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.

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.

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.

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.

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.

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.

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.

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.

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.

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.