AI 中文总结
该研究比较对称单峰分布下Student统计量的维度差异，推导了相关相位曲面，得到特定对称单峰母体，还给出固定置信度维度展开及n=6处的精确反转。
AI 中文摘要
设$q_n(r)$为$n$个独立中心化均匀变量的自正则和在$r$处的尾概率。当$r=3$时，Student统计量的双侧Edgeworth展开中首个分布敏感项消失。我们在维度$n$和$n-k$下对公共移动边界$r_n=3+\lambda/n$处的展开进行评估，在保留固定正比例观测值的删除秩上，首个非零差异一致收敛至显式相位曲面$H_\delta(\lambda)$；其零曲线统一了固定、次线性及固定比例删除，切线交点为$12/35$。通过Khintchine尺度混合表示，该比较得到一个与$n$无关、支撑紧的$C^\infty$对称单峰母体，对所有足够大的$n$，沿从下方趋近$2\{1-\Phi(\sqrt{3})\}$的名义水平，其Student尾超过等尺度均匀尾。相比之下，分位数比序表明均匀母体最大化所有偶矩及所有带非负系数的收敛偶幂级数。我们还推导了固定置信度维度展开及$n=6$处的精确反转。
英文摘要
Let $q_n(r)$ denote the tail probability at $r$ of the self-normalized sum of $n$ independent centered uniform variables. At $r=3$, the first distribution-sensitive term in the two-sided Edgeworth expansion of Student's statistic vanishes. We evaluate the expansion at the common moving boundary $r_n=3+λ/n$ in dimensions $n$ and $n-k$. Uniformly over deletion ranks retaining a fixed positive fraction of observations, the first nonzero difference converges to an explicit phase surface $H_δ(λ)$; its zero curve unifies fixed, sublinear and fixed-fraction deletions, with tangent crossing $12/35$. Through the Khintchine scale-mixture representation, this comparison yields a single compactly supported $C^\infty$ symmetric unimodal parent, independent of $n$, whose Student tail exceeds the equal-scale uniform tail for all sufficiently large $n$ along nominal levels tending to $2\{1-Φ(\sqrt3)\}$ from below. In contrast, a quantile-ratio order shows that the uniform parent maximizes every even moment and every convergent even power series with nonnegative coefficients. We also derive the fixed-confidence dimension expansion and an exact reversal at $n=6$.
CommentsSeventeen pages, no figures. Also available on Zenodo
