Bayesian sample size determination for coefficient of variation of normal distribution.

Sample size determination is an active area of research in statistics. Generally, Bayesian methods provide relatively smaller sample sizes than the classical techniques, particularly average length criterion is more conventional and gives relatively small sample sizes under the given constraints. The objective of this study is to utilize major Bayesian sample size determination techniques for the coefficient of variation of normal distribution and assess their performance by comparing the results with the freqentist approach. To this end, we noticed that the average coverage criterion is the one that provides relatively smaller sample sizes than the worst outcome criterion. By comparing with the existing frequentist studies, we show that a smaller sample size is required in Bayesian methods to achieve the same efficiency.

Comparison of Bayesian Sample Size Criteria: ACC, ALC, and WOC.

A challenge for implementing performance based Bayesian sample size determination is selecting which of several methods to use. We compare three Bayesian sample size criteria: the average coverage criterion (ACC) which controls the coverage rate of fixed length credible intervals over the predictive distribution of the data, the average length criterion (ALC) which controls the length of credible intervals with a fixed coverage rate, and the worst outcome criterion (WOC) which ensures the desired coverage rate and interval length over all (or a subset of) possible datasets. For most models, the WOC produces the largest sample size among the three criteria, and sample sizes obtained by the ACC and the ALC are not the same. For Bayesian sample size determination for normal means and differences between normal means, we investigate, for the first time, the direction and magnitude of differences between the ACC and ALC sample sizes. For fixed hyperparameter values, we show that the difference of the ACC and ALC sample size depends on the nominal coverage, and not on the nominal interval length. There exists a threshold value of the nominal coverage level such that below the threshold the ALC sample size is larger than the ACC sample size, and above the threshold the ACC sample size is larger. Furthermore, the ACC sample size is more sensitive to changes in the nominal coverage. We also show that for fixed hyperparameter values, there exists an asymptotic constant ratio between the WOC sample size and the ALC (ACC) sample size. Simulation studies are conducted to show that similar relationships among the ACC, ALC, and WOC may hold for estimating binomial proportions. We provide a heuristic argument that the results can be generalized to a larger class of models.