Inferência bayesiana objetiva e freqüentista para a probabilidade de sucesso

Abstract

This study considers two discrete distributions based on Bernoulli trials: the Binomial and the Negative Binomial. We explore credibility and confidence intervals to estimate the probability of success of each distribution. The main goal is to analyze their performance coverage probability and average range across the parametric space. We also consider point analysis of bayesian estimators and maximum likelihood estimators, whose interest is to confirm through simulation their consistency, bias and mean square error. In this paper the Objective Bayesian Inference is applied through the noninformative Bayes-Laplace prior, Haldane prior, reference prior and least favorable prior. By analyzing the prior distributions in the minimax decision theory context we verified that the least favorable prior distribution has every other considered prior distributions as particular cases when a quadratic loss function is applied, and matches the Bayes-Laplace prior in considering the quadratic weighed loss function for the Binomial model (which was never found in literature). We used the noninformative Bayes-Laplace prior and Jeffreys prior for the Negative Binomial model. Our findings show through coverage probability, average range of bayesian intervals and point estimation that the Objective Bayesian Inference has good frequentist properties for the probability of success of Binomial and Negative Binomial models. The last stage of this study discusses the presence of correlated proportions in matched-pairs (2 × 2 table) of Bernoulli with the goal of obtaining more information in relation of the considered measures for testing the occurrence of correlated proportions. In this sense the Trinomial model and the partial likelihood function were used from the frequentist and bayesian point of view. The Full Bayesian Significance Test (FBST) was used for real data sets and was shown sensitive to parameterization, however, this study was not possible for the frequentist method since distinct methods are needed to be applied to Trinomial model and the partial likelihood function.