A dimension reduction for extreme types of directed dependence

TL;DR

This paper explores a dimension reduction approach for various complex measures of directed dependence, translating them into a Markov product framework to enhance understanding and estimation.

Contribution

It introduces a method to represent extreme directed dependence measures via the Markov product, facilitating their estimation and interpretation.

Findings

Provides a new representation of dependence measures in terms of the Markov product

Enables estimation of complex dependence measures using nearest neighbor methods

Deepens understanding of the structure of extreme dependence types

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 $(Y,Y')$.