Projection as a method for increasing sensitivity and eliminating nuisance parameters

This paper examines statistical methods based upon estimating functions, i.e. functions of both the parameter and data that are designed to permit inference about an unknown parameter in a statistical model. We explore reductions of such estimating functions by projection. This reduction, analogous to the process of Rao–Blackwellization, may be used either to increase the power of a test, the efficiency of a point estimator, or alternatively to render an inference function insensitive to the value of a nuisance parameter. In the case where a complete sufficient statistic exists for a parameter of interest the methods reduce to increasing sensitivity through Rao–Blackwellization. When this same parameter is regarded as a nuisance parameter, the techniques lead us to condition on the complete sufficient statistic for this parameter. However the techniques are seen to be more widely applicable than for models permitting reduction through complete sufficiency. Examples involving mixture models will be developed.