论文标题
多元$ k $统计数据及其计算的教程
A Tutorial on Multivariate $k$-Statistics and their Computation
论文作者
论文摘要
本文档旨在使用$ k $统计量提供有关多元累积物的无偏估计的可访问教程。我们为任意命令的多元$ k $统计量提供明确的一般公式。我们还证明，使用Möbius倒置和基本的组合学，$ K $统计量是公正的。在整篇文章中，考虑了许多详细的例子。我们在讨论$ K $统计学计算的讨论中进行了结论，包括时间复杂性的挑战，我们研究了一些可能提高该计算效率的途径。本文档的目的是三倍：在不依赖诸如umbral conculus之类的专用工具的情况下，清楚地介绍了$ k $统计信息；为$ k $统计量构建一个明确的公式，以促进未来的近似值和更快的算法；并充当我们Python图书馆的伴侣纸，该纸张实现了该公式。
This document aims to provide an accessible tutorial on the unbiased estimation of multivariate cumulants, using $k$-statistics. We offer an explicit and general formula for multivariate $k$-statistics of arbitrary order. We also prove that the $k$-statistics are unbiased, using Möbius inversion and rudimentary combinatorics. Many detailed examples are considered throughout the paper. We conclude with a discussion of $k$-statistics computation, including the challenge of time complexity, and we examine a couple of possible avenues to improve the efficiency of this computation. The purpose of this document is threefold: to provide a clear introduction to $k$-statistics without relying on specialized tools like the umbral calculus; to construct an explicit formula for $k$-statistics that might facilitate future approximations and faster algorithms; and to serve as a companion paper to our Python library PyMoments, which implements this formula.
