Bayesian estimation, residual analysis and prior sensitivity study for zero-one augmented beta regression model with an application to psychometric data
Danilo Covaes Nogarotto, Caio L. N. Azevedo, Jorge Luiz Bazán
The interest on the analysis of the zero-one augmented beta regres sion (ZOABR) model has been increasing over the last years. In this paper we developed an extensive study on parameter recovery, prior sensitivity and the impact of some factors (scenarios of interest) comparing the Bayesian paradigm with the Maximum Likelihood (ML) approach. Jeffreys-rule, independence Jeffreys and improper priors were compared with usual choices. The results indicate, in a general way, that: the Bayesian approach, under the Jeffreys-rule prior, was as accurate as the ML one. In addition, as expected, the larger the sample size and the lower the variability of the data the more accurate are the parameter estimates. Also, we use the predictive distribution of the response to implement some available residual techniques (previously developed under the frequentist approach). To further illustrate the advantages of our approach, we conduct an analysis of a psychometric real data set including Bayesian residual analysis, where is showed that misleading inference can be obtained when the data is transformed. That is, when the observedzeros and ones are transformed to suitable values and the usual beta regression model is considered, instead using the ZOABR model. Finally, future developments are discussed.
Stochastic Neighborhood Structure in Bayesian Spatial Models
Aline Piroutek, R. Assunção, D. Duarte
Probabilistic Context Neighborhood: A New Tree Topology and Hypothese Tests
Aline Piroutek, D. Duarte, R. Assunção, Aluísio Pinheiro
We introduce the Probabilistic Context Neighborhood model for two dimensional lattices as an extension of the Probabilistic Context Tree model in one dimensional space preserving some of its interesting properties. This model has a variable neighborhood structure with a fixed geometry but varying radius. In this way we are able to compute the cardinality of the set of neighborhoods and use the Pseudo-Likelihood Bayesian Criterion to select an appropriate model given the data. We represent the dependence neighborhood structure as a tree making easier to understand the model complexity. We provide an algorithm to estimate the model that explores the sparse tree structure to improve computational efficiency. We also present an extension of the previous model, the Non-Homogeneous Probabilistic Context Neighborhood model, which allows a spatially changing Probabilistic Context Neighborhood as we move on the lattice.
Dynamic Control of Infeasibility for Nonlinear Programming
Abel S. Siqueira, Francisco A. M. Gomes Neto
An effective way of solving general nonlinear programming problems is the adoption of composite-step strategies that combine a step tangent to the constraints and a normal step, alternating between reducing the objective function value and the norm of the infeasibility. However, this kind of method requires the control of the iterates in order to prevent one step from destroying the progress of the other. In the Dynamic Control of Infeasibility algorithm, proposed by Bielschowsky and Gomes for equality constrained problems, the steps are controlled through the use of the so called Trust Cylinders. We present an extension of this algorithm for solving problems with general constraints. We also show numerical experiments that indicate that the new method has a performance that is comparable to well known nonlinear programing codes.
Modelling the proportion of failed courses and GPA scores for engineering major students
Hildete P. Pinheiro, Rafael P. Maia, Eufrásio A. Lima Neto, Mariana R. Motta
There is special interest on the factors which may contribute for the best academic performance of undergraduate students. Particularly, in Brazil, because of the recent quota system and affirmative action programs implemented by some universities and the Federal Government, this issue has been of great interest. We use here zero-one inflated beta models with heteroscedasticity to model the proportion of failed courses taken by Engineering major students at the State University of Campinas, Brazil. We also model the grade point average score for those students with a heteroscedastic skew t distribution. The database consists of records of 3,549 students with Engineering major who entered in the University from 2000 to 2005. The entrance exam score in each subject, some academic variables and their socioeconomic status are considered as covariates in the models. A residual analysis based on randomized quantile residuals is performed as well. Finally, we believe that the results found in this study can be useful to improve the university polices for new students since it was possible to identify student profiles with respect to their academic performance.
Censored Regression Models with Autoregressive Errors: A Likelihood-Based Perspective
Fernanda L. Schumacher, Víctor H. Lachos, Dipak K. Dey
In many studies that involve time series variables, limited or censored data are naturallycollected. This occurs, in several practical situations, for reasons such as limitations of mea-suring equipment or from experimental design. Hence, the exact true value is recorded only ifit falls within an interval range, so the responses can be either left, interval or right censored.Practitioners commonly disregard censored data cases or replace these observations with somefunction of the limit of detection, which often results in biased estimates. In this paper, wepropose an analytically tractable and efficient stochastic approximation of the EM (SAEM)algorithm to obtain the maximum likelihood estimates of the parameter of censored regressionmodels with autoregressive errors of order p. This approach permits easy and fast estimationof the parameters of autoregressive models when censoring is present and as a byproduct, en-ables predictions of unobservable values of the response variable. The observed informationmatrix is derived analytically to account for standard errors. We use simulations to investigatethe asymptotic properties of the SAEM estimates and prediction accuracy. In this simulationstudy comparisons are also made between inferences based on the censored data and thosebased on complete data obtained by crude/ad-hoc imputation methods. Finally, the method isillustrated using a meteorological time series dataset on cloud ceiling height, where the mea-surements are subject to the detection limit of the recording device. The proposed algorithmand methods are implemented in the new R package ARCensReg.
Quantile Regression for Nonlinear Mixed Effects Models: A Likelihood Based Perspective
Christian E. Galarza, Luis M. Castro, Francisco Louzada, Víctor H. Lachos
Longitudinal data are frequently analyzed using normal mixed effects models. Moreover,the traditional estimation methods are based on mean regression, which leads to non-robustparameter estimation for non-normal error distributions. Compared to the conventional meanregression approach, quantile regression (QR) can characterize the entire conditional distribu-tion of the outcome variable and is more robust to the presence of outliers and misspecificationof the error distribution. This paper develops a likelihood-based approach to analyzing QRmodels for correlated continuous longitudinal data via the asymmetric Laplace (AL) distri-bution. Exploiting the nice hierarchical representation of the AL distribution, our classicalapproach follows the Stochastic Approximation of the EM (SAEM) algorithm for deriving ex-act maximum likelihood estimates of the fixed-effects and variance components in nonlinearmixed effects models (NLMEMs). We evaluate the finite sample performance of the algorithmand the asymptotic properties of the ML estimates through empirical experiments and applica-tions to two real life datasets. The proposed SAEM algorithm is implemented in the R packageqrNLMM.
Robust Quantile Regression using a Generalized Class of Skewed Distributions
Christian E. Galarza, Víctor H. Lachos, Celso R. B. Cabral, Luis M. Castro
It is well known that the widely popular mean regression model could be inadequate if the probability distribution of the observed responses do not follow a symmetric distribution. To deal with this situation, the quantile regression turns to be a more robust alternative for accommodating outliers and the misspecification of the error distribution since it characterizesthe entire conditional distribution of the outcome variable. This paper presents a likelihood-based approach for the estimation of the regression quantiles based on a new family of skewed distributions introduced by Wichitaksorn et al. (2014). This family includes the skewed version of Normal, Student-t, Laplace, contaminated Normal and slash distribution, all with the zeroquantile property for the error term, and with a convenient and novel stochastic representation which facilitates the implementation of the EM algorithm for maximum-likelihood estimation of the pth quantile regression parameters. We evaluate the performance of the proposed EM algorithm and the asymptotic properties of the maximum-likelihood estimates through empirical experiments and application to a real life dataset. The algorithm is implemented in the R package lqr(), providing full estimation and inference for the parameters as well as simulation envelopes plots useful for assessing the goodness-of-fit.
Linear Regression Models with Finite Mixtures of Skew Heavy-Tailed Errors
Luis Benites Sánchez, Rocío Maehara, Víctor H. Lachos
We consider estimation of regression models whose error terms follow a finite mixture of scale mixtures of skew-normal (SMSN) distributions, a rich class of distributions that contains the skew-normal, skew-t, skew-slash and skew-contaminated normal distributions as proper elements. This approach allows us to model data with great flexibility, accommodating simultaneously multimodality, skewness and heavy tails. We developed a simple EM-type algorithm to perform maximum likelihood (ML) inference of theparameters of the proposed model with closed-form expression at the E-step. Furthermore, the standard errors of the ML estimates can be obtained as a byproduct. The practical utility of the new method is illustrated with the analysis of real dataset and several simulation studies. The proposed algorithm and methods are implemented in the R package FMsmsnReg().
Influence Diagnostics for Censored Linear Regression Models with Skewed and Heavy-tailed Distributions
Thalita do Bem Mattos, Víctor H. Lachos, Aldo M. Garay
The scale mixtures of skew-normal (SMSN) distributions (Lachos et al., 2010) form an attractive class of asymmetrical heavy-tailed densities that includes the skew-normal, skew-t, skew-slash, skew - contaminated normal and the entire family of cale mixtures of normal (SMN) distributions as special cases. A robust censored linear model based on the scale mixtures of skew-normal (SMSN) distributions has been recently proposed by Mattos et al.(2015), where a stochastic approximation of the EM (SAEM) algorithm is presented for iteratively computing maximum likelihood estimates of the parameters. In this paper, to examine the performance of the proposed model, case-deletion and local influence techniques are de veloped to show its robust aspect against outlying and influential observations. This is done by analyzing the sensitivity of the SAEM estimates under some usual perturbation schemes in the model or data and by inspecting some proposed diagnostic graphs. The efficacy of the method is verified through the analysis of simulated datasets and modeling a real dataset from stellar astronomy previously analyzed under normal errors.
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