The Fisherian prescription of reporting P-values as a summary of a result, as compared to the Neyman-Pearson system of acceptance or rejection of a null hypothesis, is more common in applied science. This popularity is largely due to the fact that the P-value provides a more complete, meaningful and useful evidence regarding the null hypothesis. Conventionally, P-values are defined in the context of one-sided alternatives, although there exist some ideas in the literature concerning two-sided alternatives; see e.g. [Gibbons, J.D., Pratt, J.W., 1975. P-values: Interpretation and methodology. American Statistician 24, 20-25; George, E.O., Mudholkar, G.S., 1990. P-values for two-sided tests. Biometrical Journal 32, 747-751]. This note takes an axiomatic approach for defining P-values which involves at most ordering of the alternatives but is not restricted by their nature. It also involves a correspondence between a P-value and the associated level [alpha] test for each [alpha]. A P-value turns out to be valid if and only if the associated level [alpha] test is unbiased in the traditional sense for each [alpha]. Furthermore, it is shown that the resulting optimal tests agree with those given by the Neyman-Person framework when the ordering is stochastic. Thus, a theory based on optimal P-values parallels to the Neyman-Pearson theory and bridges the two approaches to testing of hypotheses.
