Test procedures based on combination of Bayesian evidences for $H_{0}$

We introduce two procedures for testing which are based on pooling the posterior evidence for the null hypothesis provided by the full Bayesian significance test and the posterior probability for the null hypothesis. Although the proposed procedures can be used in more general situations, we focus attention in tests for a precise null hypothesis. We prove that the proposed procedure based on the linear operator is a Bayes rule. We also verify that it does not lead to the Jeffreys–Lindley paradox. For a precise null hypothesis, we prove that the procedure based on the logarithmic operator is a generalization of Jeffreys test. We apply the results to some well-known probability families. The empirical results show that the proposed procedures present good performances. As a by-product we obtain tests for normality under the skew-normal one.