When Is a Relevance Threshold Statistically Resolvable? Minimax Limits for Effect Classification

RESUMO

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 $\Delta_n$ denote a relevance threshold and $I_0$ Fisher information, and define $\lambda_n=\sqrt{nI_0}\Delta_n$. For separated negligible and meaningful parameter classes, we establish the minimax lower bound $\liminf R_n^*\ge 2\{1-\Phi(\varepsilon\kappa)\}$ when $\lambda_n\to\kappa$. If $\lambda_n\to0$, the experiments merge and consistent classification is impossible. If $\lambda_n\to\kappa\in(0,\infty)$, the problem converges to a Gaussian-shift decision problem. We characterize the exact minimax rule and risk in that limiting composite problem. As $\kappa\downarrow0$, the optimal cutoff converges to one statistical-error unit and the optimal improvement over trivial risk is $O(\kappa^2)$, while a simple relevance-boundary rule improves only at $O(\kappa^3)$; as $\kappa$ grows, that rule becomes asymptotically minimax. If $\lambda_n\to\infty$, consistent classification is attainable under a uniform estimation-resolution condition, verified for Gaussian and Bernoulli models. For $\Delta_n=dn^{-\gamma}$, the critical rate is $\gamma=1/2$. 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.