Even-sample centering in balanced top-t selection

Prove that, for every balanced binomial top-t selection problem with k=2t, gap parameter δ∈(0,1), and even sample size n≥2, the symmetric location p=(1−δ)/2 is the unique least-favorable location.

Background

The paper studies the least-favorable location along the two-level slippage family for selecting the t largest among k independent binomial populations with uniform tie breaking and separation gap δ. It proves that for each fixed k, t, and δ, the least-favorable location is eventually unique as the sample size n grows. In the balanced case k=2t, reflection symmetry forces the eventual unique minimizer to be the symmetric center p=(1−δ)/2.

Numerical results suggest a stronger finite-sample statement: in balanced problems, the symmetric center appears to remain the unique least-favorable location for every even sample size n≥2. The paper does not establish this all-even-sample result and explicitly leaves it as a conjecture for future work.