A dimension reduction for extreme types of directed dependence

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 X = (X 1, …, X p), p∈N $p\in \mathbb{N}$ , may affect a response variable Y. This includes perfect dependence of Y on X and independence between 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 X. This dimension reduction principle allows these measures to be estimated via the powerful nearest neighbor based estimation principle introduced in (Azadkia, M. and Chatterjee, S. (2021). A simple measure of conditional dependence. Ann. Stat. 49: 3070–3102). 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 (X, Y), into terms relating to its Markov product (Y, Y′).