🤖 AI Summary
Traditional Spearman’s ρ fails to capture non-monotonic dependence structures and is sensitive to marginal distributions.
Method: We propose a margin-free, unit-square-based generalized Spearman correlation coefficient, constructed within the Hilbert space of square-integrable functions. Marginal invariance is achieved via uniformity-preserving transformations and copula modeling; a novel randomized inverse transformation generates extremal singular copulas, enabling a parametric copula family that continuously interpolates non-monotonic dependence strength and supports symmetry detection.
Contributions/Results: Leveraging orthogonal expansions in Legendre polynomials and cosine bases, we derive tight analytical bounds. The sample estimator is shown to be uniformly consistent and asymptotically normal. Empirical evaluations demonstrate superior performance in exploratory dependence analysis, symmetry identification, and non-monotonic density modeling compared to existing measures.
📝 Abstract
A margin-free measure of bivariate association generalizing Spearman's rho to the case of non-monotonic dependence is defined in terms of two square integrable functions on the unit interval. Properties of generalized Spearman correlation are investigated when the functions are piecewise continuous and strictly monotonic, with particular focus on the special cases where the functions are drawn from orthonormal bases defined by Legendre polynomials and cosine functions. For continuous random variables, generalized Spearman correlation is treated as a copula-based measure and shown to depend on a pair of uniform-distribution-preserving (udp) transformations determined by the underlying functions. Bounds for generalized Spearman correlation are derived and a novel technique referred to as stochastic inversion of udp transformations is used to construct singular copulas that attain the bounds and parametric copulas with densities that interpolate between the bounds and model different degrees of non-monotonic dependence. Sample analogues of generalized Spearman correlation are proposed and their asymptotic and small-sample properties are investigated. Potential applications of the theory are demonstrated including: exploratory analyses of the dependence structures of datasets and their symmetries; elicitation of functions maximizing generalized Spearman correlation via expansions in orthonormal basis functions; and construction of tractable probability densities to model a wide variety of non-monotonic dependencies.