Further examples of maximal curves which cannot be covered by the Hermitian curve
Saeed Tafazolian, Arnoldo Teheran-Herrera, Fernando Torres
We construct examples of curves defined over the finite field Fq6 which are covered by the GK-curve. Thus such curves are maximal over Fq6 although they cannot be covered by the Hermitian curve for q > 2. We also give examples of maximal curvesthat cannot be Galois covered by the Hermitian curve over the finite field Fq2n with n > 3 odd and q > 2. We point out some applications to codes related to an array coming from telescopic semigroups.
Improving Shewhart-type Generalized Variance Control Charts for Multivariate Process Variability Monitoring using Cornish-Fisher Quantile Correction, Meijer-G Function and Other Tools
Emanuel P. Barbosa, Mario A. Gneri, Ariane Meneguetti
This paper presents an improved version of the Shewhart-type generalized variance |S| control chart for multivariate Gaussian process dispersion monitoring, based on the Cornish-Fisher quantile formula for non-normality correction of the traditional nor-mal based 3-sigma chart limits.Also, the exact sample distribution of |S| and its quantiles (chart exact limits) are obtained through the Meijer-G function (inverse Mellin-Barnes integral transform), and an auxiliary control chart based on the trace of the standardized S matrix is introduced in order to avoid non detection of certain changes in the process variance-covariance Σ matrix.The performance of the proposed CF-corrected control chart is compared, considering false alarm risk (using analytical and simulation tools), with the traditional normal based chart and with the exact distributed based chart (for dimensions d = 2 and d = 3). This study shows that the proposed control limit corrections do remove the drawback of excess of false alarm associated with the traditional normal based |S| control chart.The proposed new chart (with its corresponding auxiliary chart) is illustrated with two numerical examples.
On the curve Y n = X m + X over finite fields
Saeed Tafazolian, Fernando Torres
We show that a maximal curve over Fq2 defined by the affine equation y n = f (x), where f (x) ∈ Fq2 [x] has degree coprime to n, is such that n is a divisor of q + 1 if and only if f (x) has a root in Fq2 . In this case, all the roots of f (x) belong to Fq2 ;cf. Thm. 1.2, Thm. 4.3 in [J. Pure Appl. Algebra 212 (2008), 2513–2521]. In particular, we characterize certain maximal curves defined by equations of type y n = xm + x over finite fields.
A Mixed-Effect Model for Positive Responses Augmented by Zeros
Mariana R. Motta, Diana M. Galvis, Víctor H. Lachos, Filidor E. Vilca-Labra, Valéria Troncoso Baltar, Eliseu Verly Junior, Regina Mara Fisberg, Dirce Maria Lobo Marchioni
In this work we propose a model for positive and zero responses by means of a zero augmented mixed regression model. Under this class, we are particularly interested in studying positive responses whose distribution accommodates skewness. At the same time, responses can be zero and therefore we justify the use of a zero-augmented mixture model. We model the mean of the positive response in a logarithm scale and the mixture probability in a logit scale, both as afunction of fixed and random effects. Here, the random effects link the two random components through their joint distribution and incorporate within subject correlation due to repeated measurements and between-subject heterogeneity.An MCMC algorithm is tailored to obtain Bayesian posterior distributions of the unknown quantities of interest and Bayesian case-deletion influence diagnostics based on the q-divergence measure is performed. We motivate and illustrate the proposed methodology by means of a data set from a 24 hours dietary recall study obtained in the city of S ̃o Paulo, Brazil, and present a simulation study a to evaluate the performance of the proposed methods.Bayesian inference, gamma distribution, log-normal distribution, mixed models, random effects, usual intake, zero-augmented data.
An Ergodic Description of Ground States
Eduardo Garibaldi, Philippe Thieullen
Given a translation-invariant Hamiltonian H, a ground state on the lattice Zd is a configuration whose energy, calculated with respect to H, cannot be lowered by altering its states on a finite number of sites. The set formed by these configurations is translation-invariant. Given an observable defined on the space of configurations, a minimizing measure is a translation-invariantprobability which minimizes the average of. If 0 is the mean contribution of all interactions to the site 0, we show that any configuration of the support of a minimizing measure is necessarily a ground state.
Discrete weak-KAM methods for stationary uniquely ergodic setting
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 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.
Diagnostics for censored mixed-effects models using the multivariate t-distribution
Dipankar Bandyopadhyay, Larissa A. Matos, Luis M. Castro, Víctor H. Lachos
In biomedical studies on HIV RNA dynamics, the viral loads generate repeated measures that are often subjected to (upper and lower) detection limits, and hence these responses are either left- or right-censored. Linear and non-linear mixed-effects censored (LMEC/NLMEC) models are routinely used to analyze these longitudinal data, with normality assumptions for the random effects and residual errors. However, the derived inference may not be robust when these underlying normality assumptions are questionable, specially presence of outliers and thick-tails. Motivated by this, Matos et al. (2013b) recently proposed an exact EM-type algorithm for LMEC/NLMEC models using a multivariate Student’s-t distribution, with closed-form expressions at the E-step. In this paper, we develop influence diagnostics for LMEC/NLMEC models using multivariate Student’s-t density, based on the conditional expectation of the complete data log-likelihood which eliminates the complexity associated with the approach of Cook (1977, 1986) for censored mixed-effects models. The new methodology is illustrated through an application to a longitudinal HIV dataset using the NLMEC framework. In addition, a simulation study is presented, which explores the accuracy of the proposed measures in detecting influential observations in heavy-tailed censored data under different perturbation schemes.
Estimation Methods for Multivariate Tobit Confirmatory Factor Analysis
D. R. Costa, Víctor H. Lachos, J. L. Bazan, Caio L. N. Azevedo
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.
Models Applied to DNA Sequences with Multinomial Correlated Responses
Beatriz Cuyabano, Hildete P. Pinheiro, Aluísio Pinheiro
Multinomial multivariate models are proposed to describe the codon frequencies in DNA sequences, as well as the order and frequency that nucleotide bases have in each codon considering the dependence among the bases inside a codon. Logistic regressive models are used with diﬀerent dependence structures on the three codon positions. Also, multinomialextensions of the Bahadur’s representation are proposed to model correlated multinomial data. An application of these models to the NADH4 gene from human mitochondrial genome is presented. AIC, BIC and the leave-one-out cross validation are employed to compare the various models peformance.
Likelihood Based Inference for Quantile Regression Using the Asymmetric Laplace Distribution
Luis Benites Sánchez, Víctor H. Lachos, Filidor E. Vilca-Labra
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().
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