Lehmer-Lambert 分布

中文摘要 AI 辅助

正信号的 Lehmer 变换是其连续阶次幂和之比，作为阶次的函数，它是一条从最小观测值到最大观测值的严格递增解析曲线。将该曲线与一个逆函数为 Lambert 函数的映射复合，可得到实数轴上的一个概率律，即我们引入并研究的 Lehmer-Lambert 分布。其分布函数、密度函数和分位数函数均为闭式形式，其随机变量可精确生成，且该律在信号的增益变化和幂律校准下保持不变。其众数位于信号改变尺度的阶次处，对于样本，其尾部为指数分布，速率等于样本的两个极端对数间隙。我们推导了矩、生成函数和极限定理，并研究了 Lehmer-Lambert 分布在重度抑郁症患者脑电图分析中的应用。

英文摘要

The Lehmer transform of a positive signal is the ratio of its power sums at consecutive orders, read as a function of the order, and it is a strictly increasing analytic curve from the smallest observation to the largest. Composing this curve with a map whose inverse is the Lambert function turns it into a probability law on the real line, the Lehmer-Lambert distribution, which we introduce and study. Its distribution, density and quantile functions are closed-form, its random variates are exact, and the law is invariant under changes of gain and of power-law calibration of the signal. Its modes lie at orders at which the signal changes scale, and for a sample its tails are exponential with rates equal to the two extreme logarithmic gaps of the sample. We derive moments, generating functions and limit theorems, and we investigate applications of the Lehmer-Lambert distribution in the analysis of electroencephalograms of patients with major depressive disorder.

发表机构

• University of Toronto（多伦多大学）
• Ohio University（俄亥俄大学）

机构由 AI 辅助整理，请以论文原文为准。