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 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. Thus, despite of striking similarities, the semiparametric method falls short, respectively fails, to solve the minimax MSE estimation problem for the gross error models.
Moreover, for testing hypotheses which are defined by two closed and convex sets of tangents, we furnish a saddle point via projection on these sets. In the cases of total variation and contamination neighborhoods, the robust asymptotic tests based on least favorable pairs are recovered. Therefore, the two approaches agree in the testing context. Key Words and Phrases: Infinitesimal neighborhoods; Hellinger, total variation, contamination; semiparametric models; tangent spaces, cones, and balls; projection; influence curves; Fisher consistency; canonical influence curve; Hampel-Krasker influence curve; differentiable functionals; asymptotically linear estimators; Cramér-Rao bound; maximum mean square error; asymptotic minmax and convolution theorems; C()- and Wald tests; least favorable pairs; robust asymptotic tests; saddle points.