Directional Spearman’s Footrule Coefficients

• Directional Spearman’s Footrule Coefficients are rank-based measures defined within a copula framework that capture directional dependence by quantifying concordance and discordance.
• They extend the classical symmetric footrule by incorporating a direction indicator, thereby uncovering tail asymmetry, non-exchangeability, and subtle dependence structures.
• Nonparametric estimation using normalized ranks yields robust, asymptotically normal estimators that allow detailed analysis of local multivariate dependence.

Directional Spearman’s footrule coefficients are a family of rank-based dependence measures defined for multivariate data within the copula framework. Extending the classical (symmetric) Spearman’s footrule, these coefficients quantify the degree of concordance or discordance in specific directions, thereby detecting patterns of directional or asymmetric dependence that are invisible to traditional symmetric measures. The coefficients are parameterized by a direction indicator vector and exhibit a range of theoretical properties including reflection symmetry, sensitivity to copula structure, and robust nonparametric estimation.

1. Definition and Mathematical Formulation

Let $d \geq 2$ denote the dimension of interest, and let $C$ be a $d$-copula coupling continuous marginals $U_1,\ldots,U_d \sim U[0,1]$. For a direction vector $$\alpha = (\alpha_1, \ldots, \alpha_d) \in \{-1, 1\}^d$$, the directional Spearman footrule coefficient is defined as

$$\varphi_d^\alpha(C) = \frac{2(d+1)}{d-1} \int_0^1 \left\{ \mathbb{P}\left[ \alpha_1 U_1 > \alpha_1 u, \ldots, \alpha_d U_d > \alpha_d u \right] - \prod_{i=1}^d \mathbb{P}[\alpha_i U_i > \alpha_i u] \right\} du.$$

This quantity measures, for a specified direction $\alpha$, the deviation of the copula joint upper tail (or lower, depending on the sign pattern in $\alpha$) from the product of its univariate margins.

An alternative "min–max" form is given by

$$\varphi_d^\alpha(C) = \frac{2(d+1)}{d-1} \left\{ \mathbb{E}\left[ \left( \min_{i \in J} U_i - \max_{i \in I} U_i \right)_+ \right] - \frac{|I|! |J|!}{(d+1)!} \right\},$$

where $J = \{i: \alpha_i = 1\}$, $I = \{i: \alpha_i = -1\}$, and $(x)_+ = \max(x, 0)$.

The two extreme choices of $\alpha = \mathbf{1} = (1, \ldots, 1)$ and $\alpha = -\mathbf{1} = (-1, \ldots, -1)$ recover the classic upward and downward multivariate footrules, denoted $\varphi^+_d$ and $\varphi^-_d$, respectively (Amo et al., 24 Jan 2026).

2. Theoretical Properties

The family $\varphi_d^\alpha(C)$ satisfies the following fundamental properties:

• Consistency with Classical Footrule: The average of the upward and downward directional coefficients recovers the classical symmetric multivariate footrule:

$$\frac{ \varphi_d^+(C) + \varphi_d^-(C) }{2 } = \varphi_d(C).$$

• Null under Independence: For the product copula $\Pi_d$, all directional coefficients vanish:

$\varphi_d^\alpha(\Pi_d) = 0 \qquad \forall \alpha.$

• Maximal Dependence: For the comonotonic (Fréchet–Hoeffding upper bound) copula $M_d$,

$$\varphi_d^{\mathbf{1}}(M_d) = \varphi_d^{-\mathbf{1}}(M_d) = 1,$$

while for intermediate directions ($0 < |J| < d$),

$$\varphi_d^\alpha(M_d) = -\frac{2}{(d-1)\binom{d}{|J|}}.$$

• Summation to Zero:

$$\sum_{\alpha \in \{-1,1\}^d} \varphi_d^\alpha(C) = 0.$$

• Reflection/Survival Duality:

$$\varphi_d^\alpha(C) = \varphi_d^{-\alpha}(\widehat{C}),$$

where $\widehat C$ denotes the survival copula.

Proofs of these properties are based on the symmetry of the copula construction, combinatorial arguments, and algebraic manipulations inherent to the definition (Amo et al., 24 Jan 2026).

3. Estimation: Nonparametric Rank-Based Procedures

Given an i.i.d. sample $\{X_{ij}: i=1,\ldots,d; j=1,\ldots,n\}$ from a distribution with underlying copula $C$, the nonparametric estimator of $\varphi_d^\alpha(C)$ is constructed by replacing uniform marginals with normalized ranks: $$\widetilde\varphi_{n,d}^\alpha = \frac{2(d+1)}{(d-1)(n+1)} \left\{ \frac{1}{n} \sum_{j=1}^n \left( \min_{i \in J} R_{ij} - \max_{i \in I} R_{ij} \right)_+ - \frac{(n+1) |I|! |J|!}{(d+1)!} \right\}.$$ Here, $R_{ij}$ is the rank of $X_{ij}$ among $\{X_{i1},\ldots,X_{in}\}$. This estimator is bounded in $[-1,1]$.

These estimators can be expressed as linear combinations of lower-dimensional estimators, allowing decomposition and interpretation of dependence in subspaces. Under standard regularity conditions on $C$, the estimator is asymptotically normal and consistent: $$\sqrt{n}\left( \widetilde\varphi_{n,d}^\alpha - \varphi_d^\alpha(C) \right) \xrightarrow[]{w} N(0, \sigma_\alpha^2),$$ with the variance $\sigma_\alpha^2$ determined by the limiting Gaussian field of the empirical copula process (Amo et al., 24 Jan 2026).

4. Explicit Expressions for Standard Copula Families

Closed-form expressions for $\varphi_d^\alpha(C)$ are available for several important copula families. Sign and magnitude depend explicitly on $|J|$, the number of positive entries in $\alpha$.

Copula Family	Formula for $\varphi_d^\alpha(C)$
Farlie–Gumbel–Morgenstern (FGM)	$$\displaystyle \frac{2(d+1)(d!)^2}{(d-1)(2d+1)!} \lambda\, (-1)^{|J|}$$
Clayton	$$\displaystyle \frac{2(d+1)}{d-1} \sum_{k=0}^{|J|} (-1)^k \binom{|J|}{k} \int_0^1 \left[(|I|+k)u^{-\theta}+(1-|I|-k)\right]^{-1/\theta} du - \frac{2}{(d-1)\binom{d}{|J|}}$$
Cuadras–Augé mixture	$$\displaystyle \frac{2(d+1)}{d-1} \sum_{k=0}^{|J|} (-1)^k \binom{|J|}{k} \frac{\theta(|I|+k-1)}{(|I|+k+1)^2 - \theta[ (|I|+k)^2 -1 ]}$$

For more complex copulas such as Gumbel or Gaussian, numerical evaluation is required. The explicit directionality of $\varphi_d^\alpha(C)$ enables detection of asymmetry and tail behavior inaccessible to classical symmetric concordance measures.

5. Detection of Asymmetry and Illustrative Comparisons

Classical (symmetric) footrule coefficients, such as $\varphi_d(C)$, are insensitive to non-exchangeability and tail asymmetry. Directional footrule coefficients provide fine-grained resolution. For instance, in a 4-variate Clayton copula with $\theta=5$,

$$\varphi_4(C) \approx 0.82 \qquad \text{(classical)}, \qquad \varphi_4^{(+1,+1,+1,+1)}(C) \approx 0.91, \quad \varphi_4^{(-1,-1,-1,-1)}(C) \approx 0.77,$$

with mixed-sign directions yielding small negative values. This reveals clustering of mass in particular quadrants, a signature of tail asymmetry.

Similarly, in a Cuadras–Augé mixture with $d=4$ and $\theta=0.4$,

$$\varphi_4^{(+1,+1,+1,+1)} \approx 0.38, \quad \varphi_4^{(-1,-1,-1,-1)} \approx 0.38, \quad \varphi_4^{(\pm1,\pm1,\pm1,\mp1)} \approx 0,$$

while classical $\varphi_4 \approx 0.10$ obliterates the distinction. This suggests that directional coefficients are able to localize dependence structure and singular mass (Amo et al., 24 Jan 2026).

Monte Carlo simulations demonstrate empirical estimators closely follow the population values for moderate sample sizes ($n \gtrsim 50$), with asymptotic normality observed at $n \geq 100$.

6. Relation to Symmetric Footrule and Literature Context

Prior studies of Spearman footrule coefficients, such as in "Relation between non-exchangeability and measures of concordance of copulas" (Bukovšek et al., 2019), treated only the symmetric version, $\phi(C)$, which aggregates over all directions and ignores asymmetry. That work demonstrated theoretical bounds for $\phi(C)$ under $L_\infty$ asymmetry and compared various measures of concordance but did not introduce directional decomposition, sign conventions, or formulas for directional analysis.

The framework introduced by de Amo et al. (Amo et al., 24 Jan 2026) addresses this limitation by constructing, for the first time, direction-indexed measures, thereby enabling direct study of non-exchangeability, reflection, and tail-dependence asymmetry in high-dimensional dependence modeling.