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Biometrika Advance Access originally published online on June 9, 2008
Biometrika 2008 95(3):747-758; doi:10.1093/biomet/asn011
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© 2008 Biometrika Trust

Articles

Conditional properties of unconditional parametric bootstrap procedures for inference in exponential families

Thomas J. DiCiccio

Department of Social Statistics, Cornell University, Ithaca, New York 14853, U.S.A. tjd9{at}cornell.edu

G. Alastair Young

Department of Mathematics, Imperial College London, London, SW7 2AZ, U.K. alastair.young{at}imperial.ac.uk

Received for publication 1 November 2006. Revision received 1 October 2007.
  Abstract

Higher-order inference about a scalar parameter in the presence of nuisance parameters can be achieved by bootstrapping, in circumstances where the parameter of interest is a component of the canonical parameter in a full exponential family. The optimal test, which is approximated, is a conditional one based on conditioning on the sufficient statistic for the nuisance parameter. A bootstrap procedure that ignores the conditioning is shown to have desirable conditional properties in providing third-order relative accuracy in approximation of p-values associated with the optimal test, in both continuous and discrete models. The bootstrap approach is equivalent to third-order analytical approaches, and is demonstrated in a number of examples to give very accurate approximations even for very small sample sizes.

Key Words: Bootstrap • Conditional test • Full exponential family • Likelihood • Nuisance parameter • Signed root likelihood ratio statistic


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