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Fierro, R., & Leiva, V. (2017). A stochastic methodology for risk assessment of a large earthquake when a long time has elapsed. Stoch. Environ. Res. Risk Assess., 31(9), 2327–2336.
Abstract: We propose a stochastic methodology for risk assessment of a large earthquake when a long time has elapsed from the last large seismic event. We state an approximate probability distribution for the occurrence time of the next large earthquake, by knowing that the last large seismic event occurred a long time ago. We prove that, under reasonable conditions, such a distribution is exponential with a rate depending on the asymptotic slope of the cumulative intensity function corresponding to a nonhomogeneous Poisson process. As it is not possible to obtain an empirical cumulative distribution function of the waiting time for the next large earthquake, an estimator of its cumulative distribution function based on existing data is derived. We conduct a simulation study for detecting scenario in which the proposed methodology would perform well. Finally, a real-world data analysis is carried out to illustrate its potential applications, including a homogeneity test for the times between earthquakes.
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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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