Maximal co-ancillarity and maximal co-sufficiency

The purpose of this exposition is to provide some new perspectives on conditional inference through a notional idealised separation within the minimal sufficient statistic, allowing a geometric account of key ideas from the Fisherian position. The notional idealised separation, in terms of an ancillary statistic and what I call a maximal co-ancillary statistic, provides conceptual insight and clarifies what is sought from an approximate conditional analysis, where exact calculations may not be available. A parallel framework applies in the Fisherian assessment of model adequacy. Both aspects are discussed and illustrated geometrically through examples.