Roger Tovar-Falón

The modeling of data involving proportions, confined to a unit interval, is crucial in diverse research fields. Such data, expressing part-to-whole relationships, span from the proportion of individuals affected by diseases to the allocation of resources in economic sectors and the survival rates of species in ecology. However, modeling these data and interpreting information obtained from them present challenges, particularly when there is high zero–one inflation at the extremes of the unit interval, which indicates the complete absence or full occurrence of a characteristic or event. This inflation limits traditional statistical models, which often fail to capture the underlying distribution, leading to biased or imprecise statistical inferences. To address these challenges, we propose and derive the skew-normal zero–one inflated (SNZOI) models, a novel class of asymmetric regression models specifically designed to accommodate zero–one inflation presented in the data. By integrating a continuous-discrete mixture distribution with covariates in both continuous and discrete parts, SNZOI models exhibit superior capability compared to traditional models when describing these complex data structures. The applicability and effectiveness of the proposed models are demonstrated through case studies, including the analysis of medical data. Precise modeling of inflated proportion data unveils insights representing advancements in the statistical analysis of such studies. The present investigation highlights the limitations of existing models and shows the potential of SNZOI models to provide more accurate and precise inferences in the presence of zero–one inflation.

In problems involving binary classification, researchers often encounter data suitable for modeling dichotomous responses. These scenarios include medical diagnostics, where outcomes are classified as “disease” or “no disease”, and credit scoring in finance, determining whether a loan applicant is “high risk” or “low risk”. Dichotomous response models are also useful in many other areas for estimating binary responses. The logistic regression model is one option for modeling dichotomous responses; however, other statistical models may be required to improve the quality of fits. In this paper, a new regression model is proposed for cases where the response variable is dichotomous. This novel, non-linear model is derived from the cumulative distribution function of the proportional hazard distribution, and is suitable for modeling binary responses. Statistical inference is performed using a classical approach with the maximum likelihood method for the proposed model. Additionally, it is demonstrated that the introduced model has a non-singular information matrix. The results of a simulation study, along with an application to student dropout data, show the great potential of the proposed model in practical and everyday situations.

In this article, we introduce a novel bivariate probability distribution that is absolutely continuous. Considering the Farlie–Gumbel–Morgenstern (FGM) copula and the unit-Weibull distribution, we can obtain a bivariate unit-Weibull distribution. We evaluate the main properties of the new proposal and use two estimation methods to estimate the parameter for the bivariate probability distribution. A brief Monte Carlo simulation study is conducted to assess the behavior of the employed estimation method and the characteristics of the estimators. Ultimately, as an illustration, a real-life application is presented, demonstrating the utility of the proposal.

One of the most widely known probability distributions used to explain the probabilistic behavior of positive data is the log-normal (LN). Although the LN distribution is capable of adjusting data types, it is not always fully true that the model manages to adequately model the behavior of the response of interest since in some cases, the degree of skewness and/or kurtosis of the data are greater or less than those that the LN distribution can capture. Another peculiarity of the LN distribution is that it only fits unimodal positive data, which constitutes a limitation when dealing with data that present more than one mode (bimodality). On the other hand, the log-normal model only fits unimodal positive data and in reality there are multiple applications where the behavior of materials is bimodal. To fill this gap, this paper introduces a new probability distribution that is capable of fitting unimodal or bimodal positive data with a high or low degree of skewness and/or kurtosis. The new distribution is a generalization of the LN distribution. For the new proposal, its main properties are studied and the process of estimation of the parameters involved in the model is carried out from a classical perspective using the maximum likelihood method. An important feature of this distribution is the non-singularity of the Fisher information matrix, which guarantees the use of asymptotic theory to study the properties of the parameter estimators. A Monte Carlo type simulation study is carried out to evaluate the properties of the estimators and finally, an illustration is presented with a set of data related to the concentration of nickel in soil samples, allowing to show that the proposed distribution fits extremely well in certain situations.

It is common in many fields of knowledge to assume that the data under study have a normal distribution, which often generates mistakes in the results, since this assumption does not always coincide with the characteristics of the observations under analysis. In some cases, the data may have degrees of skewness and/or kurtosis greater than what the normal model can capture, and in others, they may present two or more modes. In this work, two new families of skewed distributions are presented that fit bimodal data with positive support. The new families were obtained from the extension of the bimodal normal distribution to the alpha-power family class. The proposed distributions were studied for their main properties, such as their probability density function, cumulative distribution function, survival function, and hazard function. The parameter estimation process was performed from a classical perspective using the maximum likelihood method. The non-singularity of Fisher’s information was demonstrated, which made it possible to find the stochastic convergence of the vector of the maximum likelihood estimators and, based on the latter, perform statistical inference via the likelihood ratio. The applicability of the proposed distributions was exemplified using real data sets.

In this paper, a new bivariate absolutely continuous probability distribution is introduced. The new distribution, which is called the bivariate unit-sinh-normal (BVUSHN) distribution, arises by applying a transformation to the bivariate Birnbaum–Saunders distribution (BVBS). The main properties of the new proposal are studied in detail. In addition, from the new distribution, the BVUSHN regression model is also introduced. For both the bivariate probability distribution and the respective associated regression model, parameter estimation is conducted from a classical approach by using the maximum likelihood method together with the two-step estimation method. A small Monte Carlo simulation study is carried out to evaluate the behavior of the used estimation method and the properties of the estimators. Finally, for illustrative purposes, two applications with real data are presented in which the usefulness of the proposals is evidenced.