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Biometrika 1994 81(3):485-499; doi:10.1093/biomet/81.3.485
© 1994 by Biometrika Trust
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Articles

Stable and invariant adjusted directed likelihoods

O. E. BARNDORFF-NIELSEN1 and S. R. CHAMBERLIN2

1 Department of Theoretical Statistics, Institute of Mathematics, Ny Munkegade, University of Aarhus DK-8000 Aarhus C, Denmark
2 Department of Mathematics and Statistics, York University North York, Ontario, Canada M3J 1P3

Received for publication 1 September 1992. Revision received 1 July 1993.
  Abstract

For the purpose of making inferences for a one-dimensional interest parameter, or constructing approximate complementary ancillaries or residuals, the directed likelihood or signed square root of the likelihood ratio statistic can be adjusted so that the resulting modified directed likelihood is under ordinary repeated sampling approximately standard normal with error of O(n–3/2), conditional on a suitable ancillary statistic and hence unconditionally. In general, suitable specification of the ancillary statistic may be difficult. We introduce two adjusted directed likelihoods which are similar to the modified directed likelihood but do not require the specification of the ancillary statistic. The error of the standard normal approximation to the distribution of these new adjusted directed likelihoods is O(n–1), conditional on any reasonable ancillary statistic, which is still an improvement over the unadjusted directed likelihoods.

Key Words: Adjusted profile likelihood • Asymptotics • Conditionality • Parameterization invariance • p*-formula


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