In this paper, we propose and derive a Birnbaum–Saunders distribution to model bimodal data. This new distribution is obtained using the product of the standard Birnbaum–Saunders distribution and a polynomial function of the fourth degree. We study the mathematical and statistical properties of the bimodal Birnbaum–Saunders distribution, including probabilistic features and moments. Inference on its parameters is conducted using the estimation methods of moments and maximum likelihood. Based on the acceptance–rejection criterion, an algorithm is proposed to generate values of a random variable that follows the new bimodal Birnbaum–Saunders distribution. We carry out a simulation study using the Monte Carlo method to assess the statistical performance of the parameter estimators. Illustrations with real-world data sets from environmental and medical sciences are provided to show applications that can be of potential use in real problems.

In this paper, we will present a new, more flexible class of distributions with a domain in the interval (0,1), which presents heavier tails than other distributions in the same domain, such as the $Beta$, Kumaraswamy, and Weibull Unitary distributions. This new distribution is obtained as a transformation of two independent random variables with a Weibull distribution, which we will call the Generalized Unitary Weibull distribution. Considering a particular case, we will obtain an alternative to the $Beta$, Kumaraswamy, and Weibull Unitary distributions. We will call this new distribution of two parameters the type 2 unitary Weibull distribution. The probability density function, cumulative probability distribution, survival function, hazard rate, and some important properties that will allow us to infer are provided. We will carry out a simulation study using the maximum likelihood method and we will analyze the behavior of the parameter estimates. By way of illustration, real data will be used to show the flexibility of the new distribution by comparing it with other distributions that are known in the literature. Finally, we will show a quantile regression application, where it is shown how the proposed distribution fits better than other competing distributions for this type of application.

In this paper, likelihood-based inference and bias correction based on Firth’s approach are developed in the modified skew-t-normal (MStN) distribution. The latter model exhibits a greater flexibility than the modified skew-normal (MSN) distribution since it is able to model heavily skewed data and thick tails. In addition, the tails are controlled by the shape parameter and the degrees of freedom. We provide the density of this new distribution and present some of its more important properties including a general expression for the moments. The Fisher’s information matrix together with the observed matrix associated with the log-likelihood are also given. Furthermore, the non-singularity of the Fisher’s information matrix for the MStN model is demonstrated when the shape parameter is zero. As the MStN model presents an inferential problem in the shape parameter, Firth’s method for bias reduction was applied for the scalar case and for the location and scale case.

The skew distribution has the characteristic of appropriately modeling asymmetric unimodal data. However, in practice, there are several cases in which the data present more than one mode. In the literature, it is possible to find a large number of authors who have studied extensions based on the skew distribution to model this type of data. In this article, a new family is introduced, consisting of a multimodal modification to the family of skew distributions. Using the methodology of the weighted version of a function, we perform the product of the density function of a family of skew distributions with a polynomial of degree 4, thus obtaining a more flexible model that allows modeling data sets, whose distribution contains at most three modes. The density function, some properties, moments, skewness coefficients, and kurtosis of this new family are presented. This study focuses on the particular cases of skew-normal and Laplace distributions, although it can be applied to any other distribution. A simulation study was carried out, to study the behavior of the model parameter estimates. Illustrations with real data, referring to medicine and environmental data, show the practical performance of the proposed model in the two particular cases presented.

In this work, we present a new generalization of the student’s t distribution. The new distribution is obtained by the quotient of two independent random variables. This quotient consists of a standard Normal distribution divided by the power of a chi square distribution divided by its degrees of freedom. Thus, the new symmetric distribution has heavier tails than the student’s t distribution and extensions of the slash distribution. We develop a procedure to use quantile regression where the response variable or the residuals have high kurtosis. We give the density function expressed by an integral, we obtain some important properties and some useful procedures for making inference, such as moment and maximum likelihood estimators. By way of illustration, we carry out two applications using real data, in the first we provide maximum likelihood estimates for the parameters of the generalized student’s t distribution, student’s t, the extended slash distribution, the modified slash distribution, the slash distribution generalized student’s t test, and the double slash distribution, in the second we perform quantile regression to fit a model where the response variable presents a high kurtosis.

This work presents a new distribution that allows modeling data from a random variable with non-negative values. The new family is defined by a stochastic representation of a scaled mixture of a random variable with a Gompertz distribution (G) and a random variable with a uniform distribution on the interval (0,1). The result of this gives rise to a new random variable with a Slash Gompertz ($SG$) distribution that is more flexible than the Gompertz distribution, that is, it better models atypical data, presenting tails heavier than the Gompertz distribution. The density and some general properties of the resulting family are studied, including its moments and kurtosis coefficient. The inference of the parameters is carried out using the method of moments and maximum likelihood. Finally, illustrations of particular cases of this family are shown, adjusting in this case two sets of real data and estimating the parameters by maximum likelihood, where it is verified that this new family of distributions fits the reliability function better than the distributions of Gompertz (G), Slash Birnbaum Saunders ($SBS$), Slash Weibull ($SW$), and Gompertz-Verhults ($GV$).