Residual and local influence analyses for unit gamma regressions

Abstract

We obtain local influence measures and residuals for the unit gamma regression model. In particular, we introduce four residuals that are based on Fisher's iterative scoring parameter estimation algorithm and develop local influence analysis based on several different perturbation schemes: cases weighting, response additive perturbation, and covariate(s) additive perturbation. An empirical application in which variables related to education and investment in research and development are used to explain the proportion of nonpoor people in a set of countries is presented and discussed. Residual and local influence analyses show that the unit gamma regression model yields a good fit to the data, even outperforming the beta regression model. The diagnostic analysis singles out countries whose data are worthy of further investigation. Our results reveal that lower poverty levels are associated with higher shares of investment in high technology. The statistical significance of such a relationship is not sensitive to atypical data points.

Log‐symmetric quantile regression models

“…However, it is an aspect to be developed and studied in the future. Although we considered a pseudo‐R 2 measure in order to select the best model, we understand that generalized leverage measures, the Cook distance, and global/local influence techniques (Tapia, Giampaoli, Leiva, & Lio, 2020; and Rocha et al, 2021) are important aspects to be taken into account in the statistical modeling. All of these and other aspects are part of future research.…”

“…Regression models with a continuous, strictly positive, and asymmetrically distributed dependent variable (response) have been widely applied in different fields, such as economics, engineering, environmental sciences, industry, and medicine, among others (Leiva, Ferreira, Gomes, & Lillo, 2015; Vanegas & Paula, 2017; Ventura, Saulo, Leiva, & Monsueto, 2019; de la Fuente‐Mella, Rojas Fuentes, & Leiva, 2020; Leiva, Sánchez, Galea, & Saulo, 2020; Sánchez, Leiva, Galea, & Saulo, 2020, 2021; and Rocha, Espinheira, & Cribari‐Neto, 2021). Nonnormal regression models have been obtained from the Birnbaum–Saunders (BS), gamma, log‐normal (log‐NO), and Weibull distributions (Johnson, Kotz, & Balakrishnan, 1994).…”

“…The proposed methodology follows a parametric approach and can be seen as an alternative to the semiparametric approach proposed in Koenker and Bassett Jr. (1978), which is distribution free. Nevertheless, the advantages of the proposed methodology over the semiparametric approach can be attributed to: (A1) the distributional flexibility provided by several members of the log-symmetric family, which can be more realistic in practice; (A2) the dispersion modeling, which provides flexibility in accommodating heteroscedasticity; and (A3) the fact that statistical analyses based on the likelihood function can be developed such as hypothesis tests and global and local influence (Rocha et al, 2021).…”

Bootstrap-based inferential improvements to the simplex nonlinear regression model

“…Other approaches for modeling limited data are the beta [ 7 ], Kumaraswamy [ 8 ], Johnson S B [ 9 ], unit gamma [ 10 ] regression models. Recently published papers show possible advantages of using the latter distribution over the beta distribution [ 11 , 12 ]. Recently, [ 13 ] proposed to the class of non-linear simplex regression models, in which they estimate the model parameters using the maximum likelihood method and derive the local influence quantities.…”

Predicting PM2.5 and PM10 Levels during Critical Episodes Management in Santiago, Chile, with a Bivariate Birnbaum-Saunders Log-Linear Model

“…In addition, diagnostics analytics is conducted by goodness-of-fit (GOF) and global/local influence techniques. GOF is used to determine which model offers a better fit to the atmospheric contamination data, whereas the local influence technique is utilized to analyze the impact of a perturbation on the overall estimation of model parameters [10,39]. Thus, model precision to predict a critical episode of atmospheric contamination is determined.…”