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We introduce the notion of p*-values (p*-variables), which generalizesp-values (p-variables) in several senses. The new notion has four naturalinterpretations: operational, probabilistic, Bayesian, and frequentist. A mainexample of a p*-value is a mid p-value, which arises in the presence ofdiscrete test statistics. A unified stochastic representation for p-values, midp-values, and p*-values is obtained to illustrate the relationship between thethree objects. We study several ways of merging arbitrarily dependent orindependent p*-values into one p-value or p*-value. Admissible calibrators ofp*-values to and from p-values and e-values are obtained with nice mathematicalforms, revealing the role of p*-values as a bridge between p-values ande-values. The notion of p*-values becomes useful in many situations even if oneis 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 classicmethods for p-values and e-values.

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