Bayesian quantile regression models for heavy tailed bounded variables using the No-U-Turn sampler

When we are interested in knowing how covariates impact different levels of the response variable, quantile regression models can be very useful, with their practical use being benefited from the increasing of computational power. The use of bounded response variables is also very common when there are data containing percentages, rates, or proportions. In this work, with the generalized Gompertz distribution as the baseline distribution, we derive two new two-parameter distributions with bounded support, and new quantile parametric mixed regression models are proposed based on these distributions, which consider bounded response variables with heavy tails. Estimation of the parameters using the Bayesian approach is considered for both models, relying on the No-U-Turn sampler algorithm. The inferential methods can be implemented and then easily used for data analysis. Simulation studies with different quantiles ( $$q=0.1$$ q = 0.1 , $$q=0.5$$ q = 0.5 and $$q=0.9$$ q = 0.9 ) and sample sizes ( $$n=100$$ n = 100 , $$n=200$$ n = 200 , $$n=500$$ n = 500 , $$n=2000$$ n = 2000 , $$n=5000$$ n = 5000 ) were conducted for 100 replicas of simulated data for each combination of settings, in the (0, 1) and [0, 1), showing the good performance of the recovery of parameters for the proposed inferential methods and models, which were compared to Beta Rectangular and Kumaraswamy regression models. Furthermore, a dataset on extreme poverty is analyzed using the proposed regression models with fixed and mixed effects. The quantile parametric models proposed in this work are an alternative and complementary modeling tool for the analysis of bounded data.