相关性阈值何时在统计上可分辨？效应分类的极小极大极限
When Is a Relevance Threshold Statistically Resolvable? Minimax Limits for Effect Classification
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中文总结 AI 辅助
本文通过极小极大框架研究相关性阈值何时可被统计分辨，提出相关性分辨指数，揭示临界速率1/2，并统一效应量、功效与实用显著性。
中文摘要 AI 辅助
统计精度与科学相关性在不同的尺度上运作。在正则问题中，抽样不确定性以 $n^{-1/2}$ 的速度收缩，而效应在科学上可忽略的幅度阈值可能是固定的，也可能随信息量变化。设 $\Delta_n$ 为相关性阈值，$I_0$ 为正则标量模型中的 Fisher 信息，定义相关性分辨指数 $\lambda_n=\sqrt{nI_0}\Delta_n$。对于分离的可忽略参数类和有意义参数类，当 $\lambda_n\to\kappa$ 时，我们建立了极小极大下界 $\liminf_{n\to\infty}R_n^*\ge 2\{1-\Phi(\varepsilon\kappa)\}$。若 $\lambda_n\to0$，实验合并，一致分类不可能。若 $\lambda_n\to\kappa\in(0,\infty)$，问题收敛到一个非退化的高斯平移决策问题：非平凡判别是可能的，但一致性不可达。若 $\lambda_n\to\infty$，在一致估计-分辨率条件下可实现一致分类。对于 $\Delta_n=dn^{-\gamma}$，临界速率为 $\gamma=1/2$。我们还证明，移动的非对称相关性区域由到其两个边界的标准化距离决定，而非仅由总宽度决定。对于异质真实效应，我们推导出显著性饱和极限，并表明在中等信息量下，点零假设显著性对科学相关效应可能最具选择性。该框架刻画了科学相关性何时在统计上可分辨，通过共同的信息尺度将效应量、功效、局部渐近理论和实际显著性联系起来。
英文摘要
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.
