Semi-parametric efficiency, distribution-freeness and invariance

Semi-parametric models typically involve a finite-dimensional parameter of interest ᵏ, along with an infinite-dimensional nuisance parameter~f. Quite often, the submodels corresponding to a fixed value of possess a group structure that induces a maximal invariant -field B (). In classical examples, where f denotes the density of some independ\-ent and identically distributed innovations, B () is the -field generated by the ranks of the residuals associated with the parameter value. It is shown that semi-parametrically efficient distribution-free inference procedures can generally be constructed from parametrically optimal ones by conditioning on B () ; this implies, for instance, that semi-parametric efficiency (at given and f) can be attained by means of rank-based methods. The same procedures, when combined with a consistent estimation of the underlying nuis\-ance density f, yield conditionally distribution-free semi-parametrically efficient inference methods, for example, semi-parametrically efficient permutation tests. Remarkably, this is achieved without any explicit tangent space or efficient score computations, and without any sample-splitting device. By means of several examples, including both i. i. d. and time-series models, we show how these results apply in models for which rank-based inference or permutation tests have so far seldom been considered.