RESUMO
Bimodal data sets are very common in different areas of knowledge. The crude birth rates data, fish length data, egg diameter data, the eruption and interruption times of the Old Faithful geyser, are examples of this type of data. In this paper, a new class of symmetric density functions for modeling bimodal data as described above are presented. From density functions with support on [ 0 , + ∞ ) , the symmetry is getting by reflecting the density function in the negative semi-axis with their respective normalization. In this way, if the primitive density function is unimodal, then the resulting density will be bimodal. We introduce asymmetry parameters and study their behavior, in particular the values of their modes and some other statistical values of interest. The cases for densities generated by Gamma, Weibull, Log-normal, and Birnbaum-Saunders densities, among others are studied. Statistical inference is performed from a classical perspective. A small simulation study to evaluate the benefits and limitations of the new proposal. In addition, an application to a data set related to the fetal weight in grams obtained through ultrasound in a sample of 500 units is also presented; the results show the great usefulness of the model in practical situations.
RESUMO
In this research article, we propose a class of models for positive and zero responses by means of a zero-augmented mixed regression model. Under this class, we are particularly interested in studying positive responses whose distribution accommodates skewness. At the same time, responses can be zero, and therefore, we justify the use of a zero-augmented mixture model. We model the mean of the positive response in a logarithmic scale and the mixture probability in a logit scale, both as a function of fixed and random effects. Moreover, the random effects link the two random components through their joint distribution and incorporate within-subject correlation because of the repeated measurements and between-subject heterogeneity. A Markov chain Monte Carlo algorithm is tailored to obtain Bayesian posterior distributions of the unknown quantities of interest, and Bayesian case-deletion influence diagnostics based on the q-divergence measure is performed. We apply the proposed method to a dataset from a 24 hour dietary recall study conducted in the city of São Paulo and present a simulation study to evaluate the performance of the proposed methods.