Testing with p*-values: Between p-values, mid 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: operational, probabilistic, Bayesian, and frequentist. A main example of a p* -value is a mid p -value, which arises in the presence of discrete test statistics. A unified stochastic representation for p -values, mid p -values, and p* -values is obtained to illustrate the relationship between the three objects. We study several ways of merging arbitrarily dependent or independent p* -values into one p -value or p*value. 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, mid p -values, or e -values. In particular, deterministic tests based on p* -values can be applied to improve some classic methods for p -values and e -values.