Adaptive Significance Levels in Tests for Linear Regression Models: The e-Value and P-Value Cases

The full Bayesian significance test (FBST) for precise hypotheses is a Bayesian alternative to the traditional significance tests based on p-values. The FBST is characterized by the e-value as an evidence index in favor of the null hypothesis (H). An important practical issue for the implementation of the FBST is to establish how small the evidence against H must be in order to decide for its rejection. In this work, we present a method to find a cutoff value for the e-value in the FBST by minimizing the linear combination of the averaged type-I and type-II error probabilities for a given sample size and also for a given dimensionality of the parameter space. Furthermore, we compare our methodology with the results obtained from the test with adaptive significance level, which presents the capital-P P-value as a decision-making evidence measure. For this purpose, the scenario of linear regression models with unknown variance under the Bayesian approach is considered.