Least-Favorable Location for Binomial Top-t Selection
Abstract
Consider k independent Bernoulli populations, each sampled n times, and select the t populations with the largest success counts, breaking ties uniformly. Classical monotonicity reduces the worst case over the preference zone with separation δ to the slippage family with levels p and p+δ, leaving only its absolute location p∈[0,1−δ] undetermined. A Gaussian approximation suggests the symmetric center pc=(1−δ)/2, and the exact two-population problem is uniquely centered there for every n≥2. For fixed k,t and δ∈(0,1), we prove that exact eventual centering holds precisely when k=2t. When k≠2t, the least-favorable location pn,k,t* satisfies pn,k,t*−pc =(k−2t)Cδn−1/2e−nΓδ{1+o(1)}, where Cδ and Γδ are explicit and positive. In either case, the least-favorable location is eventually unique. The proof writes incorrect selection as a union of pairwise misrankings and applies inclusion--exclusion, yielding a bipartite graph expansion. A single misranking has its exact maximum at the symmetric center and determines the central curvature; two-edge intersections sharing one population determine the central slope through their multiplicity imbalance; all remaining graphs have higher large-deviation rates.
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BibTeX
@misc{wu2026leastfavorable,
title = {{Least-Favorable Location for Binomial Top-$t$ Selection}},
author = {Yinhao Wu and Pinyuen Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03466},
eprint = {2609.03466},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03466},
}
RIS
TY - UNPB
TI - Least-Favorable Location for Binomial Top-t Selection
AU - Yinhao Wu
AU - Pinyuen Chen
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03466
UR - https://arxiv.org/abs/2609.03466
AB - Consider k independent Bernoulli populations, each sampled n times, and select the t populations with the largest success counts, breaking ties uniformly. Classical monotonicity reduces the worst case over the preference zone with separation δ to the slippage family with levels p and p+δ, leaving only its absolute location p∈[0,1−δ] undetermined. A Gaussian approximation suggests the symmetric center p_c=(1−δ)/2, and the exact two-population problem is uniquely centered there for every n≥2. For fixed k,t and δ∈(0,1), we prove that exact eventual centering holds precisely when k=2t. When k≠2t, the least-favorable location p_n,k,t^* satisfies p_n,k,t^*−p_c =(k−2t)C_δn^−1/2e^−nΓ_δ{1+o(1)}, where C_δ and Γ_δ are explicit and positive. In either case, the least-favorable location is eventually unique. The proof writes incorrect selection as a union of pairwise misrankings and applies inclusion--exclusion, yielding a bipartite graph expansion. A single misranking has its exact maximum at the symmetric center and determines the central curvature; two-edge intersections sharing one population determine the central slope through their multiplicity imbalance; all remaining graphs have higher large-deviation rates.
ER -