When Is a Relevance Threshold Statistically Resolvable? Minimax Limits for Effect Classification
Abstract
Statistical precision and scientific relevance operate on different scales. In regular problems, sampling uncertainty contracts at rate n−1/2, whereas the magnitude below which an effect is scientifically negligible may be fixed or may vary with information. Let Δn denote a relevance threshold and I0 Fisher information, and define λn=√(nI0)Δn. For separated negligible and meaningful parameter classes, we establish the minimax lower bound liminf Rn*≥ 2{1−Φ(εκ)} when λn→κ. If λn→0, the experiments merge and consistent classification is impossible. If λn→κ∈(0,∞), the problem converges to a Gaussian-shift decision problem. We characterize the exact minimax rule and risk in that limiting composite problem. As κ↓0, the optimal cutoff converges to one statistical-error unit and the optimal improvement over trivial risk is O(κ2), while a simple relevance-boundary rule improves only at O(κ3); as κ grows, that rule becomes asymptotically minimax. If λn→∞, consistent classification is attainable under a uniform estimation-resolution condition, verified for Gaussian and Bernoulli models. For Δn=dn−γ, the critical rate is γ=1/2. We also show that moving asymmetric relevance regions depend on standardized distances to their two boundaries rather than total width alone. For heterogeneous true effects, we derive a significance-saturation limit and show that point-null significance can be most selective for scientifically relevant effects at intermediate information. The framework characterizes when scientific relevance is statistically resolvable through a common information scale.
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BibTeX
@misc{hait2026when,
title = {{When Is a Relevance Threshold Statistically Resolvable? Minimax Limits for Effect Classification}},
author = {Subir Hait},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15811},
eprint = {2609.15811},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15811},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - When Is a Relevance Threshold Statistically Resolvable? Minimax Limits for Effect Classification
AU - Subir Hait
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.15811
UR - https://arxiv.org/abs/2609.15811
AB - Statistical precision and scientific relevance operate on different scales. In regular problems, sampling uncertainty contracts at rate n^−1/2, whereas the magnitude below which an effect is scientifically negligible may be fixed or may vary with information. Let Δ_n denote a relevance threshold and I_0 Fisher information, and define λ_n=√(nI_0)Δ_n. For separated negligible and meaningful parameter classes, we establish the minimax lower bound liminf R_n^*≥ 2{1−Φ(εκ)} when λ_n→κ. If λ_n→0, the experiments merge and consistent classification is impossible. If λ_n→κ∈(0,∞), the problem converges to a Gaussian-shift decision problem. We characterize the exact minimax rule and risk in that limiting composite problem. As κ↓0, the optimal cutoff converges to one statistical-error unit and the optimal improvement over trivial risk is O(κ^2), while a simple relevance-boundary rule improves only at O(κ^3); as κ grows, that rule becomes asymptotically minimax. If λ_n→∞, consistent classification is attainable under a uniform estimation-resolution condition, verified for Gaussian and Bernoulli models. For Δ_n=dn^−γ, the critical rate is γ=1/2. We also show that moving asymmetric relevance regions depend on standardized distances to their two boundaries rather than total width alone. For heterogeneous true effects, we derive a significance-saturation limit and show that point-null significance can be most selective for scientifically relevant effects at intermediate information. The framework characterizes when scientific relevance is statistically resolvable through a common information scale.
ER -