发表机构
Gilead Sciences; Velexi Corporation（吉利德科学; 维莱克西公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究三个观测值样本方差的精确分布，通过几何方法组织闭式层级，证明多项式母体闭式存在、有理母体非初等，并给出均匀与反正弦母体的具体公式与高效近似。
AI 中文摘要
样本方差在 $n=3$ 时的精确分布，对于均匀母体自 Rietz (1931) 起便已知，而 Royen (2007, 2008) 给出了任意有界连续母体的傅里叶级数。两者均未解决哪些母体允许有限闭式，以及它强制哪些特殊函数。在与立方体对角线对齐的坐标中，方差约束变为圆柱体，立方体截面变为具有 $S_3$ 对称性的多边形，在对角线中央带呈六边形，在任一角附近呈三角形；累积分布函数是它们交集的体积，母体密度作为权重进入。随后，闭包层级由包含母体密度的最小函数类组织：任意有界区间上的多项式母体总能在初等项中闭合，这是针对整个类的定理；在否定方向上，单个显式母体已足够，对于有理母体，我们证明累积分布函数不是初等的，障碍是不可约的双对数部分。在这两个定理之外，层级是一组特定于示例的障碍而非分类：对于代数母体，径向第一类微分在亏格二曲线上被证明非初等，指数行是一个由径向积分的贝塞尔结构支持的猜想。对于均匀母体，我们获得一个在 $Y=1/4$ 处分支的两段公式，此时方差圆盘首次在立方体中心外接六边形截面。对于奇异反正弦母体，我们推导出两个端点定律的闭式，并给出一个精度约为 $10^{-3}$ 的六项近似，而 Royen 的通用级数达到该精度约需一百项。
英文摘要
The exact distribution of the sample variance for $n=3$ is known since Rietz (1931) for the uniform parent, and Royen (2007, 2008) gave a Fourier series for any bounded continuous parent. Neither settles which parents admit a finite closed form, nor which special functions it forces. In coordinates aligned with the cube diagonal the variance constraint becomes a cylinder and the cube cross-section a polygon with $S_3$ symmetry, hexagonal over the central band of the diagonal and triangular near either corner; the CDF is the volume of their intersection, the parent density entering as a weight. A closure hierarchy is then organized by the minimal function class containing the parent density: polynomial parents on any bounded interval always close in elementary terms, a theorem for the whole class; in the negative direction a single explicit parent already suffices, and for a rational one we prove the CDF is not elementary, the obstruction being an irreducible dilogarithmic part. Beyond those two theorems the hierarchy is a set of example-specific obstructions rather than a classification: for an algebraic parent the radial first-kind differential is shown non-elementary on a genus-two curve, and the exponential row is a conjecture supported by the Bessel structure of its radial integral. For the uniform parent we obtain a two-piece formula bifurcating at $Y=1/4$, where the variance disk first circumscribes the hexagonal cross-section at the cube center. For the singular arcsine parent we derive both endpoint laws in closed form and give a six-term approximation accurate to about $10^{-3}$, an accuracy Royen's universal series reaches at about a hundred terms.
Comments40 pages, 13 figures
