Aspectos teóricos, simulação e aplicações da estimação sequencial bayesiana em distribuições multinomiais

Abstract

Sequential sampling is a statistical technique in which the sample size is not fixed. Observations are successively incorporated into the process, and at each new observation, the established stopping criterion is evaluated. The procedure is interrupted when the criterion is met, and the total number of observations made up to that point defines the sample size. The main advantage of this sampling method is that it generally results in smaller sample sizes compared to fixedsize samples, reducing costs and time. The Bayesian decision theory, associated with sequential sampling, constitutes an approach for estimating parameters of interest, which allows the inclusion of prior information that assists the sampling plan. However, the implementation of this approach presents challenges arising from the recursion involved and the definition of the stopping criterion, which become even more complex in multivariate models. Due to the inherent difficulty of the procedure, most studies in literature have been developed for discrete and univariate distributions, notably the binomial distribution. There is a gap in the development and application of sequential Bayesian estimation for multivariate distributions, such as the multinomial distribution, used to estimate proportions in problems with more than two response categories. Given this gap, this thesis aims to propose the estimation of multinomial distribution parameters through Bayesian inference using sequential sampling, exploring theoretical aspects and applications. To this end, through simulation studies, the behavior of the sequential Bayesian estimation of multinomial distribution parameters was analyzed under different scenarios. Subsequently, a Shiny application was developed in the R software, aiming to broaden the accessibility and applicability of the approach. Furthermore, the approach was applied in two real-world practical situations: in population genetics, where allele and genotype proportions were estimated, and in beekeeping, where the proportions of honeycomb cell content were estimated. In general, the results obtained demonstrate that Bayesian sequential estimation for multinomial distribution parameters is an efficient and economically advantageous alternative, capable of producing accurate estimates with smaller sample size, compared to fixed-size sampling. Furthermore, the applied studies reinforce the potential of the approach for real-world problems in different areas.