4/2016 |
Estimation in Spatial Models with Censored Response Thais S. Barbosa, Víctor H. Lachos, Larissa A. Matos, Marcos O. Prates Spatial environmental data can be subject to some upper and lower limits of detection (LOD), below orabove which the measures are not quantifiable. As a result, the responses are either left or right censored.Historically, the most common practice for analysis of such data has been to replace the censored observa-tions with some function of the limit of detection (LOD/2, 2LOD), or through data augmentation, by usingMarkov chain Monte Carlo methods. In this paper, we propose an exact estimation procedure to obtain themaximum likelihood estimates of the fixed effects and variance components, using a stochastic approxi-mation of the EM algorithm, the SAEM algorithm (Delyon et al., 1999). This approach permits easy andfast estimation of the parameters of spatial linear models when censoring is present. As a byproduct, pre-dictions of unobservable values of the response variable are possible. The proposed algorithm is appliedto a spatial dataset of depths of a geological horizon that contains both left- and right-censored data. Wealso use simulation to investigate the small sample properties of predictions and parameter estimates andthe robustness of the SAEM algorithm. In this simulation study comparisons are made between inferencesbased on the censored data and inferences based on complete data obtained by a crude/ad hoc imputationmethod (LOD/2, 2LOD). The results show that differences in inference between the two approaches can besubstantial. rp-2016-4.pdf |

3/2016 |
Moments of truncated skew-normal/independent distributions Víctor H. Lachos, Aldo M. Garay, Celso R. B. Cabral In this work we consider the problem of finding the moments of a doubly truncated member of theclass of skew-normal/independent (TSNI) distributions. We obtained a general result and then use itto derive the moments in the case of doubly truncated versions of skew-normal, skew-t, skew-slash andskew-contaminated normal distributions. Many properties of the TSNI family are studied, inference pro-cedures are developed and a simulation study is performed to assess the procedures. An application inthe context of censored regression models is also provided. rp-2016-3.pdf |

2/2016 |
Multivariate Measurement Error Models Based on Student-t Distribution under Censored Responses Larissa A. Matos, Luis M. Castro, Celso R. B. Cabral, Vı́ctor H. Lachos Measurement error models constitute a wide class of models, that include linear and nonlinear regression models. They are very useful to model many real life phenomena, particularly inthe medical and biological areas. The great advantage of these models is that, in some sense, they can be represented as mixed effects models, allowing to us the implementation of well-known techniques, like the EM-algorithm for the parameter estimation. In this paper, we consider a class of multivariate measurement error models where the observed response and/orcovariate are not fully observed, i.e., the observations are subject to certain threshold values below or above which the measurements are not quantifiable. Consequently, these observationsare considered censored. We assume a Student-t distribution for the unobserved true values of the mismeasured covariate and the error term of the model, providing a robust alternativefor parameter estimation. Our approach relies on a likelihood-based inference using the EM-algorithm. The proposed method is illustrated through simulation studies and the analysis of a real dataset. rp-2016-2.pdf |

1/2016 |
Heavy-tailed longitudinal regression models for censored data: A likelihood based perspective Larissa A. Matos, Víctor H. Lachos, Tsung-I Lin, Luis M. Castro HIV RNA viral load measures are often subjected to some upper and lower detection limits depending on the quantification assays. Hence, the responses are either left or right censored. Moreover,it is quite common to observe viral load measurements collected irregularly over time. A complica-tion arises when these continuous repeated measures have a heavy-tailed behaviour. For such data structures, we propose a robust censored linear model based on the scale mixtures of normal distributions (SMN family). To take into account the autocorrelation existing among irregularly observed measures, a damped exponential correlation structure is considered. A stochastic approximation of the EM algorithm (SAEM algorithm) is developed to obtain the maximum likelihood estimates of the model parameters. The main advantage of this new procedure allows us to estimate the parameters of interest and evaluate the log-likelihood function in an easy and fast way. Furthermore, the standard errors of the fixed effects and predictions of unobservable values of the response can be obtained asa by-product. The practical utility of the proposed methodology is exemplified using both simulatedand real data. rp-2016-1.pdf |

8/2015 |
Zero-temperature Phase Diagram for Double-Well Type Potentials in the Summable Variation Class Rodrigo Bissacot, Eduardo Garibaldi, Philippe Thieullen We study the zero-temperature limit of the Gibbs measures of a class of long-range potentials on a full shift of two symbols {0, 1}. These potentials were introduced by Walters as a natural space for the transfer operator. In our case, they are locally constant, Lipschitz continuous or, more generally, of summable variation. We assume there exists exactly two ground states: the fixed points 0 ∞ and 1 ∞ . We fully characterize, in terms of the Peierls barrier between the two ground states, the zero-temperature phase diagram of such potentials, that is, the regions of convergence or divergence of the Gibbs measures as the temperature goes to zero. rp-2015-8.pdf |

7/2015 |
Likelihood Based Inference for Censored Linear Regression Models with Scale Mixtures of Skew-Normal Distributions Thalita do Bem Mattos, Aldo M. Garay, Víctor H. Lachos In many studies the data collected are subject to some upper and lower detection limits. Hence, theresponses are either left or right censored. A complication arises when these continuous measures presentheavy tails and asymmetrical behavior, simultaneously. For such data structures, we propose a robustcensored linear model based on the scale mixtures of skew-normal (SMSN) distributions. The SMSN is anattractive class of asymmetrical heavy-tailed densities that includes the skew-normal, skew-t, skew-slash,skew-contaminated normal and the entire family of scale mixtures of normal (SMN) distributions asspecial cases. We propose a fast estimation procedure to obtain the maximum likelihood (ML) estimatesof the parameters, using a stochastic approximation of the EM (SAEM) algorithm. This approach allowsus to estimate the parameters of interest easily and quickly, obtaining as a byproduct the standard errors,predictions of unobservable values of the response and the log-likelihood function. The proposed methodsare illustrated through a real data application and several simulation studies. rp-2015-7.pdf |

6/2015 |
Influence Diagnostics in Spatial Models with Censored Response Thais S. Barbosa, Víctor H. Lachos, Dipak K. Dey Environmental data is often spatially correlated and sometimes include below detection limit observations (i.e.,censored values reported as less than a level of detection). Existing work mainly concentrates on parameter estimation using Gibbs sampling, and work conducted from a frequentist perspective in spatial censored models areelusive. In this paper, we propose an exact estimation procedure to obtain the maximum likelihood estimates of thefixed effects and variance components, using a stochastic approximation of the EM (SAEM) algorithm (Delyonet al., 1999). This approach permits estimation of the parameters of spatial linear models when censoring is presentin an easy and fast way. As a by-product, predictions of unobservable values of the response variable are possible.Motivated by this algorithm, we develop local influence measures on the basis of the conditional expectation ofthe complete-data log-likelihood function which eliminates the complexity associated with the approach of Cook(1977, 1986) for spatial censored models. Some useful perturbation schemes are discussed. The newly developedmethodology is illustrated using data from a dioxin contaminated site in Missouri. In addition, a simulation studyis presented, which explores the accuracy of the proposed measures in detecting influential observations underdifferent perturbation schemes. rp-2015-6.pdf |

5/2015 |
Robust Regression Modeling for Censored Data Based on Mixtures of Student-t Distributions Víctor H. Lachos, Luis Benites Sánchez, Celso R. B. Cabral In the framework of censored regression models, the distribution of the error terms departs significantly from normality, for instance, in the presence of heavy tails, skewness and/or atypical observations. In this paper we extend the censored linear regression model with normal errors to the case where the random errors follow a finite mixture of Student-t distributions. Thisapproach allows us to model data with great flexibility, accommodating multimodality, heavy tails and also skewness depending on the structure of the mixture components. We develop an analytically simple and efficient EM-type algorithm for iteratively computing maximum likelihood estimates of the parameters, with standard errors as a by-product. The algorithm has closed-form expressions at the E-step, that rely on formulas for the mean and variance of the truncated Student-t distributions. The efficacy of the method is verified through the analysis of simulated datasets and modeling a censored real dataset first analyzed under normal and Student-t errors. The proposed algorithm and methods are implemented in the R package CensMixReg(). rp-2015-5.pdf |

4/2015 |
Calibrated configurations for Frenkel-Kontorova type models in almost-periodic environments Eduardo Garibaldi, Samuel Petite, Philippe Thieullen The Frenkel-Kontorova model describes how an infinite chain of atoms minimizes the total energy of the system when the energy takes into account the interaction of nearest neighbors as well as the interaction with an exterior environment. An almost-periodic environment leads to consider a family of interaction energies which is stationary with respect to a minimal topological dynamical system. We introduce, in this context, the notion of calibrated configuration (stronger than the standard minimizing condition) and, for continuous superlinear interaction energies, we show the existence of these configurations for some environment of the dynamical system. Furthermore, in one dimension, we give sufficient conditions on the family of interaction energies to ensure, for any environment, the existence of calibrated configurations when the underlying dynamics is uniquely ergodic. The main mathematical tools for this study are developed in the frameworks of discrete weak KAM theory, Aubry-Mather theory and spaces of Delone sets. rp-2015-4.pdf |

3/2015 |
Bayesian Analysis of Censored Linear Regression Models with Scale Mixtures of Skew-Normal Distributions Monique B. Massuia, Aldo M. Garay, Víctor H. Lachos, Celso R. B. Cabral As is the case of many studies, the data collected are limited and an exact value is recorded only if it falls within an interval range. Hence, the resp onses can b e either left, interval or right censored. Linear (and nonlinear) regression mo dels are routinely used to analyze these typ es of data and are based on the normality assumption for the errors terms. However, those analyses might not provide robust inference when the normality assumption (or symmetry) is questionable. In this article, we develop a Bayesian framework for censored linear regression mo dels by replacing the Gaussian assumption for the randomerrors with the asymmetric class of scale mixtures of skew-normal (SMSN) distributions. The SMSN is an attractive class of asymmetrical heavy-tailed densities that includes the skew-normal, skew-t, skew-slash, the skew-contaminated normal and the entire family of scale mixtures of normal distributions as sp ecial cases. Using a Bayesian paradigm, an efficient Markov chain Monte Carlo (MCMC) algorithm is intro duced to carry out p osterior inference. The likeliho o d function is utilized to compute not only some Bayesian mo del selection measures but also to develop Bayesian case-deletion influence diagnostics based on the q-divergence measures. The prop osed Bayesian metho ds are implemented in the R package BayesCR, prop osed by the authors. The newly develop ed pro cedures are illustrated with applications using real and simulated data. rp-2015-3.pdf |

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