Research Reports

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 prove its existence for some environment of the dynamical system. Furthermore, in one dimension, we give sufficient conditions on the family of interaction energies to ensure the existence of calibrated configurations for any environment 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-2013-19.pdf

We propose two methods for estimating multivariate Tobit Confirmatory Factor Analysis (TCFA) with covariates, one from Bayesian and another from likelihood based perspectives. TCFA is particularly useful in analysis of multivariate data with censored information. In contrast with previous developments that utilize Monte Carlo simulations for maximum likelihood estimation, an exact EM-type algorithm is proposed, which uses closed form expressions at the E-step that rely on the mean and variance of a truncated multinormal distribution and can be computed using available software. Through simulation studies, we compare the performance of the proposed algorithm when the censored pattern is ignored for different levels of censoring. Our results suggest that this algorithm has excellent performance, since it recovered the true parameters of the TCFA model much better than did the traditional CFA model. In addition, by considering a hierarchical formulation of the models, we also explore the estimation of the parameters via MCMC techniques by using proper priors. A Bayesian case deletion influence diagnostic based on the q-divergence measure and model selection criteria is also developed and applied to analyze a real dataset from an education assessment. In addition, a simulation study is conducted to compare the performance of the proposed method with the traditional CFA model.

rp-2013-17.pdf

To make inferences about the shape of a population distribution, the widely popular mean regression model, for example, is inadequate if the distribution is not approximately Gaussian (or symmetric). Compared to conventional mean regression (MR), quantile regression (QR)can characterize the entire conditional distribution of the outcome variable, and is more robust to outliers and misspecification of the error distribution. We present a likelihood-based approach to the estimation of the regression quantiles based on the asymmetric Laplace distribution (ALD), a choice that turns out to be natural in this context. The ALD has a nice hierarchical representation which facilitates the implementation of the EM algorithm for maximum-likelihood estimation of the parameters at the pth level with the observed information matrix as a byproduct. Inspired by the EM algorithm, we develop case-deletion diagnostics analysis for QR models, following the approach of Zhu et al. (2001). This is because the observed data log–likelihood function associated with the proposed model is somewhat complex (e.g., not differentiable at zero) and by using Cook’s well-known approach it can be very difficult to obtain case-deletion measures. The techniques are illustrated with both simulated and real data. In particular, in an empirical comparison, our approach out-performed other common classic estimators under a wide array of simulated data models and is flexible enough to easily accommodate changes in their assumed distribution. The proposed algorithm and methods are implemented in the R package ALDqr().

rp-2013-15.pdf

In the framework of censored regression models the random errors are routinely assumed to have a normal distribution, mainly for mathematical convenience. However, this method has been criticized in the literature because of its sensitivity to deviations from the normality assumption. In practice, data such as income or viral load in AIDS studies, often violate this assumption because of heavy tails. Here, we first establish a new link between the cen-sored regression model and a recently studied class of symmetric distributions, which extend the normal one by the inclusion of kurtosis, called scale mixtures of normal (SMN) distributions. The Student-t, Pearson type VII, slash, contaminated normal, among others distributions, are contained in this class. Choosing a member of this class can be a good alternative to model this kind of data, because they have been shown its flexibility in several applications. In this work, 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 certain truncated SMN distributions. The proposed algorithm is implemented in the R package SMNCensReg. Applications with simulated and a real data are reported, illustrating the usefulness of the new methodology.

rp-2013-13.pdf