EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6144 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Novel Discrete Probability Distribution with Theoretical and Inferential Insights: Cutting-Edge Approaches to Sustainable Dispersion Data Modeling Mohamed S. Eliwa1,∗, Weed E. Alghanem1, Manahel A. Alenizi1 1 Department of Statistics and Operations Research, College of Science, Qassim University, Saudi Arabia Abstract. As real-world data becomes increasingly complex, there is a growing need for advanced probability models to support sustainable discrete data analysis. This study presents a novel and flexible extension of the discrete Gompertz distribution, developed within the framework of the half-logistic model. Named the discrete Gompertz half-logistic (DGzHLo) model, this new formu- lation enhances the adaptability of existing discrete distributions to better capture intricate data patterns. To understand its theoretical foundation, key mathematical and statistical properties are examined, including the probability mass function, cumulative distribution function, reliabil- ity function, and hazard rate function. Measures such as dispersion, skewness, and kurtosis offer insights into the model’s behavior, while entropy and order statistics further reveal its structural characteristics. The model’s parameters are estimated using the maximum likelihood estimation method, and a thorough simulation study assesses the accuracy and efficiency of the estimators across various sample sizes. To illustrate its practical relevance, the model is applied to three real-world datasets, demonstrating its superior flexibility and robustness in capturing complex data structures compared to existing models. These findings underscore the DGzHLo model’s effectiveness in advancing discrete data modeling and analysis. 2020 Mathematics Subject Classifications: 62E15, 62F10 Key Words and Phrases: Statistical model, Order statistics, Reliability theory, Estimation, Simulation, Data analysis 1. Introduction Sustainable dispersion data modeling has become increasingly essential in response to the exponential surge in real-world data, driven by technological advancements and digital innovation. The growing complexity of data structures presents significant chal- lenges, as conventional probability models often fail to capture the intricate dependencies, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6144 Email address: m.eliwa@qu.edu.sa (M. S. Eliwa) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 2 of 26 stochastic variations, and nonlinear trends observed across diverse fields such as engineer- ing, epidemiology, finance, and actuarial sciences. This limitation underscores the need for advanced probabilistic frameworks that provide greater flexibility and adaptability for robust data analysis. Among the well-explored models in statistical and applied probabil- ity research is the Gompertz (Gz) distribution a continuous probability distribution that extends the exponential model by introducing an additional shape parameter. This mod- ification enhances its ability to model a wide range of real-world phenomena, particularly those characterized by dynamic hazard rates. Due to its structural advantages, the Gz distribution has been extensively applied across scientific and practical domains, partic- ularly in modeling lifetime and failure rates. In demography and actuarial science, it is frequently employed to describe adult lifespan distributions, offering critical insights into mortality patterns and life expectancy. Similarly, in biology and gerontology, researchers utilize the Gz distribution for survival analysis, especially in aging studies and biological lifespan modeling. Formally, a random variable T is said to follow the Gz model with a shape parameter ϖ > 0 and scale parameter τ > 0 if its cumulative distribution function (CDF) is given by L(t;ϖ, τ) = 1− e− ϖ τ (eτt−1); t > 0. (1) This formulation underscores the versatility of the Gz distribution in modeling diverse data patterns, particularly in contexts characterized by increasing failure rates and non-constant hazard functions. Beyond its conventional applications, the Gz distribution has been increasingly utilized in computational sciences to model software failure rates, contributing to software reliability assessment. For a comprehensive exploration of its applications, see Cordeiro et al. [5], Roozegar et al.[23], Mazucheli et al. [8], Eliwa et al. [19], Bantan et al. [22], Wang and Guo [10], Alizadeh et al. [13], Hassan and Abdelghaffar [26], among others. Building upon the fundamental structure of the Gz distribution, Alizadeh et al. [12] introduced the Gz-G family using the methodological framework developed by Alzaatreh et al. [1]. The latter proposed the transformed-transformer family, a general approach for constructing new probability distributions with enhanced adaptability. Accordingly, a random variable Y follows the Gz-G family if its CDF is given by Θ(y;ϖ, τ,Ω) = 1− e− ϖ τ {[1−G(y;Ω)]−τ−1}; y > 0, (2) where Ω is a vector of parameters (1 × m;m = 1, 2, 3, ...), and G(y;Ω) is the baseline CDF. The reliability function (RF) of the Gz-G family can be formulated as Θ(y;ϖ, τ,Ω) = e− ϖ τ {[1−G(y;Ω)]−τ−1}; y > 0. (3) The probability density function (PDF) corresponding to Equation (2) can be written as ψ(y;ϖ, τ,Ω) = ϖg(y;Ω) [ G(y;Ω) ]−(τ+1) e− ϖ τ {[1−G(y;Ω)]−τ−1}; y > 0, (4) where g(y;Ω) is the baseline PDF. Several researchers have built upon the framework introduced by Alzaatreh et al. [1] to develop both univariate and bivariate probability M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 3 of 26 distribution families. Significant contributions in this area include studies by El-Morshedy and Eliwa [15], Alizadeh et al. [11], Abdelall et al. [29], Alsolmi [17], Atchade et al. [18], among others. These works have played a crucial role in advancing probability modeling by introducing more flexible and adaptable distributions. In recent years, there has been an increasing focus on discretizing continuous probabil- ity distributions, particularly for applications where measuring lifetimes on a continuous scale is impractical. This issue commonly arises in areas such as reliability analysis and quality control, where product or system lifetimes are naturally recorded in discrete units like days, cycles, or events. To address this, various discrete probability models have been proposed by researchers such as Roy [25], Afify et al. [30], Abd EL-Hady et al. [3], Eliwa et al. [20], Gómez-Déniz and Caldeŕın-Ojeda [4], Yousof et al. [6], Nakagawa and Osaki [27], Ibrahim et al. [16], Hamdi et al. [28], among others. Despite these advancements, there remains considerable scope for developing new discrete probability distributions tailored to specific data structures and real-world scenarios. This study introduces a new discrete probability model, termed the discrete Gompertz half-logistic (DGzHLo) model, derived from the discrete version of the Gz-G family originally formulated by Eliwa et al. [21]. The CDF of the discrete version of the Gz-G (DGz-G) family is given by HZ(z; γ, τ,Ω) = 1− γ 1 τ {[1−G(z+1;Ω)]−τ−1}; z ∈ N0, (5) where γ = e−ϖ, 0 < γ < 1, τ > 0 and N0 = {0, 1, 2, 3, ...}. Incorporating the DGz- G structure with the half-logistic (HLo) framework enhances the model’s flexibility in capturing discrete lifetime data, where the CDF of the HLo model is given by G(s; θ) = 1− e−θs 1 + e−θs ; s ≥ 0, (6) where θ > 0. The motivation for introducing the DGzHLo model stems from the need to: • Generating models with varying skewness characteristics (negative, positive, or sym- metric) to capture diverse data distributions in sustainability studies. • Accommodating increasing hazard rate functions, particularly in engineering and medical contexts, to improve predictions and decision-making in sustainable devel- opment. • Modeling over-dispersed, equi-dispersed, and under-dispersed data to provide more accurate insights into environmental and economic sustainability metrics. • Accounting for different levels of kurtosis, including leptokurtic and platykurtic dis- tributions, to better represent the variability observed in sustainability data. • Handling zero-inflation and non-inflation data, particularly under extreme obser- vations, to improve the robustness of models in the context of rare but impactful sustainability events. M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 4 of 26 This paper is organized as follows: Section 2 introduces the DGzHLo distribution, out- lining its formulation and key properties. Section 3 examines its statistical and reliability features, while Section 4 discusses parameter estimation using the maximum likelihood es- timation method. A simulation study assessing the performance of the estimators across various sample sizes is presented in Section 5. Section 6 demonstrates the practical appli- cation of the DGzHLo model through the analysis of three real-world datasets, comparing its performance to other existing distributions. Finally, Section 7 provides a summary of the main findings and suggests potential avenues for future research. 2. Mathematical Framework and Visual Depiction Recall, Equations (5) and (6), the CDF and RF of the DGzHLo can be formulated as FZ(z; γ, τ, θ) = 1− γ 1 τ {[ 1− 1−e−θ(z+1) 1+e−θ(z+1) ]−τ −1 } ; z ∈ N0, (7) FZ(z; γ, τ, θ) = γ 1 τ {[ 1− 1−e−θ(z+1) 1+e−θ(z+1) ]−τ −1 } ; z ∈ N0, (8) respectively. The probability mass function (PMF) corresponding to Equation (7) can be expressed as fz(z; γ, τ, θ) = F (z; γ, τ, θ)− F (z + 1; γ, τ, θ) = γ− 1 τ γ 1 τ [ 1− 1−e−θz 1+e−θz ]−τ − γ 1 τ [ 1− 1−e−θ(z+1) 1+e−θ(z+1) ]−τ  ; z ∈ N0. (9) For further details on the survival discretization approach, refer to Roy [25]. The hazard rate function (HRF) can be formulated as h(z; γ, τ, θ) = 1− γ 1 τ {[ 1− 1−e−θ(z+1) 1+e−θ(z+1) ]−τ − [ 1− 1−e−θz 1+e−θz ]−τ } ; z ∈ N0, (10) where h(z; γ, τ,Ω) = fz(z;γ,τ,θ) FZ(z−1;γ,τ,θ) . The PMF in Equation (9) is log-concave for all pa- rameter values of the model. Specifically, the ratio f(z+1;γ,τ,θ) f(z;γ,τ,θ) is a decreasing function of z for all parameter values. As a result, the distribution is strongly unimodal, possesses all of its moments, and the hazard rate functions (HRFs) are increasing. Figures 1 and 2 illustrate the PMF and HRF of the DGzHLo distribution for different values of the model parameters. As shown in Figures 1 and 2, the PMF of the DGzHLo distribution proves to be a valuable statistical tool for analyzing unimodal data with varying degrees of skew- ness. This characteristic makes the PMF particularly beneficial for sustainable dispersion data, where it can accommodate different forms of data distributions, ensuring accurate modeling of real-world phenomena across multiple domains. Additionally, the HRF of the DGzHLo distribution exhibits an increasing trend, enhancing its applicability in diverse M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 5 of 26 5 10 15 20 0 .0 0 0 .0 2 0 .0 4 0 .0 6 0 .0 8 γ = 0.9, τ = 0.3, θ = 0.5 z P M F 0 20 40 60 80 100 0 .0 0 0 0 .0 1 0 0 .0 2 0 γ = 0.9, τ = 0.6, θ = 0.1 z P M F 5 10 15 20 0 .0 0 0 .0 2 0 .0 4 0 .0 6 γ = 0.7, τ = 0.8, θ = 0.2 z P M F 2 4 6 8 10 0 .0 0 0 .1 0 0 .2 0 γ = 0.4, τ = 0.5, θ = 0.4 z P M F 2 4 6 8 10 0 .0 2 0 .0 6 0 .1 0 0 .1 4 γ = 0.8, τ = 0.8, θ = 0.4 z P M F 2 4 6 8 10 0 .0 0 0 .1 0 0 .2 0 γ = 0.1, τ = 0.8, θ = 0.2 z P M F Figure 1: The PMF of the DGzHLo distribution. 5 10 15 20 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 γ = 0.9, τ = 0.3, θ = 0.5 z H R F 0 20 40 60 80 100 0 .0 0 .2 0 .4 0 .6 0 .8 γ = 0.9, τ = 0.6, θ = 0.1 z H R F 2 4 6 8 10 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 γ = 0.4, τ = 0.5, θ = 0.4 z H R F Figure 2: The HRF of the DGzHLo distribution. fields, including engineering and medicine. The increasing HRF is especially useful for modeling lifetime data, reliability analysis, and risk assessment, where the hazard rate tends to increase over time such as in the analysis of engineering systems, medical condi- tions, or treatment effectiveness. Therefore, the DGzHLo distribution provides a robust and adaptable framework for analyzing and interpreting data in both sustainability and health-related fields. 3. Different Statistical Properties 3.1. Moments, cumulants, and associated measures Moments and cumulants are essential statistical tools that provide valuable insights into the shape, variability, and relationships within a probability distribution. Moments, including the mean, variance, and index of dispersion, offer information about the central M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 6 of 26 tendency and spread of the distribution. Cumulants, in contrast, capture higher-order features such as skewness and kurtosis, providing a deeper understanding of the distri- bution’s characteristics. These measures are integral to statistical modeling, reliability analysis, and parameter estimation. In this section, we will derive the moments, cumu- lants, and other related measures for the DGzHLo model. These quantities are crucial for gaining a comprehensive understanding of the model’s key properties and its practical ap- plications. Let Z be a non-negative random variable following the DGzHLo distribution, denoted as Z ∼ DGzHLo(z; γ, τ, θ). The rth moment of Z can be expressed as µ′r = E(Zr) = ∞∑ z=0 zrfz(z; γ, τ, θ) = ∞∑ z=1 [zr − (z − 1)r]FZ(z − 1; γ, τ, θ) = γ− 1 τ ∞∑ z=1 [zr − (z − 1)r] γ 1 τ [ 1− 1−e−θz 1+e−θz ]−τ . (11) Using Equation (11), the mean (µ′1) and variance, say Var(Z), can be respectively listed as µ′1 = γ− 1 τ ∞∑ z=1 γ 1 τ [ 1− 1−e−θz 1+e−θz ]−τ , Var(Z) = γ− 1 τ ∞∑ z=1 (2z − 1) γ 1 τ [ 1− 1−e−θz 1+e−θz ]−τ − (µ′1) 2. (12) The dispersion index (DsI) is the ratio of variance to the mean, serving as a tool to assess whether a model is suitable for over-dispersed or under-dispersed data. It is widely used in ecology as a standard metric to evaluate patterns of clustering (over-dispersion) or repulsion (under-dispersion). When DsI > 1, the distribution is considered over-dispersed, whereas DsI < 1 indicates under-dispersion. The DsI for the DGzHLo model is expressed as DsI(Z) = ∑∞ z=1 (2z − 1) γ 1 τ [ 1− 1−e−θz 1+e−θz ]−τ ∑∞ z=1 γ 1 τ [ 1− 1−e−θz 1+e−θz ]−τ − ∞∑ z=1 γ 1 τ [ 1− 1−e−θz 1+e−θz ]−τ . (13) Alternatively, the moment generating function (MGF) can be expressed as MZ(t) = ∞∑ z=0 eztfz(z; γ, τ, θ) = γ− 1 τ [ ∞∑ z=0 eztγΛ(z;τ) − ∞∑ z=0 eztγΛ(z+1;τ) ] = γ− 1 τ [ ( γΛ(0;τ) + etγΛ(1;τ) + e2tγΛ(2;τ) + e3tγΛ(3;τ) + ... ) M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 7 of 26 − ( γΛ(1;τ) + etγΛ(2;τ) + e2tγΛ(3;τ) + e3tγΛ(4;τ) + ... ) ] = γ− 1 τ [ 1 + ∞∑ z=1 ( ezt − e(z−1)t ) γΛ(z;τ) ] , (14) where Λ(z; τ) = 1 τ [ 1− 1− e−θz 1 + e−θz ]−τ . The first four derivatives of Equation (14), with respect to t at t = 0, yield the first four moments about the origin, i.e. E(Zr) = dr dtr MZ(t)|t=0. Moreover, utilizing Equation (11) or (14), the skewness, say Sk(Z) and kurtosis, say Ku(Z), can be reported as Sk(Z) = (µ′3 − 3µ′2µ ′ 1 + 2µ′31 )/(Var(Z)) 3/2, Ku(Z) = (µ′4 − 4µ′3µ ′ 1 + 6µ′2µ ′2 1 − 3µ′41 )/(Var(Z)) 2, respectively. In probability theory, cumulants, denoted as kn, are a set of quantities that offer an alternative to the moments of a probability model. In certain cases, working with cumulants can simplify the theoretical treatment of problems compared to using moments. The cumulant generating function (CGF) is defined as the logarithm of the MGF. Therefore, the cumulants kn can be derived from the moments as follows kn = dn dtn logMZ(t)|t=0; n = 1, 2, 3, .... (15) Additionally, the cumulants are related to the moments through the following recursive formula kn = µ′n − n−1∑ m=1 ( n− 1 m− 1 ) µ′n−mkm. (16) The first cumulant corresponds to the mean, the second cumulant represents the variance, and the third cumulant is related to the third central moment. However, starting from the fourth cumulant, the direct relationship with central moments no longer holds, and higher-order cumulants no longer directly correspond to the central moments. For the DGzHLo distribution, the rth moment cannot be expressed in a closed form. Therefore, we utilize Maple software to investigate and compute some of its statistical properties. Table 1 presents descriptive statistics derived from the DGzHLo model for different pa- rameter values of θ and γ with τ = 0.5, highlighting that the DGzHLo model provides a strong framework for modeling sustainability data with varying kurtosis, skewness, and M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 8 of 26 dispersion characteristics. This makes it an invaluable tool across a wide range of fields, from engineering to social sciences, where sustainability is a key focus. Table 1. Some descriptive statistics using the DGzHLo model. Measure θ ↓ γ → 0.1 0.3 0.5 0.7 0.9 0.5 0.7444 1.4266 2.2267 3.4741 6.5495 Mean 1.5 0.0488 0.2107 0.4477 0.8444 1.8556 2.0 0.0080 0.0801 0.2365 0.5210 1.2703 0.5 0.8149 1.7661 3.0468 5.2333 10.719 Var 1.5 0.0465 0.1752 0.3374 0.6008 1.2373 2.0 0.0079 0.0738 0.1859 0.3448 0.7183 0.5 1.0946 1.2379 1.3682 1.5063 1.6366 DsI 1.5 0.9524 0.8316 0.7536 0.7114 0.6667 2.0 0.9919 0.9206 0.7860 0.6617 0.5655 0.5 1.1576 0.8163 0.5891 0.3497 −0.0376 Sk 1.5 4.1937 1.5999 0.9045 0.4929 0.0216 2.0 11.0353 3.0963 1.3395 0.6264 0.0731 0.5 3.9593 3.1603 2.7533 2.4635 2.3412 Ku 1.5 18.646 4.0924 2.8785 2.4486 2.3122 2.0 122.778 10.6124 3.0893 2.4472 2.2957 The DGzHLo model proves to be highly versatile in analyzing a wide range of data types, including both negatively and positively skewed data, and symmetric data with varying levels of kurtosis. As seen in Table 1, this model can be adapted to handle data dis- tributions with leptokurtic (high kurtosis) and platykurtic (low kurtosis) characteristics, making it an excellent tool for capturing the nuances of real-world data. One of the major advantages of the DGzHLo model lies in its ability to handle overdispersed and underdispersed data, which are common in many fields, including economics, engineer- ing, and epidemiology. Overdispersion occurs when the variance exceeds the mean, while underdispersion is observed when the variance is smaller than the mean. The DGzHLo model’s flexibility in capturing these features makes it particularly useful for data sets where traditional models, like the Poisson or negative binomial distributions, may fall short. Furthermore, the DGzHLo model can also accommodate equidispersed data, where the variance is approximately equal to the mean. This is crucial for applications in various scientific and engineering domains where data often follow such distributions, and accurate modeling of these types can lead to better decision-making and more robust predictions. In the context of sustainable development and engineering sustainability data modeling, the DGzHLo model offers a significant advantage. Sustainable data often exhibit complex patterns, with varying dispersion, skewness, and kurtosis, especially in long-term sustain- ability studies where factors influencing the data may change over time. The model’s capacity to handle different dispersion levels and its application to a wide range of data types ensures that it can provide insights into the sustainability of processes, resources, M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 9 of 26 and systems. 3.2. Entropy Entropy is a measure of the uncertainty or unpredictability associated with a random variable, denoted as Z. It quantifies the level of disorder or information content within the distribution of Z (see Rényi, [24]). In the context of sustainable dispersion data, entropy plays a vital role in assessing the variability and uncertainty in sustainability metrics, such as resource consumption, environmental impact, or social sustainability indicators. By capturing the degree of spread or concentration of data, entropy helps to quantify the uncertainty inherent in these measures, which is essential for making informed decisions about sustainable practices. The Rényi entropy of the discrete random variable can be expressed as Iη(Z) = 1 1− η log ∞∑ z=0 fηz (z; γ, τ, θ), where η ∈ ]0,∞[ and η ̸= 1. The Shannon entropy can be defined by E [− log f(Z; γ, τ, θ)]. It is observed that the Shannon entropy can be calculated as a special case of the Rényi entropy when η → 1. For the DGzHLo model, the Iη(Z) can be formulated as Iη(Z) = 1 1− η −η τ log γ + log ∞∑ z=0 γ 1 τ [ 1− 1−e−θz 1+e−θz ]−τ − γ 1 τ [ 1− 1−e−θ(z+1) 1+e−θ(z+1) ]−τ η . (17) The DGzHLo model, with its ability to handle diverse dispersion characteristics, including overdispersion, underdispersion, and equidispersion, can be effectively applied to sustain- ability data. It provides a robust framework for modeling the uncertainty and variation in sustainability metrics, considering various levels of kurtosis and skewness. In fields such as environmental science, economics, and social policy, where sustainability data often exhibit complex dispersion patterns, the DGzHLo model offers a flexible and powerful tool for analyzing these uncertainties. Specifically, entropy in sustainable dispersion data can be used to understand how the probability distribution of a sustainability measure is spread out or concentrated. This can provide valuable insights into the stability or variability of the data, helping stakeholders make decisions that align with long-term sus- tainability goals. For example, entropy could be used to assess the unpredictability of environmental changes or the effectiveness of sustainable practices, guiding policymakers and businesses toward strategies that mitigate risk and uncertainty. 3.3. Mean time to failure, mean time between failure and availability In the context of discrete random variables, the DGzHLo model offers a valuable frame- work for evaluating system performance based on reliability metrics such as mean time to failure (MTTF), mean time between failures (MTBF), and availability (Av). MTTF represents the average time until the first failure of a non-repairable system, and for the DGzHLo model, it can be estimated using the expected failure time derived from the M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 10 of 26 distribution’s parameters. MTBF, which applies to repairable systems, indicates the av- erage time between two successive failures, and the DGzHLo model helps estimate this by analyzing failure rates across multiple cycles. When applied to sustainable dispersion data, the DGzHLo model’s capacity to handle varying dispersion characteristics including overdispersion, underdispersion, and equidis- persion becomes particularly useful. For instance, it can help analyze and model the variability in the time between system failures or the time until a certain sustainability threshold is reached. This is crucial in sustainability assessments, where understanding the lifespan and reliability of systems (such as renewable energy infrastructures or sustain- able production systems) is essential. The DGzHLo model, therefore, not only provides insights into the reliability and performance of such systems but also helps quantify the uncertainty and variation inherent in sustainability metrics, offering a robust tool for long- term planning and decision-making in sustainable practices. If T ∼ DGzHLo(t; γ1, τ1, θ1), then the MTBF is given as MTBF(T ) = −t ln(γ 1 τ1 {[ 1− 1−e−θ1(t+1) 1+e−θ1(t+1) ]−τ1 −1 } 1 ) ; t > 0, (18) whereas if T ∼ DGzHLo(t; γ2, τ2, θ2), then the MTTF can be listed as MTTF(T ) = γ − 1 τ2 2 ∞∑ t=1 γ 1 τ2 [ 1− 1−e−θ2t 1+e−θ2t ]−τ2 2 ; t > 0. (19) In the context of sustainability dispersion data, availability measures the proportion of time a system is operational, factoring in both its operational time and the downtime caused by repairs. For systems characterized by the DGzHLo distribution, availability can be assessed by analyzing the system’s uptime and the time between failures, both of which are influenced by the model’s parameters. This evaluation is particularly useful in sustainability assessments, where understanding system reliability and performance is essential for optimizing resource usage and ensuring long-term sustainability. The DGzHLo model provides a framework to quantify and manage these dynamics, offering valuable insights into system availability in sustainable practices, where Av(T ) = −1 t γ− 1 τ2 2 ∞∑ z=1 γ 1 τ2 [ 1− 1−e−θ2z 1+e−θ2z ]−τ2 2  ln(γ 1 τ1 {[ 1− 1−e−θ1(t+1) 1+e−θ1(t+1) ]−τ1 −1 } 1 )  ; t > 0. When utilized within the DGzHLo framework, these metrics provide a comprehensive understanding of a system’s reliability and performance, particularly in scenarios where failure times are discrete and repairs are necessary. Availability represents the probability that a component is operational at a given time and can be expressed as the ratio of MTTF to MTBF. This approach allows for a detailed analysis of system behavior, especially in systems that require maintenance and have discrete failure events. M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 11 of 26 3.4. Order statistics and L-moment statistics Order statistics (OS) play a significant role in various areas of statistical theory and practice. Consider a random sample Z1, Z2, ...,Zn drawn from the DGzHLo(z; τ, γ, θ) model, and let Z1:n, Z2:n,...,Zn:n represent the corresponding order statistics. The CDF for the ith order statistic Zi:n, for an integer value of z, can be expressed as follows Fi:n(z; γ, τ, θ) = n∑ k=i ( n k ) [Fi(z; γ, τ, θ)] k [1− Fi(z; γ, τ, θ)] n−k = n∑ k=i n−k∑ j=0 ∆j (n,k) [Fi(z; γ, τ, θ)] k+j = n∑ k=i n−k∑ j=0 ∆j (n,k) 1− γ 1 τ {[ 1− 1−e−θ(z+1) 1+e−θ(z+1) ]−τ −1 } k+j , (20) where ∆j (n,k) = (−1)j ( n k )( n− k j ) . The corresponding PMF of the ith OS, as given by Equation (9), can be represented as fi:n(z; γ, τ,Ω) = n∑ k=i n−k∑ j=0 k+j∑ m=0 ∆ (m,j) (n,k) f(z; γ m, τ, θ), (21) where ∆ (m,j) (n,k) = (−1)j+m ( n k )( n− k j )( k + j m ) . Thus, the uth moment of Zi:n can be written as Ψu i:n = E(Zu i:n) = ∞∑ z=0 n∑ k=i n−k∑ j=0 k+j∑ m=0 ∆ (m,j) (n,k) z uf(z; γm, τ, θ). (22) L-moments (LMs) are calculated as linear combinations of order statistics (OS). Introduced by Hosking and Wallis [9], LMs serve to summarize both theoretical distributions and observed data samples. LM statistics are widely used to compute sample statistics for data from specific regions or to assess the homogeneity or heterogeneity of proposed groupings of sites. Let Z(i|n) be ith largest observations in sample of size n, then the LMs can be take the form λ∗r = 1 r r−1∑ s=0 (−1)s ( r − 1 s ) E (Zr−s:r) . (23) From Equation (23), we get λ∗1 = E(Z1:1), λ ∗ 2 = 1 2E (Z2:2 + Z1:2), λ ∗ 3 = 1 3 [E (Z3:3 − Z2:3)− E (Z2:3 + Z1:3)] and λ∗4 = 1 4 {E [(Z4:4 − Z3:4) + (Z2:4 − Z1:4)]− 2E (Z3:4 − Z2:4)}. Then, we can define some statistical measures such as LM of mean, LM coefficient of variation, M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 12 of 26 LM coefficient of Sk and LM coefficient of ku in the form λ∗1, λ∗ 2 λ∗ 1 , λ∗ 3 λ∗ 2 and λ∗ 4 λ∗ 2 , respectively. The implementation of L-moments in the DGzHLo model involves using linear combi- nations of order statistics to capture key features of the distribution, such as location, dispersion, skewness, and kurtosis. These moments are particularly useful for modeling discrete probability distributions, as they provide a robust way to characterize the distri- bution’s shape, especially in the presence of heavy tails or skewed data. In the context of the DGzHLo model, L-moments are calculated from the order statistics of the data, which helps in estimating the model’s parameters. By comparing the sample L-moments to the theoretical L-moments of the DGzHLo model, the parameters are adjusted to best fit the data. This method enhances the model’s ability to accurately represent sustainability- related or other real-world data while maintaining robustness to extreme values. 4. Maximum Likelihood Estimation In this section, we estimate the unknown parameters of the DGzHLo model using the maximum likelihood method (MLE). Suppose Z1, Z2, ...,Zn be a random sample from the DGzHLo model. Then, the log-likelihood function (L) can be listed as L = −1 τ ln(γ) +n i=1 ln γ 1 τ {[ 1− 1−e−θzi 1+e−θzi ]−τ } − γ 1 τ {[ 1− 1−e−θ(zi+1) 1+e−θ(zi+1) ]−τ } . (24) The MLEs of the parameters γ, τ and θ can be derived by solving the nonlinear likelihood equations obtained by differentiating (24) with respect to γ, τ and θ. The components of the score vector, V(γ, τ, θ) = (∂L∂γ , ∂L ∂τ , ∂L ∂θ ) T are Vγ = −n τγ + 1 τγ n i=1 g2(zi)− g2(zi + 1) g1(zi) , (25) Vτ = −n ln(γ) τ2 − ln(γ) τ2 n i=1 g2(zi) [ τ ln(1− 1−e−θzi 1+e−θzi ) + 1 ] − g2(zi + 1) [ τ ln(1− 1−e−θ(zi+1) 1+e−θ(zi+1) ) + 1 ] g1(zi) (26) and Vθ = n i=1 g2(zi) [ 1− 1−e−θzi 1+e−θzi ]−1 [G(zi; θ)]θ − g2(zi + 1) [ 1− 1−e−θ(zi+1) 1+e−θ(zi+1) ]−1 [G(zi + 1; θ)]θ g1(zi) , (27) where [G(zi; θ)]θ = ∂ ∂θ [ 1− e−θzi 1 + e−θzi ] , g1(zi) = γ 1 τ {[ 1− 1−e−θzi 1+e−θzi ]−τ } − γ 1 τ {[ 1− 1−e−θ(zi+1) 1+e−θ(zi+1) ]−τ } , M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 13 of 26 and g2(zi) = γ 1 τ {[ 1− 1−e−θzi 1+e−θzi ]−τ }{[ 1− 1− e−θzi 1 + e−θzi ]−τ } . Setting the Equations (25-27) to zero and solving them gives the maximum likelihood estimates (MLEs) for the DGzHLo family parameters. Since these equations cannot be solved analytically, an iterative method such as the Newton-Raphson algorithm is needed to solve them numerically. In R, this can be implemented using packages like stats or optim, which provide functions for numerical optimization and solving such equations. 5. Simulation Results: Estimator Behavior This section presents the results from simulations conducted to evaluate the perfor- mance of the estimators for the parameters of the DGzHLo model. Using synthetic data generated from the proposed model with known parameters, we assess how accurately the maximum likelihood estimates (MLEs) perform in terms of bias, variance, and efficiency. The study considers various sample sizes and different parameter values to examine the robustness and consistency of the estimators under different conditions. The evaluation incorporates additional criteria such as mean-squared errors (MSEs), mean relative error (MRE) and other relevant performance metrics to provide a comprehensive understand- ing of the MLEs’ accuracy and reliability. These findings offer valuable insights into the practical applicability of the estimators for real-world data. The assessment is based on a thorough simulation study, which includes a range of scenarios to explore the behavior of the DGzHLo model’s estimators. (i) Generate 1000 samples of size n = 20, 50, 150, 300, 500 from three Scheme, Schema I: DGzHLo(0.1, 0.5, 0.7), Schema II: DGzHLo(0.3, 0.7, 0.3) and Schema III: DGzHLo(0.5, 0.2, 0.9). (ii) Compute the MLEs for the 1000 samples, say âj and b̂j for j = 1, 2, ..., 1000. (iii) Compute the biase and MSEs, and MREs where bias(ϖ) = 1 1000 1000∑ j=1 (ϖ̂j −ϖj) , MSE(ϖ) = 1 1000 1000∑ j=1 (ϖ̂j −ϖj) 2 , and MRE(ϖ) = 1 1000 1000∑ j=1 ∣∣∣∣ϖ̂j −ϖj ϖj ∣∣∣∣ . The empirical results are shown in Figures 3-5. M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 14 of 26 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n B ia s 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n B ia s 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n B ia s γ τ θ 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 n M S E 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 n M S E 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 n M S E γ τ θ 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 n M R E 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 n M R E 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 n M R E γ τ θ Figure 3: The parameter biases, MSE, and MRE for Schema I. 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n B ia s 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n B ia s 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n B ia s γ τ θ 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 n M S E 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 n M S E 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 n M S E γ τ θ 100 200 300 400 500 0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 n M R E 100 200 300 400 500 0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 n M R E 100 200 300 400 500 0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 n M R E γ τ θ Figure 4: The parameter biases, MSE, and MRE for Schema II. M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 15 of 26 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .2 n B ia s 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .2 n B ia s 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .2 n B ia s γ τ θ 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n M S E 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n M S E 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n M S E γ τ θ 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 n M R E 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 n M R E 100 200 300 400 500 0 .0 0 .2 0 .4 0 .6 0 .8 n M R E γ τ θ Figure 5: The parameter biases, MSE, and MRE for Schema III. The simulation results highlight the excellent performance of the estimators in terms of bias, MSE, and MRE for larger sample sizes. In Schemas I, II, and III, the bias of the estimators decreases as the sample size increases, with bias values approaching zero, indicating that the estimators become nearly unbiased as the sample size grows. In Schema I, the estimators exhibit minimal bias across all sample sizes, and both the MSE and MRE steadily decrease as n increases, suggesting improved accuracy and precision with larger datasets. Similarly, in Schemas II and III, the bias remains consistently low and continues to decrease as sample sizes increase, demonstrating the robustness of the model’s estimators. The MSE and MRE also show a clear reduction as n grows, indicating that the estimators become more efficient with larger samples. Overall, across all schemas, the estimators perform exceptionally well, showing low bias, MSE, and MRE, especially as sample sizes increase. These findings underscore the effectiveness and reliability of the MLE method for the DGzHLo model in practical applications, highlighting the robustness of the estimation process and its suitability for modeling real-world data. 6. Sustainable Data Analysis Sustainable dispersion data modeling using discrete probability distributions helps manage variability in count data within engineering applications, especially in reliability analysis, system maintenance, and quality control. By applying discrete distributions, it M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 16 of 26 effectively captures fluctuations in failure rates and component lifespans. These models account for overdispersion, where the variance surpasses the mean, a typical feature of engineering systems with high variability. In this section, we demonstrate the practical significance of the DGzHLo distribution through two real data applications. The per- formance of the fitted models is assessed using multiple criteria, including −L, Akaike Information Criterion (AIC), Corrected Akaike Information Criterion (CAIC), Chi-square (χ2) along with its P-value, and the Kolmogorov-Smirnov (K-S) test with its correspond- ing P-value. The DGzHLo distribution will be compared against several competing models presented in Table 2. Table 2. The competitive models of the DGzHLo distribution. Distribution Abbreviation Discrete Weibull DW Exponentiated discrete Weibull EDW Discrete inverse Weibull DIW Discrete exponential Geo Discrete generalized exponential type II DGEx-II Discrete Rayleigh DR Discrete inverse Rayleigh DIR Discrete Lindley DLi Exponentiated discrete Lindley EDLi Discrete Lindley type II DLi-II Discrete extension Xgamma DExg Discrete log-logistic DLLc Discrete Lomax DLo Two- parameter discrete Burr type XII DB-XII Discrete Pareto DPa Discrete inverted Nadarajah–Haghighi DINH Poisson Poi 6.1. Dataset I: Computer breakdowns Industrial engineering focuses on optimizing complex systems and processes to enhance efficiency and productivity across various sectors. It involves analyzing workflows, reduc- ing waste, and ensuring the effective utilization of resources. A key aspect of industrial engineering is managing machine performance, where engineers monitor and address is- sues such as computer breakdowns that occur over time. These breakdowns can disrupt production schedules and cause downtime, significantly impacting overall productivity. To mitigate this, industrial engineers employ predictive maintenance strategies and data analysis to anticipate equipment failures, thus minimizing downtime and improving the reliability of manufacturing systems. This section presents a dataset that tracks the num- ber of computer breakdowns over 128 consecutive weeks of operation (see Hand et al., [2]). Figure 6 displays the distribution of dataset I. The results show a moderate right- M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 17 of 26 ward skew, likely due to some extreme observations, as well as a relatively high level of variability. The kurtosis suggests that the distribution has slightly heavier tails than a normal distribution, indicating some overdispersion. Table 3 presents the goodness-of-fit 0 20 40 60 80 100 0 5 1 0 1 5 2 0 Scatter | Scores Height S c o re s −2 −1 0 1 2 0 5 1 0 1 5 2 0 Normal Q−Q Plot Theoretical Quantiles S a m p le Q u a n ti le s 0 5 10 15 20 Strip Plot z 0 5 10 15 20 Box Plot z 0 2 4 6 8 10 12 0 .0 0 0 .1 0 0 .2 0 Relative Frequencies z 0 5 1 0 1 5 2 0 1 Violin Plot Figure 6: Non-parametric plots for dataset I. test results for dataset I at a significance level of 0.05, evaluating the performance of sev- eral statistical models applied to the data. Figure 7 visually complements these findings by illustrating how each model fits the observed data. It compares the predicted values of each model with the actual data, highlighting any discrepancies and providing a clear overview of the overall fit. Although the DGzHLo, DExg, and Geo models all perform adequately in modeling the data, the DGzHLo model proves to be the most effective. It provides the best fit and demonstrates superior accuracy in capturing the underlying distribution of the data, offering a more precise representation than the other models. M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 18 of 26 Table 3. The goodness-of-fit test for dataset I. X OF DGzHLo DExg DINH DIW DB-XII Geo Poi DPa DR 0 15 16.32 19.04 12.40 9.69 19.89 25.53 2.31 47.81 3.64 1 19 19.47 16.47 30.16 33.09 36.59 20.43 9.27 19.19 10.30 2 23 18.35 16.06 19.94 23.10 19.05 16.36 18.61 10.76 15.31 3 14 15.86 15.23 12.78 14.48 10.69 13.09 24.91 7.02 18.04 4 15 13.23 13.61 8.72 9.54 6.83 10.49 25.01 5.00 18.44 5 10 10.81 11.52 6.29 6.64 4.76 8.39 20.09 3.77 16.92 6 8 8.67 9.33 4.73 4.84 3.52 6.72 13.44 2.97 14.17 7 4 6.82 7.28 3.69 3.65 2.72 5.38 7.71 2.40 10.94 8 6 5.26 5.52 2.95 2.85 2.17 4.31 3.87 1.99 7.84 9 2 3.97 4.09 2.41 2.27 1.77 3.46 1.74 1.69 5.23 10 3 2.92 2.97 2.01 1.85 1.48 2.75 0.69 1.45 3.26 11 3 2.10 2.12 1.69 1.53 1.25 2.21 0.25 1.26 1.89 12 2 1.49 1.49 1.46 1.28 1.08 1.77 0.08 1.11 1.04 +13 4 2.73 3.27 18.76 13.183 16.19 7.11 0.01 21.59 0.99 Total 128 128 128 128 128 128 128 128 128 128 −l 310.657 318.953 331.931 330.446 342.581 320.703 384.974 369.766 347.148 MLEγ 0.869 0.497 1.244 0.076 0.785 0.801 4.016 0.509 0.972 MLEτ 0.045 0.513 1.634 1.235 3.309 − − − − MLEθ 1.432 − − − − − − − − χ2 4.788 7.093 20.199 18.709 40.183 11.651 88.995 94.971 49.897 P.value 0.686 0.527 0.003 0.005 0 0.234 0 0 0 M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 19 of 26 0 2 4 6 8 10 0 2 0 4 0 DGzHLo z P M F 0 2 4 6 8 10 0 2 0 4 0 z P M F Obs Exp 0 2 4 6 8 10 0 2 0 4 0 DExg z P M F 0 2 4 6 8 10 0 2 0 4 0 z P M F Obs Exp 0 2 4 6 8 10 0 2 0 4 0 DINH z P M F 0 2 4 6 8 10 0 2 0 4 0 z P M F Obs Exp 0 2 4 6 8 10 0 2 0 4 0 DIW z P M F 0 2 4 6 8 10 0 2 0 4 0 z P M F Obs Exp 0 2 4 6 8 10 0 2 0 4 0 DB−XII z P M F 0 2 4 6 8 10 0 2 0 4 0 z P M F Obs Exp 0 2 4 6 8 10 0 2 0 4 0 Geo z P M F 0 2 4 6 8 10 0 2 0 4 0 z P M F Obs Exp 0 2 4 6 8 10 0 2 0 4 0 Poi z P M F 0 2 4 6 8 10 0 2 0 4 0 z P M F Obs Exp 0 2 4 6 8 10 0 2 0 4 0 DPa z P M F 0 2 4 6 8 10 0 2 0 4 0 z P M F Obs Exp 0 2 4 6 8 10 0 2 0 4 0 DR z P M F 0 2 4 6 8 10 0 2 0 4 0 z P M F Obs Exp Figure 7: The fitted PMFs for dataset I. 6.2. Dataset II: Failure times This dataset represents the failure times (in weeks) of 50 devices subjected to a life test, as discussed in Bebbington et al. [14], focusing on sustainable dispersion data. In this analysis, we will compare the fit of the DGzHLo distribution with several competing models to assess its performance. Non-parametric plots, shown in Figure 8, provide a visual representation of the data’s distribution. It is important to note that the data ex- hibits asymmetry, with some outlier observations that may influence the model fitting and interpretation. These characteristics will be considered when evaluating the effectiveness of the DGzHLo distribution compared to other models. M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 20 of 26 0 20 40 60 80 100 0 4 8 Scatter | Scores Height S c o re s −2 −1 0 1 2 0 4 8 Normal Q−Q Plot Theoretical Quantiles S a m p le Q u a n ti le s 0 2 4 6 8 10 Box Plot z 0 4 8 1 Violin Plot Figure 8: Non-parametric plots for dataset II. The MLEs, along with their corresponding standard errors (Std-er), and the goodness- of-fit statistics are presented in Tables 4 and 5 at a significance level of 0.05, respectively. Table 4. The MLEs with their corresponding Std-er for data set II. Model ↓ Parameter → γ τ θ MLE Std-er MLE Std-er MLE Std-er DGzHLo 0.153 0.028 3.211 0.117 0.009 0.001 EDW 0.989 0.164 1.139 3.227 0.784 3.053 DW 0.981 0.011 1.023 0.131 − − DIW 0.018 0.013 0.582 0.061 − − DLi-II 0.969 0.005 0.058 0.027 − − EDLi 0.972 0.005 0.480 0.087 − − DLLc 1.0 0.321 0.439 0.062 − − DPa 0.739 0.032 − − − − Table 5. The goodness-of-fit statistics for data set II. Statistic ↓ Model → DGzHLo EDW DW DIW DLi-II EDLi DLLc DPa −L 235.0 240.2 241.6 261.9 240.6 240.3 294.9 275.9 AIC 476.0 486.7 487.2 527.8 485.2 484.6 593.8 553.7 CAIC 476.6 487.2 487.5 528.1 485.4 484.8 594.0 553.8 K-S 0.171 0.195 0.187 0.258 0.186 0.195 0.535 0.335 P-value 0.106 0.045 0.061 0.0026 0.064 0.045 < 0.001 < 0.001 M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 21 of 26 Regarding Table 5, it is evident that the DGzHLo, EDW, DW, DLi-II, and EDLi models all perform reasonably well in analyzing this data, with the DGzHLo model showing a P- value greater than 0.05. However, our goal is always to identify the best model for optimal data evaluation. Based on the −L, AIC, CAIC, K-S statistics, and their corresponding P-values, we conclude that the DGzHLo model provides the best fit among all the tested models. This is evident as it has the smallest values for −L, AIC, CAIC, and K-S statistics, along with the highest P-value. Figure 9 further supports the findings presented in Table 5, where ”PP” refers to the probability-probability plot. 0 20 40 60 80 0 .0 1 0 .0 3 0 .0 5 HRF Plot z H R F 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 PP Plot Obs E x p 0 20 40 60 80 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 z F (z ) 0 20 40 60 80 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 Empirical CDF F (z ) Figure 9: The HRF, PP plot, and estimated CDFs for dataset II. It is evident that the dataset, dataset II, is best modeled by the DGzHLo distribution. 6.3. Dataset III: Electronic components This data, reported in Lawless [7], presents the failure times for a sample of 15 elec- tronic components in an accelerated life test, focusing on sustainable dispersion data. Non-parametric plots are shown in Figure 10. The data is asymmetric, without any ex- treme observations. For this dataset, we will compare the fit of the DGzHLo distribution with several competitive models. The MLEs along with their corresponding Std-er, as well as the goodness-of-fit statistics, are presented in Tables 6 and 7, respectively. M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 22 of 26 2 4 6 8 10 12 14 0 2 0 4 0 6 0 Scatter | Scores Height S c o re s −1 0 1 0 2 0 4 0 6 0 Normal Q−Q Plot Theoretical Quantiles S a m p le Q u a n ti le s 0 10 20 30 40 50 60 Box Plot z 0 2 0 4 0 6 0 1 Violin Plot Figure 10: Non-parametric plots for dataset III. Table 6. The MLEs with their corresponding Std-er for data set III. Model ↓ Parameter → γ τ θ MLE Std-er MLE Std-er MLE Std-er DGzHLo 0.959 0.019 0.037 0.026 0.550 0.004 DGzEx 0.587 0.023 0.588 0.041 0.039 0.002 DEx 0.965 0.009 − − − − DGEx-II 0.956 0.013 1.491 0.535 − − DR 0.999 2.58× 10−4 − − − − DIR 1.8× 10−7 0.055 − − − − DIW 2.2× 10−4 7.75× 10−4 0.875 0.164 − − DLo 0.012 0.039 104.506 84.409 − − DB-XII 0.975 0.051 13.367 27.785 − − DPa 0.720 0.061 − − − − Table 7. The goodness-of-fit statistics for data set III. Model Statistic DGzHLo DGzEx Geo DGEx-II DR DIR DIW DLo DB-XII DPa −L 62.934 63.804 65.000 64.420 66.394 89.096 68.703 65.864 75.724 77.402 AIC 131.868 133.608 134.000 134.839 134.788 180.192 141.406 135.728 155.448 156.805 CAIC 134.049 135.789 136.308 135.839 136.096 180.499 142.406 136.728 156.448 157.112 K-S 0.116 0.120 0.177 0.129 0.216 0.698 0.209 0.205 0.388 0.405 P-value 0.974 0.963 0.673 0.937 0.433 9.1× 10−7 0.482 0.491 0.015 0.009 M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 23 of 26 Regarding Table 7, the DGzHLo distribution emerges as the best model among all the tested models. Figure 11 further supports the results presented in Table 7. It is clear that the dataset likely originates from the DGzHLo model. 0 10 20 30 40 50 60 0 .0 2 0 .0 4 0 .0 6 0 .0 8 HRF Plot z H R F 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 PP Plot Obs E x p 0 10 20 30 40 50 60 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 z F (z ) 0 10 20 30 40 50 60 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 Empirical CDF F (z ) Figure 11: The HRF, P-P plot, estimated CDFs for dataset III. 7. Concluding Remarks and Future Work This article proposes a new discrete model, the DGzHLo, and examines its statistical properties. The DGzHLo model is found to offer flexibility in modeling data with vari- ous skewness types, including negative, positive, and symmetric shapes. This versatility makes it particularly useful for modeling sustainability dispersion data, where variabil- ity often follows diverse patterns. The DGzHLo model accommodates increasing hazard rate functions, making it highly applicable for both engineering and medical applications, where such rate functions are crucial for reliability analysis and predicting system failures. Additionally, the model can handle over-dispersion, equi-dispersion, and under-dispersion, accurately capturing a wide range of data variability often observed in real-world appli- cations. Another key feature of the DGzHLo model is its ability to account for different levels of kurtosis, including leptokurtic and platykurtic distributions. This ability makes it useful in handling data with varying tail behaviors, often encountered in sustainability studies. Furthermore, the model effectively handles zero-inflation and non-inflation data, particularly in the presence of extreme observations, a common challenge in many fields, including environmental studies and economics. The parameters of the DGzHLo model were estimated using the MLE method. A simulation study was carried out to evaluate the model’s performance, and the results demonstrated that the MLE method effectively esti- mated the model parameters with high accuracy. The flexibility of the DGzHLo model was further showcased through the analysis of three distinct datasets, where it outperformed other models in terms of both fit and predictive accuracy. This confirmed its potential as a reliable tool for analyzing complex sustainability dispersion data. Looking ahead, future research will focus on extending the DGzHLo model to include time-varying parameters and exploring its application in additional fields such as environmental monitoring, climate change studies, and economics. Further refinements to estimation techniques, particularly through the use of Bayesian methods, will aim to improve the model’s robustness and com- M. S. Eliwa, W. E. Alghanem, M. A. Alenizi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6144 24 of 26 putational efficiency. Additionally, integrating the DGzHLo model with machine learning approaches will be explored to enable real-time predictive analytics across various indus- tries, further enhancing its applicability and utility in complex data analysis. Acknowledgements The authors gratefully acknowledge Qassim University, represented by the Deanship of Graduate Studies and Scientific Research, on the financial support for this research under the number (QU-J-UG-2-2025-52613) during the academic year 1446 AH / 2024 AD. • Data Availability Statement: The data sets are available in the paper. • Conflicts of Interest: The authors declare no conflict of interests. References [1] A. Alzaatreh, C. Lee, and F. Famoye. A new method for generating families of con- tinuous distributions. Metron, 71:63–79, 2013. [2] D. J. Hand, F. Daly, K. M. Conway, D. Lunn, and E. Ostrowski. A handbook of small data sets, Chapman & Hall. DOI: 10.1201/9780429246579, 1993. [3] E. Abd EL-Hady, M. A. Hegazy, and A. A. EL-Helbawy. 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