Tail probabilities from observed likelihoods

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).