An ordering for the strength of functional dependence

TL;DR

The paper introduces a bold new global dependence order, the conditional convex order, to quantify the strength of functional dependence of a scalar Y on a random vector X. It axiomatizes the desired properties (including independence and perfect dependence as extrema, information monotonicity, and conditional independence) and develops a Schur-order-based and a dimension-reduction characterization, enabling practical verification across models. This order underpins monotonicity results for Chatterjee's xi and related measures, and supports the construction of new ccx-consistent dependence metrics that inherit key invariance and extremal properties. The authors demonstrate verification in additive error models, multivariate normal settings, and copula-based models, and contrast ccx with concordance and other orderings, showing its broader applicability and theoretical appeal. The work provides a principled, verifiable framework for comparing dependences in high-dimensional settings and yields tools for model-free inference and variable selection that respect the strength of functional dependence.

Abstract

We introduce a new dependence order that satisfies eight natural axioms that we propose for a global dependence order. Its minimal and maximal elements characterize independence and perfect dependence. Moreover, it characterizes conditional independence, satisfies information monotonicity, and exhibits several invariance properties. Consequently,it is an ordering for the strength of functional dependence of a random variable Y on a random vector X. As we show, various dependence measures, such as Chatterjee's rank correlation, are increasing in this order. We characterize our ordering by the Schur order and by the concordance order, and we verify it in models such as the additive error model, the multivariate normal distribution, and various copula-based models.

Figures (5)

• Figure 1: The red curve depicts the piecewise linear integrated decreasing rearrangement $x \mapsto \int_{0}^x (\eta_{Y|X}^v)^\ast(u) \mathsf{\,d} u$, $v\in (0,1-q]$, of the Bernoulli-distributed random vector $(Y,X)$ discussed in Example \ref{['Ex.Bernoulli']}. For any Bernoulli-distributed random vector $(Y',X')$ with $(Y,X) \preccurlyeq_{ccx} (Y',X')$ or, equivalently, $\eta_{Y|X}^v \prec_S \eta_{Y'|X'}^v$ for Lebesgue almost all $v\in (0,1-q]$, its integrated decreasing rearrangement (blue curve) lies above the red curve and hence must have a slope greater than $\alpha\vee \beta$ in the interval starting at $0$ and a slope less than $\alpha\wedge \beta$ in the interval ending at $1$. Due to the marginal constraint both the red and blue functions equal $1-q$ at point $1$.
• Figure 2: Example for the construction of the bivariate SI random vector $(q_{Y|X}^{\uparrow U}(V),U)$ in Proposition \ref{['lemtrafSI']}\ref{['lemtrafSI1']} when a discrete bivariate random vector $(Y,X)$ mapping into a grid $\{a_1,\ldots,a_4\}\times \{b_1,\ldots,b_4\}$ is given: The top left matrix illustrates the mass distribution of $(Y,X)$; the bottom left matrix describes the associated conditional distribution functions $\eta_{Y|X}^v\,;$ the bottom right matrix defines the conditional distribution function $v \mapsto F_u(q_Y(v))$ that is defined in \ref{['definreacdf']} and decreasing in $u$; the top right matrix describes the mass distribution of the SI random vector $(q_{Y|X}^{\uparrow U}(V),U)$ defined by \ref{['defyu']}.
• Figure 3: Visualization of the concordance order (left plot) and the conditional convex order (right plot) in the case of bivariate copulas. Here, $W$ and $M$ denote the lower and upper Fréchet copula and $\Pi$ the independence copula. While $W$ and $M$ are the uniquely determined minimal and maximal elements in concordance order, the conditional convex order has the independence copula as global minimal element and all perfectly dependent copulas as global maximal elements.
• Figure 4: Left plot: Densities and means of the conditional distributions $F_{Y}(Y)|X = q_{X}(u)$ (solid) and $F_{Y'}(Y')|X' = q_{X'}(u)$ (dashed) in Example \ref{['exdisp']} for $\sigma = 1$ and $\sigma' = 2$ and for $u = 0.1\,.$ The densities are determined by a kernel estimation based on a sample of size $10^7\,.$ Right plot: Densities from the left plot shifted by the mean. Since the supports are non-nested intervals satisfying \ref{['eqsuppcondncx']}, they are not comparable in convex order.
• Figure 5: Left: the distribution function of a random variable $Y\,;$ right: the distribution function of a random variable $Y'$ which is a shift of $Y$ given by $Y' = Y+2$ if $Y\leq 0.5\,,$ and $Y'= Y+1.5$ if $Y>0.5\,;$ see Example \ref{['exdiscY']}. Since $Y$ and $Y'$ have the same strength of functional dependence on a random vector $\bold{X}$ in terms of Chatterjee's rank correlation (i.e., $\xi(Y|\bold{X}) = \xi(Y'|\bold{X})$), they should be equally ranked in a suitable global dependence order.

Theorems & Definitions (43)

• Remark 1.1
• Definition 1.2: Conditional convex order
• Remark 1.3
• Theorem 1.4: Fundamental properties of $\preccurlyeq_{ccx}$
• Remark 1.5
• Theorem 1.6: Characterization of $\preccurlyeq_{ccx}$ by bivariate concordance order
• Remark 1.7
• Theorem 1.8: Additive error models
• Remark 1.9
• Lemma 2.1: Hardy-Littlewood-Polya theorem