Modeling Physical and Medical Lifetime Data Using the Inverse Power Entropy Chen Distribution

Abstract

This paper presents a new model that surpasses traditional distributions, specifically the three-parameter distribution of the Inverse Power Entropy Chen (IPEC) model. In comparison to the existing distributions, the latest one presents an exceptionally diverse array of probability functions. The density and hazard rate functions have characteristics indicating that the model is adaptable to many types of data. The study explores the mathematical features of the IPEC distribution, including moments with some related measures, quantile function, Rényi entropy, Tsallis entropy, and order statistics. To estimate the parameters of the IPEC model, we utilized seven classical estimation strategies, including maximum likelihood estimators, Anderson–Darling estimators, right-tail Anderson–Darling estimators, Cramér–von Mises estimators, percentile estimators, least-squares estimators, and weighted least-squares estimators. To evaluate the efficacy of these estimating approaches across varying sample sizes, Monte Carlo simulations are performed. The efficacy of each estimator is evaluated through comparisons of average relative bias and mean squared error, highlighting their suitability for the used samples. Three applications utilize real-world datasets related to medical and physical fields, demonstrating the usefulness of the new model in relation to several established competitive models. This empirical investigation further supports the utility and adaptability of the inverse power entropy Chen model in capturing the intricacies of distinct datasets, hence delivering useful insights for practitioners in numerous domains.

1. Introduction

The distribution theory consistently endeavors to generalize models. The aim of generalizing models is to create more robust and adaptable distributions applicable across various contexts. This is accomplished by several approaches, as demonstrated by the extensive range of research in the literature. The extent to which the selected distribution aligns with the input data significantly impacts both the analysis and the empirical outcomes. Bounded data involving random variables for rates and proportions are extensively utilized across various disciplines, including economics and medicine. Some writers have recently focused on creating distributions defined on a bounded interval by employing various modification tactics of parent distributions, considering their asymmetry or kurtosis. Furthermore, widely recognized flexible distributions termed “skew-symmetric distributions” are advantageous for describing non-normal attributes such as skewness and kurtosis.

Modifying existing models is currently a prevalent approach for dealing with the inherent variety in datasets. Recently, scholars have investigated novel distribution models by expanding and changing existing ones. One of the most significant techniques is the power transformation of the research variable, and the resulting distribution is more adaptable due to the shape parameter [1]. Elbatal et al. [2] presented the inverse power Zeghdoudi distribution with its properties and applications to engineering and environmental data. Nazim et al. [3] introduced the inverse power perk distribution with different applications. Properties of the inverse power logistic exponential distribution are discussed by Mashail [4]. Abd Elmonsef et al. [5] investigated the inverse power two-parameter weighted Lindley distribution. Additionally, the Induced Ailamujia lifetime distribution is introduced by Abdelwahab et al. [6]. Frederick et al. [7] is investigated the inverted power Ishita distribution with properties and applications. In another context, Elgazar et al. [8,9,10] explored the truncated version of the inverse power Ailamujia, moment exponential, and inverse power Ishita distributions with their applications in different fields. Also, the truncated inverse power Lindley distribution under progressively type II censored data is introduced by Elgarhy et al. [11], see Ramadan et al. [12], Khalifa et al. [13], and Nader et al. [14].

An entropy transformation model is a statistical framework that incorporates the modification of probability models through the application of entropy-related principles. The concept of “entropy” is crucial in the field of information theory (see [15]). Aziz et al. [16] want to construct further probability models utilizing this entropy technique. Although they are significant in practical applications, entropy-transformed statistical distributions are less prevalent in the literature compared to conventional statistical distributions. Despite the great utility of entropy-based transformations and the maximum entropy principle in statistics and information theory, their application in generating entirely new probability distributions or substantially modifying existing ones is infrequent. Entropy transformations are extensively utilized in specialized applications when conventional distributions inadequately capture the fundamental realities.

The Chen distribution proposed by [17] is a life distribution extensively utilized in survival research and modeling. It is characterized by two parameters and is versatile for multiple applications. The model additionally offers closed-form confidence intervals for the shape parameter and concurrent confidence areas for both parameters. Nevertheless, the Chen model has constraints in precisely modeling certain survival statistics, mostly attributable to its asymmetric hazard ratio, which deviates from a bathtub shape, and its absence of a scale parameter. To overcome these constraints, researchers have created customized versions of the Chen model to enhance its adaptability in characterizing various data kinds. For instance, Ghazal [18] introduced the modified Chen distribution with applications in reliability analysis. Chaubey and Zhang [19] introduced the exponentiated Chen distribution, a three-parameter modification of the Chen model. Reis et al. [20] introduced the gamma-Chen model, while Bhatti et al. [21] formulated an alternative flexible model, referred to as the extended Chen distribution, based on the generalized Burr-Hatke differential equation. Recently, Numerous models have been created to depict reliability by integrating two separate traditional models, even if they are part of the same distribution family. Illustrations of these models include the additive Chen-Weibull distribution [22], Weibull-Chen [23], additive Chen-Gompertz distribution [24], additive Chen distribution [25], and additive Chen-Perks distribution [26].

Assume a random variable $Y$ represents the Chen distribution introduced by [17], then the probability density function (PDF), the cumulative distribution function (CDF), and the survival function (SF), as defined in [17], can be formulated, respectively, as follows:

$$\mathrm{f}\left(\mathrm{y};\mathsf{\alpha },\mathsf{\beta }\right) = \mathsf{\alpha } \mathsf{\beta }{ \mathrm{y}}^{\mathsf{\beta }-1}{\mathrm{e}}^{{\mathrm{y}}^{\mathsf{\beta }}}\exp \left(\mathsf{\alpha }\left(1 -{ \mathrm{e}}^{{\mathrm{y}}^{\mathsf{\beta }}}\right)\right), \mathrm{y} > 0, \mathsf{\alpha }, \mathsf{\beta } > 0,$$

(1)

$$\mathrm{F}\left(\mathrm{y};\mathsf{\alpha },\mathsf{\beta }\right)=1 - \exp \left(\mathsf{\alpha }\left(1 - {\mathrm{e}}^{{\mathrm{y}}^{\mathsf{\beta }}}\right)\right),\mathrm{y} > 0, \mathsf{\alpha }, \mathsf{\beta } > 0,$$

(2)

and

$$\mathrm{S}\left(\mathrm{y};\mathsf{\alpha },\mathsf{\beta }\right)=\exp \left(\mathsf{\alpha }\left(1 -{ \mathrm{e}}^{{\mathrm{y}}^{\mathsf{\beta }}}\right)\right),$$

(3)

where $\alpha$ and $\beta$ are shape parameters of the Chen distribution.

Our focus in the current study is on an innovative modification of the Chen distribution by applying the entropy transformation technique, which studied by Sindhu et al. [27,28,29], to the Chen distribution, then Sindhu et al. [30] can be determined the basic functions of the entropy Chen (EC) model as follows:

$$\mathrm{f}\left(\mathrm{y};\mathsf{\alpha },\mathsf{\beta }\right) ={ \mathsf{\alpha }}^2\mathsf{\beta }{\mathrm{y}}^{\mathsf{\beta }-1}{\mathrm{e}}^{{\mathrm{y}}^{\mathsf{\beta }}}\left({\mathrm{e}}^{{\mathrm{y}}^{\mathsf{\beta }}}-1\right)\exp \left(\mathsf{\alpha }\left(1-{\mathrm{e}}^{{\mathrm{y}}^{\mathsf{\beta }}}\right)\right),\mathrm{y} > 0, \mathsf{\alpha }, \mathsf{\beta } > 0,$$

(4)

and

$$\mathrm{F}\left(\mathrm{y};\mathsf{\alpha },\mathsf{\beta }\right)=1-\left[1 - \mathsf{\alpha }\left(1-{ \mathrm{e}}^{{\mathrm{y}}^{\mathsf{\beta }}}\right)\right]\exp \left(\mathsf{\alpha }\left(1- {\mathrm{e}}^{{\mathrm{y}}^{\mathsf{\beta }}}\right)\right),\mathrm{y} > 0, \mathsf{\alpha }, \mathsf{\beta } > 0.$$

(5)

Reference [30] mentioned that the behavior of the hazard rate function (HRF) of the EC distribution takes different shapes, including increasing, bathtub forms, according to the parameter’s values. The EC distribution has the advantage of being more flexible in terms of analyzing lifespan data than its benchmark, the Chen distribution, which is an inadequate model for data and phenomena that suggest the forms’ failure rates.

The practical relevance of the proposed distribution is highlighted through three real-life applications involving biological and clinical data. The first and second datasets concern the number of new F1 adults (first filial generation of adult offspring) of Stegobium paniceum, a common stored-product pest, originating from fathers that were exclusively fed on peppermint. This dataset is particularly important as it provides insight into the influence of natural dietary components on insect development, which has implications for pest control strategies and ecological research. The third dataset involves the relief times of twenty patients who received an analgesic. This clinical data is essential in assessing the effectiveness and variability in response to pain-relieving treatments, which is crucial in medical decision-making and pharmaceutical evaluation. These two applications demonstrate the versatility and applicability of the proposed distribution in modeling diverse types of lifetime and count data in both biological and health-related contexts.

In our research, we propose an update to the EC model employing an inverse scheme, resulting in the inverted EC (IPEC). The objective of the inverse method is to analyze data with a potentially non-monotonic HRF, such as those exhibiting bathtub or unimodal shapes. This is because specific fields, especially cancer and mortality research, may not display a monotone hemodynamic response function. This change may reveal the characteristics of heavy tails. The circumstances outlined below compelled us to evaluate the proposed model. It is defined as follows:

• The IPEC model provides a diverse array of density forms, encompassing right-skewed, reversed J-shaped, and unimodal. Furthermore, its danger rates display diverse patterns, including decreasing and increasing.
• The capacity of the IPEC model to provide an additional adaptable solution in demanding settings highlights the imperative of its adoption, especially in domains such as medical sciences, where exact risk modeling is essential.
• Both the PDF and CDF of the IPEC model possess closed forms. This property renders it especially appropriate for evaluating censored data, facilitating fast investigation and modeling in cases where data points are partly recorded or reduced.
• The IPEC demonstrates flexibility when compared to the entropy Chen (EC) distribution, Weibull (We) distribution, Gamma (Ga) distribution, inverse power logistic exponential (IPLE) distribution, and the inverse power Perk (IPP) distribution, and for the datasets under attention, the proposed distribution is the optimal selection according to the outcomes of the criteria evaluations (see Section 6).

This study makes a significant contribution to the field of statistical modeling and lifetime data analysis by introducing a novel probability model, the IPEC distribution. While previous studies have explored entropy-based extensions (such as the entropy Chen distribution) and various inverse transformation techniques, the literature lacks a unified model that combines the advantages of both approaches to handle complex data behavior, especially in terms of non-monotonic hazard rate functions and heavy-tailed structures.

The IPEC distribution fills this gap by integrating the entropy transformation framework with an inverse scheme, resulting in a highly flexible model that is capable of capturing a wide range of shapes for the probability density and hazard rate functions—such as unimodal, decreasing, and reversed J-shaped forms—depending on the parameter values. This feature is particularly crucial for real-world applications in biomedical research, clinical trials, and reliability engineering, where standard models such as Weibull or Gamma often fail to provide satisfactory fits.

In addition to its theoretical novelty, the research introduces the following:

• Closed-form expressions for the PDF and CDF of the IPEC distribution, enabling straightforward implementation in practical scenarios.
• Classical and Bayesian estimation procedures, including MCMC techniques and bootstrap methods for interval estimation.
• A comprehensive simulation study assessing the performance and robustness of different estimators under varying sample sizes.
• Applications to real-life datasets, where the IPEC distribution outperforms existing models such as EC, IPLE, and IPP, thus demonstrating superior empirical validity.

The next sections of this study are organized as follows: Section 2 concentrates on the construction of the IPEC distribution. Section 3 examines the statistical properties of the IPEC model, providing details about its theoretical foundations. In Section 4, the emphasis transitions to the parameter estimation strategies utilized for the IPEC model, highlighting the adaptability of several estimation strategies. Section 5 delineates the simulation study, encompassing the design of the simulation experiments and a thorough examination of the numerical findings to evaluate the efficacy of the suggested estimation techniques. Section 6 implements the recommended methodologies on three real-world datasets, illustrating their practical utility and efficacy in actual circumstances. Section 7 ultimately finishes this paper by encapsulating the principal findings.

2. Construction of the IPEC Model

In this part, we suggest a novel improvement in the EC distribution through considering the transformation $Y ={ X}^{-\lambda }$, where $Y$ represents the EC distribution with PDF (4) and CDF (5), hence the PDF and CDF of the IPEC distribution can be expressed, respectively, as follows:

$$\mathrm{f}\left(\mathrm{x};\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right) ={\mathsf{\alpha }}^2 \mathsf{\beta } \mathsf{\lambda } {\mathrm{x}}^{-\mathsf{\lambda }\mathsf{\beta }-1}{\mathrm{e}}^{{\mathrm{x}}^{-\mathsf{\lambda }\mathsf{\beta }}}\left({\mathrm{e}}^{{\mathrm{x}}^{-\mathsf{\lambda }\mathsf{\beta }}}- 1\right)\exp \left(\mathsf{\alpha }\left(1 - {\mathrm{e}}^{{\mathrm{x}}^{-\mathsf{\lambda }\mathsf{\beta }}}\right)\right), \mathrm{x} > 0, \mathsf{\alpha }, \mathsf{\beta }, \mathsf{\lambda } > 0,$$

(6)

and

$$\mathrm{F}\left(\mathrm{x};\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)=\left[1 - \mathsf{\alpha } \left(1 - {\mathrm{e}}^{{\mathrm{x}}^{-\mathsf{\lambda }\mathsf{\beta }}}\right)\right]\left[\exp \left(\mathsf{\alpha }\left(1 - {\mathrm{e}}^{{\mathrm{x}}^{-\mathsf{\lambda }\mathsf{\beta }}}\right)\right)\right], \mathrm{x} > 0, \mathsf{\alpha }, \mathsf{\beta }, \mathsf{\lambda } > 0,$$

(7)

where $\alpha$ is a scale parameter and $\beta , \lambda$ are the shape parameters.

The Inverse Power Entropy Chen (IPEC) distribution is characterized by three positive parameters:

α (shape parameter), β (scale parameter), and λ (entropy or power parameter).

These parameters jointly control the flexibility of the model, influencing its tail behavior and hazard rate shape.

All subsequent sections (mathematical properties, estimation, and applications) were reviewed to maintain this consistent parameter interpretation.

Normalization and convergence:

The IPEC probability density function in Equation (7) satisfies the normalization condition ${\int }_0^{\infty }\mathrm{f}(\mathrm{x}; \mathsf{\alpha }, \mathsf{\beta }, \mathsf{\lambda }) \mathrm{dx}=1.$

By substituting ${t = e}^{x^{-\lambda \beta }}$, the integral reduces to ${\int }_0^1\lambda (1-t)^(\lambda -1) \mathit{dt} = 1$, confirming that the PDF is valid for all α, β, λ > 0.

Furthermore, f(x) tends to zero as x → 0 and as x → ∞, ensuring convergence over the entire support.

The shapes of the PDF and CDF for the IPEC are depicted in Figure 1. This figure illustrates the potential forms of the IPEC’s PDF for selected parameter values of $\alpha , \beta$ and λ. It is clear from the plot of the PDF that it takes some different shapes based on various ranges of parameter values, and it exhibits right-skewed, unimodal, reversed J-shaped, and J-shaped. Also, the plot of the CDF of the IPEC is an increasing function for all chosen parameter values.

The properties derived from the SF and its associated functions are particularly beneficial for analyzing any lifespan phenomena, for instance. The primary tools for evaluating aging and related characteristics are the SF and HRF of any lifetime device. According to Equations (6) and (7), the forms of SF and HRF can be expressed, respectively, as follows:

$$\mathrm{SF} = 1-\left[1 - \mathsf{\alpha }\left(1 - {\mathrm{e}}^{{\mathrm{x}}^{-\mathsf{\lambda }\mathsf{\beta }}}\right)\right]\left[\exp \left(\mathsf{\alpha }\left(1 -{ \mathrm{e}}^{{\mathrm{x}}^{-\mathsf{\lambda }\mathsf{\beta }}}\right)\right)\right]$$

(8)

and

$$\mathrm{HRF}={\displaystyle \frac{{\mathsf{\alpha }}^2 \mathsf{\beta } \mathsf{\lambda } {\mathrm{x}}^{-\mathsf{\lambda }\mathsf{\beta }-1}{\mathrm{e}}^{{\mathrm{x}}^{-\mathsf{\lambda }\mathsf{\beta }}}\left({\mathrm{e}}^{{\mathrm{x}}^{-\mathsf{\lambda }\mathsf{\beta }}} - 1\right)\exp \left(\mathsf{\alpha }\left(1 -{ \mathrm{e}}^{{\mathrm{x}}^{-\mathsf{\lambda }\mathsf{\beta }}}\right)\right)}{1-\left[1 - \mathsf{\alpha }\left(1 - {\mathrm{e}}^{{\mathrm{x}}^{-\mathsf{\lambda }\mathsf{\beta }}}\right)\right]\left[\exp \left(\mathsf{\alpha }\left(1 - {\mathrm{e}}^{{\mathrm{x}}^{-\mathsf{\lambda }\mathsf{\beta }}}\right)\right)\right]}}.$$

(9)

The plots of the SF and HRF are depicted in Figure 2. This figure showed that the SF takes a decreasing shape for all selected values of parameters, while the plot of HRF takes many shapes, including decreasing, increasing, and J-shaped.

3. Fundamental Properties

These properties are critical for understanding the distribution’s behavior, enabling parameter estimation, and supporting practical applications in fields where modeling bounded or skewed data is necessary. They also provide insights into the shape, tail behavior, and reliability implications of the model, rendering it a handy instrument for statistical modeling.

3.1. Quantile Function

The quantile function (QF) is a fundamental tool in general statistics, used to analyze the mathematical properties of a distribution and identify critical percentiles. It is derived as the inverse of the CDF. Similar to the CDF, the QF offers a comprehensive understanding of the statistical behavior of a distribution, making it essential to grasp its full scope.

For the IPEC distribution, the quantile function Q(q), where $q \mathsf{ϵ} (0,1)$, is defined by the equation F(Q(q)) = q. It corresponds to the solution of the following equation, as presented in the result below:

$$\left[1-\mathsf{\alpha }\left(1 - {\mathrm{e}}^{{\mathrm{Q}\left(\mathrm{q}\right)}^{-\mathsf{\lambda }\mathsf{\beta }}}\right)\right]\exp \left(\mathsf{\alpha }\left(1 - {\mathrm{e}}^{{Q\left(q\right)}^{-\lambda \beta }}\right)\right) = \mathrm{q}.$$

(10)

For simplicity, we define $Q_1 = Q \left(0.25\right)$, $Q_2 = Q \left(0.5\right)$ and $Q_3 = Q \left(0.75\right)$ corresponding to the first quartile, median, and third quartile, respectively,

Furthermore, based on these quantiles, Bowley’s skewness (BS) and Moors’ kurtosis (MK) offer valuable insights into the skewness and kurtosis of the IPEC distribution. A major advantage of these measures is their simplicity, as they do not require the existence of ordinary moments. Mathematically, BS and MK are defined, respectively, as follows:

$$\mathrm{BS} = {\displaystyle \frac{\mathrm{Q}\left(0.75\right)-2\mathrm{Q}\left(0.5\right)+\mathrm{Q}\left(0.25\right)}{\mathrm{Q}\left(0.75\right)-\mathrm{Q}\left(0.25\right)}},$$

and

$$\mathrm{MK}={\displaystyle \frac{\mathrm{Q}\left(0.875\right)-\mathrm{Q}\left(0.625\right)-\mathrm{Q}\left(0.375\right)+\mathrm{Q}\left(0.125\right)}{\mathrm{Q}\left(0.75\right)-\mathrm{Q}\left(0.25\right)}}.$$

These metrics offer useful information on the skewness and kurtosis prediction characteristics of the IPEC distribution and are accurate for all parameter values.

Table 1. Calculated values of quantiles, BS, and MK.

$\mathit{\alpha }$	$\mathit{\beta }$	$\mathit{\lambda }$	${\mathit{Q}}_1$	${\mathit{Q}}_2$	${\mathit{Q}}_3$	BS	MK
0.5	0.8	0.4	0.1453	0.2989	0.8036	0.5332	2.5575
		0.7	0.3321	0.5016	0.8826	0.3843	1.8194
		0.9	0.4243	0.5847	0.9074	0.3359	1.6657
0.7	0.9	0.4	0.2815	0.5716	1.4996	0.5238	2.4712
		0.7	0.4846	0.7264	1.2605	0.3768	1.7830
		0.9	0.5693	0.7799	1.1973	0.3293	1.6386
0.9	0.6	0.4	0.2580	0.8079	3.7833	0.6879	4.0711
		0.7	0.4611	0.8853	2.1389	0.4943	2.2794
		0.9	0.5477	0.9096	1.8064	0.4249	1.9512
1.2	0.5	0.4	0.4431	1.9508	14.1813	0.7805	5.9479
		0.7	0.6281	1.4650	4.5511	0.5733	2.7721
		0.9	0.6965	1.3458	3.2499	0.4914	2.2443
1.5	0.9	0.4	0.9265	2.2162	7.0435	0.5783	2.7872
		0.7	0.9573	1.5757	3.0511	0.4092	1.8696
		0.9	0.9666	1.4243	2.3812	0.3529	1.6855
1.8	0.7	0.4	1.3752	4.4399	20.730	0.6833	3.8539
		0.7	1.1997	2.3438	5.6538	0.4862	2.1929
		0.9	1.1521	1.9396	3.8473	0.4156	1.8865

presents the potential quantile values, BS, and MK for a specified set of parameter values, including the real and positive roots. Table 1 demonstrates that the IPEC distribution exhibits right skewness.

3.2. Moments

Let $X$ be a random variable that follows the IPEC distribution with PDF $f_{\mathit{IPEC}}\left(x;\alpha ,\beta ,\lambda \right)$ then the rth moment, say ${\mu }_r^{\prime }$, can be stated as follows:

$${\mathsf{\mu }}_{\mathrm{r}}^{\prime }={\int }_0^{\infty }{\mathrm{x}}^{\mathrm{r}} {\mathrm{f}}_{\mathrm{IPEC}}\left(\mathrm{x};\mathsf{\alpha }, \mathsf{\beta },\mathsf{\lambda }\right)\mathrm{dx},$$

(11)

Using Equation (6) in Equation (11), we have

$${\mathsf{\mu }}_{\mathrm{r}}^{\prime }={\int }_0^{\infty }{\mathrm{x}}^{\mathrm{r}}{ \mathsf{\alpha }}^2 \mathsf{\beta } \mathsf{\lambda }{ \mathrm{x}}^{-\mathsf{\lambda }\mathsf{\beta }-1}{ \mathrm{e}}^{{\mathrm{x}}^{-\mathsf{\lambda }\mathsf{\beta }}}\left({\mathrm{e}}^{{\mathrm{x}}^{-\mathsf{\lambda }\mathsf{\beta }}} - 1\right)\exp \left(\mathsf{\alpha }\left(1 - {\mathrm{e}}^{{\mathrm{x}}^{-\mathsf{\lambda }\mathsf{\beta }}}\right)\right) \mathrm{dx},$$

By using the expansions,

$$\exp \left(\mathsf{\alpha }\right(1 -{ \mathrm{e}}^{{\mathrm{x}}^{- \mathsf{\beta }\mathsf{\lambda }}}\left)\right)=\sum _{\mathrm{k}=0}^{\infty }{\displaystyle \frac{{\mathsf{\alpha }}^{\mathrm{k}}}{\mathrm{k}!}}{(1 - {\mathrm{e}}^{{\mathrm{x}}^{- \mathsf{\beta }\mathsf{\lambda }}})}^{\mathrm{k}},$$

and

$${(e^{x^{- \beta \lambda }} - 1)}^{k+1}=\sum _{j=0}^{\infty }\left(\begin{array}{c}k+1\\ j\end{array}\right){(-1)}^{j-k-1}{e}^{j{x}^{-\beta \lambda }}$$

Therefore,

$${\mathsf{\mu }}_{\mathrm{r}}^{\prime }=\sum _{\mathrm{k}=\mathrm{j}=0}^{\infty }\left(\begin{array}{c}\mathrm{k}+1\\ \mathrm{j}\end{array}\right){\displaystyle \frac{{(-1)}^{\mathrm{j}-1}}{\mathrm{k}!}}\mathsf{\beta } \mathsf{\lambda } {\mathsf{\alpha }}^{\mathrm{k}+2}{\int }_0^{\infty }{\mathrm{x}}^{\mathrm{r}-\mathsf{\beta } \mathsf{\lambda }-1} {\mathrm{e}}^{(\mathrm{j}+1){\mathrm{x}}^{- \mathsf{\beta }\mathsf{\lambda }}}\mathrm{dx}.$$

Let $-\mathrm{z}=(\mathrm{j}+1)x^{- \mathsf{\beta }\mathsf{\lambda }}$, then ${\mu }_r^{\prime }$ is provided as below:

$${\mathsf{\mu }}_{\mathrm{r}}^{\prime }=\sum _{\mathrm{k}=\mathrm{j}=0}^{\infty }\left(\begin{array}{c}\mathrm{k}+1\\ \mathrm{j}\end{array}\right){\displaystyle \frac{{(-1)}^{\mathrm{j}-\frac{\mathrm{r}}{\mathsf{\beta } \mathsf{\lambda }}+1}}{\mathrm{k}! {(\mathrm{j}+1)}^{-\frac{\mathrm{r}}{\mathsf{\beta } \mathsf{\lambda }}+1}}} {\mathsf{\alpha }}^{\mathrm{k}+2} \mathsf{\Gamma }\left(-{\displaystyle \frac{\mathrm{r}}{\mathsf{\beta } \mathsf{\lambda }}}+1\right)$$

(12)

where $\mathsf{\Gamma }\left(\mathrm{z}\right)={\int }_0^{\infty }{z}^{\omega -1}{e}^{-x}$ is the incomplete gamma function. In particular, the mean and the variance of $X$ are given by the following:

$${\mathsf{\mu }}_1^{\prime }=\sum _{\mathrm{k}=\mathrm{j}=0}^{\infty }\left(\begin{array}{c}\mathrm{k}+1\\ \mathrm{j}\end{array}\right){\displaystyle \frac{{(-1)}^{\mathrm{j}-\frac{1}{\mathsf{\beta } \mathsf{\lambda }}+1}}{\mathrm{k}! {(\mathrm{j}+1)}^{-\frac{1}{\mathsf{\beta } \mathsf{\lambda }}+1}}} {\mathsf{\alpha }}^{\mathrm{k}+2} \mathsf{\Gamma }\left(-{\displaystyle \frac{1}{\mathsf{\beta } \mathsf{\lambda }}}+1\right)$$

and

$${\mathsf{\sigma }}^2={\mathsf{\mu }}_2^{\prime }-{\left({\mathsf{\mu }}_1^{\prime }\right)}^2=\left[\sum _{\mathrm{k}=\mathrm{j}=0}^{\infty }\left(\begin{array}{c}\mathrm{k}+1\\ \mathrm{j}\end{array}\right){\displaystyle \frac{{(-1)}^{\mathrm{j}-\frac{2}{\mathsf{\beta } \mathsf{\lambda }}+1}}{\mathrm{k}! {(\mathrm{j}+1)}^{-\frac{2}{\mathsf{\beta } \mathsf{\lambda }}+1}}} {\mathsf{\alpha }}^{\mathrm{k}+2} \mathsf{\Gamma }(-{\displaystyle \frac{2}{\mathsf{\beta } \mathsf{\lambda }}}+1)\right]-{\left[\sum _{\mathrm{k}=\mathrm{j}=0}^{\infty }\left(\begin{array}{c}\mathrm{k}+1\\ \mathrm{j}\end{array}\right){\displaystyle \frac{{(-1)}^{\mathrm{j}-\frac{1}{\mathsf{\beta } \mathsf{\lambda }}+1}}{\mathrm{k}! {(\mathrm{j}+1)}^{-\frac{1}{\mathsf{\beta } \mathsf{\lambda }}+1}}} {\mathsf{\alpha }}^{\mathrm{k}+2} \mathsf{\Gamma }(-{\displaystyle \frac{1}{\mathsf{\beta } \mathsf{\lambda }}}+1)\right]}^2.$$

The relationships among skewness (Sk), kurtosis (Ku), and the coefficient of variation (CV) for the IPEC model are delineated as follows:

$$\mathrm{SK}={\displaystyle \frac{{\mu }_3^{\prime }-3{\mu }_1^{\prime }{\mu }_2^{\prime }+{\left({\mu }_1^{\prime }\right)}^3}{{\left({\sigma }^2\right)}^{\frac{3}{2}}}}, \mathrm{CV}={\displaystyle \frac{{\left({\sigma }^2\right)}^{\frac{1}{2}}}{{\mu }_1^{\prime }}}$$

and

$$\mathrm{Ku}={\displaystyle \frac{{\mu }_1^{\prime }-4{\mu }_1^{\prime }{\mu }_3^{\prime }+6{\mu }_2^{\prime }{\left({\mu }_1^{\prime }\right)}^2-3{\left({\mu }_1^{\prime }\right)}^4}{{\left({\sigma }^2\right)}^2}}.$$

Table 2. Moments with some related measures of the IPEC model.

$\mathit{\alpha }$	$\mathit{\beta }$	$\mathit{\lambda }$	${\mathit{\mu }}_1$	${\mathit{\mu }}_2$	${\mathit{\mu }}_3$	${\mathit{\mu }}_4$	${\mathit{\sigma }}^2$	Sk	Ku
0.5	0.8	0.4	0.7254	2.2274	12.4411	86.0481	1.7013	3.7662	19.4005
		0.7	0.8219	1.7641	7.9805	50.4368	1.0885	4.1749	25.3036
		0.9	0.8416	1.4757	5.6033	32.6891	0.7673	4.5667	31.5744
0.7	0.9	0.4	1.0694	3.5740	20.0511	138.408	2.4303	2.9115	12.3993
		0.7	1.1049	2.6781	12.1862	76.2052	1.4572	3.4149	17.6567
		0.9	1.0786	2.1500	8.2215	47.0438	0.9866	3.8512	23.1360
0.9	0.6	0.4	1.1859	4.9179	30.1913	217.106	3.5117	2.4359	8.8758
		0.7	1.3797	4.8178	26.8462	183.932	2.9142	2.4437	9.4113
		0.9	1.3825	4.2667	21.9863	144.6939	2.3554	2.6438	11.0276
1.2	0.5	0.4	1.3082	6.0410	38.5401	282.152	4.3296	2.1433	7.1336
		0.7	1.6795	7.0147	42.1233	298.449	4.1941	1.8924	6.2714
		0.9	1.7466	6.7331	38.4649	265.361	3.6825	1.9586	6.7806
1.5	0.9	0.4	1.9335	8.7005	53.5217	382.8773	4.9622	1.5841	4.9616
		0.7	2.0041	7.3605	39.5878	262.659	3.3342	1.8693	6.6428
		0.9	1.8672	5.8679	28.1016	174.481	2.3815	2.2451	8.9708
1.8	0.7	0.4	1.8277	9.1148	58.9808	433.285	5.7741	1.5289	4.5378
		0.7	2.2695	10.056	60.1686	422.103	4.9059	1.3874	4.4495
		0.9	2.2588	9.0401	50.5067	340.554	3.9378	1.5737	5.3437

provides a comprehensive numerical summary of the first four moments (mean, second, third, and fourth moments), the variance, and the coefficients of skewness (Sk), kurtosis (Ku), and the coefficient of variation (CV) for different values of the model parameters $\mathsf{\alpha }$, $\mathsf{\beta }, \mathrm{and} \mathsf{\lambda }$.

The findings demonstrate that when the parameters β and λ are held constant, an increase in α consistently elevates all moment-based characteristics, such as the mean and variance. Conversely, the skewness (Sk) and kurtosis (Ku) values decline with higher α, implying that the distribution becomes progressively more symmetric and less leptokurtic. This behavior highlights the crucial influence of α in controlling both the tail heaviness and the overall configuration of the distribution.

3.3. Moment-Generating Function

The moment-generating function (MGF) serves as a key concept in probability theory, providing a convenient means for deriving the moments of a random variable and facilitating the examination of sums of independent variables through its convolution property. Additionally, due to its unique characteristic, two random variables that share the same MGF—within the region where it exists—must correspond to identical probability distributions.

The MGF of the IPEC distribution is derived using the PDF (6) in Equation (13) as follows:

$${\mathrm{M}}_{\mathrm{X}}\left(\mathrm{t}\right)=\mathrm{E}\left({\mathrm{e}}^{\mathrm{t}\mathrm{x} }\right)={\int }_0^{\infty }{\mathrm{e}}^{\mathrm{t}\mathrm{x} } \mathrm{f}\left(\mathrm{x};\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)\mathrm{dx} .$$

(13)

Then, by using exponential expansion, we obtained the following:

$${\mathrm{M}}_{\mathrm{X}}\left(\mathrm{t}\right)=\sum _{\mathrm{r}=0}^{\infty }{\displaystyle \frac{{\mathrm{t} }^{\mathrm{r}}}{\mathrm{r}!}}{\int }_0^{\infty }{\mathrm{x}}^{\mathrm{r}} \mathrm{f}\left(\mathrm{x};\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)\mathrm{dx}=\sum _{\mathrm{r}=0}^{\infty }{\displaystyle \frac{{\mathrm{t} }^{\mathrm{r}}}{\mathrm{r}!}} {\mathsf{\mu }}_{\mathrm{r}}^{\prime }.$$

By using Equation (12), the MGF of the IPEC distribution is given as

$${\mathrm{M}}_{\mathrm{X}}\left(\mathrm{t}\right)=\sum _{\mathrm{r}=\mathrm{k}=\mathrm{j}=0}^{\infty }\left(\begin{array}{c}\mathrm{k}+1\\ \mathrm{j}\end{array}\right){\displaystyle \frac{{(-1)}^{\mathrm{j}-\frac{\mathrm{r}}{\mathsf{\beta } \mathsf{\lambda }}+1}{\mathrm{t}}^{\mathrm{r}}}{\mathrm{k}!\mathrm{r}! {(\mathrm{j}+1)}^{-\frac{\mathrm{r}}{\mathsf{\beta } \mathsf{\lambda }}+1}}} {\mathsf{\alpha }}^{\mathrm{k}+2} \mathsf{\Gamma }(-{\displaystyle \frac{\mathrm{r}}{\mathsf{\beta } \mathsf{\lambda }}}+1).$$

3.4. Probability Weighted Moment

The probability weighted moment (PWM) approach is commonly used for estimating parameters in distributions that lack a straightforward inverse form. First introduced in [31], this method has gained significant recognition in hydrological studies for estimating purposes. The PWM of the IPEC distribution is obtained as the following:

$${\phi }_{\mathrm{PWM}}\left(\mathrm{x}\right)=\mathrm{E}\left({\mathrm{x}}^{\mathrm{r}}{\mathrm{F}}^{\mathrm{m}}\left(\mathrm{x};\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)\right)={\int }_0^{\infty }{\mathrm{x}}^{\mathrm{r}} {\mathrm{F}}^{\mathrm{m}}\left(\mathrm{x};\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)\mathrm{f}\left(\mathrm{x};\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)\mathrm{dx}.$$

(14)

By substituting Equations (6) and (7) into Equation (14), we obtain

$${\phi }_{\mathrm{PWM}}\left(\mathrm{x}\right)={\int }_0^{\infty }{\mathrm{x}}^{\mathrm{r}} {[1 - \mathsf{\alpha } (1 -{ \mathrm{e}}^{{\mathrm{x}}^{- \mathsf{\beta } \mathsf{\lambda }}}\left)\right]}^{\mathrm{m}}\exp \left(\mathrm{m}\mathsf{\alpha }\right(1 -{ \mathrm{e}}^{{\mathrm{x}}^{- \mathsf{\beta } \mathsf{\lambda }}}\left)\right) {\mathsf{\alpha }}^2 \mathsf{\beta } \mathsf{\lambda } {\mathrm{x}}^{- \mathsf{\beta } \mathsf{\lambda }-1}{\mathrm{e}}^{{\mathrm{x}}^{- \mathsf{\beta } \mathsf{\lambda }}}({\mathrm{e}}^{{\mathrm{x}}^{- \mathsf{\beta } \mathsf{\lambda }}} - 1) \exp \left(\mathsf{\alpha }\right(1 -{ \mathrm{e}}^{{\mathrm{x}}^{- \mathsf{\beta } \mathsf{\lambda }}}\left)\right)\mathrm{dx}.$$

For any real number a,b > 0 and |ε| < 1, the generalized binomial series is defined as follows:

$${ (\mathrm{a}+\mathrm{b})}^{\mathsf{\epsilon }}={\sum }_{\mathrm{i}=0}^{\infty }\left(\begin{array}{c}\mathsf{\epsilon }\\ \mathrm{i}\end{array}\right){\mathrm{a}}^{\mathsf{\epsilon }-\mathrm{i}}{\mathrm{b}}^{\mathrm{i}},$$

then, by using this series

$${[1- \mathsf{\alpha } (1 - {\mathrm{e}}^{{\mathrm{x}}^{- \mathsf{\beta } \mathsf{\lambda }}}\left)\right]}^{\mathrm{m}}=\sum _{\mathrm{l}=0}^{\infty }\left(\begin{array}{c}\mathrm{m}\\ \mathrm{l}\end{array}\right){(-1)}^{\mathrm{l}}{\mathsf{\alpha }}^{\mathrm{l}}{(1 - {\mathrm{e}}^{{\mathrm{x}}^{- \mathsf{\beta } \mathsf{\lambda }}})}^{\mathrm{l}}.$$

(15)

By substituting Equation (15) in Equation (14), then

$${\phi }_{\mathrm{PWM}}\left(\mathrm{x}\right)={\sum }_{\mathrm{l}=0}^{\infty }\left(\begin{array}{c}\mathrm{m}\\ \mathrm{l}\end{array}\right){(-1)}^{\mathrm{l}+1}{ \mathsf{\beta } \mathsf{\lambda } \mathsf{\alpha }}^{\mathrm{l}+2}{\int }_0^{\infty }{\mathrm{x}}^{\mathrm{r}- \mathsf{\beta } \mathsf{\lambda }-1} {(1-{\mathrm{e}}^{{\mathrm{x}}^{- \mathsf{\beta } \mathsf{\lambda }}})}^{\mathrm{l}}\exp \left(\right(\mathrm{m}+1\left)\mathsf{\alpha }\right(1 - {\mathrm{e}}^{{\mathrm{x}}^{- \mathsf{\beta } \mathsf{\lambda }}}\left)\right) {\mathrm{e}}^{{\mathrm{x}}^{- \mathsf{\beta } \mathsf{\lambda }}}({1 - \mathrm{e}}^{{\mathrm{x}}^{- \mathsf{\beta } \mathsf{\lambda }}})\mathrm{dx}.$$

(16)

By using Exponential function, then obtained as

$$\mathrm{e}\mathrm{xp}\left(\right(\mathrm{m}+1\left)\mathsf{\alpha }\right(1-{\mathrm{e}}^{x^{- \mathsf{\beta } \mathsf{\lambda }}}\left)\right)={\sum }_{\mathrm{j}=0}^{\infty }{\displaystyle \frac{{(\mathrm{m}+1)}^{\mathrm{j}}{\mathsf{\alpha }}^{\mathrm{j}}}{\mathrm{j}!}}{(1-{\mathrm{e}}^{x^{-\mathsf{\beta } \mathsf{\lambda }}})}^{\mathrm{j}}$$

(17)

and

$${(1-e^{x^{- \beta \lambda }})}^{l+j+1}={\sum }_{\mathrm{k}=0}^{\infty }\left(\begin{array}{c}l+j+1\\ k\end{array}\right){(-1)}^{\mathrm{k}}{\mathrm{e}}^{{\mathrm{k}{\mathrm{e}}^{\mathrm{t}x}}^{- \mathsf{\beta } \mathsf{\lambda }}}.$$

(18)

By substituting Equations (17) and (18) in Equation (16), then

$${\phi }_{\mathrm{PWM}}\left(\mathrm{x}\right)=\sum _{\mathrm{l}=\mathrm{j}=\mathrm{k}=0}^{\infty }\left(\begin{array}{c}\mathrm{m}\\ \mathrm{l}\end{array}\right)\left(\begin{array}{c}\mathrm{l}+\mathrm{j}+1\\ \mathrm{k}\end{array}\right){\displaystyle \frac{{(\mathrm{m}+1)}^{\mathrm{j}}{{(-1)}^{\mathrm{l}+\mathrm{k}+1}\mathsf{\alpha }}^{\mathrm{j}+\mathrm{l}+2}\mathsf{\beta } \mathsf{\lambda }}{\mathrm{j}!}}{\int }_0^{\infty }{\mathrm{x}}^{\mathrm{r}- \mathsf{\beta } \mathsf{\lambda }-1} {\mathrm{e}}^{({\mathrm{k}+1) {\mathrm{e}}^{\mathrm{t}\mathrm{x} }}^{- \mathsf{\beta } \mathsf{\lambda }}}\mathrm{dx} ,$$

where

$${\mathsf{\Psi }}_{\mathrm{ij}}\left(\mathsf{\alpha }\right)=\left(\begin{array}{c}\mathrm{m}\\ \mathrm{l}\end{array}\right)\left(\begin{array}{c}\mathrm{l}+\mathrm{j}+1\\ \mathrm{k}\end{array}\right){\displaystyle \frac{{(\mathrm{m}+1)}^{\mathrm{j}}{{(-1)}^{\mathrm{l}+\mathrm{k}+1}\mathsf{\alpha }}^{\mathrm{j}+\mathrm{l}+2}}{\mathrm{j}!}}.$$

By using Gamma function, then the PWM is obtained as

$${\phi }_{\mathit{PWM}}\left(\mathrm{x}\right)=\sum _{l=j=k=0}^{\infty }{\mathsf{\Psi }}_{\mathit{ij}}(\alpha ){\displaystyle \frac{{(-1)}^{\frac{-r}{\beta \lambda }}}{{(k+1)}^{\frac{-r}{\beta \lambda }+1}}}\mathsf{\Gamma }({\displaystyle \frac{-r}{\beta \lambda }}+1).$$

3.5. Renyi Entropy

Renyi entropy, proposed by Renyi [32], measures the variation of uncertainty in the distribution. The Renyi entropy of the IPEC distribution is defined as follows:

$$\begin{gathered} \mathrm{RE}\left(\mathsf{\eta }\right)={\displaystyle \frac{1}{1-\mathsf{\eta }}}\log \left[{\int }_0^{\infty }{\mathrm{f}}^{\mathsf{\eta }}\left(\mathrm{x}\right)\mathrm{dx}\right] \\ ={\displaystyle \frac{1}{1-\mathsf{\eta }}}\log \left[{\int }_0^{\infty }{{(-1)}^{\mathsf{\eta }}\mathsf{\alpha }}^{2\mathsf{\eta }} {\mathsf{\beta }}^{\mathsf{\eta }} {\mathsf{\lambda }}^{\mathsf{\eta }} {\mathrm{x}}^{-\mathsf{\eta } \mathsf{\beta } \mathsf{\lambda }-\mathsf{\eta }}{ \mathrm{e}}^{{\mathsf{\eta }\mathrm{x}}^{- \mathsf{\beta } \mathsf{\lambda }}}{({\mathrm{e}}^{{\mathrm{x}}^{- \mathsf{\beta } \mathsf{\lambda }}} - 1)}^{\mathsf{\eta }}\exp \left(\mathsf{\eta }\mathsf{\alpha }\right(1 -{ \mathrm{e}}^{{\mathrm{x}}^{- \mathsf{\beta } \mathsf{\lambda }}}\left)\right) \mathrm{dx}\right], \end{gathered}$$

By using Exponential function, then

$$\mathrm{e}\mathrm{xp}(\mathsf{\eta } \mathsf{\alpha } (1 - {\mathrm{e}}^{{\mathrm{x}}^{- \mathsf{\beta } \mathsf{\lambda }}}\left)\right)={\sum }_{\mathrm{j}=0}^{\infty }{\displaystyle \frac{{\mathsf{\eta }}^{\mathrm{j}}{\mathsf{\alpha }}^{\mathrm{j}}}{\mathrm{j}!}}{(1 -{ \mathrm{e}}^{{\mathrm{x}}^{- \mathsf{\beta } \mathsf{\lambda }}})}^{\mathrm{j}}.$$

Then,

$$\mathrm{RE}\left(\mathsf{\eta }\right)={\displaystyle \frac{1}{1-\mathsf{\eta }}}\log \left[{\int }_0^{\infty }\sum _{\mathrm{j}=0}^{\infty }{\displaystyle \frac{{\left(-1\right)}^{\mathsf{\eta }+\mathrm{j}}{\mathsf{\eta }}^{\mathrm{j}}{\mathsf{\alpha }}^{\mathrm{j}+2\mathsf{\eta }}}{\mathrm{j}!}}{\left(1 - {\mathrm{e}}^{{\mathrm{x}}^{- \mathsf{\beta } \mathsf{\lambda }}}\right)}^{\mathrm{j}} {\mathsf{\beta }}^{\mathsf{\eta }}{\mathsf{\lambda }}^{\mathsf{\eta }} {\mathrm{x}}^{-\mathsf{\eta } \mathsf{\beta } \mathsf{\lambda }-\mathsf{\eta }}{\mathrm{e}}^{{\mathsf{\eta }\mathrm{x}}^{- \mathsf{\beta } \mathsf{\lambda }}}{\left({\mathrm{e}}^{{\mathrm{x}}^{- \mathsf{\beta } \mathsf{\lambda }}} - 1\right)}^{\mathrm{j}+\mathsf{\eta }} \mathrm{dx}\right].$$

For any real number a,b > 0 and |ε| < 1, the generalized binomial series is defined as follows:

$${(\mathrm{a}+\mathrm{b})}^{\mathsf{\epsilon }}=\sum _{\mathrm{i}=0}^{\infty }\left(\begin{array}{c}\mathsf{\epsilon }\\ \mathrm{i}\end{array}\right){\mathrm{a}}^{\mathsf{\epsilon }-\mathrm{i}}{\mathrm{b}}^{\mathrm{i}}.$$

Then,

$$\mathrm{RE}\left(\mathsf{\eta }\right)={\displaystyle \frac{1}{1-\mathsf{\eta }}}\log \left[{\int }_0^{\infty }\sum _{\mathrm{j}=\mathrm{i}=0}^{\infty }\left(\begin{array}{c}\mathrm{j}+\mathsf{\eta }\\ \mathrm{i}\end{array}\right){\displaystyle \frac{{{(-1)}^{\mathrm{i}+\mathsf{\eta }}\mathsf{\eta }}^{\mathrm{j}}{\mathsf{\alpha }}^{\mathrm{j}+2\mathsf{\eta }}}{\mathrm{j}!}} {\mathsf{\beta }}^{\mathsf{\eta }}{\mathsf{\lambda }}^{\mathsf{\eta }} {\mathrm{x}}^{-\mathsf{\eta } \mathsf{\beta } \mathsf{\lambda }-\mathsf{\eta }}{\mathrm{e}}^{{(\mathrm{i}+\mathsf{\eta })\mathrm{x}}^{- \mathsf{\beta } \mathsf{\lambda }}}\mathrm{dx}\right],$$

where $\mathsf{\Omega }(\alpha ,\beta ,\lambda )=\left(\begin{array}{c}\mathrm{j}+\mathsf{\eta }\\ \mathrm{i}\end{array}\right){\displaystyle \frac{{{(-1)}^{\mathrm{i}+\mathsf{\eta }}\mathsf{\eta }}^{\mathrm{j}}{\mathsf{\alpha }}^{\mathrm{j}+2\mathsf{\eta }}}{\mathrm{j}!}} {\mathsf{\beta }}^{\mathsf{\eta }}{\mathsf{\lambda }}^{\mathsf{\eta }}$.

By using Gamma function, then the Renyi entropy is developed follows:

$$\mathrm{RE}\left(\mathsf{\eta }\right)={\displaystyle \frac{1}{1-\mathsf{\eta }}}\log \left[\sum _{\mathrm{j}=\mathrm{i}=0}^{\infty }\mathsf{\Omega }(\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }){\left({\displaystyle \frac{-1}{\mathrm{i}+\mathsf{\eta }}}\right)}^{\mathsf{\eta }+\mathsf{\eta }\mathsf{\beta }\mathsf{\lambda }} {\left(\mathsf{\eta }\mathsf{\beta }\right)}^{-1}\mathsf{\Gamma }\left({\displaystyle \frac{\mathsf{\eta }+\mathsf{\eta }\mathsf{\beta }\mathsf{\lambda }-1}{\mathsf{\beta }\mathsf{\lambda }}}\right)\right].$$

(19)

3.6. Tsallis Entropy

Tsallis [33] introduced entropy called Tsallis entropy for generalizing the standard statistical mechanics which is defined as

$$\mathrm{TE}\left(\mathsf{\rho }\right)={\displaystyle \frac{1}{1-\mathsf{\rho }}}\log \left[1-{\int }_0^{\infty }{\mathrm{f}}^{\mathsf{\rho }}\left(\mathrm{x}\right) \mathrm{dx}\right].$$

For any real number a,b > 0 and |ε| < 1, the generalized binomial series is defined as follows:

$${(\mathrm{a}+\mathrm{b})}^{\mathsf{\epsilon }}=\sum _{\mathrm{i}=0}^{\infty }\left(\begin{array}{c}\mathsf{\epsilon }\\ \mathrm{i}\end{array}\right){\mathrm{a}}^{\mathsf{\epsilon }-\mathrm{i}}{\mathrm{b}}^{\mathrm{i}}.$$

Therefore, Tsallis entropy is given as

$$\mathrm{TE}\left(\mathsf{\rho }\right)={\displaystyle \frac{1}{1-\mathsf{\rho }}}\log \left[1-\sum _{\mathrm{j}=\mathrm{i}=0}^{\infty }\Delta (\alpha ,\beta ,\lambda ){\left({\displaystyle \frac{-1}{i+\rho }}\right)}^{\rho +\rho \beta \lambda } {(\rho \beta )}^{-1}\mathsf{\Gamma }\left({\displaystyle \frac{\rho +\rho \beta \lambda -1}{\beta \lambda }}\right)\right],$$

(20)

where $\Delta (\alpha ,\beta ,\lambda )=\left(\begin{array}{c}\mathrm{j}+\mathsf{\rho }\\ \mathrm{i}\end{array}\right){\displaystyle \frac{{{(-1)}^{\mathrm{i}+\mathsf{\rho }}\mathsf{\rho }}^{\mathrm{j}}{\mathsf{\alpha }}^{\mathrm{j}+2\mathsf{\rho }}}{\mathrm{j}!}} {\mathsf{\beta }}^{\mathsf{\rho }}{\mathsf{\lambda }}^{\mathsf{\rho }}$.

3.7. Havrada–Charavat’s Entropy

Havrda–Charvát Entropy (also known as Tsallis entropy in a modified form) is a generalized entropy measure particularly for applications in information theory and statistical mechanics where classical Shannon entropy may not be adequate.

$$\mathrm{HCE}(\mathsf{\delta })={\displaystyle \frac{1}{1-\mathsf{\delta }}}\log \left[{\int }_0^{\infty }{\mathrm{f}}^{\mathsf{\delta }}\left(\mathrm{x}\right) \mathrm{dx}-1\right].$$

For any real number a,b > 0 and |ε| < 1, the generalized binomial series is defined as follows:

$${(\mathrm{a}+\mathrm{b})}^{\mathsf{\epsilon }}=\sum _{\mathrm{i}=0}^{\infty }\left(\begin{array}{c}\mathsf{\epsilon }\\ \mathrm{i}\end{array}\right){\mathrm{a}}^{\mathsf{\epsilon }-\mathrm{i}}{\mathrm{b}}^{\mathrm{i}}.$$

Therefore, Havrada–Charavat’s entropy is given as

$$\mathrm{HCE}(\mathsf{\delta })={\displaystyle \frac{1}{1-\mathsf{\delta }}}\log \left[\sum _{\mathrm{j}=\mathrm{i}=0}^{\infty }\Delta (\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }){\left({\displaystyle \frac{-1}{\mathrm{i}+\mathsf{\delta }}}\right)}^{\mathsf{\rho }+\mathsf{\rho }\mathsf{\beta }\mathsf{\lambda }} {(\mathsf{\delta }\mathsf{\beta })}^{-1}\mathsf{\Gamma }\left({\displaystyle \frac{\mathsf{\delta }+\mathsf{\rho }\mathsf{\beta }\mathsf{\lambda }-1}{\mathsf{\beta }\mathsf{\lambda }}}\right)-1\right],$$

(21)

In the following derivations, interchanges between infinite summation and integration are justified under the Dominated Convergence Theorem (DCT) and Fubini–Tonelli Theorem, since the terms of the expanded series are nonnegative and the integrands are absolutely integrable for all α, β, λ > 0. Therefore, the manipulations involving series expansions and integrations in Section 3.2, Section 3.3, Section 3.4 and Section 3.5 are mathematically valid.

To confirm the accuracy and stability of the derived expressions for the moments and entropies, we compared the analytical values obtained from the closed-form equations with numerical integration and Monte Carlo simulation results.

For several combinations of $\alpha , \beta , \lambda > 0$, the analytical and numerical results agreed up to a relative error less than 10⁻⁵, demonstrating that the formulas are numerically reliable and stable for computational applications.

3.8. Order Statistics

Assume that $X_1, X_2, \dots ,X_n$ is a random sample of size $n$ drawn from the IPEC distribution with PDF and CDF as defined in Equations (6) and (7), respectively. Next, order statistics are indicated by $X_{\left(1\right)}, X_{\left(2\right)}, \dots ,X_{\left(n\right)};$ where $X_{\left(1\right) } = \mathit{min}(X_{\left(1\right)}, X_{\left(2\right)}, \dots ,X_{\left(n\right)})$ and $X_{(n) }=\mathit{max}(X_{\left(1\right)}, X_{\left(2\right)}, \dots ,X_{(n) })$.

The ${\kappa }^{\mathrm{th}}$ order statistics of PDF is obtained as

$$\begin{gathered} {\mathrm{f}}_{{\mathrm{x}}_{\left(\mathsf{\kappa }\right)}}(\mathrm{x};\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda })={\displaystyle \frac{\mathrm{n}!}{(\mathsf{\kappa }-1)!(\mathrm{n}-\mathsf{\kappa })!}}\mathrm{f}(\mathrm{x};\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }){\left[\mathrm{F}(\mathrm{x};\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda })\right]}^{\mathsf{\kappa }-1}{\left[1-\mathrm{F}(\mathrm{x};\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda })\right]}^{\mathrm{n}-\mathsf{\kappa }} \\ ={\displaystyle \frac{n!}{\left(\kappa -1\right)!\left(n-\kappa \right)!}}\sum _{i=0}^{n-\kappa }\left(\begin{array}{c}n-\kappa \\ i\end{array}\right){\left(-1\right)}^{i}f\left(x;\alpha ,\beta ,\lambda \right){\left[F\left(x;\alpha ,\beta ,\lambda \right)\right]}^{\kappa +i-1} \end{gathered}$$

(22)

If k = n in Equation (22) the pd f of the nth order statistic Y(n) for the IPEC distribution as

$${\mathrm{f}}_{{\mathrm{x}}_{\left(\mathrm{n}\right)}}(\mathrm{x};\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda })={\displaystyle \frac{\mathrm{n}!}{\left(\mathrm{n}-1\right)!}}{\left(-1\right)}^{\mathrm{i}}\mathrm{f}\left(\mathrm{x};\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right){\left[\mathrm{F}\left(\mathrm{x};\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)\right]}^{\mathrm{n}+\mathrm{i}-1}.$$

For k = 1 in Equation (22), the pd f of the first order statistic Y(1), for the IPEC distribution as

$${\mathrm{f}}_{{\mathrm{x}}_{\left(1\right)}}(\mathrm{x};\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda })={\displaystyle \frac{\mathrm{n}!}{\left(\mathrm{n}-1\right)!}}\sum _{\mathrm{i}=0}^{\mathrm{n}-1}\left(\begin{array}{c}\mathrm{n}-1\\ \mathrm{i}\end{array}\right){\left(-1\right)}^{\mathrm{i}}\mathrm{f}\left(\mathrm{x};\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right){\left[\mathrm{F}\left(\mathrm{x};\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)\right]}^{\mathrm{i}}.$$

4. Methods of Estimation

This part discusses estimation of the parameters of the IPEC model by using seven traditional techniques, including maximum likelihood estimators (MLE), Anderson–Darling estimators (ADE), right-tail Anderson–Darling estimators (RADE), Cramér–von Mises estimators (CME), percentile estimators (PE), least-squares estimators (LSE), and weighted least-squares estimators (WLSE). These estimation approaches entail optimizing an objective function to identify the most suitable estimator, whether by maximization or minimization additional information regarding the employed estimating methods, refer to [34,35].

4.1. Maximum Likelihood Method

Let $x_1, x_2,\dots , x_n$ be a random sample of size $n$ from the IPEC distribution with PDF in Equation (6), thus the likelihood function L is specified by the following:

$$\mathrm{L}={\prod }_{\mathrm{i}=1}^{\mathrm{n}}\mathrm{f}\left({\mathrm{x}}_{\mathrm{i}};\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)={\prod }_{\mathrm{I}=1}^{\mathrm{n}}{\mathsf{\alpha }}^2 \mathsf{\beta } \mathsf{\lambda } {{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }-1}{ \mathrm{e}}^{{{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}}\left({\mathrm{e}}^{{{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}} -1\right)\exp \left(\mathsf{\alpha }\left(1 -{ \mathrm{e}}^{{{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}}\right)\right).$$

(23)

Therefore, the log likelihood ($l$) function can be expressed as follows:

$$\begin{gathered} \mathrm{l}=\sum _{\mathrm{i}=1}^{\mathrm{n}}\ln \left(\mathrm{f}\left({\mathrm{x}}_{\mathrm{i}};\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)\right)=\sum _{\mathrm{i}=1}^{\mathrm{n}}\ln \left({\mathsf{\alpha }}^2\mathsf{\beta }\mathsf{\lambda }{{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }-1}{\mathrm{e}}^{{{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}}\left({\mathrm{e}}^{{{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}}-1\right)\exp \left(\mathsf{\alpha }\left(1-{\mathrm{e}}^{{{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}}\right)\right)\right), \\ =\sum _{\mathrm{i}=1}^{\mathrm{n}}2\ln \left(\mathsf{\alpha }\right)+\ln \left(\mathsf{\beta }\right)+\ln \left(\mathsf{\lambda }\right)-\left(\mathsf{\lambda }\mathsf{\beta }+1\right)\ln \left({\mathrm{x}}_{\mathrm{i}}\right)+\ln \left({\mathrm{e}}^{{{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}}-1\right)+\mathsf{\alpha }\left(1-{\mathrm{e}}^{{{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}}\right)+{{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}, \\ =2\mathrm{n}\ln \left(\mathsf{\alpha }\right)+\mathrm{n}\ln \left(\mathsf{\beta }\right)+\mathrm{n}\ln \left(\mathsf{\lambda }\right)-\left(\mathsf{\lambda }\mathsf{\beta }+1\right)\sum _{\mathrm{i}=1}^{\mathrm{n}}\ln \left({\mathrm{x}}_{\mathrm{i}}\right)+\sum _{\mathrm{i}=1}^{\mathrm{n}}\ln \left({\mathrm{e}}^{{{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}}-1\right)+\mathsf{\alpha }\sum _{\mathrm{i}=1}^{\mathrm{n}}\left(1-{\mathrm{e}}^{{{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}}\right)+\sum _{\mathrm{i}=1}^{\mathrm{n}}\left({{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}\right). \end{gathered}$$

By differentiating the function $\left(l\right)$ about the parameters α, β and λ, respectively, we can ascertain:

$${\displaystyle \frac{\partial \mathrm{l}}{\partial \mathsf{\alpha }}}={\displaystyle \frac{2\mathrm{n}}{\mathsf{\alpha }}}+\sum _{\mathrm{i}=1}^{\mathrm{n}}\left(1-{\mathrm{e}}^{{{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}}\right),$$

(24)

$${\displaystyle \frac{\partial \mathrm{l}}{\partial \mathsf{\beta }}}={\displaystyle \frac{\mathrm{n}}{\mathsf{\beta }}}-\mathsf{\lambda }\sum _{\mathrm{i}=1}^{\mathrm{n}}\ln \left({\mathrm{x}}_{\mathrm{i}}\right)-\sum _{\mathrm{i}=1}^{\mathrm{n}}{\displaystyle \frac{\mathsf{\lambda } {{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}{\mathrm{e}}^{{{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}}\ln \left({\mathrm{x}}_{\mathrm{i}}\right)}{\left({\mathrm{e}}^{{{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}}-1\right)}}+\mathsf{\alpha }\sum _{\mathrm{i}=1}^{\mathrm{n}}\mathsf{\lambda }{ {\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}{\mathrm{e}}^{{{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}}\ln \left({\mathrm{x}}_{\mathrm{i}}\right)-\sum _{\mathrm{i}=1}^{\mathrm{n}}\mathsf{\lambda }{{ \mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}\ln \left({\mathrm{x}}_{\mathrm{i}}\right),$$

(25)

$${\displaystyle \frac{\partial \mathrm{l}}{\partial \mathsf{\lambda }}}={\displaystyle \frac{\mathrm{n}}{\mathsf{\lambda }}}-\mathsf{\beta }\sum _{\mathrm{i}=1}^{\mathrm{n}}\ln \left({\mathrm{x}}_{\mathrm{i}}\right)-\sum _{\mathrm{i}=1}^{\mathrm{n}}{\displaystyle \frac{\mathsf{\beta } {{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}{\mathrm{e}}^{{{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}}\ln \left({\mathrm{x}}_{\mathrm{i}}\right)}{\left({\mathrm{e}}^{{{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}}-1\right)}}+\mathsf{\alpha }\sum _{\mathrm{i}=1}^{\mathrm{n}}\mathsf{\beta } {{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}{\mathrm{e}}^{{{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}}\ln \left({\mathrm{x}}_{\mathrm{i}}\right)-\sum _{\mathrm{i}=1}^{\mathrm{n}}\mathsf{\beta }{{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}\ln \left({\mathrm{x}}_{\mathrm{i}}\right),$$

(26)

By equating Equations (24)–(26) to zero, we observe the absence of explicit solutions; therefore, we must utilize nonlinear numerical methods to derive the ML estimators of the parameters of the IPEC distribution.

Under regularity conditions, these estimators are consistent, asymptotically unbiased, and normally distributed as $n\to \infty$.

Similar theoretical properties apply to the alternative estimation approaches (ADE, RADE, MPS, CME, LSE, and WLSE), whose objective functions are defined analogously and optimized numerically using standard algorithms.

4.2. Cramer–Von–Mises Method

The CM estimators for the IPEC parameters α, β, and λ are determined by minimizing the next formula:

$$\begin{gathered} \mathrm{CM}\left(\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)={\displaystyle \frac{1}{12\mathrm{n}}}+\sum _{\mathrm{i}=1}^{\mathrm{n}}{\left[\mathrm{F}\left({\mathrm{x}}_{\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)-{\displaystyle \frac{2\mathrm{i}-1}{2\mathrm{n}}}\right]}^2 \\ ={\displaystyle \frac{1}{12\mathrm{n}}}+\sum _{\mathrm{i}=1}^{\mathrm{n}}{\left[\left[1-\mathsf{\alpha }\left(1-{\mathrm{e}}^{{{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}}\right)\right]\left[\exp \left(\mathsf{\alpha }\left(1-{\mathrm{e}}^{{{\mathrm{x}}_{\mathrm{i}}}^{-\mathsf{\lambda }\mathsf{\beta }}}\right)\right)\right]-{\displaystyle \frac{2\mathrm{i}-1}{2\mathrm{n}}}\right]}^2. \end{gathered}$$

Or by solving the following nonlinear Equations with respect to α, β, and λ:

$$\sum _{\mathrm{i}=1}^{\mathrm{n}}\left[\mathrm{F}\left({\mathrm{x}}_{\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)-{\displaystyle \frac{2\mathrm{i}-1}{2\mathrm{n}}}\right] {∆}_1\left({\mathrm{x}}_{\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)=0 ,$$

$$\sum _{\mathrm{i}=1}^{\mathrm{n}}\left[\mathrm{F}\left({\mathrm{x}}_{\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)-{\displaystyle \frac{2\mathrm{i}-1}{2\mathrm{n}}}\right] {∆}_2\left({\mathrm{x}}_{\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)=0 ,$$

$$\sum _{\mathrm{i}=1}^{\mathrm{n}}\left[\mathrm{F}\left({\mathrm{x}}_{\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)-{\displaystyle \frac{2\mathrm{i}-1}{2\mathrm{n}}}\right] {∆}_3\left({\mathrm{x}}_{\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)=0 ,$$

where ${∆}_1\left(.|\alpha ,\beta ,\lambda \right)$, ${∆}_2\left(.|\alpha ,\beta ,\lambda \right)$, and ${∆}_3\left(.|\alpha ,\beta ,\lambda \right)$ are the first derivatives of the CDF of the IPEC model.

4.3. Anderson–Darling and Right-Tail Anderson–Darling Methods

Finding sample distributions that deviate from the normality without the need for statistical testing was suggested by Anderson and Darling (AD) [36]. The AD estimates ${\widehat{\alpha }}_{\mathit{ADE}}$, ${\widehat{\beta }}_{\mathit{ADE}}$ and ${\widehat{\lambda }}_{\mathit{ADE}}$ of the parameters α, β, and λ determined by minimizing, with respect to α, β, and λ the function:

$$\mathrm{AD}\left({\mathrm{x}}_{\mathrm{i}};\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)=-\mathrm{n}-{\displaystyle \frac{1}{\mathrm{n}}}\sum _{\mathrm{i}=1}^{\mathrm{n}}\left(2\mathrm{i}-1\right)\left[\log \mathrm{F}\left({\mathrm{x}}_{\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)+\log \overline{\mathrm{F}}\left({\mathrm{x}}_{\mathrm{n}+1-\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)\right].$$

These estimates can also be derived by resolving the nonlinear equations:

$$\sum _{\mathrm{i}=1}^{\mathrm{n}}\left(2\mathrm{i}-1\right)\left[{\displaystyle \frac{{∆}_1\left({\mathrm{x}}_{\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}{\mathrm{F}\left({\mathrm{x}}_{\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}}-{\displaystyle \frac{{∆}_1\left({\mathrm{x}}_{\mathrm{n}+1-\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}{\overline{\mathrm{F}}\left({\mathrm{x}}_{\mathrm{n}+1-\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}}\right]=0 ,$$

$$\sum _{\mathrm{i}=1}^{\mathrm{n}}\left(2\mathrm{i}-1\right)\left[{\displaystyle \frac{{∆}_2\left({\mathrm{x}}_{\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}{\mathrm{F}\left({\mathrm{x}}_{\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}}-{\displaystyle \frac{{∆}_2\left({\mathrm{x}}_{\mathrm{n}+1-\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}{\overline{\mathrm{F}}\left({\mathrm{x}}_{\mathrm{n}+1-\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}}\right]=0 ,$$

and

$$\sum _{\mathrm{i}=1}^{\mathrm{n}}\left(2\mathrm{i}-1\right)\left[{\displaystyle \frac{{∆}_3\left({\mathrm{x}}_{\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}{\mathrm{F}\left({\mathrm{x}}_{\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}}-{\displaystyle \frac{{∆}_3\left({\mathrm{x}}_{\mathrm{n}+1-\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}{\overline{\mathrm{F}}\left({\mathrm{x}}_{\mathrm{n}+1-\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}}\right]=0 .$$

The right-tail Anderson–Darling estimators ${\widehat{\alpha }}_{\mathit{RTADE}}$, ${\widehat{\beta }}_{\mathit{RTADE}}$ and ${\widehat{\lambda }}_{\mathit{RTADE}}$ of the parameters α, β, and λ are obtained by minimizing, with respect to α, β, and λ, the function:

$$\mathrm{RTAD}\left(\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)={\displaystyle \frac{\mathrm{n}}{2}}-2\sum _{\mathrm{i}=1}^{\mathrm{n}}\mathrm{F}\left({\mathrm{x}}_{\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)-{\displaystyle \frac{1}{\mathrm{n}}}\sum _{\mathrm{i}=1}^{\mathrm{n}}\left(2\mathrm{i}-1\right)\log \overline{\mathrm{F}}\left({\mathrm{x}}_{\mathrm{n}+1-\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right).$$

These estimators can also be obtained by solving the nonlinear equations:

$$-2\sum _{\mathrm{i}=1}^{\mathrm{n}}{\displaystyle \frac{{∆}_1\left({\mathrm{x}}_{\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}{\mathrm{F}\left({\mathrm{x}}_{\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}}+{\displaystyle \frac{1}{\mathrm{n}}}\sum _{\mathrm{i}=1}^{\mathrm{n}}\left(2\mathrm{i}-1\right){\displaystyle \frac{{∆}_1\left({\mathrm{x}}_{\mathrm{n}+1-\mathrm{i}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}{\overline{\mathrm{F}}\left({\mathrm{x}}_{\mathrm{n}+1-\mathrm{i}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}}=0,$$

$$-2\sum _{\mathrm{i}=1}^{\mathrm{n}}{\displaystyle \frac{{∆}_2\left({\mathrm{x}}_{\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}{\mathrm{F}\left({\mathrm{x}}_{\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}}+{\displaystyle \frac{1}{\mathrm{n}}}\sum _{\mathrm{i}=1}^{\mathrm{n}}\left(2\mathrm{i}-1\right){\displaystyle \frac{{∆}_2\left({\mathrm{x}}_{\mathrm{n}+1-\mathrm{i}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}{\overline{\mathrm{F}}\left({\mathrm{x}}_{\mathrm{n}+1-\mathrm{i}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}}=0,$$

and

$$-2\sum _{\mathrm{i}=1}^{\mathrm{n}}{\displaystyle \frac{{∆}_3\left({\mathrm{x}}_{\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}{\mathrm{F}\left({\mathrm{x}}_{\mathrm{i}:\mathrm{n}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}}+{\displaystyle \frac{1}{\mathrm{n}}}\sum _{\mathrm{i}=1}^{\mathrm{n}}\left(2\mathrm{i}-1\right){\displaystyle \frac{{∆}_3\left({\mathrm{x}}_{\mathrm{n}+1-\mathrm{i}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}{\overline{\mathrm{F}}\left({\mathrm{x}}_{\mathrm{n}+1-\mathrm{i}}|\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)}}=0.$$

4.4. Percentile Method

Utilizing a closed-form distribution function allows for the estimation of a distribution parameter by plotting a linear representation against the percentile points. This approach for determining the parameters of the Weibull distribution was proposed by Kao [37] and Kao [38]. Given $n$ random samples $x_1, x_2,\dots , x_n$ from the distribution function, where $x_{\left(k\right)}<\dots <x_{\left(n\right)}$ denotes ordered samples, then the estimates of the parameters α, β, and λ can be derived by minimizing the following formula:

$$\mathrm{P}\left(\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)=\sum _{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{x}}_{\left(\mathrm{i}\right)}-\mathrm{QF}\right)}^2 ,$$

(27)

where ${\mathrm{u}}_{\mathrm{i}}=\mathrm{i}/(\mathrm{n}+1)$ is an unbiased estimator of $\mathrm{F}\left({\mathrm{x}}_{\left(\mathrm{i}\right)};\mathsf{\alpha },\mathsf{\beta },\mathsf{\lambda }\right)$, and the percentile estimates can be derived by differentiating Equation (27) with respect to α, β, and λ, respectively, and equating the resulting equations to zero, we can obtain the values of estimated parameters.

4.5. Least-Squares and Weighted Least-Squares Methods

Swain et al. [39] proposed utilizing the LS and WLS estimates to ascertain the characteristics of the beta distribution. This study utilizes the same methods to analyze the IPEC distribution. The LS estimates of the parameters α, β, and λ of the IPEC distribution, denoted as ${\widehat{\alpha }}_{\mathit{LSE}}$, ${\widehat{\beta }}_{\mathit{LSE}}$ and ${\widehat{\lambda }}_{\mathit{LSE}}$ corresponding, can be derived by minimizing the following formula:

$$\sum _{\mathrm{j}=1}^{\mathrm{n}}{\left[\mathrm{F}\left({\mathrm{X}}_{\left(\mathrm{j}\right)}\right)-{\displaystyle \frac{\mathrm{j}}{\mathrm{n}+1}}\right]}^2= \sum _{\mathrm{j}=1}^{\mathrm{n}}{\left[\left[1-\mathsf{\alpha }\left(1-{\mathrm{e}}^{{{\mathrm{x}}_{\mathrm{j}}}^{-\mathsf{\lambda }\mathsf{\beta }}}\right)\right]\left[\exp \left(\mathsf{\alpha }\left(1-{\mathrm{e}}^{{{\mathrm{x}}_{\mathrm{j}}}^{-\mathsf{\lambda }\mathsf{\beta }}}\right)\right)\right]-{\displaystyle \frac{\mathrm{j}}{\mathrm{n}+1}}\right]}^2,$$

with respect to α, β, and λ.

The WLS estimates of the unknown parameters can be derived by minimizing the following function:

$$\sum _{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{w}}_{\mathrm{j}}{\left[\mathrm{F}\left({\mathrm{X}}_{\left(\mathrm{j}\right)}\right)-{\displaystyle \frac{\mathrm{j}}{\mathrm{n}+1}}\right]}^2,$$

(28)

with respect to α, β, and λ. The weights $w_j$ are equal to ${\displaystyle \frac{1}{V\left(X_{\left(j\right)}\right)}}={\displaystyle \frac{{\left(n+1\right)}^2\left(n+2\right)}{j\left(n-j+1\right)}}$. Thus, in this case, the WLS estimators of α, β, and λ can be obtained by minimizing the following formula:

$$\sum _{\mathrm{j}=1}^{\mathrm{n}}{\displaystyle \frac{{\left(\mathrm{n}+1\right)}^2\left(\mathrm{n}+2\right)}{\mathrm{n}-\mathrm{j}+1}} {\left[\left[1-\mathsf{\alpha }\left(1-{\mathrm{e}}^{{{\mathrm{x}}_{\mathrm{j}}}^{-\mathsf{\lambda }\mathsf{\beta }}}\right)\right]\left[\exp \left(\mathsf{\alpha }\left(1-{\mathrm{e}}^{{{\mathrm{x}}_{\mathrm{j}}}^{-\mathsf{\lambda }\mathsf{\beta }}}\right)\right)\right]-{\displaystyle \frac{\mathrm{j}}{\mathrm{n}+1}}\right]}^2,$$

with respect to α, β, and λ.

5. Simulation Analysis

In this section, we perform a Monte Carlo simulation analysis to validate the conclusions presented in the previous sections. The inverse of the Equation CDF was employed to generate random numbers for the simulation investigation. To achieve our goals, samples of sizes 25, 50, 75, and 100 were generated, considering the three contingencies outlined below. All calculations were conducted using the R programming language. The subsequent technique was employed.

• The quantile function defined in Equation (10) was employed to generate random numbers for the simulation investigation.
• The results of the initial phase are utilized to compute the estimations $\widehat{\mathsf{\alpha }}$, $\widehat{\mathsf{\beta }},$ and $\widehat{\mathsf{\lambda }}$ for the parameters α, β, and λ, respectively.
• Execute steps 1 to 3, $N$ times to represent diverse samples, where $N$ equals 1000.
• Given $\widehat{\omega }=\left(\widehat{\alpha },\widehat{ \beta }, \widehat{\lambda }\right)$, and $\omega =\left(\alpha , \beta , \lambda \right)$ compute the MLE, LSE, WLSE, CME, ADE, RADE, and PE estimates, relative biases, and mean square errors (MSEs).
• To ensure reproducibility of the numerical analysis, all simulations were conducted using standardized procedures based on the pseudo-code presented in Appendix A.

For each parameter combination, N = 1000 replications were performed with sample sizes n = 25, 50, 75, and 100.

The bias and mean squared error (MSE) of each estimator were computed. The results reported in

Table 3. Simulation results of the IPEC model for α = 0.6, β = 0.5, and λ = 0.4.

			For α			For β			For λ
n	Methods	Average	MSE	RBIAS	Average	MSE	RBIAS	Average	MSE	RBIAS	⅀Ranks
25	MLE	0.5905	0.01432	0.01585	0.5286	0.00291	0.05726	0.3954	0.00211	0.01152	171
	LSE	0.6124	0.01514.5	0.02076	0.5196	0.00525	0.03911	0.3839	0.00386	0.04006	28.56
	WLSE	0.6091	0.01453	0.01514	0.5213	0.00493	0.04272	0.3853	0.00333	0.03675	204
	CME	0.5948	0.01596	0.00872	0.5285	0.00514	0.05705	0.3969	0.00365	0.00751	235
	ADE	0.6058	0.01411	0.00973	0.5267	0.00606	0.05333	0.3856	0.00302	0.03604	193
	RADE	0.6030	0.01514.5	0.00501	0.5271	0.00312	0.05424	0.3912	0.00354	0.02213	18.52
	PE	1.0094	1.48237	0.68237	0.6234	0.45557	0.24677	0.5352	0.37407	0.33817	427
50	MLE	0.5851	0.00692	0.02465	0.5273	0.00181	0.05464	0.3919	0.00141	0.02022	151
	LSE	0.5941	0.00735	0.00991	0.5259	0.00255	0.05181	0.3846	0.00206	0.03846	245.5
	WLSE	0.5912	0.00681	0.01473	0.5292	0.00233	0.05846	0.3841	0.00163	0.03755	213
	CME	0.5851	0.00766	0.02486	0.5264	0.00244	0.05292	0.3939	0.00195	0.01511	245.5
	ADE	0.5913	0.00714	0.01452	0.5282	0.00366	0.05635	0.3865	0.00152	0.03374	234
	RADE	0.5897	0.00703	0.01714	0.5272	0.00212	0.05443	0.3895	0.00184	0.02623	192
	PE	1.0678	1.38527	0.77977	0.6826	1.32127	0.36527	0.5953	0.82607	0.48837	427
75	MLE	0.5965	0.00431	0.00586	0.5244	0.00131	0.04885	0.3874	0.00081	0.03142	161
	LSE	0.6029	0.00495	0.00505	0.5209	0.00215	0.04171	0.3847	0.00156	0.03815	276
	WLSE	0.6006	0.00463.5	0.00102	0.5218	0.00173	0.04362	0.3857	0.00113	0.03564	17.52
	CME	0.5970	0.00506	0.00494	0.5234	0.00256	0.04693	0.3897	0.00145	0.02581	255
	ADE	0.6012	0.00452	0.00203	0.5245	0.00173	0.04896	0.3833	0.00102	0.04186	224
	RADE	0.6003	0.00463.5	0.00051	0.5243	0.00173	0.04864	0.3859	0.00134	0.03503	18.53
	PE	1.1043	1.81817	0.84067	0.6313	0.20717	0.26267	0.4985	0.34837	0.24637	427
100	MLE	0.5949	0.00341	0.00846	0.5232	0.00101	0.04651	0.3866	0.00071	0.03351	111
	LSE	0.6001	0.00405	0.00021	0.5245	0.00155.5	0.04892	0.3817	0.00115.5	0.04575	245
	WLSE	0.5981	0.00373	0.00313	0.5254	0.00143	0.05085	0.3825	0.00093	0.04374	213
	CME	0.5956	0.00416	0.00735	0.5251	0.00155.5	0.05024	0.3863	0.00115.5	0.03432	286
	ADE	0.5986	0.00362	0.00242	0.5264	0.00143	0.05286	0.3813	0.00082	0.04676	213
	RADE	0.5972	0.00384	0.00474	0.5245	0.00143	0.04913	0.3858	0.00104	0.03563	213
	PE	1.2434	2.08927	1.07247	0.6216	0.25547	0.24337	0.5730	0.47037	0.43267	427

,

Table 4. Simulation results of the IPEC model for α = 0.8, β = 0.6, and λ = 0.5.

			For α			For β			For λ
n	Methods	Average	MSE	RBIAS	Average	MSE	RBIAS	Average	MSE	RBIAS	⅀Ranks
25	MLE	0.7767	0.02566	0.02916	0.6506	0.00532.5	0.08436	0.4966	0.00531	0.00681	22.54.5
	LSE	0.7968	0.02252	0.00401	0.6468	0.00564	0.07794	0.4670	0.00715	0.06596	223
	WLSE	0.7933	0.02241	0.00832	0.6333	0.00441	0.00651	0.4848	0.00654	0.03044	131
	CME	0.7856	0.02494	0.01805	0.6424	0.00532.5	0.07062	0.4964	0.00623	0.00712	18.52
	ADE	0.7909	0.02333	0.01144	0.6461	0.00585.5	0.07683	0.4797	0.00562	0.04065	22.54.5
	RADE	0.7925	0.02505	0.00943	0.6484	0.00585.5	0.08075	0.4903	0.00846	0.01953	27.56
	PE	0.8682	0.24227	0.08537	0.6745	0.16187	0.12427	0.5482	0.11737	0.09657	427
50	MLE	0.7972	0.01013	0.00354	0.6344	0.00271	0.05733	0.4858	0.00241	0.02841	131
	LSE	0.8044	0.01034.5	0.00556	0.6350	0.00355.5	0.05844	0.4725	0.00435	0.05516	316
	WLSE	0.8028	0.00991	0.00343	0.6324	0.00333	0.05391	0.4780	0.00343	0.04395	163
	CME	0.7987	0.01086	0.00161	0.6373	0.00355.5	0.06226	0.4839	0.00384	0.03222	24.54
	ADE	0.8027	0.01002	0.00332	0.6325	0.00302	0.05412	0.4784	0.00302	0.04334	142
	RADE	0.8033	0.01034.5	0.00425	0.6353	0.00344	0.05895	0.4815	0.00456	0.03693	27.55
	PE	1.0527	0.74397	0.31597	0.6862	0.19487	0.14377	0.5491	0.19867	0.09827	427
75	MLE	0.7966	0.00643.5	0.00426	0.6331	0.00221	0.05523	0.4828	0.00181	0.03431	15.52.5
	LSE	0.8014	0.00655	0.00184	0.6342	0.00296	0.05694	0.4735	0.00305.5	0.05306	30.56
	WLSE	0.7999	0.00631.5	0.00011	0.6323	0.00263	0.05392	0.4777	0.00243	0.04465	15.52.5
	CME	0.7976	0.00676	0.00305	0.6362	0.00285	0.06036	0.4807	0.00284	0.03862	285
	ADE	0.8003	0.00631.5	0.00042	0.6319	0.00242	0.05311	0.4778	0.00212	0.04434	12.51
	RADE	0.8007	0.00643.5	0.00093	0.6345	0.00274	0.05765	0.4787	0.00305.5	0.04253	244
	PE	1.1215	0.91217	0.40197	0.6789	0.20597	0.13167	0.5646	0.16027	0.12927	427
100	MLE	0.7972	0.00481	0.00356	0.6311	0.00181	0.05184	0.4813	0.00131	0.03732.5	15.51.5
	LSE	0.8005	0.00525	0.00073	0.6325	0.00245	0.05415	0.4758	0.00225.5	0.04826	29.55.5
	WLSE	0.7992	0.00503.5	0.00104	0.6304	0.00213	0.05072	0.4793	0.00173	0.04154	19.54
	CME	0.7976	0.00536	0.00295	0.6342	0.00266	0.05716	0.4813	0.00214	0.03732.5	29.55.5
	ADE	0.7998	0.00492	0.00031.5	0.6309	0.00192	0.05173	0.4777	0.00152	0.04445	15.51.5
	RADE	0.7997	0.00503.5	0.00031.5	0.6304	0.00234	0.05061	0.4826	0.00225.5	0.03481	16.53
	PE	1.1409	0.89347	0.42617	0.6964	0.15637	0.16077	0.5400	0.12807	0.08017	427

,

Table 5. Simulation results of the IPEC model for α = 1, β = 0.75, and λ = 0.6.

			For α			For β			For λ
n	Methods	Average	MSE	RBIAS	Average	MSE	RBIAS	Average	MSE	RBIAS	⅀Ranks
25	MLE	1.0534	0.02966	0.05346	0.7891	0.00442	0.05213	0.6031	0.00531	0.00521	193
	LSE	1.0462	0.02451.5	0.04621	0.8050	0.00964	0.07335	0.5510	0.01096	0.08166	23.54
	WLSE	1.0463	0.02451.5	0.04632	0.7826	0.00533	0.04342	0.5695	0.00773	0.05096	17.52
	CME	1.0482	0.02815	0.04823	0.8182	0.01756	0.09096	0.5780	0.01065	0.03663	286
	ADE	1.0514	0.02734	0.05144	0.7819	0.00291	0.04271	0.5824	0.00572	0.02942	141
	RADE	1.0525	0.02693	0.05255	0.7949	0.01195	0.05994	0.5774	0.00924	0.03774	255
	PE	1.0511	0.20517	0.05117	0.9492	0.13577	0.26567	0.4465	0.08147	0.25597	427
50	MLE	1.0094	0.01344	0.00941	0.7885	0.00391	0.05133.5	0.5776	0.00371	0.03731	11.51
	LSE	1.0117	0.01313	0.01175	0.7938	0.00575	0.05855	0.5570	0.00725	0.07167	306
	WLSE	1.0115	0.01291	0.01153	0.7885	0.00494	0.05133.5	0.5656	0.00563	0.05746	20.53
	CME	1.0114	0.01395	0.01142	0.7949	0.00586	0.05996	0.5729	0.00624	0.04523	265
	ADE	1.0116	0.01302	0.01164	0.7855	0.00402	0.04732	0.5692	0.00492	0.05135	172
	RADE	1.0151	0.01426	0.01516	0.7841	0.00463	0.04551	0.5774	0.00776	0.03762	244
	PE	1.0626	0.33227	0.06267	0.8553	0.27637	0.14057	0.5719	0.10767	0.04694	397
75	MLE	1.0053	0.00842	0.00531	0.7871	0.00281	0.04942	0.5786	0.00281	0.03562	91
	LSE	1.0059	0.00874	0.00593	0.7905	0.00414	0.05404	0.5656	0.00485.5	0.05747	27.55.5
	WLSE	1.0061	0.00842	0.00614	0.7908	0.00373	0.05445	0.5678	0.00373	0.05376	233
	CME	1.0055	0.00915.5	0.00552	0.7944	0.00476	0.05926	0.5741	0.00444	0.04324	27.55.5
	ADE	1.0066	0.00842	0.00665	0.7875	0.00312	0.05003	0.5701	0.00302	0.04985	192
	RADE	1.0087	0.00915.5	0.00876	0.7861	0.00455	0.04821	0.5751	0.00485.5	0.04143	264
	PE	1.0817	0.36007	0.08177	0.8481	0.23057	0.13087	0.5817	0.11697	0.03051	367
100	MLE	1.0150	0.00301	0.01501	0.7829	0.00151	0.04393	0.5971	0.00181	0.00481	81
	LSE	1.0193	0.00375	0.01935	0.7917	0.00346	0.05567	0.5746	0.00303	0.04237	336
	WLSE	1.0179	0.00312	0.01792	0.7823	0.00274	0.04312	0.5871	0.00314	0.02145	192
	CME	1.0191	0.00386	0.01914	0.7858	0.00295	0.04775.5	0.5881	0.00325.5	0.01983	295
	ADE	1.0188	0.00323	0.01883	0.7837	0.00213	0.04494	0.5858	0.00252	0.02376	214
	RADE	1.0216	0.00364	0.02166	0.7784	0.00182	0.03791	0.5941	0.00325.5	0.00982	20.53
	PE	1.1058	0.32707	0.10587	0.7858	0.07477	0.04775.5	0.5872	0.05197	0.02134	37.57

,

Table 6. Simulation results of the IPEC model for α = 1.2, β = 0.8, and λ = 0.5.

			For α			For β			For λ
n	Methods	Average	MSE	RBIAS	Average	MSE	RBIAS	Average	MSE	RBIAS	⅀Ranks
25	MLE	1.3842	0.06695	0.15356	0.7789	0.00482	0.02634	0.6119	0.05936	0.22387	306.5
	LSE	1.3559	0.05032	0.12993	0.8343	0.00291	0.04295	0.5068	0.03155	0.01372	181.5
	WLSE	1.3547	0.04851	0.12892	0.7591	0.02696	0.05116	0.5542	0.00991	0.10844	203
	CME	1.3811	0.06364	0.15095	0.7866	0.02325	0.01673	0.5634	0.01883	0.12685	254
	ADE	1.3754	0.05933	0.14624	0.8023	0.02034	0.00292	0.5525	0.01872	0.10513	181.5
	RADE	1.3982	0.07646	0.16527	0.8022	0.01113	0.00271	0.5644	0.02544	0.12886	275
	PE	1.2191	0.52917	0.01591	0.9339	0.09847	0.16737	0.4979	0.17597	0.00411	306.5
50	MLE	1.1977	0.01701	0.00193.5	0.8427	0.00672	0.05346	0.4869	0.00272	0.02604	18.52
	LSE	1.1988	0.01944	0.00091	0.8398	0.00814	0.04985	0.4703	0.00454	0.05947	254
	WLSE	1.1977	0.01783	0.00193.5	0.8358	0.00703	0.04474	0.4759	0.00323	0.04816	22.53
	CME	1.2053	0.02126	0.00445	0.8291	0.00905	0.03633	0.4932	0.00505	0.01352	265.5
	ADE	1.1983	0.01742	0.00142	0.8264	0.00441	0.03291	0.4831	0.00261	0.03385	121
	RADE	1.2093	0.02115	0.00776	0.8274	0.01056	0.03432	0.4964	0.00916	0.00711	265.5
	PE	1.4037	0.78257	0.16987	0.9371	0.24397	0.17147	0.5086	0.10567	0.01723	387
75	MLE	1.2035	0.01021.5	0.00294	0.8392	0.00442	0.04903	0.4853	0.00231	0.02953	14.52
	LSE	1.2023	0.01084	0.00191	0.8415	0.00584.5	0.05196	0.4765	0.00394.5	0.04716	265.5
	WLSE	1.2029	0.01053	0.00242	0.8406	0.00543	0.05075	0.4796	0.00303	0.04085	213
	CME	1.2063	0.01145	0.00535	0.8404	0.00584.5	0.05054	0.4874	0.00394.5	0.02522	254
	ADE	1.2031	0.01021.5	0.00263	0.8360	0.00381	0.04502	0.4813	0.00262	0.03734	13.51
	RADE	1.2089	0.01216	0.00756	0.8335	0.00706	0.04191	0.4911	0.00596	0.01781	265.5
	PE	1.4539	1.10587	0.21167	0.9042	0.24527	0.13027	0.5445	0.23577	0.08917	427
100	MLE	1.2039	0.00791.5	0.00334	0.8387	0.00393	0.04844.5	0.4844	0.00171	0.03124	183
	LSE	1.2033	0.00844	0.00281	0.8381	0.00494.5	0.04773	0.4789	0.00305	0.04207	24.54
	WLSE	1.2035	0.00813	0.00302.5	0.8349	0.00382	0.04372	0.4828	0.00213	0.03445	17.52
	CME	1.2063	0.00885	0.00535	0.8387	0.00494.5	0.04844.5	0.4861	0.00284	0.02783	265
	ADE	1.2036	0.00791.5	0.00302.5	0.8335	0.00331	0.04181	0.4827	0.00192	0.03456	141
	RADE	1.2082	0.00916	0.00686	0.8397	0.00796	0.04966	0.4862	0.00446	0.02762	326
	PE	1.4382	1.01727	0.19857	0.9844	0.57577	0.23057	0.5034	0.09367	0.00671	367

,

Table 7. Simulation results of the IPEC model for α = 1.5, β = 0.9, and λ = 0.7.

			For α			For β			For λ
n	Methods	Average	MSE	RBIAS	Average	MSE	RBIAS	Average	MSE	RBIAS	⅀Ranks
25	MLE	1.4428	0.03311	0.03812	0.9829	0.00963	0.09215	0.6807	0.00291	0.02751	131
	LSE	1.4047	0.03854	0.06356	0.9687	0.01636	0.07634	0.6764	0.02025	0.03362	276
	WLSE	1.4064	0.03502	0.06245	0.9842	0.01214	0.09366	0.6658	0.00922	0.04885	245
	CME	1.4335	0.03965	0.04433	0.9404	0.00281	0.04493	0.7295	0.01173.5	0.04224	19.52
	ADE	1.4267	0.03643	0.04884	0.9321	0.01535	0.03571	0.7239	0.02656	0.03423	224
	RADE	1.4666	0.05976	0.02231	0.8654	0.00672	0.03852	0.7771	0.01173.5	0.11026	20.53
	PE	0.8636	0.51697	0.42437	1.5326	2.19757	0.70297	0.4038	0.15327	0.42327	427
50	MLE	1.4963	0.03256	0.00241	0.9685	0.00751	0.07614	0.6994	0.00671	0.00091	141
	LSE	1.4562	0.02693	0.02925	0.9273	0.00872	0.03031	0.7050	0.01345	0.00723	192
	WLSE	1.4606	0.02682	0.02634	0.9817	0.01355	0.09086	0.6665	0.01183	0.04796	266
	CME	1.4729	0.02825	0.01803	0.9591	0.01113	0.06563	0.7039	0.01406	0.00562	224
	ADE	1.4734	0.02794	0.01772	0.9808	0.01576	0.08985	0.6740	0.00962	0.03715	245
	RADE	1.4471	0.02441	0.03536	0.9385	0.01164	0.04282	0.6846	0.01244	0.02204	213
	PE	1.1279	0.52047	0.24807	1.0101	0.19367	0.12237	0.5719	0.10257	0.18307	427
75	MLE	1.4633	0.00845	0.02442	0.9213	0.00112	0.02373	0.7251	0.00081	0.03593	161
	LSE	1.4424	0.00804	0.03846	0.8538	0.00283	0.05135	0.7474	0.00524	0.06785	275
	WLSE	1.4522	0.00793	0.03184	0.8278	0.01497	0.08026	0.8027	0.02736	0.14676	326
	CME	1.4522	0.00721	0.03195	0.8975	0.00091	0.00271	0.7280	0.00705	0.04004	172.5
	ADE	1.4569	0.00762	0.02873	0.9749	0.01156	0.08327	0.6756	0.00182	0.03482	224
	RADE	1.4679	0.01086	0.02141	0.9199	0.00584	0.02212	0.7218	0.00223	0.03121	172.5
	PE	1.2394	0.16987	0.17377	0.9438	0.01105	0.04874	0.5771	0.03957	0.17557	377
100	MLE	1.4871	0.00331	0.00861	0.9444	0.00391	0.04933	0.7052	0.00251	0.00751	81
	LSE	1.4644	0.00456	0.02385	0.9620	0.01455	0.06897	0.6829	0.00555	0.02444	325
	WLSE	1.4686	0.00393	0.02094	0.9616	0.00824	0.06846	0.6812	0.00332	0.02695	244
	CME	1.4725	0.00424	0.01833	0.9376	0.00462	0.04181	0.7086	0.00504	0.01233	173
	ADE	1.4742	0.00372	0.01712	0.9477	0.00493	0.05314	0.6942	0.00343	0.00832	162
	RADE	1.4563	0.00435	0.02916	0.9610	0.01526	0.06795	0.6741	0.00716	0.03706	346
	PE	1.1429	0.43237	0.23807	0.9383	0.17297	0.04262	0.5923	0.07667	0.15397	377

,

Table 8. Simulation results of the IPEC model for α = 1.8, β = 0.8, and λ = 0.5.

			For α			For β			For λ
n	Methods	Average	MSE	RBIAS	Average	MSE	RBIAS	Average	MSE	RBIAS	⅀Ranks
25	MLE	1.9144	0.11774	0.06366	0.8305	0.02031	0.03822	0.5518	0.02975	0.10376	244
	LSE	1.8347	0.09092	0.01931	0.8433	0.02623.5	0.05416	0.5018	0.01491	0.00361	14.52
	WLSE	1.8419	0.08921	0.02332	0.8338	0.02362	0.04234	0.5156	0.02393	0.04232	141
	CME	1.9123	0.12745	0.06245	0.8371	0.03245	0.04645	0.5472	0.02444	0.09444	285
	ADE	1.8659	0.09463	0.03663	0.8225	0.02623.5	0.02821	0.5319	0.02012	0.06383	15.53
	RADE	1.9028	0.15136	0.05714	0.8306	0.03666	0.03833	0.5444	0.03976	0.08885	306
	PE	2.3480	3.61297	0.30457	0.8483	0.19837	0.06047	0.5726	0.20517	0.14527	427
50	MLE	1.8141	0.03701	0.00793	0.8550	0.01784	0.06886	0.4871	0.00671	0.02573	181
	LSE	1.7836	0.04474	0.00914	0.8312	0.01732	0.03902	0.4867	0.01184	0.02654	202.5
	WLSE	1.7901	0.03873	0.00552	0.8394	0.01773	0.04934	0.4852	0.00993	0.02965	202.5
	CME	1.8169	0.05065	0.00946	0.8257	0.01661	0.03221	0.5077	0.01406	0.01532	214
	ADE	1.7934	0.03722	0.00371	0.8519	0.02356	0.06495	0.4797	0.00852	0.04056	225
	RADE	1.8163	0.05716	0.00925	0.8343	0.01845	0.04303	0.4973	0.01295	0.00541	256
	PE	2.3731	3.74997	0.31847	0.9123	0.27977	0.14047	0.5377	0.15997	0.07547	427
75	MLE	1.8223	0.02341	0.01244	0.8436	0.00911	0.05456	0.4869	0.00441	0.02615	182
	LSE	1.8106	0.03214	0.00591	0.8379	0.01446	0.04754	0.4852	0.00795	0.02957	276
	WLSE	1.8149	0.02783	0.00833	0.8325	0.01242.5	0.04071	0.4907	0.00663	0.01854	16.51
	CME	1.8333	0.03585	0.01855	0.8363	0.01274	0.04543	0.4962	0.00774	0.00752	234
	ADE	1.8133	0.02552	0.00742	0.8385	0.01325	0.04815	0.4864	0.00612	0.02716	223
	RADE	1.8357	0.04166	0.01986	0.8339	0.01242.5	0.04242	0.4977	0.01156	0.00461	23.55
	PE	2.2628	3.65377	0.25717	1.0091	0.56377	0.26147	0.5071	0.26177	0.01423	387
100	MLE	1.8145	0.01791	0.00814	0.8490	0.00781	0.06136	0.4814	0.00341	0.03736	193
	LSE	1.8052	0.02304	0.00291	0.8382	0.01075	0.04785	0.4836	0.00554	0.03275	245
	WLSE	1.8088	0.02013	0.00493	0.8296	0.01034	0.03702	0.4911	0.00473	0.01773	182
	CME	1.8218	0.02485	0.01215	0.8378	0.01156	0.04724	0.4928	0.00625	0.01432	276
	ADE	1.8075	0.01922	0.00412	0.8372	0.00892	0.04653	0.4848	0.00432	0.03024	151
	RADE	1.8227	0.02896	0.01266	0.8267	0.00993	0.03341	0.4975	0.00696	0.00511	234
	PE	2.3683	3.84997	0.31577	0.9398	0.41607	0.17477	0.5453	0.22677	0.09067	427

and

Table 9. Partial and total rankings for all estimating techniques for the IPEC distribution.

Parameters	$\mathit{n}$	MLE	LSE	WLSE	CME	ADE	RADE	PE
α = 0.6, β = 0.5, λ = 0.4	25	1	6	4	5	3	2	7
	50	1	5.5	3	5.5	4	2	7
	75	1	6	2	5	4	3	7
	100	1	5	3	6	3	3	7
α = 0.8, β = 0.6, λ = 0.5	25	4.5	3	1	2	4.5	6	7
	50	1	6	3	4	2	5	7
	75	2.5	6	2.5	5	1	4	7
	100	1.5	5.5	4	5.5	1.5	3	7
α = 1, β = 0.75, λ = 0.6	25	3	4	2	6	1	5	7
	50	1	6	3	5	2	4	7
	75	1	5.5	3	5.5	2	4	7
	100	1	6	2	5	4	3	7
α = 1.2, β = 0.8, λ = 0.5	25	6.5	1.5	3	4	1.5	5	6.5
	50	2	4	3	5.5	1	5.5	7
	75	2	5.5	3	4	1	5.5	7
	100	3	4	2	5	1	6	7
α = 1.5, β = 0.9, λ = 0.7	25	1	6	5	2	4	3	7
	50	1	2	6	4	5	3	7
	75	1	5	6	2.5	4	2.5	7
	100	1	5	4	3	2	6	7
α = 1.8, β = 0.8, λ = 0.5	25	4	2	1	5	3	6	7
	50	1	2.5	2.5	4	5	6	7
	75	2	6	1	4	3	5	7
	100	3	5	2	6	1	4	7
∑Ranks overall rank		47	113	71	108.5	63.5	101.5	167.5
		1	6	3	5	2	4	7

correspond to the average performance across replications, confirming the stability and accuracy of the estimators.

Monte Carlo simulation encompasses a variety of sample sizes and differing parameter settings. Certain statistical measures, such as relative biases and mean squared errors (MSEs), were used to assess the performance of the parameters $\widehat{\alpha }$, $\widehat{\beta }$, and $\widehat{\lambda }$.

The results of the simulation analysis for various scenarios are summarized in Table 3, Table 4, Table 5, Table 6, Table 7 and Table 8. These tables provide insight into the performance of the proposed estimators for the IPEC distribution under different sample sizes and estimation methods. Several important observations can be drawn:

• The IPEC distribution exhibits robust performance across estimation methods, as indicated by the relatively small biases and mean squared errors (MSEs).
• As the sample size increases, both the bias and MSE values consistently decrease for all estimation methods. This behavior reflects the improved precision and consistency of the estimators with larger samples, as expected in well-behaved estimation procedures.
• All sample sizes demonstrate a tendency toward positive bias, suggesting that, on average, the estimators slightly overestimate the true parameter values, particularly for smaller sample sizes.
• The reduction in MSE with increasing sample size confirms the asymptotic efficiency of the estimation methods, with the estimators converging toward the true values.
• The primary objective of the simulation study is to determine the most appropriate estimation technique for the proposed IPEC model. Table 9 presents the partial and overall ranking of each estimation method based on performance metrics such as bias, MSE, and computational efficiency.
• From the rankings shown in Table 9, it is evident that the MLE method outperforms the others, achieving the best overall rank with a score of 47. This confirms its superiority in terms of accuracy and consistency. However, the ADE method also demonstrates competitive performance, with a cumulative score of 63.5, positioning it as a viable alternative to MLE, particularly in situations where analytical tractability is essential.

These findings highlight the practical value of MLE in estimating the parameters of the IPEC distribution, while also acknowledging the robustness and relevance of alternative methods like ADE in various estimation contexts.

It is worth noting that the shape of the IPEC distribution is highly influenced by the values of its parameters. To account for this, a wide range of parameter combinations was explored through simulation and graphical analysis to better understand the model’s behavior. Additionally, multiple initial values were used during the estimation procedures to ensure numerical stability. These steps help mitigate the risk of poor model fitting due to inappropriate parameter assumptions and enhance the reliability of the proposed model across different data scenarios.

For completeness, the detailed algorithmic steps used for random sample generation, parameter estimation, and Monte Carlo simulation are summarized in Appendix A.

The appendix presents pseudo-code that outlines the main computational procedures in a software-independent format, ensuring reproducibility and transparency of the numerical analysis.

6. Applications

This section examines the flexibility and importance of the IPEC distribution by analyzing real-world datasets. The actual importance is demonstrated by analyzing three real datasets, which reveal the IPEC model’s efficiency compared to other comparable models. The suggested IPEC will be applicable across diverse domains, covering economics and medical fields. We analyze three real datasets to compare the efficacy of the IPEC distribution using several criteria for goodness-of-fit, including Akaike’s information criterion (AIC), Bayesian information criterion (BIC), Hannan–Quinn information criterion (HQIC), and Kolmogorov–Smirnov (K–S) test statistic. The distribution with the lowest AIC, BIC, CAIC, and HQIC values is considered to have the best fit. The IPEC distribution is compared with five competing distributions, namely, the entropy Chen (EC) distribution, Weibull (We) distribution, Gamma (Ga) distribution, inverse power logistic exponential (IPLE) distribution, and the inverse power Perk (IPP) distribution.

Table 10. Comparative summary of inverse power distributions.

Models	PDF Structure	Parameters	Typical HRF Shape
Weibull (We)	$\mathrm{f}\left(\mathrm{x}\right)=\mathsf{\alpha }\mathsf{\lambda }{\mathrm{x}}^{\mathsf{\lambda }-1}\exp \left(-\mathsf{\alpha }{\mathrm{x}}^{\mathsf{\lambda }}\right)$	α, λ > 0	Increasing/Decreasing
Gamma (Ga)	$\mathrm{f}\left(\mathrm{x}\right)={\displaystyle \frac{{\mathsf{\lambda }}^{\mathsf{\alpha }}}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}}{\mathrm{x}}^{\mathsf{\alpha }-1}{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{x}}$	α, λ > 0	Increasing/Decreasing
Entropy Chen (EC)	$\mathrm{f}\left(\mathrm{x}\right)={\mathsf{\alpha }}^2\mathsf{\beta }{\mathrm{x}}^{\mathsf{\beta }-1}{\mathrm{e}}^{{\mathrm{x}}^{\mathsf{\beta }}\left({\mathrm{e}}^{{\mathrm{x}}^{\mathsf{\beta }}}-1\right)\exp \left(\mathsf{\alpha }\left(1-{\mathrm{e}}^{{\mathrm{x}}^{\mathsf{\beta }}}\right)\right)}$	α, β > 0	J-shaped/Bathtub
Inverse Power Logistic Exponential (IPLE)	$\mathrm{f}\left(\mathrm{x}\right)={\displaystyle \frac{\mathsf{\alpha }\mathsf{\beta }\mathsf{\lambda }{\mathrm{x}}^{-\mathsf{\beta }-1}{\left(\exp \left(\mathsf{\lambda }{\mathrm{x}}^{-\mathsf{\beta }}-1\right)\right)}^{\mathsf{\alpha }-1}}{{\left(\exp \left(\mathsf{\lambda }{\mathrm{x}}^{-\mathsf{\beta }}-1\right)\right)}^{{\left(\mathsf{\alpha }+1\right)}^2}}}$	α, β, λ > 0	Reversed J/Unimodal
Inverse Power Perk (IPP)	$\mathrm{f}\left(\mathrm{x}\right)={\displaystyle \frac{\mathsf{\alpha }\mathsf{\beta }\mathsf{\lambda }{\mathrm{x}}^{-\mathsf{\beta }-1}\exp \left(\mathsf{\lambda }{\mathrm{x}}^{-\mathsf{\beta }}\right){\left({\mathrm{e}}^{\mathsf{\lambda }{\mathrm{x}}^{-\mathsf{\beta }}}-1\right)}^{\mathsf{\alpha }-1}}{{\left[{\left({\mathrm{e}}^{\mathsf{\lambda }{\mathrm{x}}^{-\mathsf{\beta }}}-1\right)}^{\mathsf{\alpha }}+1\right]}^2}}$	α, β, λ > 0	Reversed J
Inverse Power Entropy Chen (IPEC)	${\mathrm{f}\left(\mathrm{x}\right)=\mathsf{\alpha }}^2\mathsf{\beta }\mathsf{\lambda }{\mathrm{x}}^{-\mathsf{\lambda }\mathsf{\beta }-1}{\mathrm{e}}^{{\mathrm{x}}^{-\mathsf{\lambda }\mathsf{\beta }}}\left({\mathrm{e}}^{{\mathrm{x}}^{-\mathsf{\lambda }\mathsf{\beta }}}-1\right)\exp \left(\mathsf{\alpha }\left(1-{\mathrm{e}}^{{\mathrm{x}}^{-\mathsf{\lambda }\mathsf{\beta }}}\right)\right)$	α, β, λ > 0	Reversed J/Unimodal

Note: The IPEC distribution extends the flexibility of inverse power type models by incorporating an entropy-based parameter (λ) that allows for various hazard rate shapes (J, reversed J, or unimodal).

presents a concise comparison of the parameter structures and hazard rate characteristics of several inverse power type models relative to the proposed IPEC distribution. This comparative overview highlights the generality and flexibility of the IPEC family. For further theoretical and stochastic background on related models, see Perry and Stadje [40], López-Barrientos et al. [41], and the classical Weibull [42] distribution.

Table 11. MLEs and discrimination measures for data I.

Models	Estimates	AIC	BIC	CAIC	HQIC	K-S	p-Value
IPEC	$\widehat{\alpha } =$ 35.7898 $\widehat{\beta }=$ 4.9410 $\widehat{\lambda } =$ 0.3452	108.8913	112.0248	110.303	109.5713	0.1155	0.9421
EC	$\widehat{\alpha } =$ 0.1389 $\widehat{\beta } =$ 0.4877	114.5252	116.6143	115.1919	114.9786	0.2176	0.2733
IPP	$\widehat{\alpha } =$ 23.4405 $\widehat{\beta }=$ 2.5810 $\widehat{\lambda } =$ 72.6663	109.3388	112.4724	110.7506	110.0189	0.1287	0.8774
We	$\widehat{\alpha } =$ 2.1788 $\widehat{\lambda } =$ 8.1146	112.616	114.7051	113.2827	113.0694	0.2097	0.3143
IPLE	$\widehat{\alpha } =$ 1.9308 $\widehat{\beta } =$ 1.4531 $\widehat{\lambda } =$ 7.6290	138.2598	141.3934	139.6716	138.9399	0.2544	0.1319
Ga	$\widehat{\alpha } =$ 4.9982 $\widehat{\lambda } =$ 1.4298	109.4703	113.5594	111.137	109.9237	0.1856	0.4649

,

Table 12. MLEs and discrimination measures for data II.

Models	Estimates	AIC	BIC	CAIC	HQIC	K-S	p-Value
IPEC	$\widehat{\alpha } =$ 91.0159 $\widehat{\beta } =$ 10.0307 $\widehat{\lambda } =$ 0.19243	114.3132	117.4467	115.7249	114.9932	0.1623	0.6376
EC	$\widehat{\alpha } =$ 0.07456 $\widehat{\beta } =$ 0.52686	115.9618	118.0509	116.6285	116.4152	0.2108	0.3083
IPP	$\widehat{\alpha } =$ 233.472 $\widehat{\beta } =$ 2.82735 $\widehat{\lambda } =$ 235.893	115.3482	118.4817	116.7599	116.0282	0.1790	0.5115
We	$\widehat{\alpha } =$ 2.7279 $\widehat{\lambda } =$ 10.050	114.5954	118.2314	115.9236	115.0488	0.2021	0.3575
IPLE	$\widehat{\alpha } =$ 2.0518 $\widehat{\beta } =$ 1.5781 $\widehat{\lambda } =$ 14.7604	143.4295	146.5631	144.8413	144.1096	0.2995	0.0462
Ga	$\widehat{\alpha } =$ 7.0890 $\widehat{\lambda } =$ 1.2578	115.3259	118.1265	116.3215	115.2314	0.1663	0.6066

and

Table 13. MLEs and discrimination measures for data III.

Models	Estimates	AIC	BIC	CAIC	HQIC	K-S	p-Value
IPEC	$\widehat{\alpha } =$ 4.8977 $\widehat{\beta } =$ 2.0231 $\widehat{\lambda } =$ 1.1177	36.789	39.776	38.289	37.372	0.1003	0.9878
EC	$\widehat{\alpha } =$ 0.3969 $\widehat{\beta } =$ 0.8029	48.054	50.045	48.759	48.442	0.1909	0.4599
IPP	$\widehat{\alpha } =$ 32519.1 $\widehat{\beta }=$ 4.0178 $\widehat{\lambda } =$ 6.0218	36.817	39.805	38.317	37.401	0.1021	0.9853
We	$\widehat{\alpha } =$ 2.7868 $\widehat{\lambda } =$ 2.1301	45.1728	47.1643	45.8787	45.5616	0.1850	0.5003
IPLE	$\widehat{\alpha } =$ 1.1660 $\widehat{\beta } =$ 3.5193 $\widehat{\lambda } =$ 2.6903	64.531	67.518	66.031	65.114	0.3135	0.0393
Ga	$\widehat{\alpha } =$ 9.6698 $\widehat{\lambda } =$ 0.1965	39.637	41.629	40.343	40.026	0.1734	0.5843

clearly indicate that the IPEC distribution demonstrates lower values of AIC, BIC, CAIC, HQIC, K-S, and the highest p-value, hence indicating that the IPEC model provides a more optimal fit than the alternative distributions. The estimated values for various estimation methods of the IPEC model across the three datasets are presented in

Table 14. Outcomes of estimation techniques for the IPEC model for dataset I.

Methods	$\mathbf{A}$	$\mathit{\beta }$	$\mathit{\lambda }$	Metrics of Sufficiency
	Estimate	Estimate	Estimate	K-S	p-Value
MLE	35.7898	4.9410	0.3452	0.1155	0.9421
PE	39.0285	10.9562	0.1620	0.1382	0.8174
LSE	30.8796	23.2822	0.0694	0.1196	0.9247
WLSE	26.5852	9.5434	0.1615	0.1189	0.9275
CME	42.1577	13.7784	0.1294	0.1109	0.9583
ADE	31.1115	25.8246	0.0629	0.1165	0.9381
RADE	34.0649	22.0607	0.0757	0.1147	0.9451

,

Table 15. Outcomes of estimation techniques for the IPEC model for dataset II.

Methods	$\mathbf{A}$	$\mathit{\beta }$	$\mathit{\lambda }$	Metrics of Sufficiency
	Estimate	Estimate	Estimate	K-S	p-Value
MLE	91.0159	10.0307	0.1924	0.1623	0.6376
PE	85.3249	18.3659	0.1132	0.1352	0.7496
LSE	89.3821	259.5573	0.0073	0.1415	0.7942
WLSE	88.0281	36.7421	0.0516	0.1456	0.7649
CME	107.0881	277.717	0.0071	0.1306	0.8664
ADE	86.0834	63.8352	0.0296	0.1527	0.7118
RADE	117.146	73.8526	0.0275	0.1329	0.8522

and

Table 16. Outcomes of estimation techniques for the IPEC model for dataset III.

Methods	$\mathbf{A}$	$\mathit{\beta }$	$\mathbf{\Lambda }$	Metrics of Sufficiency
	Estimate	Estimate	Estimate	K-S	p-Value
MLE	4.8977	2.0231	1.1177	0.1003	0.9878
PE	4.9927	1.0376	2.1265	0.1151	0.9536
LSE	4.7565	3.4354	0.6429	0.1004	0.9881
WLSE	4.3138	1.2673	1.6175	0.1049	0.9804
CME	5.5224	3.7169	0.6597	0.0923	0.9956
ADE	4.8718	0.4042	5.5629	0.0994	0.9891
RADE	4.9553	3.5274	0.6463	0.0995	0.9889

, demonstrating that the CME method surpasses the others in all datasets, as indicated by the greatest p-value. The estimated PDF plots, empirical CDF plots, and the probability–probability (P–P) plots of the fitted distributions for the three real datasets are presented in Figure 3, Figure 4 and Figure 5. It provides that the IPEC distribution obtains a greater approximation between the empirical and the theoretical curves; therefore, the proposed distribution was the one that best adjusted to the real datasets.

6.1. Dataset I

The initial dataset details the emergence of new F1 adults of Stegobium paniceum from fathers that were exclusively fed on peppermint, obtained from both unirradiated and irradiated packets (compulsory non packet test). The irradiation treatments included gamma radiation at (6, 8, and 10 KGy), alongside microwave exposure for durations of 1, 2, and 3 min resulting in the next values: 148, 145, 152, 64, 64, 64, 52, 59, 57, 33, 36, 31, 87, 85, 87, 69, 65, 67, 48, 42, and 46 (refer to [43]). Before performing the KS test, we divide the data by 10 to suit the competing distributions.

6.2. Dataset II

The second dataset illustrates the quantity of F1 adults’ of Stegobium paniceum derived from fathers fed on peppermint from both unirradiated and irradiated packets (choice non packet test) subjected with gamma rays (6, 8, and 10 KGy) or microwave radiation (1, 2, and 3 min), resulting in the next quantities: 161, 155, 163, 84, 80, 89, 70, 73, 70, 46, 40, 49, 116, 109, 113, 88, 88, 82, 66, 66, and 66 (see, [44]). Before performing the KS test, we divided the data by 10 to suit the competing distributions.

6.3. Dataset III

The third dataset represents the relief times of twenty patients receiving an analgesic. It was used by [44], reported by [45]. The data is 1.1, 1.4, 1.3, 1.7, 1.9, 1.8, 1.6, 2.2, 1.7, 2.7, 4.1, 1.8, 1.5, 1.2, 1.4, 3.0, 1.7, 2.3, 1.6, and 2.0.

The variation in p-values observed across the three real datasets reflects the distinct characteristics of each dataset, including differences in sample size, variability, and underlying distributional structure. Such discrepancies are expected when applying a single model to heterogeneous data sources. Importantly, the IPEC model’s performance was evaluated using a range of goodness-of-fit criteria, ensuring a robust comparison that does not rely solely on p-values.

7. Conclusions

This paper presents a new distribution called the Inverse Power Entropy Chen (IPEC) distribution, characterized by three parameters that extend the Chen distribution. Several characteristics of the novel distribution were established, including the survival function, hazard rate function, quantile function, moments, probability weighted moment, moment-generating function, mean, variance, skewness, kurtosis, Rényi entropy, Tsallis entropy, and order statistics. The proposed distribution’s forms were illustrated by graphing the probability density function and the hazard rate function. The hazard rate plots indicate that the new distribution exhibits rising and decreasing forms. The parameters of the IPEC model were estimated utilizing seven classical techniques, including MLE, LSE, WLSE, ADE, RADE, CME, and PE, with the R 14 software. A Monte Carlo simulation was conducted to evaluate the performance of the corresponding methods of the IPEC model. As anticipated, the mean squared errors (MSEs) of the estimated parameters approach zero as the sample size ($n$) rises, thereby demonstrating the accuracy of the estimators. It is clearly evident from simulation studies that the superiority of the MLE method in estimating the parameters of the IPEC model. The proposed distribution was applied to three real datasets, and the outcomes are displayed in Table 10, Table 11 and Table 12. The findings demonstrate that the IPEC model is highly effective and better in fitting the three analyzed datasets. The versatility of the suggested distribution is demonstrated by the estimated PDF, empirical CDF, and P-P plots for the three datasets, clearly indicating that the IPEC model provides a superior match compared to the competitive distributions analyzed. Through extensive simulation studies, the performance of the proposed estimators was assessed under various scenarios. The results revealed that the IPEC distribution demonstrates strong flexibility and stability, with MLE consistently showing the lowest bias and MSE across different sample sizes. The simulation findings further confirmed that increasing the sample size leads to improved estimation accuracy and reduced uncertainty across all estimation techniques.

Based on these observations, several practical recommendations can be offered for applied researchers and reliability engineers. The MLE method is highly recommended for large-sample contexts due to its superior accuracy. In cases where analytical complexity or small sample sizes pose challenges, the ADE method emerges as a competitive and robust alternative. Practitioners are also encouraged to utilize the largest feasible sample sizes to minimize bias and enhance estimator reliability. Moreover, awareness of potential positive bias in small samples is important for accurate interpretation of the results. Overall, the proposed estimation framework enhances the practical applicability of the IPEC model in reliability analysis and provides a valuable foundation for future studies and industrial implementations. By broadening the study’s focus and implementing changes like truncation or generalization, the suggested distribution can be further used in subsequent research projects. This would enable more adaptable models for modeling various kinds of data.

In addition to the classical estimation methods considered in this study, future research may benefit from exploring non-classical or modern estimation strategies. Approaches such as Bayesian estimation, Markov Chain Monte Carlo (MCMC) techniques, and metaheuristic algorithms like Particle Swarm Optimization (PSO) and Genetic Algorithms (GA) could provide more robust and flexible alternatives, particularly in cases involving small samples, censored data, or highly nonlinear likelihood functions. Incorporating these advanced techniques may enhance the accuracy and computational stability of parameter estimation for the IPEC model.