Lehmer 变换

中文摘要 AI 辅助

Lehmer 变换将正数据集编码为一条解析曲线，即其连续阶幂和之比作为阶的函数。当阶扫过实直线时，该曲线生成样本极值之间的每个位置统计量，其对数是指数倾斜系综自由能的单位增量；该曲线的统计理论源于这一恒等式。曲线的斜率是 Jeffreys 散度，算术-几何差距是相对熵，并且对独立因子的乘法性使得乘法噪声的反卷积成为逐点除法。一个定量刚性定理将可辨识性的前沿定位于曲线的几何增长处，具有共振对数周期族的相异定律在超越该前沿后共享同一条曲线。经验曲线是次序统计量的精确重新坐标化，因此是最小充分的，其影响函数、精确主导偏差和极限理论得到了完整发展。在重尾情况下，曲线经历相变，其两个拐点相距一个单位，位于尾部指数及其单位位移处，这是对基于阈值的估计器的全样本诊断补充。

英文摘要

The Lehmer transform encodes a positive dataset as a single analytic curve, the ratio of its power sums at consecutive orders read as a function of the order. As the order sweeps the real line the curve generates every location statistic between the sample extremes, and its logarithm is the unit increment of the free energy of an exponentially tilted ensemble; the statistical theory of the curve flows from that identity. The slope of the curve is a Jeffreys divergence, the arithmetic-geometric gap is a relative entropy, and multiplicativity over independent factors makes deconvolution of multiplicative noise a pointwise division. A quantitative rigidity theorem locates the frontier of identifiability at geometric growth of the curve, with resonant log-periodic families of distinct laws sharing one curve beyond it. The empirical curve is an exact re-coordinatization of the order statistic, hence minimal sufficient, and its influence function, exact leading bias and limit theory are developed in full. At heavy tails the curve undergoes a phase transition whose two knees, one unit apart, sit at the tail index and its unit shift, a whole-sample diagnostic complementing threshold-based estimators.

发表机构

• University of Toronto（多伦多大学）

机构由 AI 辅助整理，请以论文原文为准。