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 $Δ_n$ denote a relevance threshold and $I_0$ Fisher information, and define $λ_n=\sqrt{nI_0}Δ_n$. For separated negligible and meaningful parameter classes, we establish the minimax lower bound $\liminf R_n^*\ge 2\{1-Φ(\varepsilonκ)\}$ when $λ_n\toκ$. If $λ_n\to0$, the experiments merge and consistent classification is impossible. If $λ_n\toκ\in(0,\infty)$, the problem converges to a Gaussian-shift decision problem. We characterize the exact minimax rule and risk in that limiting composite problem. As $κ\downarrow0$, 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 $λ_n\to\infty$, 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.