Figure 8 suggests a bathtub hrf shape, which is accurately picked by our model.

8.2. Repair Time Data

The data consists of $n=46$ active repair times (in hours) for an airborne communication transceiver as referenced by Tlhaloganyang et al. [25].

The asymptotic 95% confidence intervals (estimate ± margin of error) for the parameters are: $\lambda \in [0.9471\pm 0.5781]$, $\delta \in [4.8750\times {10}^3\pm 0.0012]$, $v\in [113.0410\pm 5.4845]$, $\theta \in [0.4956\pm 0.1050]$.

From the results presented in

Table 5. Repair time data: parameter estimates and GoF statistics.

			Estimates						GoF Statistics
Model	$\mathit{\lambda }$	$\mathit{\delta }$	$\mathit{v}$	$\mathit{\theta }$	−2log(L)	AIC	CAIC	BIC	${\mathit{W}}^{\ast }$	${\mathit{A}}^{\ast }$	K-S	p-Value
ExpH-W	0.9471	4.8750 $\times {10}^3$	13.0410	0.4956	197.8110	205.8110	206.7866	213.1256	0.0270	0.1883	0.0639	0.9918
	(0.2949)	(6.2641 $\times {10}^{-4}$)	(2.7982)	(5.3574 $\times {10}^{-2}$)
	$\delta$	v	$\theta$
HW	$3.1373\times {10}^{10}$	$2.9971\times {10}^1$	$3.0764\times {10}^{-1}$		$222.0694$	$228.0694$	$228.6408$	$233.5553$	$0.1005$	$0.6748$	$0.2653$	$0.0031$
	$(4.2941\times {10}^{-11})$	$\left(1.1653\right)$	$(3.9643\times {10}^{-2})$
	$\delta$	a	b	$\lambda$
RBTIIEHLTLW	5.0497	4.9008 $\times {10}^5$	1.3699 $\times {10}^2$	4.2824 $\times {10}^{-2}$	204.9325	212.9325	213.9081	220.247	0.0928	0.6568	0.1167	0.5585
	(0.3531)	(6.2536 $\times {10}^{-7}$)	(4.4681 $\times {10}^{-2}$)	(3.4922 $\times {10}^{-3}$)
	a	b	$\alpha$	$\beta$
WLx	4.7329 $\times {10}^2$	2.4865	3.0636 $\times {10}^{-2}$	0.2239	200.6681	208.6681	209.6437	215.9826	0.0569	0.3680	0.0949	0.8015
	(2.3553 $\times {10}^{-4}$)	(1.4675)	(1.8319 $\times {10}^{-2}$)	(0.5191)
	$\alpha$	$\beta$	$\gamma$	$\theta$
OLLEW	0.0012	0.0462	2.4555	14.8432	202.4335	210.4335	211.4091	217.7481	0.0758	0.4579	0.0941	0.8105
	(0.0093)	(0.0110)	(1.0908)	(0.0654)
	$\alpha$	$\beta$	$\delta$	$\theta$
EHLWP	6.3669 $\times {10}^{-2}$	6.1644	8.0412 $\times {10}^2$	9.5572	200.5537	208.5537	209.5293	215.8683	0.0573	0.3482	0.0923	0.8276
	(1.6357 $\times {10}^{-2}$)	(0.2137)	(1.5626 $\times {10}^{-3}$)	(6.3483)
	$\beta$	$\lambda$	$\theta$	$\gamma$
ELOLLW	1.2322 $\times {10}^{-7}$	3.5026 $\times {10}^{-2}$	6.7826	0.8985	208.9394	216.9394	217.915	224.254	0.1298	0.9009	0.1204	0.5170
	(1.5449)	(1.2110 $\times {10}^{-2}$)	(6.9530 $\times {10}^{-5}$)	(9.5760 $\times {10}^{-2}$)
	$\theta$	$\delta$	$\beta$	v
HWP	4.8635 $\times {10}^{-8}$	1.3634 $\times {10}^2$	0.5314	8.9756	200.4499	208.4499	209.4255	215.7645	0.034	0.2407	0.1138	0.5912
	(2.3895 $\times {10}^{-2}$)	(90.5960)	(5.0737 $\times {10}^{-2}$)	(2.1548)
	a	b	$\alpha$	$\beta$
KumW	$13.8461$	$29.4165$	$4.5439$	$0.1572$	$204.1452$	$212.1452$	$213.1208$	$219.4597$	$0.0879$	$0.5957$	$0.1248$	$0.4705$
	$\left(3.2934\right)$	$\left(31.1143\right)$	$\left(4.3328\right)$	$\left(0.0366\right)$

, the ExpH-W distribution dominates the competing models for the repair time data since it has the lowest of all the GoF statistics and highest p value of the K-S statistic.

The profile log-likelihood plots in Figure 9 show that the parameters of the ExpH-W distribution are identifiable for repair time data. Figure 10 demonstrates how the ExpH-W model fit the repair times data compared to the selected models. The close alignment between the fitted K-M survival curve and the empirical line, as well as between the ECDF and its empirical line in Figure 11, indicates that the model provides a strong fit to the repair time data. The TTT scaled plot in Figure 12 suggests an inverted bathtub hrf plot, which is accurately captured by the Exp-H-W model.

9. Conclusions

This work introduces a novel family of distributions that builds upon the Harris-G framework. Several special cases derived from the proposed model demonstrate enhanced flexibility, particularly in capturing complex failure rate behaviors. The model applies to a bathtub followed by an upside bathtub, a bathtub, and increasing and decreasing hrfs. There are several estimation techniques that include ML, OLS, WLS, AD, and CVM. Monte Carlo simulation studies were considered using these various estimation techniques and the ML method was rated the best technique for estimating parameters of the ExpH-G FoD. The performance of the proposed model was assessed against several competing models using two real-world datasets. The proposed model outperformed the selected competing models, thereby suggesting that researchers may consider selecting the ExpH-G family in place of the selected competing models. Despite its flexibility, the Exp–Harris–Weibull distribution has limitations, particularly with small sample sizes.