Neighborhoods as nuisance parameters? Robustness vs. semiparametrics

Deviations from the center within a robust neighborhood may naturally be considered an infinite dimensional nuisance parameter. Thus, in principle, 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. We derive such a semiparametric influence curve by nonlinear projection on the tangent balls arising in robust statistics. This semiparametric IC is compared with the robust IC that minimizes maximum weighted mean square error of asymptotically linear estimators over infinitesimal neighborhoods. For Hellinger balls, the two coincide (with the classical one). In the total variation model, the semiparametrie 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. Thus, despite of striking similarities, the semiparametric method falls short, or fails, to solve the robust MSE problem for gross error models.