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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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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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Fierro, R., Leiva, V., & Balakrishnan, N. (2015). Statistical Inference on a Stochastic Epidemic Model. Commun. Stat.-Simul. Comput., 44(9), 2297–2314.
Abstract: In this work, we develop statistical inference for the parameters of a discrete-time stochastic SIR epidemic model. We use a Markov chain for describing the dynamic behavior of the epidemic. Specifically, we propose estimators for the contact and removal rates based on the maximum likelihood and martingale methods, and establish their asymptotic distributions. The obtained results are applied in the statistical analysis of the basic reproduction number, a quantity that is useful in establishing vaccination policies. In order to evaluate the population size for which the results are useful, a numerical study is carried out. Finally, a comparison of the maximum likelihood and martingale estimators is conducted by means of Monte Carlo simulations.
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Fierro, R., Leiva, V., & Moller, J. (2015). The Hawkes Process With Different Exciting Functions And Its Asymptotic Behavior. J. Appl. Probab., 52(1), 37–54.
Abstract: The standard Hawkes process is constructed from a homogeneous Poisson process and uses the same exciting function for different generations of offspring. We propose an extension of this process by considering different exciting functions. This consideration may be important in a number of fields; e.g. in seismology, where main shocks produce aftershocks with possibly different intensities. The main results are devoted to the asymptotic behavior of this extension of the Hawkes process. Indeed, a law of large numbers and a central limit theorem are stated. These results allow us to analyze the asymptotic behavior of the process when unpredictable marks are considered.
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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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Khosravi, M., Leiva, V., Jamalizadeh, A., & Porcu, E. (2016). On a nonlinear Birnbaum-Saunders model based on a bivariate construction and its characteristics. Commun. Stat.-Theory Methods, 45(3), 772–793.
Abstract: The Birnbaum-Saunders (BS) distribution is an asymmetric probability model that is receiving considerable attention. In this article, we propose a methodology based on a new class of BS models generated from the Student-t distribution. We obtain a recurrence relationship for a BS distribution based on a nonlinear skew-t distribution. Model parameters estimators are obtained by means of the maximum likelihood method, which are evaluated by Monte Carlo simulations. We illustrate the obtained results by analyzing two real data sets. These data analyses allow the adequacy of the proposed model to be shown and discussed by applying model selection tools.
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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., Ferreira, M., Gomes, M. I., & Lillo, C. (2016). Extreme value Birnbaum-Saunders regression models applied to environmental data. Stoch. Environ. Res. Risk Assess., 30(3), 1045–1058.
Abstract: Extreme value models are widely used in different areas. The Birnbaum-Saunders distribution is receiving considerable attention due to its physical arguments and its good properties. We propose a methodology based on extreme value Birnbaum-Saunders regression models, which includes model formulation, estimation, inference and checking. We further conduct a simulation study for evaluating its performance. A statistical analysis with real-world extreme value environmental data using the methodology is provided as illustration.
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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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Leiva, V., Marchant, C., Ruggeri, F., & Saulo, H. (2015). A criterion for environmental assessment using Birnbaum-Saunders attribute control charts. Environmetrics, 26(7), 463–476.
Abstract: Assessing environmental risk is useful for preventing adverse effects on human health in highly polluted cities. We design a criterion for environmental monitoring based on an attribute control chart for the number of dangerous contaminant levels when the concentration to be monitored follows a Birnbaum-Saunders distribution. This distribution is being widely applied to environmental data. We provide a novel justification for its usage in environmental sciences. The control coefficient and the minimum inspection concentration for the designed criterion are determined to yield the specified in-control average run length, whereas the out-of-control case is obtained according to a shift in the target mean. A simulation study is conducted to evaluate the proposed criterion, which reports its performance to provide earlier alerts of out-of-control processes. An application with real-world environmental data is carried out to validate its coherence with what is reported by the health authority. Copyright (c) 2015 John Wiley & Sons, Ltd.
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Leiva, V., Ruggeri, F., Saulo, H., & Vivanco, J. F. (2017). A methodology based on the Birnbaum-Saunders distribution for reliability analysis applied to nano-materials. Reliab. Eng. Syst. Saf., 157, 192–201.
Abstract: The Birnbaum-Saunders distribution has been widely studied and applied to reliability studies. This paper proposes a novel use of this distribution to analyze the effect on hardness, a material mechanical property, when incorporating nano-particles inside a polymeric bone cement. A plain variety and two modified types of mesoporous silica nano-particles are considered. In biomaterials, one can study the effect of nano-particles on mechanical response reliability. Experimental data collected by the authors from a micro-indentation test about hardness of a commercially available polymeric bone cement are analyzed. Hardness is modeled with the Birnbaum-Saunders distribution and Bayesian inference is performed to derive a methodology, which allows us to evaluate the effect of using nano-particles at different loadings by the R software. (C) 2016 Elsevier Ltd. All rights reserved.
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Leiva, V., Santos-Neto, M., Cysneiros, F. J. A., & Barros, M. (2016). A methodology for stochastic inventory models based on a zero-adjusted Birnbaum-Saunders distribution. Appl. Stoch. Models. Bus. Ind., 32(1), 74–89.
Abstract: The Birnbaum-Saunders (BS) distribution is receiving considerable attention. We propose a methodology for inventory logistics that allows demand data with zeros to be modeled by means of a new discrete-continuous mixture distribution, which is constructed by using a probability mass at zero and a continuous component related to the BS distribution. We obtain some properties of the new mixture distribution and conduct a simulation study to evaluate the performance of the estimators of its parameters. The methodology for stochastic inventory models considers also financial indicators. We illustrate the proposed methodology with two real-world demand data sets. It shows its potential, highlighting the convenience of using it by improving the contribution margins of a Chilean food industry. Copyright (c) 2015 John Wiley & Sons, Ltd.
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Leiva, V., Saulo, H., Leao, J., & Marchant, C. (2014). A family of autoregressive conditional duration models applied to financial data. Comput. Stat. Data Anal., 79, 175–191.
Abstract: The Birnbaum-Saunders distribution is receiving considerable attention due to its good properties. One of its extensions is the class of scale-mixture Birnbaum-Saunders (SBS) distributions, which shares its good properties, but it also has further properties. The autoregressive conditional duration models are the primary family used for analyzing high-frequency financial data. We propose a methodology based on SBS autoregressive conditional duration models, which includes in-sample inference, goodness-of-fit and out-of-sample forecast techniques. We carry out a Monte Carlo study to evaluate its performance and assess its practical usefulness with real-world data of financial transactions from the New York stock exchange. (C) 2014 Elsevier B.V. All rights reserved.
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Leiva, V., Tejo, M., Guiraud, P., Schmachtenberg, O., Orio, P., & Marmolejo-Ramos, F. (2015). Modeling neural activity with cumulative damage distributions. Biol. Cybern., 109(4-5), 421–433.
Abstract: Neurons transmit information as action potentials or spikes. Due to the inherent randomness of the inter-spike intervals (ISIs), probabilistic models are often used for their description. Cumulative damage (CD) distributions are a family of probabilistic models that has been widely considered for describing time-related cumulative processes. This family allows us to consider certain deterministic principles for modeling ISIs from a probabilistic viewpoint and to link its parameters to values with biological interpretation. The CD family includes the Birnbaum-Saunders and inverse Gaussian distributions, which possess distinctive properties and theoretical arguments useful for ISI description. We expand the use of CD distributions to the modeling of neural spiking behavior, mainly by testing the suitability of the Birnbaum-Saunders distribution, which has not been studied in the setting of neural activity. We validate this expansion with original experimental and simulated electrophysiological data.
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Lillo, C., Leiva, V., Nicolis, O., & Aykroyd, R. G. (2018). L-moments of the Birnbaum-Saunders distribution and its extreme value version: estimation, goodness of fit and application to earthquake data. J. Appl. Stat., 45(2), 187–209.
Abstract: Understanding patterns in the frequency of extreme natural events, such as earthquakes, is important as it helps in the prediction of their future occurrence and hence provides better civil protection. Distributions describing these events are known to be heavy tailed and positive skew making standard distributions unsuitable for modelling the frequency of such events. The Birnbaum-Saunders distribution and its extreme value version have been widely studied and applied due to their attractive properties. We derive L-moment equations for these distributions and propose novel methods for parameter estimation, goodness-of-fit assessment and model selection. A simulation study is conducted to evaluate the performance of the L-moment estimators, which is compared to that of the maximum likelihood estimators, demonstrating the superiority of the proposed methods. To illustrate these methods in a practical application, a data analysis of real-world earthquake magnitudes, obtained from the global centroid moment tensor catalogue during 1962-2015, is carried out. This application identifies the extreme value Birnbaum-Saunders distribution as a better model than classic extreme value distributions for describing seismic events.
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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. (2016). A Multivariate Log-Linear Model for Birnbaum-Saunders Distributions. IEEE Trans. Reliab., 65(2), 816–827.
Abstract: Univariate Birnbaum-Saunders models have been widely applied to fatigue studies. Calculation of fatigue life is of great importance in determining the reliability of materials. We propose and derive new multivariate generalized Birnbaum-Saunders regression models. We use the maximum likelihood method and the EM algorithm to estimate their parameters. We carry out a simulation study to evaluate the performance of the corresponding maximum likelihood estimators. We illustrate the new models with real-world multivariate fatigue data.
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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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Rojas, F., & Leiva, V. (2016). Inventory management in food companies with statistically dependent demand. Acad.-Rev. Latinoam. Adm., 29(4), 450–485.
Abstract: Purpose – The objective of this paper is to propose a methodology based on random demand inventory models and dependence structures for a set of raw materials, referred to as “components”, used by food services that produce food rations referred to as “menus”. Design/methodology/approach – The contribution margins of food services that produce menus are optimised using random dependent demand inventory models. The statistical dependence between the demand for components and/or menus is incorporated into the model through the multivariate Gaussian (or normal) distribution. The contribution margins are optimised by using probabilistic inventory models for each component and stochastic programming with a differential evolution algorithm. Findings – When compared to the non-optimised system previously used by the company, the (average) expected contribution margin increases by 18.32 per cent when using a continuous review inventory model for groceries and uniperiodic models for perishable components (optimised system). Research limitations/implications – The multivariate modeling can be improved by using (a) other non-Gaussian (marginal) univariate probability distributions, by means of the copula method that considers more complex statistical dependence structures; (b) time-dependence, through autoregressive time-series structures and moving average; (c) random modelling of lead-time; and (d) demands for components with values equal to zero using zero-inflated or adjusted probability distribution. Practical implications – Professional management of the supply chain allows the users to register data concerning component identification, demand, and stock levels to subsequently be used with the proposed methodology, which must be implemented computationally. Originality/value – The proposed multivariate methodology allows it to describe demand dependence structures through inventory models applicable to components used to produce menus in food services.
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Sanchez, L., Leiva, V., Caro-Lopera, F. J., & Cysneiros, F. J. A. (2015). On matrix-variate Birnbaum-Saunders distributions and their estimation and application. Braz. J. Probab. Stat., 29(4), 790–812.
Abstract: Diverse phenomena from the real-world can be modeled using random matrices, allowing matrix-variate distributions to be considered. The normal distribution is often employed in this modeling, but usually the mentioned random matrices do not follow such a distribution. An asymmetric non-normal model that is receiving considerable attention due to its good properties is the Birnbaum-Saunders (BS) distribution. We propose a statistical methodology based on matrix-variate BS distributions. This methodology is implemented in the statistical software R. A simulation study is conducted to evaluate its performance. Finally, an application with real-world matrix-variate data is carried out to illustrate its potentiality and suitability.
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