Tail probabilities from observed likelihoods

Abstract

SUMMARY An exponential model not in standard form is fully characterized by an observed likelihood function and its first sample space derivative, up to one-one transformations of the observable variable. This property is used to modify the Lugannani & Rice (1980) tail probability approximation to make it parameterization invariant. Then, for general continuous models a version of tangent exponential model is defined, and used to derive a general tail probability approximation that uses only the observed likelihood and its first sample-space derivative. The analysis extends from density functions to distribution functions the tangent exponential model methods of Fraser (1988). A related tail probability approximation has been reported (Barndorff-Nielsen, 1988b) in the discussion to Reid (1988).