## The Annals of Statistics

### Asymptotic Lognormality of $P$-Values

#### 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.

#### Article information

Source
Ann. Statist., Volume 10, Number 1 (1982), 44-64.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176345689

Digital Object Identifier
doi:10.1214/aos/1176345689

Mathematical Reviews number (MathSciNet)
MR642718

Zentralblatt MATH identifier
0484.62038

JSTOR
Lambert, Diane; Hall, W. J. Asymptotic Lognormality of $P$-Values. Ann. Statist. 10 (1982), no. 1, 44--64. doi:10.1214/aos/1176345689. https://projecteuclid.org/euclid.aos/1176345689
• See Correction: D. Lambert, W. J. Hall. Corrections: Asymptotic Lognormality of $P$-Values. Ann. Statist., Volume 11, Number 1 (1983), 348--348.