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Biometrika 1990 77(1):77-95;
© 1990 by Biometrika Trust
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Approximations of marginal tail probabilities and inference for scalar parameters

THOMAS J. DICICCIO, CHRISTOPHER A. FIELD and D. A. S. FRASER

Department of Statistics, Stanford University Stanford, Cal 94305, U.S.A.
Department of Mathematics, York University North York, Ontario, M3J 1P3, Canada

Received for publication 1 March 1989. Revision received 1 August 1989.
  Abstract

In many situations, inference for a scalar parameter in the presence of nuisance parameters requires integration of either a joint density of pivotal quantities or a joint posterior density. For such inference, accurate approximations of marginal tail probabilities are useful to avoid high-dimensional integrals. Two tail probability approximations are developed in this paper. Numerical results given for conditional inference in location-scale and linear regression models show the approximations to be generally accurate even for small sample sizes.

Key Words: Bayesian inference • Conditional inference • Linear regression model • Location-scale model • Lugannani-Rice formula • Saddlepoint approximation • Signed root likelihood ratio statistic • Type II censoring


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