# Relatórios de Pesquisa

 11/2012 Influence Diagnostics for Student-t Censored Linear Regression ModelsMonique B. Massuia, Celso R. B. Cabral, Larissa A. Matos, Víctor H. Lachos 10/2012 A multiple group Item Response Theory model with centred skew normal latent trait distributions under a Bayesian frameworkJosé R. S. Santos, Caio L. N. Azevedo, Heleno BolfarineThe multiple group IRT model (MGM) provides a useful framework for analyzing item response data from clustered respondents. In the MGM, the selected groups of respondents are of specific interest such thatgroup-specific population distributions need to be defined. An usual assumption for parameter estimation for this model, is to assume that the latent traits are random variables which follow possibly differentsymmetric normal distributions. However, many works suggest that this assumption does not apply in many cases. Furthermore, when this assumption does not hold, parameter estimates tend to be biased and misleadinginference can result. Therefore, it is important to model the distribution of the latent traits properly. In this paper we present an alternative latent traits modeling, for multiple group framework, based on theso-called skew-normal distribution. We name it SMGIRT model (skew multiple group IRT model). It extends the approach proposed by some authors in the literature. We use the centred parameterization. This approach ensures model identifiability. We propose and compare, concerning convergence issues, two MCMC algorithms for parameter estimation. A simulation study was performed in order to assess parameter recovery for the proposed modeland the selected algorithm concerning convergence issues. The results reveals that our proposed algorithm recovers properly all model parameters. Furthermore, we analyzed a real data set which presentsindication of asymmetry concerning the latent traits distribution. The results obtained by using our approach confirmed the presence of negative asymmetry of the latent traits distribution. Moreover, our modeloutperforms the usual symmetric normal MGM, leading to different conclusions concerning parameter estimation. 9/2012 $|S|$ Control Chart for Multivariate Process Variability Monitoring Based on Cornish-Fisher Correction and Meijer-G FunctionEmanuel P. Barbosa, Mario A. Gneri, Ariane MeneguettiThis paper presents an improved version of the generalized variance |S| control chart for multivariate process dispersion monitoring, based on the Cornish-Fisher formula for non-normality correctionof the normal 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, and an auxiliary control chart basedon the trace of V (standardized S matrix) is introduced. The performance of this corrected control chart is compared (in terms of false alarm risk) with the traditional normal based chart and the exactdistribution based chart (for dimensions d = 2 and d = 3). This study shows that the control limits corrections do remove the drawback of excess of false alarm associated with the traditional normalbased |S| control chart. The proposed new chart is illustrated with two numerical examples. 8/2012 Averaging Methods for Studying the Periodic Orbits of Discontinuous Differential SystemsJaume Llibre, Douglas D. Novaes, Marco A. TeixeiraThe main objective of this work is to extend the averaging method for studying the periodic orbits of a class of differential equations with discontinuous second member. Thus, overall results are presented to ensure the existence of limit cycles of such systems. Certainly these results represent new insights in averaging, in particular its relation with non smooth dynamical systems theory. An application is presented in careful detail. 7/2012 Perturbed Damped Pendulum: Finding Periodic SolutionsDouglas D. NovaesUsing the equation of motion of the damped pendulum, we introduce the averaging method on the study of periodic solutions of dynamical systems with small perturbation. We provide sufficient conditions for the existence of periodic solutions of the perturbed damped pendulum with small oscillations having equations of motion$\ddot{\T}=-a\T-b\dot{\T}+\e f(t,\T,\dot{\T}),$where $a>0,\,b>0$ and $\e$ are real parameters, with $a=g/l$, $g$ the acceleration of the gravity, $l$ the length of the rod and $b$ the damping coefficient. Here the parameters $b$ and $\e$ are small and the smooth function $f$ is $T$--periodic in $t$. The averaging theory provides a useful means to study dynamical systems, accessible to Master and PhD students. 6/2012 On The Periodic Solutions of a Generalized Smooth and Non-Smooth Perturbed Planar Double Pendulum with Small OscillationsJaume Llibre, Douglas D. Novaes, Marco A. TeixeiraWe provide sufficient conditions for the existence of periodic solutions of the smooth and non-smooth perturbed planar double pendulum with small oscillations having equations of motion$\begin{array}{l}\ddot{\T}_{1}=-a\T_{1}+\T_{2}+\e \left(F_1(t,\T_1,\dot {\T}_1,\T_2, \dot{\T}_2)+F_2(t,\T_1,\dot {\T}_1,\T_2, \dot {\T}_2){\rmsgn}(\dot{\T_{1}})\right),\\\ddot{\T}_{2}=b\T_{1}-b\T_{2}+\e \left(F_3(t,\T_1,\dot {\T}_1,\T_2, \dot{\T}_2)+F_4(t,\T_1,\dot {\T}_1,\T_2, \dot {\T}_2){\rmsgn}(\dot{\T_{2}})\right),\end{array}$where $a>1,b>0$ and $\e$ are real parameters. Here the parameter $\e$ is small and the smooth functions $F_i$ for $i=1,2,3,4$ define the perturbation which are periodic functions in $t$ and in resonance $p_{i}$:$q_{i}$ with some of the periodic solutions of the unperturbed double pendulum, being $p_{i}$ and $q_{i}$ relatively prime positive integers. 5/2012 On The Periodic Solutions of a Perturbed Double PendulumJaume Llibre, Douglas D. Novaes, Marco A. TeixeiraWe provide sufficient conditions for the existence of periodic solutions of the planar perturbed double pendulum with small oscillations having equations of motion$\begin{array}{l}\ddot{\T}_{1}=-2a\T_{1}+a\T_{2}+\e F_1(t,\T_1,\dot \T_1,\T_2, \dot \T_2),\\\ddot{\T}_{2}=2a\T_{1}-2a\T_{2}+\e F_2(t,\T_1,\dot \T_1,\T_2, \dot\T_2),\end{array}$where $a$ and $\e$ are real parameters. The two masses of the unperturbed double pendulum are equal, and its two stems have the same length $l$. In fact $a=g/l$ where $g$ is the acceleration ofthe gravity. Here the parameter $\e$ is small and the smooth functions $F_1$ and $F_2$ define the perturbation which are periodic functions in $t$ and in resonance $p$:$q$ with some of the periodicsolutions of the unperturbed double pendulum, being $p$ and $q$ positive integers relatively prime. 4/2012 Minimização irrestrita usando gradientes conjugados e regiões de confiançaJohn Lenon C. Gardenghi, Sandra A. SantosThis work focus on the conjugate gradient method to solve the trust region sub-problem for unconstrained minimization. We aim to describe an intuitive and detailed study about this method, starting from an introduction to methods of conjugate directions, some necessary requisites and tools for understanding the conjugate gradient method and its integration with the trust region strategy for unconstrained minimization. The computational implementation of the method using the CAS Maxima enabled the numerical experiments, which validated the study and the implementation done and allowed a comparison between conjugate gradient and Leverberg-Marquardt for least squares problems. 3/2012 Bayesian general multivariate latent variable modeling of longitudinal item response dataCaio L. N. Azevedo, Jean-Paul Fox, Dalton F. AndradeLongitudinal item response data are characterized by examinees that are assessed at different time points or measurement occasions such that time-specific measurements are nested within examinees. Besides the usual nesting of response observations within examinees, the time-specific latent traits are also nested within examinees. In the well-known hierarchical modeling approach, the complex dependencies due to the nested structure of the data are commonly modeled by introducing random effects such that observations and latent traits are conditionally independently distributed. However, the implied compound symmetry structure is often not sufficient to model the complex time-heterogenous dependencies.Therefore, a Bayesian general multivariate item response modeling framework is proposed that accounts for the complex within-examinee latent trait dependencies. Flexible parametric covariance structures are considered to modelspecific within-examinee dependencies. Furthermore, it can handle many measurement occasions, different item response functions, and different latent trait population distributions, and generalizes some works of current literature. Due to identification rules and restricted parametric covariance structures, a conditional modeling approach is pursued to specify proper priors for the unrestricted parameters and to implement an efficient MCMC algorithm, by conditioning on baseline population parameters. The study is motivated by a large-scale longitudinal research program of the Brazilian Federal government to improve the teaching quality and general structure of schools for primary education. It is shown that the growth in math achievement can be accurately measured when accounting for complex dependencies over grades using time-heterogenous covariances structures. 2/2012 A note on identification and metric issues for skew IRT modelsCaio L. N. Azevedo, Heleno Bolfarine, Dalton F. AndradeThe skew-normal distribution (SND) is a flexible family of densities which preserves some useful properties of the original normal distribution. Some stochastic representations for the SND have beenproposed in the literature. The Henze (H) and Sahu, Branco and Dey (SBD) are the two most used ones. On the other hand, the centered parametrization is useful for inference purposes. The main goals ofthis article are: establish a link between the standard H and SDB skew-normal distributions and use this result to model the latent traits for IRT models. We proved that standard H and SDB distributions are related to each other through a function of the asymmetry parameter and also that they are exactly the same under centered parametrization (CP). Using these results, we showed that the common density obtained through the CP is useful to model the latent traits for unidimensional IRT models. This approach allows to represent asymmetric latent traits behavior and ensures the model identification as well.