Sufficient conditions for asymptotic lognormality of exact and approximate, unconditional and conditional P-values are established. It is pointed out that the mean, which is half the Bahadur slope, and the standard deviation of the asymptotic distribution of the log transformed P-value together, but not the mean alone, permit approximation of both the level and power of the test. This provides a method of discriminating between tests that have Bahadur efficiency one. The asymptotic distributions of the log transformed P-values of the common one- and two-sample tests for location are derived and compared.
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Lambert et al. (1982) studied this question.