On comparing two methods for approximate conditional inference

Abstract

Approximate conditional inference for a real parameter in the presence of nuisance parameters was examined from a sample-space differential viewpoint in Fraser and Reid (1988) and a conditional inference procedure was proposed. Conditional likelihood-based inference in the same setting was discussed in Cox and Reid (1987), where emphasis was placed on orthogonalizing the nuisance parameter to the parameter of interest. In this paper the sample-space partitions of the two methods are examined for the case that the minimal sufficient statistic has the same dimension as the parameter space. The methods are identical if observed and expected information gives the same orthogonality; an example indicates how they can differ more generally. A specially chosen reparameterization provides some geometrical insight to the methods and allows a comparison in terms of score functions and locally defined orthogonal parameters.