Research Reports

The 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.

rp-2012-10.pdf

The 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.

rp-2012-8.pdf

We 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.

rp-2012-6.pdf