Asymptotic representations for Spearman's footrule correlation coefficient

This study addressed the theoretical challenges faced by rank-dependence structures in Spearman's footrule correlation coefficient. Two asymptotic representations were proposed to approximate its null distribution. The first approach simplifies dependencies by substituting empirical distribution functions with their population counterparts. The second employs Hájek projection technique to decompose the initial form into a sum of independent components, establishing rigorous asymptotic normality. Simulation experiments combined with real-world data analyses validated both representations, demonstrating their excellent approximation to the limiting normal distribution under the independence hypothesis.