Directional footrule-coefficients

TL;DR

This work extends Spearman's footrule to multivariate directional dependence within the copula framework, defining $oldsymbol\varphi_d^oldsymbol\beta(C)=\frac{2(d+1)}{d-1}\int_0^1\Big(\mathbb{P}[\alpha_1U_1>\alpha_1u,\dots,\alpha_dU_d>\alpha_du]-\prod_{i=1}^d\mathbb{P}[\alpha_iU_i>\alpha_iu]\Big)\mathrm{d}u$ for direction $\boldsymbol\alpha\in\{-1,1\}^d$. The paper proves compatibility with the classical footrule in extreme directions, provides a representation as a linear combination of lower-dimensional coefficients, and derives symmetry/reflection properties. It introduces nonparametric rank-based estimators $\widetilde{\varphi}_{n,d}^\alpha$ based on pseudo-observations, establishes their asymptotic unbiasedness and consistency, and demonstrates finite-sample performance via simulations for Clayton, Cuadras-Augé, and FGM copulas. The directional coefficients enable detection of directional dependence patterns that symmetric measures miss, with potential applications in risk, reliability, and environmental multivariate analysis. Overall, the approach offers a rigorous, interpretable, and flexible tool for directional dependence analysis in copula-based models.

Abstract

Rank-based dependence measures such as Spearman's footrule are robust and invariant, but they often fail to capture directional or asymmetric dependence in multivariate settings. This paper introduces a new family of directional Spearman's footrule coefficients for multivariate data, defined within the copula framework to clearly separate marginal behavior from dependence structure. We establish their main theoretical properties, showing full consistency with the classical footrule, including behavior under independence and extreme dependence, as well as symmetry and reflection properties. Nonparametric rank-based estimators are proposed and their asymptotic consistency is discussed. Explicit expressions for several known families of copulas illustrate the ability of the proposed coefficients to detect directional dependence patterns undetected by classical measures.

Figures (6)

• Figure 1: Values of $\varphi_4^\alpha(C_\theta^{CL})$ for Clayton $4$-copulas as functions of the parameter $\theta$, for all possible values of $|J|$. The curves for $|J|=0$ and $|J|=4$ (corresponding to $\alpha=-\mathbf{1}$ and $\alpha=\mathbf{1}$, respectively) exhibit positive dependence that increases with $\theta$, while intermediate values of $|J|$ display negative directional dependence.
• Figure 2: Values of $\varphi_4^{\alpha}$ for Cuadras-Augé $4$-copulas as a function of $\theta$, for all possible values of $\alpha$.
• Figure 3: Values of the different estimators of the directional footrule-coefficients as functions of the parameter $\theta$ for $3$-dimensional Clayton family copulas.
• Figure 4: Empirical distribution of $\widetilde{\varphi}_4^{(1,1,1,1)}$ for the Cuadras-Augé $4$-copula with parameter values $\theta=\{0.4,0.8\}$ and sample sizes $n=\{20,50,100,500\}$. Horizontal red line shows the value of $\varphi_4^{(1,1,1,1)}$.
• Figure 5: Empirical distribution of $\widetilde{\varphi}_4^{(1,-1,1,1)}$ for the Cuadras-Augé $4$-copula with parameter values $\theta=\{0.4,0.8\}$ and sample sizes $n=\{20,50,100,500\}$. Horizontal red line shows the value of $\varphi_4^{(1,-1,1,1)}$.

Theorems & Definitions (22)

• Theorem 1: Sklar
• Definition 1: PD$(\alpha)$ dependence concept
• Definition 2: PD$(\alpha)$ order
• Theorem 2
• proof
• Remark 1
• Corollary 3
• proof
• Theorem 4
• proof