AI 中文总结
本文提出椭圆最优置信区间，通过闭式优化投影椭圆区域构造二项比例差的最短得分区间，保证覆盖且不欠覆盖，并精确量化极端比例下的过度覆盖。
AI 中文摘要
构造两个独立二项比例之差的置信区间涉及一个未被估计量识别的干扰方向。单样本 Wilson 得分区间反转标量得分检验，但没有直接的双变量类比来隔离差值：反转联合正态近似在单位正方形内产生一个椭圆区域，而估计量 \\(p_1-p_2\\) 是一维的。我们将椭圆最优（EO）置信区间定义为该区域上 \\(p_1-p_2\\) 的范围，并以闭式求解所得优化问题，在六个互斥且穷尽的情形中获得显式边界。该解具有紧凑特征：EO 区间是通过在干扰方差上最大化而非估计它而获得的得分区间。因此，它是通过投影椭圆区域获得的最短区间，并继承其覆盖保证。我们推导出精确的覆盖过剩量，\\(2[\Phi(z\mathcal{R})-\Phi(z)]\\)，其中 \\(\mathcal{R}\\) 是最不利标准差与真实标准差之比。该过剩量在参数空间中的一条显式线上消失，以 \\(\alpha\\) 为界，并且在样本量的比例缩放下不变。二项覆盖率的精确枚举表明，Wald 区间（其方差估计量在平衡分配下存在 \\(1-1/n\\) 的向下偏差）几乎在所有地方低于名义覆盖率。EO 区间在正态近似下从不欠覆盖，并且总是产生可容许的非退化边界。其代价是当两个比例极端时过度覆盖，我们对此进行了精确量化。
英文摘要
Constructing a confidence interval for the difference between two independent binomial proportions involves a nuisance direction that is not identified by the estimand. The one-sample Wilson score interval inverts a scalar score test, but has no direct bivariate analogue isolating the difference: inverting the joint normal approximation yields an elliptical region in the unit square, whereas the estimand \(p_1-p_2\) is one-dimensional. We define the Elliptically Optimal (EO) confidence interval as the range of \(p_1-p_2\) over this region and solve the resulting optimization problem in closed form, obtaining explicit bounds in six mutually exclusive and exhaustive cases. The solution admits a compact characterization: the EO interval is the score interval obtained by maximizing over the nuisance variance rather than estimating it. It is therefore the shortest interval obtained by projecting the elliptical region, and inherits its coverage guarantee. We derive the exact coverage excess, \(2[Φ(z\mathcal{R})-Φ(z)]\), where \(\mathcal{R}\) is the ratio of the least-favourable to the true standard deviation. The excess vanishes on an explicit line through the parameter space, is bounded by \(α\), and is invariant under proportional scaling of the sample sizes. Exact enumeration of the binomial coverage shows that the Wald interval, whose variance estimator is downward biased by a factor \(1-1/n\) under balanced allocation, falls below nominal coverage almost everywhere. The EO interval never under-covers under the normal approximation and always yields admissible, non-degenerate bounds. Its price is over-coverage when both proportions are extreme, which we quantify exactly.
