Lower bound for empirical fourth-moment minimizers

Establish a lower bound specific to minimizers of the empirical fourth-moment contrast, controlling fluctuations of the contrast and their supremum over the searched set, including the dependence on search dimension.

Background

The paper derives sufficient sample-size bounds for recovering the population direction by minimizing an empirical fourth-moment contrast and separately gives information-theoretic two-point arguments. However, the authors emphasize that the sufficient \varsigma^{-8} scaling arises from the analysis of the fourth-moment procedure rather than from an information-theoretic lower bound for the full inference problem.

The unresolved problem is to prove a procedure-specific lower bound for empirical minimizers of the fourth-moment contrast. Such a result would need to analyze the random fluctuations of the contrast uniformly over the searched directions, while retaining the search-dimension dependence that drives the high-dimensional failure mode.