Title: A dimension reduction for extreme types of directed dependence Show Chinese title

Title: 极端方向依赖的维数约简

Abstract: In recent years, a variety of novel measures of dependence have been introduced being capable of characterizing diverse types of directed dependence, hence diverse types of how a number of predictor variables $\mathbf{X} = (X_1, \dots, X_p)$, $p \in \mathbb{N}$, may affect a response variable $Y$. This includes perfect dependence of $Y$ on $\mathbf{X}$ and independence between $\mathbf{X}$ and $Y$, but also less well-known concepts such as zero-explainability, stochastic comparability and complete separation. Certain such measures offer a representation in terms of the Markov product $(Y,Y')$, with $Y'$ being a conditionally independent copy of $Y$ given $\mathbf{X}$. This dimension reduction principle allows these measures to be estimated via the powerful nearest neighbor based estimation principle introduced in [4]. To achieve a deeper insight into the dimension reduction principle, this paper aims at translating the extreme variants of directed dependence, typically formulated in terms of the random vector $(\mathbf{X},Y)$, into the Markov product $(Y,Y')$. Show Chinese abstract

Abstract: 近年来，人们引入了各种新型的依赖性度量方法，能够刻画不同类型的单向依赖关系，因此也能描述多个预测变量 $\mathbf{X} = (X_1, \dots, X_p)$, $p \in \mathbb{N}$ 如何影响响应变量 $Y$。这包括 $Y$ 对 $\mathbf{X}$ 的完全依赖以及 $\mathbf{X}$ 和 $Y$ 之间的独立性，还包括一些不太为人所知的概念，例如零可解释性、随机可比性和完全分离。 某些此类度量以马尔可夫积$(Y,Y')$的形式表示，其中$Y'$是在给定$\mathbf{X}$条件下$Y$的条件独立副本。 这种维度约简原则允许这些度量通过[4]中引入的强大的基于最近邻的估计原理来估计。 为了更深入地理解这一维度约简原则，本文旨在将通常以随机向量$(\mathbf{X},Y)$表述的有向依赖的极端变体转化为马尔可夫积$(Y,Y')$。