Ancillaries and Conditional Inference

Abstract

Sufficiency has long been regarded as the primary reduction procedure to simplify a statistical model, and the assessment of the procedure involves an implicit global repeated sampling principle. By contrast, conditional procedures are almost as old and yet appear only occasionally in the central statistical literature. Recent likelihood theory examines the form of a general large sample statistical model and finds that certain natural conditional procedures provide, in wide generality, the definitive reduction from the initial variable to a variable of the same dimension as the parameter, a variable that can be viewed as directly measuring the parameter. We begin with a discussion of two intriguing examples from the literature that compare conditional and global inference methods, and come quite extraordinarily to opposite assessments concerning the appropriateness and validity of the two approaches. We then take two simple normal examples, with and without known scaling, and progressively replace the restrictive normal location assumption by more general distributional assumptions. We find that sufficiency typically becomes inapplicable and that conditional procedures from large sample likelihood theory produce the definitive reduction for the analysis. We then examine the vector parameter case and find that the elimination of nuisance parameters requires a marginalization step, not the commonly proffered conditional calculation that is based on exponential model structure. Some general conditioning and modelling criteria are then introduced. This is followed by a survey of common ancillary examples, which are then assessed for conformity to the criteria. In turn, this leads to a discussion of the place for the global repeated sampling principle in statistical inference. It is argued that the principle in conjunction with various optimality criteria has been a primary factor in the long-standing attachment to the sufficiency approach and in the related neglect of the conditioning procedures based directly on available evidence.