Hyperbolic Covariance and its Applications in Independence Test

Abstract

This study introduces an innovative nonlinear association measure specifically designed for complex data analysis: hyperbolic covariance, inspired by hyperbolic geometry. This measure is crafted using a characteristic covariance kernel within hyperbolic spaces and features a crucial property: independence-zero equivalence. This property guarantees that the hyperbolic covariance between two random vectors is zero if and only if independent. Building on this foundation, we propose a novel test statistic for independence testing, detailing its asymptotic behaviors under null and alternative hypotheses. Furthermore, we establish that our test attains the optimal minimax rate of convergence, which is proportional to

\sqrt{n}

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. Through a series of case studies, we demonstrate the superior testing power of our proposed method in various contexts, effectively handling nonlinear relationships.

Keywords

Hyperbolic covariance / Independence test / Hyperboloid model / Positive-definite kernel / Independence-zero equivalence / 62G10 / 62H15

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Funding

National Aerospace Science Foundation of China(2171216, 12231017, 71921001, 71991474)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature