Asymptotic Lognormality of $P$-Values
The Annals of Statistics · 1982 · Volume 10 · Issue 1
Abstract
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.
Keywords
$p$-value, 62F20, 62G20, Bahadur efficiency, one-sample tests, slope, two-sample tests
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