Testing with p*-values: Between p-values and e-values

We introduce the notion of p*-values (p*-variables), which generalizes p-values (p-variables) in several senses. The new notion has four natural interpretations: probabilistic, operational, Bayesian, and frequentist. The simplest interpretation of a p*-value is the average of several p-values. We show that there are four equivalent definitions of p*-values. The randomized p*-test is proposed, which is a randomized version of the simple p-test. Admissible calibrators of p*-values to and from p-values and e-values are obtained with nice mathematical forms, revealing the role of p*-values as a bridge between p-values and e-values. The notion of p*-values becomes useful in many situations even if one is only interested in p-values and e-values. In particular, tests based on p*-values can be applied to improve several classic methods for p-values and e-values.