Barros, M., Galea, M., Leiva, V., & Santos-Neto, M. (2018). Generalized Tobit models: diagnostics and application in econometrics. J. Appl. Stat., 45(1), 145–167.
Abstract: The standard Tobit model is constructed under the assumption of a normal distribution and has been widely applied in econometrics. Atypical/extreme data have a harmful effect on the maximum likelihood estimates of the standard Tobit model parameters. Then, we need to count with diagnostic tools to evaluate the effect of extreme data. If they are detected, we must have available a Tobit model that is robust to this type of data. The family of elliptically contoured distributions has the Laplace, logistic, normal and Student-t cases as some of its members. This family has been largely used for providing generalizations of models based on the normal distribution, with excellent practical results. In particular, because the Student-t distribution has an additional parameter, we can adjust the kurtosis of the data, providing robust estimates against extreme data. We propose a methodology based on a generalization of the standard Tobit model with errors following elliptical distributions. Diagnostics in the Tobit model with elliptical errors are developed. We derive residuals and global/local influence methods considering several perturbation schemes. This is important because different diagnostic methods can detect different atypical data. We implement the proposed methodology in an R package. We illustrate the methodology with real-world econometrical data by using the R package, which shows its potential applications. The Tobit model based on the Student-t distribution with a small quantity of degrees of freedom displays an excellent performance reducing the influence of extreme cases in the maximum likelihood estimates in the application presented. It provides new empirical evidence on the capabilities of the Student-t distribution for accommodation of atypical data.
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Garcia-Papani, F., Uribe-Opazo, M. A., Leiva, V., & Aykroyd, R. G. (2017). Birnbaum-Saunders spatial modelling and diagnostics applied to agricultural engineering data. Stoch. Environ. Res. Risk Assess., 31(1), 105–124.
Abstract: Applications of statistical models to describe spatial dependence in geo-referenced data are widespread across many disciplines including the environmental sciences. Most of these applications assume that the data follow a Gaussian distribution. However, in many of them the normality assumption, and even a more general assumption of symmetry, are not appropriate. In non-spatial applications, where the data are uni-modal and positively skewed, the Birnbaum-Saunders (BS) distribution has excelled. This paper proposes a spatial log-linear model based on the BS distribution. Model parameters are estimated using the maximum likelihood method. Local influence diagnostics are derived to assess the sensitivity of the estimators to perturbations in the response variable. As illustration, the proposed model and its diagnostics are used to analyse a real-world agricultural data set, where the spatial variability of phosphorus concentration in the soil is considered-which is extremely important for agricultural management.
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Leao, J., Leiva, V., Saulo, H., & Tomazella, V. (2017). Birnbaum-Saunders frailty regression models: Diagnostics and application to medical data. Biom. J., 59(2), 291–314.
Abstract: In survival models, some covariates affecting the lifetime could not be observed or measured. These covariates may correspond to environmental or genetic factors and be considered as a random effect related to a frailty of the individuals explaining their survival times. We propose a methodology based on a Birnbaum-Saunders frailty regression model, which can be applied to censored or uncensored data. Maximum-likelihood methods are used to estimate the model parameters and to derive local influence techniques. Diagnostic tools are important in regression to detect anomalies, as departures from error assumptions and presence of outliers and influential cases. Normal curvatures for local influence under different perturbations are computed and two types of residuals are introduced. Two examples with uncensored and censored real-world data illustrate the proposed methodology. Comparison with classical frailty models is carried out in these examples, which shows the superiority of the proposed model.
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Leiva, V., Liu, S. Z., Shi, L., & Cysneiros, F. J. A. (2016). Diagnostics in elliptical regression models with stochastic restrictions applied to econometrics. J. Appl. Stat., 43(4), 627–642.
Abstract: We propose an influence diagnostic methodology for linear regression models with stochastic restrictions and errors following elliptically contoured distributions. We study how a perturbation may impact on the mixed estimation procedure of parameters in the model. Normal curvatures and slopes for assessing influence under usual schemes are derived, including perturbations of case-weight, response variable, and explanatory variable. Simulations are conducted to evaluate the performance of the proposed methodology. An example with real-world economy data is presented as an illustration.
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Liu, S. Z., Leiva, V., Ma, T. F., & Welsh, A. (2016). Influence diagnostic analysis in the possibly heteroskedastic linear model with exact restrictions. Stat. Method. Appl., 25(2), 227–249.
Abstract: The local influence method has proven to be a useful and powerful tool for detecting influential observations on the estimation of model parameters. This method has been widely applied in different studies related to econometric and statistical modelling. We propose a methodology based on the Lagrange multiplier method with a linear penalty function to assess local influence in the possibly heteroskedastic linear regression model with exact restrictions. The restricted maximum likelihood estimators and information matrices are presented for the postulated model. Several perturbation schemes for the local influence method are investigated to identify potentially influential observations. Three real-world examples are included to illustrate and validate our methodology.
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Marchant, C., Leiva, V., Cysneiros, F. J. A., & Vivanco, J. F. (2016). Diagnostics in multivariate generalized Birnbaum-Saunders regression models. J. Appl. Stat., 43(15), 2829–2849.
Abstract: Birnbaum-Saunders (BS) models are receiving considerable attention in the literature. Multivariate regression models are a useful tool of the multivariate analysis, which takes into account the correlation between variables. Diagnostic analysis is an important aspect to be considered in the statistical modeling. In this paper, we formulate multivariate generalized BS regression models and carry out a diagnostic analysis for these models. We consider the Mahalanobis distance as a global influence measure to detect multivariate outliers and use it for evaluating the adequacy of the distributional assumption. We also consider the local influence approach and study how a perturbation may impact on the estimation of model parameters. We implement the obtained results in the R software, which are illustrated with real-world multivariate data to show their potential applications.
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Santos-Neto, M., Cysneiros, F. J. A., Leiva, V., & Barros, M. (2016). Reparameterized Birnbaum-Saunders regression models with varying precision. Electron. J. Stat., 10(2), 2825–2855.
Abstract: We propose a methodology based on a reparameterized Birnbaum-Saunders regression model with varying precision, which generalizes the existing works in the literature on the topic. This methodology includes the estimation of model parameters, hypothesis tests for the precision parameter, a residual analysis and influence diagnostic tools. Simulation studies are conducted to evaluate its performance. We apply it to two real-world case-studies to show its potential with the R software.
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