Vasicek Quantile and Mean Regression Models for Bounded Data: New Formulation, Mathematical Derivations, and Numerical Applications

Another unit Burr XII quantile regression model based on the different reparameterization applied to dropout in Brazilian undergraduate courses

In many practical situations, there is an interest in modeling bounded random variables in the interval (0, 1), such as rates, proportions, and indexes. It is important to provide new continuous models to deal with the uncertainty involved by variables of this type. This paper proposes a new quantile regression model based on an alternative parameterization of the unit Burr XII (UBXII) distribution. For the UBXII distribution and its associated regression, we obtain score functions and observed information matrices. We use the maximum likelihood method to estimate the parameters of the regression model, and conduct a Monte Carlo study to evaluate the performance of its estimates in samples of finite size. Furthermore, we present general diagnostic analysis and model selection techniques for the regression model. We empirically show its importance and flexibility through an application to an actual data set, in which the dropout proportion of Brazilian undergraduate animal sciences courses is analyzed. We use a statistical learning method for comparing the proposed model with the beta, Kumaraswamy, and unit-Weibull regressions. The results show that the UBXII regression provides the best fit and the most accurate predictions. Therefore, it is a valuable alternative and competitive to the well-known regressions for modeling double-bounded variables in the unit interval.

The unit generalized half-normal quantile regression model: formulation, estimation, diagnostics, and numerical applications

In this paper, we propose and derive a new regression model for response variables defined on the open unit interval. By reparameterizing the unit generalized half-normal distribution, we get the interpretation of its location parameter as being a quantile of the distribution. In addition, we can evaluate effects of the explanatory variables in the conditional quantiles of the response variable as an alternative to the Kumaraswamy quantile regression model. The suitability of our proposal is demonstrated with two simulated examples and two real applications. For such data sets, the obtained fits of the proposed regression model are compared with that provided by a Kumaraswamy regression model.

An overview on parametric quantile regression models and their computational implementation with applications to biomedical problems including COVID-19 data

Quantile regression allows us to estimate the relationship between covariates and any quantile of the response variable rather than the mean. Recently, several statistical distributions have been considered for quantile modeling. The objective of this study is to provide a new computational package, two biomedical applications, one of them with COVID-19 data, and an up-to-date overview of parametric quantile regression. A fully parametric quantile regression is formulated by first parameterizing the baseline distribution in terms of a quantile. Then, we introduce a regression-based functional form through a link function. The density, distribution, and quantile functions, as well as the main properties of each distribution, are presented. We consider 18 distributions related to normal and non-normal settings for quantile modeling of continuous responses on the unit interval, four distributions for continuous response, and one distribution for discrete response. We implement an R package that includes estimation and model checking, density, distribution, and quantile functions, as well as random number generators, for distributions using quantile regression in both location and shape parameters. In summary, a number of studies have recently appeared applying parametric quantile regression as an alternative to the distribution-free quantile regression proposed in the literature. We have reviewed a wide body of parametric quantile regression models, developed an R package which allows us, in a simple way, to fit a variety of distributions, and applied these models to two examples with biomedical real-world data from Brazil and COVID-19 data from US for illustrative purposes. Parametric and non-parametric quantile regressions are compared with these two data sets.