Beta regression is often used to model the relationship between a dependent variable that assumes values on the open interval (0, 1) and a set of predictor variables. An important challenge in beta regression is to find residuals whose distribution is well approximated by the standard normal distribution. Two previous works compared residuals in beta regression, but the authors did not include the quantile residual.Using Monte Carlo simulation techniques, this paper studies the behavior of certain residuals in beta regression in several scenarios. Overall, the results suggest that the distribution of the quantile residual is better approximated by the standard normal distribution than that of the other residuals in most scenarios. Three applications illustrate the effectiveness of the quantile residual.

This study aimed to determine the levels of fumonisins produced by Fusarium verticillioides and FUM gene expression on Bt (Bacillus thuringiensis) and non-Bt maize, post harvest, during different periods of incubation. Transgenic hybrids 30F35 YG, 2B710 Hx and their isogenic (30F35 and 2B710) were collected from the field and a subset of 30 samples selected for the experiments. Maize samples were sterilized by gamma radiation at a dose of 20 kGy. Samples were then inoculated with F. verticillioides and analyzed under controlled conditions of temperature and relative humidity for fumonisin B1 and B2 (FB1 and FB2) production and FUM1, FUM3, FUM6, FUM7, FUM8, FUM13, FUM14, FUM15, and FUM19 expression. 2B710 Hx and 30F35 YG kernel samples were virtually intact when compared to the non-Bt hybrids that came from the field. Statistical analysis showed that FB1 production was significantly lower in 30F35 YG and 2B710 Hx than in the 30F35 and 2B710 hybrids (P < 0.05). However, there was no statistical difference for FB2 production (P > 0.05). The kernel injuries observed in the non-Bt samples have possibly facilitated F. verticillioides penetration and promoted FB1 production under controlled conditions. FUM genes were expressed by F. verticillioides in all of the samples. However, there was indication of lower expression of a few FUM genes in the Bt hybrids; and a weak association between FB1 production and the relative expression of some of the FUM genes were observed in the 30F35 YG hybrid.

Credit cards are a financial product with special characteristics. Dividing the amount paid by the customer in a given month by the total bill results in a variable that is partly discrete and partly continuous, which we call the relative payment amount (RPA). This variable is discrete at 0, c and 1, and it is continuous in the open interval ( c, 1). The 0< c<1 value is known and is given by the ratio between the value of the minimum payment and the full amount, and this value is not fixed for all customers. Thus, in practice, the RPA is a variable whose support of its distribution is non-constant across population units. In this work, we propose a regression model for the RPA. The model allows all of the unknown parameters of the conditional distribution of the response variable to be modelled as a function of the explanatory variables, and it also accounts for the non-constant known parameter c. The estimation of the parameters of this model is discussed, diagnostic analysis is addressed and closed-form expressions for the score function and for the Fisher’s information matrix are provided. Moreover, some results related to the non-constant nature of c are obtained, simulation studies are performed and an application using real credit card data is presented.

Zero adjusted regression models are used to fit variables that are discrete at zero and continuous at some interval of the positive real numbers. Diagnostic analysis in these models is usually performed using the randomized quantile residual, which is useful for checking the overall adequacy of a zero adjusted regression model. However, it may fail to identify some outliers. In this work, we introduce a residual for outlier identification in zero adjusted regression models. Monte Carlo simulation studies and an application suggest that the residual introduced here has good properties and detects outliers that are not identified by the randomized quantile residual.