This study introduced a two-parameter zero-inflated discrete random variable distribution designed to model failure profiles in zero-inflated, dispersed datasets, commonly found in biological engineering and reliability analysis. The proposed distribution combined traditional count models, such as Poisson, Lindley, or negative binomial, with a probability mass at zero, providing a robust framework for addressing excess zeros and the underlying dispersion of data. The mathematical foundation of the distribution was derived with an emphasis on its statistical and reliability properties. The probability mass function was applicable to datasets with asymmetric dispersion and varying kurtosis structures. In addition, the hazard rate function was used to analyze failure rate behaviors, capturing patterns such as increasing, decreasing, and bathtub-shaped failure rates, often encountered in real-world datasets. Also, characterization of the proposed distribution was explored based on conditional expectation and the hazard rate function. Parameter estimation techniques were proposed, alongside computational simulations, to identify the most consistent estimators for data modeling. The goodness of fit of the proposed model was rigorously evaluated by comparing it with existing count models, demonstrating its superior ability to model zero-inflated, overdispersed data. Finally, the practical application of the new distribution was demonstrated using real-life biological engineering datasets, highlighting its effectiveness and flexibility in modeling complex zero-inflated data across various failure profiles and reliability contexts.

The intertwining relationship between sustainability and discrete probability distributions found its significance in decision-making processes and risk assessment frameworks. Count data modeling and its practical applications have gained attention in numerous research studies. This investigation focused on a particular discrete distribution characterized by a single parameter obtained through the survival discretization method. Statistical attributes of this distribution were accurately explicated using generalized hypergeometric functions. The unveiled characteristics highlighted its suitability for analyzing data displaying "right-skewed" asymmetry and possessing extended "heavy" tails. Its failure rate function effectively addressed scenarios marked by a consistent decrease in rates. Furthermore, it proved to be a valuable tool for probabilistic modeling of over-dispersed data. The study introduced various estimation methods such as maximum product of spacings, Anderson-Darling, right-tail Anderson-Darling, maximum likelihood, least-squares, weighted least-squares, percentile, and Cramer-Von-Mises, offering comprehensive explanations. A ranking simulation study was conducted to evaluate the performance of these estimators, employing ranking techniques to identify the most effective estimator across different sample sizes. Finally, real-world sustainability engineering and medical datasets were analyzed to demonstrate the significance and application of the newly introduced model.

This study introduces a novel and adaptable bimodal class of two-piece skew-normal distributions, specifically designed to accommodate datasets with up to two modes. This paper comprehensively examines the analytical attributes of the suggested model, encompassing its cumulative distribution function, moments, moment generating function, Rényi entropy, reliability metrics with aging intensity, and other critical statistical properties. The model effectively captures asymmetric behavior, accommodating both negative and positive skewness, and is particularly well suited for leptokurtic data characterized by an increasing hazard rate. Furthermore, it adeptly manages both over- and under-dispersed data, enhancing its relevance across many domains. To improve its adaptability, extensions of the distribution concerning location and scale are also devised. Parameter estimation is conducted using the maximum likelihood approach. A detailed simulation study assesses the performance of the estimators, illustrating their asymptotic consistency and efficiency. The practical utility of the suggested distribution is demonstrated through applications to real-world datasets, where it regularly outperforms multiple existing rival models in goodness-of-fit. Finally, a likelihood ratio test is utilized to statistically validate the superiority of the presented model compared to its nested alternatives.

This paper introduces a novel and highly flexible two-parameter discrete distribution designed for modeling complex count data. We comprehensively derive its statistical, reliability, and actuarial properties, establishing key metrics including moments, entropy, stochastic orders, and risk measures such as value-at-risk and tail-value-at-risk. The proposed model is particularly adept at capturing right-skewed, overdispersion data characterized by outliers and varying kurtosis. Notably, its hazard rate function accommodates diverse shapes, including increasing, decreasing, unimodal, bathtub, and J-shaped, while asymptotically approaching a constant to exhibit geometric-like memoryless properties. Model parameters are estimated via the maximum likelihood method for both complete and censored datasets. Additionally, we develop computationally efficient Monte Carlo simulation strategies leveraging these versatile hazard profiles. Empirical applications across actuarial science, clinical nephrology, and agricultural entomology demonstrate the model's superior efficacy in capturing extreme values when compared to existing competing distributions.