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ABSTRACT Sampling is an essential step in estimating a parameter: thus, cost and time associated to this step should be minimized. Sequential sampling is characterized by using samples of variable sizes given as a function of observations, and sequential sampling provides a smaller sample size than a fixed-size sample in most cases. In addition, the Bayesian decision theory can be incorporated into sequential sampling to perform parameter estimation because it allows the inclusion of a priori information about the parameter of interest, which optimizes the procedure. However, the great challenge to performing the Bayesian sequential estimation in establishing the stopping criteria. Most studies in this area investigate binomial distributions, while few analyze multinomial distributions. This study aimed to define the stopping criteria for the Bayesian sequential estimation of the parameters of multinomial distributions with conjugate Dirichlet priors. The proposed methodology was applied to a set of X-ray test data for quality control of maize seed lots. This test uses conventional sampling techniques in which a sample has a fixed size with 200 seeds. The influence of two priors on the stopping criteria was evaluated, one uniform and one conjugate, with hyperparameters based on reference information from the literature. The results indicated a reduction in the sample size in most lots evaluated.

ABSTRACT: Tetrazolium tests use conventional sampling techniques in which a sample has a fixed size. These tests may be improved by sequential sampling, which does not work with fixed-size samples. When data obtained from an experiment are analyzed sequentially the analysis can be terminated when a particular decision has been made, and thus, there is no need to pre-establish the number of seeds to assess. Bayesian statistics can also help, if we have sufficient knowledge about coffee production in the area to construct a prior distribution. Therefore, we used the Bayesian sequential approach to estimate the percentage of viable coffee seeds submitted to tetrazolium testing, and we incorporated priors with information from other analyses of crops from previous years. We used the Beta prior distribution and, using data obtained from sample lots of Coffea arabica, determined its hyperparameters with a histogram and O’Hagan's methods. To estimate the lowest risk, we computed the Bayes risks, which provided us with a basis for deciding whether or not we should continue the sampling process. The results confirm that the Bayesian sequential estimation can indeed be used for the tetrazolium test: the average percentage of viability obtained with the conventional frequentist method was 88 %, whereas that obtained with the Bayesian method with both priors was 89 %. However, the Bayesian method required, on average, only 89 samples to reach this value while the traditional estimation method needed as many as 200 samples.