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Neighborhoods as Nuisance Parameters? Robustness vs. Semiparametrics

Deviations from the center within a robust neighborhood of a parametric model distribution may naturally be considered an infinite dimensional nuisance parameter. Thus, the semiparametric method may be tried, which is to compute the scores function for the main parameter minus its orthogonal projection on the closed linear tangent space for the nuisance parameter, and then rescale for Fisher consistency. In this paper, we derive such a semiparametric influence curve by nonlinear projection on the tangent balls arising in robust statistics. This semiparametric influence curve is then compared with the optimally robust influence curve that minimizes maximum weighted mean square error of the corresponding asymptotically linear estimators over infinitesimal neighborhoods. For Hellinger balls, the two coincide (with the classical one). In the total variation model, the semiparametric IC solves the robust MSE problem for a particular bias weight. In the case of contamination neighborhoods, the semiparametric IC is bounded only from above. Due to an interchange of truncation and linear combination, the discrepancy increases with the dimension. While there is coincidence for Hellinger balls, at least clipping is achieved for total variation and contamination neighborhoods, but the semiparametric method in general falls short to solve the minimax MSE estimation problem for the gross error models. The semiparametric approach is carried further to testing contaminated hypotheses. In the one-sided case, for testing hypotheses defined by any two closed convex sets of tangents, a saddle point is furnished by projection on the set of differences of these sets. For total variation and contamination neighborhoods, we thus recover the robust asymptotic tests based on least favorable pairs. So the two approaches agree in the testing context.