In this work, we introduce a regression model for double-bounded variables in the interval (0, 1) following a Kumaraswamy distribution. The model resembles a generalized linear model, in which the response's median is modeled by a regression structure through the asymmetric Aranda-Ordaz parametric link function. We consider the maximum likelihood approach to estimate the regression and the link function parameters altogether. We study large sample properties of the proposed maximum likelihood approach, presenting closed-form expressions for the score vector as well as the observed and Fisher information matrices. We briefly present and discuss some diagnostic tools. We provide numeric evaluation of the finite sample inferences to show the performance of the estimators. Finally, to exemplify the usefulness of the methodology, we present and explore an empirical application.
Beta regressions are widely used for modeling random variables that assume values in the standard unit interval, (0, 1), such as rates, proportions, and income concentration indices. Parameter estimation is typically performed via maximum likelihood, and hypothesis testing inferences on the model parameters are commonly performed using the likelihood ratio test. Such a test, however, may deliver inaccurate inferences when the sample size is small. It is thus important to develop alternative testing procedures that are more accurate when the sample contains only few observations. In this paper, we consider the beta regression model with parametric mean link function and derive two modified likelihood ratio test statistics for that class of models. We provide simulation evidence that shows that the new tests usually outperform the standard likelihood ratio test in samples of small to moderate sizes. We also present and discuss two empirical applications.
This paper proposes a control chart useful for detecting small shifts in the mean of a double‐bounded process, such as fractions and proportions, in the presence of control variables. For this purpose, we consider the cumulative sum (CUSUM) control chart applied to different residuals of the beta regression model. We conduct an extensive Monte Carlo simulation study to evaluate and compare the performance of the proposed control chart with two other control charts in the literature in terms of run length analysis. The numerical results show that the proposed control chart is more sensitive to detect changes in the process than its competitors and that the quantile residual is the most suitable residual to be used in our proposal. Finally, based on the quantile residual, we present and discuss applications to real and simulated data to show the applicability of the proposed control chart.
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