This paper presents a new two-parameter discrete probability distribution called the discrete modified weighted exponential (DM-WE) distribution. A thorough examination of its mathematical and structural aspects is performed, resulting in closed-form formulas for essential statistical attributes. This encompasses the form, unimodality, and characteristics of both the hazard rate and reversed hazard rate functions. Additionally, essential foundational aspects, including parameter identifiability, asymptotic tail behavior, mode, quantile function, random data generation, and moments, are derived. The DM-WE distribution demonstrates a discontinuous growing failure rate and features a highly adaptable probability mass function that can represent various distributional forms. The examination of dispersion, zero-modification indices, and entropy metrics indicates the model is specifically tailored for over-dispersed, leptokurtic, and zero-inflated count data environments. Numerical investigations show that the coefficients of skewness and kurtosis exhibit a pronounced bathtub-shaped pattern within the explored parameter grid, indicating significant variability in tail thickness and asymmetry. Order statistics and their descriptive characteristics are also determined. The maximum likelihood estimation framework is established for parameter estimation, along with its asymptotic features. A thorough Monte Carlo simulation analysis is conducted to assess and compare the finite-sample performance and efficacy of the estimators. The practical applicability and superiority of the DM-WE distribution are proved using a comprehensive model comparison methodology and goodness-of-fit criteria applied to real-world datasets, establishing its effectiveness over existing rival discrete models.
Citation: Mohamed S. Eliwa, Mahmoud El-Morshedy, Afrah Al-Bossly, Mohamed El-Dawoody, M. E. Sobh. The discrete modified weighted exponential distribution: mathematical theory with applications[J]. AIMS Mathematics, 2026, 11(11): 36469-36516. doi: 10.3934/math.20261411
This paper presents a new two-parameter discrete probability distribution called the discrete modified weighted exponential (DM-WE) distribution. A thorough examination of its mathematical and structural aspects is performed, resulting in closed-form formulas for essential statistical attributes. This encompasses the form, unimodality, and characteristics of both the hazard rate and reversed hazard rate functions. Additionally, essential foundational aspects, including parameter identifiability, asymptotic tail behavior, mode, quantile function, random data generation, and moments, are derived. The DM-WE distribution demonstrates a discontinuous growing failure rate and features a highly adaptable probability mass function that can represent various distributional forms. The examination of dispersion, zero-modification indices, and entropy metrics indicates the model is specifically tailored for over-dispersed, leptokurtic, and zero-inflated count data environments. Numerical investigations show that the coefficients of skewness and kurtosis exhibit a pronounced bathtub-shaped pattern within the explored parameter grid, indicating significant variability in tail thickness and asymmetry. Order statistics and their descriptive characteristics are also determined. The maximum likelihood estimation framework is established for parameter estimation, along with its asymptotic features. A thorough Monte Carlo simulation analysis is conducted to assess and compare the finite-sample performance and efficacy of the estimators. The practical applicability and superiority of the DM-WE distribution are proved using a comprehensive model comparison methodology and goodness-of-fit criteria applied to real-world datasets, establishing its effectiveness over existing rival discrete models.
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Figures(14) / Tables(19)
Mohamed S. Eliwa, Mahmoud El-Morshedy, Afrah Al-Bossly, Mohamed El-Dawoody, M. E. Sobh. The discrete modified weighted exponential distribution: mathematical theory with applications[J]. AIMS Mathematics, 2026, 11(11): 36469-36516. doi: 10.3934/math.20261411
The PMF plots of the DM-WE distribution for various values of $ \alpha $ and $ \beta $
The HRF and RHRF plots of the DM-WE distribution for various values of $ \alpha $ and $ \beta $
Mean, variance, DI, ZI, CS, and CK of the DM-WE distribution over the full parameter space $ (\alpha, \beta) $
CS and CK versus $ \alpha $ for five values of $ \beta $; markers show the minima
Simulation performance metrics across sample size
Monte Carlo simulation meta-analysis: error decomposition and interval coverage matrix
Exploratory data analysis of the COVID-19 daily deaths dataset, reflecting a right-skewed distribution, discrete nature, and an increasing hazard behavior
Comparative visual analysis verifying the goodness-of-fit for the proposed DM-WE against competing distributions (COVID-19 dataset)
Comprehensive diagnostic plots confirming the precision of the DM-WE distribution fit to the COVID-19 dataset
Graphical validation of MLEs properties for the COVID-19 dataset
Exploratory data analysis identifying the nonparametric characteristics of the European corn borer dataset
Graphical comparison emphasizing the lowest cumulative error achieved by the DM-WE distribution (corn borer dataset)
Diagnostic goodness-of-fit plots verifying the robustness of the DM-WE distribution on the corn borer dataset
Graphical validation of the MLE properties associated with the corn borer dataset estimations
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