[Paper Review] Neighborhoods as Nuisance Parameters? Robustness vs. Semiparametrics

Helmut Rieder|arXiv (Cornell University)|Dec 4, 2014

Advanced Statistical Methods and Models18 references3 citations

TL;DR

This paper investigates whether semiparametric methods—specifically, projection of scores onto tangent spaces—can yield optimally robust estimators in the presence of model deviations modeled as infinite-dimensional nuisance parameters. It shows that while semiparametric influence curves coincide with optimally robust ones under Hellinger neighborhoods, they fall short under total variation and contamination neighborhoods, where clipping effects emerge but do not fully achieve minimax mean square error optimality.

ABSTRACT

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.

Motivation & Objective

• To explore whether semiparametric methods can be applied to robust neighborhood models where model deviations act as infinite-dimensional nuisance parameters.
• To assess whether the semiparametric approach—projecting scores onto tangent spaces—yields estimators that are optimal in minimizing maximum weighted mean square error.
• To investigate the connection between semiparametric influence curves and optimally robust influence curves under different types of neighborhood models (Hellinger, total variation, contamination).
• To extend the semiparametric framework to robust testing, particularly for one-sided hypotheses under contaminated models.
• To determine whether the semiparametric method recovers known robust asymptotic tests based on least favorable pairs for total variation and contamination neighborhoods.

Proposed method

• Derives the semiparametric influence curve via nonlinear projection of the score function onto tangent balls arising in robust statistics, rather than linear tangent spaces.
• Applies the standard semiparametric recipe: compute the score function for the main parameter, subtract its orthogonal projection onto the closed linear tangent space of the nuisance parameter, and rescale for Fisher consistency.
• Considers the case where the nuisance tangent set is not a linear space, leading to nonlinear projection directly onto the closed balls (e.g., Hellinger, total variation, contamination balls).
• Uses the Cramér–Rao bound and asymptotic minimax theory to compare the performance of semiparametric influence curves against the optimally robust influence curve that minimizes maximum weighted mean square error.
• Applies the method to hypothesis testing by constructing saddle points via projection onto the set of differences of tangent cones for two closed convex sets of tangents.
• Demonstrates that for total variation and contamination neighborhoods, the resulting semiparametric influence curves are clipped versions of the scores, resembling the Hampel–Krasker influence curve.

Experimental results

Research questions

• RQ1 Can the semiparametric method—based on projection onto tangent spaces—produce optimally robust influence curves for neighborhood models?
• RQ2 Under what conditions does the semiparametric influence curve coincide with the optimally robust influence curve that minimizes maximum weighted mean square error?
• RQ3 Does the semiparametric approach recover known robust asymptotic tests based on least favorable pairs for contamination and total variation neighborhoods?
• RQ4 What happens to the semiparametric influence curve when the nuisance tangent set is not a linear space, but a closed ball?
• RQ5 Is the semiparametric method sufficient to solve the minimax mean square error estimation problem in gross error models?

Key findings

• For Hellinger neighborhoods, the semiparametric influence curve derived via nonlinear projection coincides exactly with the classically optimal influence curve.
• For total variation and contamination neighborhoods, the semiparametric influence curve is a clipped version of the score function, capturing essential features of the optimally robust influence curve.
• Despite clipping, the semiparametric method does not fully achieve minimax mean square error optimality in gross error models, indicating a fundamental limitation.
• The semiparametric approach recovers the robust asymptotic tests based on least favorable pairs for total variation and contamination neighborhoods, particularly through projection onto the set of differences of tangent cones.
• The influence curve derived via nonlinear projection on tangent balls satisfies Fisher consistency and is asymptotically linear, ensuring valid inference under regularity conditions.
• The uniqueness of the least favorable tangent pair under contamination and total variation neighborhoods suggests that the limiting Hellinger distance between any two least favorable probability pairs tends to zero as sample size increases.