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.
From design of experiments to analysis of variance of multivariate data: a tutorial review on ANOVA simultaneous component analysis
University of Granada（格拉纳达大学）；Datharsis（达塔西斯）；University of Copenhagen（哥本哈根大学）；Stellenbosch University（斯坦伦布什大学）；South African Grape and Wine Research Institute (SAGWRI)（南非葡萄与葡萄酒研究所）；University of Amsterdam（阿姆斯特丹大学）
Tutorial: a priori estimation of sample size, effect size, and statistical power for cluster analysis, latent class analysis, and multivariate mixture models
School of Psychological Science, University of Bristol（布里斯托大学心理科学学院）
The Delta-Method and Influence Function in Medical Statistics: a Reproducible Tutorial
Approximate statistical inference via determination of the asymptotic distribution of a statistic is routinely used for inference in applied medical statistics (e.g. to estimate the standard error of the marginal or conditional risk ratio). One method for variance estimation is the classical Delta-method but there is a knowledge gap as this method is not routinely included in training for applied medical statistics and its uses are not widely understood. Given that a smooth function of an asymptotically normal estimator is also asymptotically normally distributed, the Delta-method allows appro…
由Bai et al.（2010）提出的多元非线性Granger因果关系在检测两组变量之间的动态相互关系方面发挥着重要作用。遵循Hiemstra和Jones（1994）提出的Hiemstra-Jones（HJ）检验思想，他们试图利用多元$U$ -统计量的渐进性质建立其检验统计量的中心极限定理（CLT）。然而，Bai等人（2016）重新审视了HJ检验，发现HJ给出的检验统计量不是$U$ -统计量的函数，这意味着Hiemstra和Jones（1994）提出的CLT和Bai等人（2010）扩展的CLT都不适用于统计推断。本文重新估计了新的检验统计量的概率，并重新建立了新的检验统计量的CLT。数值模拟表明，我们的新的估计是一致的，我们的新的测试表现出体面的大小和权力。
