Improved testing procedures below the established separation threshold

Determine whether a testing procedure different from the proposed bootstrap U-statistic test can improve the separation rate for general U-statistic-based testing problems beyond the sufficient consistency threshold determined by the Hilbert–Schmidt norm of the Hájek covariance operator.

Background

The paper derives a sufficient condition for consistency of its bootstrap test at signals larger than a covariance-dependent separation radius. It then explains that general minimax optimality is difficult to formulate because the relevant threshold depends strongly on the chosen kernel. Although a minimax lower bound is subsequently established for the specific vector of pairwise Kendall’s tau coefficients, the broader question of whether another testing procedure can improve the rate for general U-statistic testing remains unresolved.