Maximal Ancillarity, Semiparametric Efficiency, and the Elimination of Nuisances

Restricting statistical experiments via nuisance-ancillary $\sigma$-fields yields nuisance-free experiments. However, a moot point with ancillarity is that maximal ancillary $\sigma$-fields are typically not unique. There are exceptions, though, among which the limiting experiments in a locally asymptotically normal (LAN) context. Building on this, we address the maximal ancillarity uniqueness problem by adopting a H\'ajek-Le Cam asymptotic perspective and define the concept of sequences of locally asymptotically maximal nuisance-ancillary $\sigma$-fields. We then show that any semiparametrically efficient procedure admits versions that are measurable with respect to such $\sigma$-fields while enjoying strict finite-sample nuisance-ancillarity, hence eliminating the nuisance without the hassle of estimating it. This is in sharp contrast with classical tangent space projections, which also achieve semiparametric efficiency but only enjoy asymptotic nuisance-ancillarity -- at the price, moreover, of adequately estimating the nuisance. When the nuisance is the density of some noise or innovation driving the data-generating process of a LAN experiment, we show that a sequence of locally asymptotically maximal nuisance-ancillary $\sigma$-fields is generated by the so-called center-outward residual ranks and signs based on measure transportation results. Restricting local experiments to such $\sigma$-fields yields sequences of finite-sample nuisance-free (here, distribution-free) restrictions of the original local LAN experiments that nevertheless achieve the semiparametric efficiency bounds of the original ones.