New Advances in Distribution Theory and Its Applications

In recent years, the prolific development of new statistical distributions via various techniques has often led to increased complexity without necessarily enhancing model flexibility or parameter interpretability, a trend that has sometimes diminished the prac-tical relevance of distribution theory. This Special Issue therefore features contributions on meaningful advancements in statistical distribution theory, with a focus on models that are both flexible and interpretable. By emphasizing these priorities, the Special Issue aims to connect theoretical developments with practical applications, so that new models are not only mathematically robust but also accessible and useful for applied work across diverse fields. This collection presents different contributions reflecting the evolving land-scape of distribution theory. Balancing theoretical development with real-world rele-vance, the articles propose models designed to address the challenges of contemporary data analysis, where applicability emerges as a fundamental aspect.

The Study by Contribution 1 introduces a novel discrete analog of the Hjorth distri-bution. By transforming this continuous distribution into the discrete domain, the authors maintain important theoretical properties such as flexibility and tractability, while in-creasing its applicability to count data commonly encountered in practice, also in the pres-ence of censored data. The paper thoroughly derives key statistical properties and com-pares the new distribution with existing discrete models. Moreover, the authors demon-strate the model’s practical utility through applications to real datasets, highlighting its ability to provide a better fit and deeper insight into discrete phenomena.

Contribution 2 proposes two bivariate extensions of the weighted discretized Fré-chet–Weibull distribution. The two proposed models are generated by using minimum and maximum operators to capture complex dependence structures between two discrete random variables. The paper rigorously derives the mathematical foundations of the pro-posed models, exploring their distributional and dependence properties. A comprehen-sive statistical analysis based on real datasets underscores the practical relevance and flex-ibility of these models in representing bivariate discrete data.

Contribution 3 introduces a new quantile regression model based on the unit ratio-Weibull (URW) distribution, aimed at identifying the factors influencing the COVID-19 mortality rate in Latin America. By examining socio-economic, health system, and demo-graphic variables, the authors identify key factors driving the mortality rate differences. Their findings provide crucial insights for policymakers aiming to improve public health preparedness and responses during pandemics, making a significant contribution to the intersection of statistical modeling and public health analysis.

Contribution 4 develops a novel class of models designed to address the common problem of count data exhibiting an excess of zeros and ones. The proposed models inte-grate a continuous–discrete mixture distribution with covariates, allowing them to accurately represent the complex dynamics of data. The authors formalize the properties, derive estimation procedures, and validate the models through applications to empirical datasets.

Contribution 5 develops surveys a wide range of wrapped distributions for modeling circular data, such as angles or time-of-day measurements. Covering 45 continuous and 10 discrete wrapped distributions, the paper systematically presents their probability functions, cumulative distributions, trigonometric moments, and key descriptive statistics including mean direction and resultant length. The review also proposes some applica-tions obtained by using an R package (version 4.4.1) that facilitates fitting these models, thus serving as a foundational resource for researchers and practitioners working with circular data.

Various stochastic representations for the zero–one-inflated Poisson Lindley distri-bution have been studied by Contribution 6. The authors describe four different stochastic representations, provide explicit formulas for moments and conditional distributions, and propose some hypothesis tests to investigate the presence of one-inflation in addition to a fixed-rate parameter.

A simulation study is used to investigate the hypotheses and its corresponding like-lihood ratio tests, suggesting that all tests are powerful and able to properly handle type I error rates, under a reasonable sample size.

A flexible two-parameter distribution to model data bounded within the unit interval is proposed by Contribution 7. Derived from the exponential distribution, the unit expo-nential distribution can capture both positive and negative skewness. The paper discusses the mathematical properties, including moments and hazard functions, and develops maximum likelihood estimation methods. Applications to environmental and engineer-ing datasets highlight the model’s superior fitting capabilities compared to existing alter-natives.

Contribution 8 introduces the NODAL G-classes, new flexible families of distribu-tions useful for modeling a wide variety of hazard shapes—including increasing, decreas-ing, bathtub, and J-shaped hazards—in both continuous and discrete contexts. The au-thors provide the theoretical foundation, propose maximum likelihood estimation tech-niques, and validate their models on diverse real datasets, showing improved fit if com-pared with well-established competitive models.

Contribution 9 develops a general framework for unit distributions and introduce the Unit-Dagum models, a class of distributions for data defined on the unit interval. Two new distributions within this class are obtained by applying different transformations to Dagum random variable. Moreover, by considering the possibility of reparametrizing the distributions to express them in terms of indicators of interest, a regression approach for response variables defined on the unit interval is explored. The resulting models appear to be highly competitive when compared with the most commonly used regression mod-els for this type of data, such as Beta regression.

Contribution 10 proposes a novel bivariate Poisson model derived from the bivariate Bernoulli distribution that facilitates both positive and negative correlations, an advance-ment over some traditional bivariate Poisson models which typically only capture posi-tive dependence. The authors explore the statistical properties of this model, develop maximum likelihood and moment-based estimation methods, and explore its applicabil-ity on healthcare utilization data. Their results highlight the model’s improved flexibility and accuracy in capturing dependence structures in correlated count data.

Finally, Contribution 11 introduces the ExpKum-G class, a finite mixture of exponen-tiated Kumaraswamy-G distributions. Focusing on the exponentiated Kumaraswamy–Weibull sub-model, the authors derive several statistical properties and propose both maximum likelihood and Bayesian estimation methods under progressive Type II censoring. Simulation studies demonstrate the effectiveness of these estimation strategies, showing good performance in terms of bias and mean squared error. The application to bladder cancer survival data confirms the usefulness of the proposed model under differ-ent censoring schemes.

We would like to express our sincere gratitude to all the authors who have contributed original works in line with the objectives of this Special Issue. Advancing methodological developments in the field of distribution theory, while pursuing goals such as model flexibility and parameter interpretability, is by no means a simple task. For this reason, we deeply appreciate the studies and proposals presented in this Special Issue.

We would also like to extend our thanks to those authors whose manuscripts, though not accepted through the review process, demonstrated genuine interest in the topics addressed and showed commendable commitment and active participation.