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Biometrika Advance Access originally published online on January 31, 2008
Biometrika 2008 95(1):1-16; doi:10.1093/biomet/asm093
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© 2008 Biometrika Trust

Articles

Studentization and deriving accurate p-values

D.A.S. Fraser

Department of Statistics, University of Toronto, Toronto, M5S 3G3, Canada dfraser{at}utstat.toronto.edu

Judith Rousseau

CEREMADE, University Paris Dauphine, 75016 Paris, France rousseau{at}ceremade.dauphine.fr

Received for publication 1 January 2005. Revision received 1 August 2007.
  Abstract

We have a statistic for assessing an observed data point relative to a statistical model but find that its distribution function depends on the parameter. To obtain the corresponding p-value, we require the minimally modified statistic that is ancillary; this process is called Studentization. We use recent likelihood theory to develop a maximal third-order ancillary; this gives immediately a candidate Studentized statistic. We show that the corresponding p-value is higher-order Un(0, 1), is equivalent to a repeated bootstrap version of the initial statistic and agrees with a special Bayesian modification of the original statistic. More importantly, the modified statistic and p-value are available by Markov chain Monte Carlo simulations and, in some cases, by higher-order approximation methods. Examples, including the Behrens–Fisher problem, are given to indicate the ease and flexibility of the approach.

Key Words: Ancillary • Bayesian • Behrens–Fisher problem • Bootstrap • Conditioning • Departure measure • Likelihood • p-value • Studentization


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