Semi-parametric efficiency, distribution-freeness and invariance

Semi-parametric models typically involve a finite-dimensional parameter of interest θ∈Θᵏ, along with an infinite-dimensional nuisance parameter~f. Quite often, the submodels corresponding to a fixed value of θ possess a group structure that induces a maximal invariant σ-field B(θ). In classical examples, where f denotes the density of some independ\-ent and identically distributed innovations, B(θ) is the σ-field generated by the ranks of the residuals associated with the parameter value θ. It is shown that semi-parametrically efficient distribution-free inference procedures can generally be constructed from parametrically optimal ones by conditioning on B(θ); this implies, for instance, that semi-parametric efficiency (at given θ 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.